Properties

Label 288.4.d.d.145.5
Level $288$
Weight $4$
Character 288.145
Analytic conductor $16.993$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,4,Mod(145,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.145");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 288.d (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: \(16.9925500817\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8248384.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 145.5
Root \(-0.641412 + 1.89436i\) of defining polynomial
Character \(\chi\) \(=\) 288.145
Dual form 288.4.d.d.145.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.15486i q^{5} -27.4175 q^{7} +O(q^{10})\) \(q+9.15486i q^{5} -27.4175 q^{7} -20.5252i q^{11} -32.0471i q^{13} +111.764 q^{17} -129.764i q^{19} +9.16510 q^{23} +41.1885 q^{25} +41.0606i q^{29} +187.606 q^{31} -251.003i q^{35} -114.127i q^{37} -282.915 q^{41} -89.3870i q^{43} -54.6464 q^{47} +408.717 q^{49} -726.878i q^{53} +187.905 q^{55} +216.579i q^{59} -754.222i q^{61} +293.387 q^{65} -379.433i q^{67} +302.080 q^{71} -504.396 q^{73} +562.748i q^{77} -301.780 q^{79} +599.003i q^{83} +1023.18i q^{85} +277.528 q^{89} +878.651i q^{91} +1187.97 q^{95} -765.905 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 28 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 28 q^{7} - 52 q^{17} + 328 q^{23} - 106 q^{25} + 636 q^{31} - 236 q^{41} - 408 q^{47} + 654 q^{49} - 1024 q^{55} + 1744 q^{65} - 1704 q^{71} + 956 q^{73} + 44 q^{79} + 220 q^{89} + 5104 q^{95} - 2444 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 9.15486i 0.818836i 0.912347 + 0.409418i \(0.134268\pi\)
−0.912347 + 0.409418i \(0.865732\pi\)
\(6\) 0 0
\(7\) −27.4175 −1.48040 −0.740202 0.672385i \(-0.765270\pi\)
−0.740202 + 0.672385i \(0.765270\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 20.5252i − 0.562598i −0.959620 0.281299i \(-0.909235\pi\)
0.959620 0.281299i \(-0.0907652\pi\)
\(12\) 0 0
\(13\) − 32.0471i − 0.683713i −0.939752 0.341857i \(-0.888944\pi\)
0.939752 0.341857i \(-0.111056\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 111.764 1.59452 0.797258 0.603639i \(-0.206283\pi\)
0.797258 + 0.603639i \(0.206283\pi\)
\(18\) 0 0
\(19\) − 129.764i − 1.56684i −0.621494 0.783419i \(-0.713474\pi\)
0.621494 0.783419i \(-0.286526\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 9.16510 0.0830893 0.0415447 0.999137i \(-0.486772\pi\)
0.0415447 + 0.999137i \(0.486772\pi\)
\(24\) 0 0
\(25\) 41.1885 0.329508
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 41.0606i 0.262923i 0.991321 + 0.131461i \(0.0419669\pi\)
−0.991321 + 0.131461i \(0.958033\pi\)
\(30\) 0 0
\(31\) 187.606 1.08694 0.543468 0.839430i \(-0.317111\pi\)
0.543468 + 0.839430i \(0.317111\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 251.003i − 1.21221i
\(36\) 0 0
\(37\) − 114.127i − 0.507093i −0.967323 0.253546i \(-0.918403\pi\)
0.967323 0.253546i \(-0.0815970\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −282.915 −1.07766 −0.538828 0.842416i \(-0.681133\pi\)
−0.538828 + 0.842416i \(0.681133\pi\)
\(42\) 0 0
\(43\) − 89.3870i − 0.317009i −0.987358 0.158505i \(-0.949333\pi\)
0.987358 0.158505i \(-0.0506673\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −54.6464 −0.169596 −0.0847978 0.996398i \(-0.527024\pi\)
−0.0847978 + 0.996398i \(0.527024\pi\)
\(48\) 0 0
\(49\) 408.717 1.19159
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 726.878i − 1.88386i −0.335815 0.941928i \(-0.609012\pi\)
0.335815 0.941928i \(-0.390988\pi\)
\(54\) 0 0
\(55\) 187.905 0.460675
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 216.579i 0.477900i 0.971032 + 0.238950i \(0.0768033\pi\)
−0.971032 + 0.238950i \(0.923197\pi\)
\(60\) 0 0
\(61\) − 754.222i − 1.58309i −0.611114 0.791543i \(-0.709278\pi\)
0.611114 0.791543i \(-0.290722\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 293.387 0.559849
\(66\) 0 0
\(67\) − 379.433i − 0.691868i −0.938259 0.345934i \(-0.887562\pi\)
0.938259 0.345934i \(-0.112438\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 302.080 0.504933 0.252467 0.967606i \(-0.418758\pi\)
0.252467 + 0.967606i \(0.418758\pi\)
\(72\) 0 0
\(73\) −504.396 −0.808700 −0.404350 0.914604i \(-0.632502\pi\)
−0.404350 + 0.914604i \(0.632502\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 562.748i 0.832872i
\(78\) 0 0
\(79\) −301.780 −0.429784 −0.214892 0.976638i \(-0.568940\pi\)
−0.214892 + 0.976638i \(0.568940\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 599.003i 0.792158i 0.918216 + 0.396079i \(0.129629\pi\)
−0.918216 + 0.396079i \(0.870371\pi\)
\(84\) 0 0
\(85\) 1023.18i 1.30565i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 277.528 0.330538 0.165269 0.986248i \(-0.447151\pi\)
0.165269 + 0.986248i \(0.447151\pi\)
\(90\) 0 0
\(91\) 878.651i 1.01217i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1187.97 1.28298
\(96\) 0 0
\(97\) −765.905 −0.801710 −0.400855 0.916141i \(-0.631287\pi\)
−0.400855 + 0.916141i \(0.631287\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 201.253i 0.198272i 0.995074 + 0.0991360i \(0.0316079\pi\)
−0.995074 + 0.0991360i \(0.968392\pi\)
\(102\) 0 0
\(103\) −682.440 −0.652843 −0.326421 0.945224i \(-0.605843\pi\)
−0.326421 + 0.945224i \(0.605843\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 457.252i − 0.413123i −0.978434 0.206562i \(-0.933773\pi\)
0.978434 0.206562i \(-0.0662274\pi\)
\(108\) 0 0
\(109\) − 625.812i − 0.549926i −0.961455 0.274963i \(-0.911334\pi\)
0.961455 0.274963i \(-0.0886655\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −981.151 −0.816805 −0.408402 0.912802i \(-0.633914\pi\)
−0.408402 + 0.912802i \(0.633914\pi\)
\(114\) 0 0
\(115\) 83.9052i 0.0680365i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3064.29 −2.36053
\(120\) 0 0
\(121\) 909.717 0.683484
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1521.43i 1.08865i
\(126\) 0 0
\(127\) 808.055 0.564593 0.282296 0.959327i \(-0.408904\pi\)
0.282296 + 0.959327i \(0.408904\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 1110.85i − 0.740884i −0.928856 0.370442i \(-0.879206\pi\)
0.928856 0.370442i \(-0.120794\pi\)
\(132\) 0 0
\(133\) 3557.80i 2.31955i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 466.765 0.291084 0.145542 0.989352i \(-0.453507\pi\)
0.145542 + 0.989352i \(0.453507\pi\)
\(138\) 0 0
\(139\) 351.773i 0.214654i 0.994224 + 0.107327i \(0.0342292\pi\)
−0.994224 + 0.107327i \(0.965771\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −657.773 −0.384656
\(144\) 0 0
\(145\) −375.904 −0.215291
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 1290.49i − 0.709540i −0.934954 0.354770i \(-0.884559\pi\)
0.934954 0.354770i \(-0.115441\pi\)
\(150\) 0 0
\(151\) −1175.51 −0.633521 −0.316761 0.948505i \(-0.602595\pi\)
−0.316761 + 0.948505i \(0.602595\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1717.51i 0.890022i
\(156\) 0 0
\(157\) 1092.09i 0.555148i 0.960704 + 0.277574i \(0.0895303\pi\)
−0.960704 + 0.277574i \(0.910470\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −251.284 −0.123006
\(162\) 0 0
\(163\) 3626.97i 1.74286i 0.490519 + 0.871430i \(0.336807\pi\)
−0.490519 + 0.871430i \(0.663193\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 45.8012 0.0212228 0.0106114 0.999944i \(-0.496622\pi\)
0.0106114 + 0.999944i \(0.496622\pi\)
\(168\) 0 0
\(169\) 1169.98 0.532536
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2455.02i 1.07891i 0.842014 + 0.539455i \(0.181370\pi\)
−0.842014 + 0.539455i \(0.818630\pi\)
\(174\) 0 0
\(175\) −1129.28 −0.487805
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1026.28i 0.428533i 0.976775 + 0.214267i \(0.0687362\pi\)
−0.976775 + 0.214267i \(0.931264\pi\)
\(180\) 0 0
\(181\) − 3699.05i − 1.51905i −0.650477 0.759526i \(-0.725431\pi\)
0.650477 0.759526i \(-0.274569\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1044.82 0.415226
\(186\) 0 0
\(187\) − 2293.98i − 0.897071i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5108.93 1.93544 0.967721 0.252023i \(-0.0810960\pi\)
0.967721 + 0.252023i \(0.0810960\pi\)
\(192\) 0 0
\(193\) −1414.13 −0.527417 −0.263709 0.964602i \(-0.584946\pi\)
−0.263709 + 0.964602i \(0.584946\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2816.66i − 1.01867i −0.860567 0.509337i \(-0.829891\pi\)
0.860567 0.509337i \(-0.170109\pi\)
\(198\) 0 0
\(199\) 948.556 0.337896 0.168948 0.985625i \(-0.445963\pi\)
0.168948 + 0.985625i \(0.445963\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 1125.78i − 0.389232i
\(204\) 0 0
\(205\) − 2590.05i − 0.882424i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2663.43 −0.881499
\(210\) 0 0
\(211\) − 4487.28i − 1.46406i −0.681271 0.732032i \(-0.738572\pi\)
0.681271 0.732032i \(-0.261428\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 818.326 0.259578
\(216\) 0 0
\(217\) −5143.68 −1.60910
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 3581.72i − 1.09019i
\(222\) 0 0
\(223\) 4590.98 1.37863 0.689315 0.724462i \(-0.257912\pi\)
0.689315 + 0.724462i \(0.257912\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2897.47i 0.847189i 0.905852 + 0.423594i \(0.139232\pi\)
−0.905852 + 0.423594i \(0.860768\pi\)
\(228\) 0 0
\(229\) 34.6293i 0.00999288i 0.999988 + 0.00499644i \(0.00159042\pi\)
−0.999988 + 0.00499644i \(0.998410\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1054.02 0.296355 0.148178 0.988961i \(-0.452659\pi\)
0.148178 + 0.988961i \(0.452659\pi\)
\(234\) 0 0
\(235\) − 500.280i − 0.138871i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −654.700 −0.177192 −0.0885962 0.996068i \(-0.528238\pi\)
−0.0885962 + 0.996068i \(0.528238\pi\)
\(240\) 0 0
\(241\) 3194.00 0.853707 0.426854 0.904321i \(-0.359622\pi\)
0.426854 + 0.904321i \(0.359622\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3741.74i 0.975719i
\(246\) 0 0
\(247\) −4158.57 −1.07127
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 5042.90i − 1.26815i −0.773273 0.634074i \(-0.781382\pi\)
0.773273 0.634074i \(-0.218618\pi\)
\(252\) 0 0
\(253\) − 188.115i − 0.0467459i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5166.64 1.25403 0.627016 0.779007i \(-0.284276\pi\)
0.627016 + 0.779007i \(0.284276\pi\)
\(258\) 0 0
\(259\) 3129.08i 0.750702i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7366.11 −1.72705 −0.863524 0.504308i \(-0.831748\pi\)
−0.863524 + 0.504308i \(0.831748\pi\)
\(264\) 0 0
\(265\) 6654.47 1.54257
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 7877.80i − 1.78557i −0.450484 0.892784i \(-0.648749\pi\)
0.450484 0.892784i \(-0.351251\pi\)
\(270\) 0 0
\(271\) 5399.92 1.21041 0.605206 0.796069i \(-0.293091\pi\)
0.605206 + 0.796069i \(0.293091\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 845.402i − 0.185381i
\(276\) 0 0
\(277\) − 4416.07i − 0.957892i −0.877844 0.478946i \(-0.841019\pi\)
0.877844 0.478946i \(-0.158981\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8068.94 −1.71300 −0.856499 0.516148i \(-0.827365\pi\)
−0.856499 + 0.516148i \(0.827365\pi\)
\(282\) 0 0
\(283\) − 5241.13i − 1.10089i −0.834870 0.550447i \(-0.814457\pi\)
0.834870 0.550447i \(-0.185543\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7756.81 1.59537
\(288\) 0 0
\(289\) 7578.21 1.54248
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6372.75i 1.27065i 0.772246 + 0.635324i \(0.219133\pi\)
−0.772246 + 0.635324i \(0.780867\pi\)
\(294\) 0 0
\(295\) −1982.75 −0.391322
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 293.715i − 0.0568093i
\(300\) 0 0
\(301\) 2450.76i 0.469301i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6904.79 1.29629
\(306\) 0 0
\(307\) − 3810.22i − 0.708342i −0.935181 0.354171i \(-0.884763\pi\)
0.935181 0.354171i \(-0.115237\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8106.73 −1.47810 −0.739052 0.673648i \(-0.764726\pi\)
−0.739052 + 0.673648i \(0.764726\pi\)
\(312\) 0 0
\(313\) −559.983 −0.101125 −0.0505625 0.998721i \(-0.516101\pi\)
−0.0505625 + 0.998721i \(0.516101\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5828.98i 1.03277i 0.856357 + 0.516385i \(0.172723\pi\)
−0.856357 + 0.516385i \(0.827277\pi\)
\(318\) 0 0
\(319\) 842.776 0.147920
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 14503.0i − 2.49835i
\(324\) 0 0
\(325\) − 1319.97i − 0.225289i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1498.26 0.251070
\(330\) 0 0
\(331\) 2847.98i 0.472928i 0.971640 + 0.236464i \(0.0759885\pi\)
−0.971640 + 0.236464i \(0.924011\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3473.66 0.566526
\(336\) 0 0
\(337\) −10127.8 −1.63707 −0.818537 0.574454i \(-0.805214\pi\)
−0.818537 + 0.574454i \(0.805214\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 3850.65i − 0.611508i
\(342\) 0 0
\(343\) −1801.78 −0.283636
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10148.2i 1.56999i 0.619505 + 0.784993i \(0.287333\pi\)
−0.619505 + 0.784993i \(0.712667\pi\)
\(348\) 0 0
\(349\) 9515.96i 1.45954i 0.683695 + 0.729768i \(0.260372\pi\)
−0.683695 + 0.729768i \(0.739628\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2813.56 −0.424223 −0.212111 0.977245i \(-0.568034\pi\)
−0.212111 + 0.977245i \(0.568034\pi\)
\(354\) 0 0
\(355\) 2765.50i 0.413457i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2427.25 −0.356839 −0.178419 0.983955i \(-0.557098\pi\)
−0.178419 + 0.983955i \(0.557098\pi\)
\(360\) 0 0
\(361\) −9979.71 −1.45498
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 4617.67i − 0.662192i
\(366\) 0 0
\(367\) −5021.46 −0.714219 −0.357109 0.934063i \(-0.616238\pi\)
−0.357109 + 0.934063i \(0.616238\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 19929.1i 2.78887i
\(372\) 0 0
\(373\) − 3182.40i − 0.441765i −0.975300 0.220882i \(-0.929106\pi\)
0.975300 0.220882i \(-0.0708937\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1315.87 0.179764
\(378\) 0 0
\(379\) 5868.93i 0.795426i 0.917510 + 0.397713i \(0.130196\pi\)
−0.917510 + 0.397713i \(0.869804\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 7350.18 0.980618 0.490309 0.871549i \(-0.336884\pi\)
0.490309 + 0.871549i \(0.336884\pi\)
\(384\) 0 0
\(385\) −5151.88 −0.681985
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 13009.1i 1.69560i 0.530317 + 0.847800i \(0.322073\pi\)
−0.530317 + 0.847800i \(0.677927\pi\)
\(390\) 0 0
\(391\) 1024.33 0.132487
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 2762.76i − 0.351923i
\(396\) 0 0
\(397\) 4877.88i 0.616659i 0.951280 + 0.308330i \(0.0997700\pi\)
−0.951280 + 0.308330i \(0.900230\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5552.33 −0.691446 −0.345723 0.938337i \(-0.612366\pi\)
−0.345723 + 0.938337i \(0.612366\pi\)
\(402\) 0 0
\(403\) − 6012.23i − 0.743153i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2342.49 −0.285289
\(408\) 0 0
\(409\) 6989.27 0.844981 0.422491 0.906367i \(-0.361156\pi\)
0.422491 + 0.906367i \(0.361156\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 5938.03i − 0.707485i
\(414\) 0 0
\(415\) −5483.79 −0.648647
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 10461.0i − 1.21970i −0.792518 0.609849i \(-0.791230\pi\)
0.792518 0.609849i \(-0.208770\pi\)
\(420\) 0 0
\(421\) − 4648.55i − 0.538139i −0.963121 0.269070i \(-0.913284\pi\)
0.963121 0.269070i \(-0.0867162\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4603.40 0.525406
\(426\) 0 0
\(427\) 20678.8i 2.34360i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12490.7 1.39595 0.697975 0.716122i \(-0.254085\pi\)
0.697975 + 0.716122i \(0.254085\pi\)
\(432\) 0 0
\(433\) 9446.37 1.04842 0.524208 0.851590i \(-0.324362\pi\)
0.524208 + 0.851590i \(0.324362\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 1189.30i − 0.130188i
\(438\) 0 0
\(439\) −2793.60 −0.303716 −0.151858 0.988402i \(-0.548526\pi\)
−0.151858 + 0.988402i \(0.548526\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7601.37i 0.815241i 0.913151 + 0.407621i \(0.133641\pi\)
−0.913151 + 0.407621i \(0.866359\pi\)
\(444\) 0 0
\(445\) 2540.73i 0.270657i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10708.8 −1.12557 −0.562785 0.826603i \(-0.690270\pi\)
−0.562785 + 0.826603i \(0.690270\pi\)
\(450\) 0 0
\(451\) 5806.89i 0.606287i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8043.92 −0.828802
\(456\) 0 0
\(457\) 233.840 0.0239356 0.0119678 0.999928i \(-0.496190\pi\)
0.0119678 + 0.999928i \(0.496190\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 981.307i − 0.0991410i −0.998771 0.0495705i \(-0.984215\pi\)
0.998771 0.0495705i \(-0.0157853\pi\)
\(462\) 0 0
\(463\) 14082.7 1.41356 0.706782 0.707431i \(-0.250146\pi\)
0.706782 + 0.707431i \(0.250146\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9286.49i 0.920188i 0.887870 + 0.460094i \(0.152184\pi\)
−0.887870 + 0.460094i \(0.847816\pi\)
\(468\) 0 0
\(469\) 10403.1i 1.02424i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1834.68 −0.178349
\(474\) 0 0
\(475\) − 5344.79i − 0.516286i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19409.3 −1.85143 −0.925715 0.378222i \(-0.876536\pi\)
−0.925715 + 0.378222i \(0.876536\pi\)
\(480\) 0 0
\(481\) −3657.46 −0.346706
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 7011.76i − 0.656469i
\(486\) 0 0
\(487\) −12124.8 −1.12818 −0.564091 0.825712i \(-0.690773\pi\)
−0.564091 + 0.825712i \(0.690773\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 5100.69i − 0.468820i −0.972138 0.234410i \(-0.924684\pi\)
0.972138 0.234410i \(-0.0753159\pi\)
\(492\) 0 0
\(493\) 4589.10i 0.419235i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8282.26 −0.747505
\(498\) 0 0
\(499\) 85.2797i 0.00765058i 0.999993 + 0.00382529i \(0.00121763\pi\)
−0.999993 + 0.00382529i \(0.998782\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12287.2 −1.08918 −0.544592 0.838701i \(-0.683316\pi\)
−0.544592 + 0.838701i \(0.683316\pi\)
\(504\) 0 0
\(505\) −1842.45 −0.162352
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 450.441i 0.0392248i 0.999808 + 0.0196124i \(0.00624322\pi\)
−0.999808 + 0.0196124i \(0.993757\pi\)
\(510\) 0 0
\(511\) 13829.2 1.19720
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 6247.64i − 0.534571i
\(516\) 0 0
\(517\) 1121.63i 0.0954141i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15088.1 1.26876 0.634378 0.773023i \(-0.281256\pi\)
0.634378 + 0.773023i \(0.281256\pi\)
\(522\) 0 0
\(523\) − 17719.4i − 1.48149i −0.671789 0.740743i \(-0.734474\pi\)
0.671789 0.740743i \(-0.265526\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20967.6 1.73314
\(528\) 0 0
\(529\) −12083.0 −0.993096
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9066.62i 0.736808i
\(534\) 0 0
\(535\) 4186.08 0.338280
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 8388.98i − 0.670388i
\(540\) 0 0
\(541\) − 12244.5i − 0.973074i −0.873660 0.486537i \(-0.838260\pi\)
0.873660 0.486537i \(-0.161740\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5729.22 0.450299
\(546\) 0 0
\(547\) 7822.46i 0.611452i 0.952119 + 0.305726i \(0.0988992\pi\)
−0.952119 + 0.305726i \(0.901101\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5328.19 0.411957
\(552\) 0 0
\(553\) 8274.05 0.636254
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16555.5i 1.25938i 0.776845 + 0.629692i \(0.216819\pi\)
−0.776845 + 0.629692i \(0.783181\pi\)
\(558\) 0 0
\(559\) −2864.60 −0.216743
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 12580.7i − 0.941766i −0.882196 0.470883i \(-0.843935\pi\)
0.882196 0.470883i \(-0.156065\pi\)
\(564\) 0 0
\(565\) − 8982.30i − 0.668829i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2657.93 −0.195828 −0.0979141 0.995195i \(-0.531217\pi\)
−0.0979141 + 0.995195i \(0.531217\pi\)
\(570\) 0 0
\(571\) − 17669.0i − 1.29496i −0.762081 0.647481i \(-0.775822\pi\)
0.762081 0.647481i \(-0.224178\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 377.497 0.0273786
\(576\) 0 0
\(577\) 14617.7 1.05467 0.527334 0.849658i \(-0.323192\pi\)
0.527334 + 0.849658i \(0.323192\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 16423.1i − 1.17271i
\(582\) 0 0
\(583\) −14919.3 −1.05985
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 4096.53i − 0.288044i −0.989574 0.144022i \(-0.953996\pi\)
0.989574 0.144022i \(-0.0460036\pi\)
\(588\) 0 0
\(589\) − 24344.5i − 1.70305i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 21988.3 1.52269 0.761343 0.648349i \(-0.224540\pi\)
0.761343 + 0.648349i \(0.224540\pi\)
\(594\) 0 0
\(595\) − 28053.1i − 1.93288i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 20767.7 1.41660 0.708302 0.705909i \(-0.249461\pi\)
0.708302 + 0.705909i \(0.249461\pi\)
\(600\) 0 0
\(601\) 5382.61 0.365326 0.182663 0.983176i \(-0.441528\pi\)
0.182663 + 0.983176i \(0.441528\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 8328.33i 0.559661i
\(606\) 0 0
\(607\) −11165.4 −0.746607 −0.373304 0.927709i \(-0.621775\pi\)
−0.373304 + 0.927709i \(0.621775\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1751.26i 0.115955i
\(612\) 0 0
\(613\) 16413.5i 1.08146i 0.841195 + 0.540731i \(0.181852\pi\)
−0.841195 + 0.540731i \(0.818148\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 51.5882 0.00336607 0.00168303 0.999999i \(-0.499464\pi\)
0.00168303 + 0.999999i \(0.499464\pi\)
\(618\) 0 0
\(619\) 6349.55i 0.412294i 0.978521 + 0.206147i \(0.0660925\pi\)
−0.978521 + 0.206147i \(0.933907\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −7609.11 −0.489330
\(624\) 0 0
\(625\) −8779.94 −0.561916
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 12755.3i − 0.808567i
\(630\) 0 0
\(631\) 13379.1 0.844078 0.422039 0.906578i \(-0.361315\pi\)
0.422039 + 0.906578i \(0.361315\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 7397.63i 0.462309i
\(636\) 0 0
\(637\) − 13098.2i − 0.814709i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20406.3 1.25741 0.628705 0.777644i \(-0.283585\pi\)
0.628705 + 0.777644i \(0.283585\pi\)
\(642\) 0 0
\(643\) 19415.1i 1.19076i 0.803446 + 0.595378i \(0.202998\pi\)
−0.803446 + 0.595378i \(0.797002\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −8167.12 −0.496264 −0.248132 0.968726i \(-0.579817\pi\)
−0.248132 + 0.968726i \(0.579817\pi\)
\(648\) 0 0
\(649\) 4445.31 0.268866
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7444.93i 0.446160i 0.974800 + 0.223080i \(0.0716111\pi\)
−0.974800 + 0.223080i \(0.928389\pi\)
\(654\) 0 0
\(655\) 10169.7 0.606662
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 23780.4i 1.40569i 0.711342 + 0.702846i \(0.248088\pi\)
−0.711342 + 0.702846i \(0.751912\pi\)
\(660\) 0 0
\(661\) − 2528.90i − 0.148809i −0.997228 0.0744046i \(-0.976294\pi\)
0.997228 0.0744046i \(-0.0237056\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −32571.2 −1.89933
\(666\) 0 0
\(667\) 376.324i 0.0218461i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −15480.5 −0.890640
\(672\) 0 0
\(673\) 16733.7 0.958447 0.479224 0.877693i \(-0.340918\pi\)
0.479224 + 0.877693i \(0.340918\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 24191.5i − 1.37335i −0.726966 0.686673i \(-0.759070\pi\)
0.726966 0.686673i \(-0.240930\pi\)
\(678\) 0 0
\(679\) 20999.2 1.18685
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 13965.2i − 0.782376i −0.920311 0.391188i \(-0.872064\pi\)
0.920311 0.391188i \(-0.127936\pi\)
\(684\) 0 0
\(685\) 4273.17i 0.238350i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −23294.4 −1.28802
\(690\) 0 0
\(691\) − 8685.63i − 0.478172i −0.970998 0.239086i \(-0.923152\pi\)
0.970998 0.239086i \(-0.0768479\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3220.43 −0.175767
\(696\) 0 0
\(697\) −31619.7 −1.71834
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 25942.2i − 1.39775i −0.715243 0.698876i \(-0.753684\pi\)
0.715243 0.698876i \(-0.246316\pi\)
\(702\) 0 0
\(703\) −14809.6 −0.794532
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 5517.86i − 0.293522i
\(708\) 0 0
\(709\) 5487.75i 0.290687i 0.989381 + 0.145343i \(0.0464287\pi\)
−0.989381 + 0.145343i \(0.953571\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1719.43 0.0903128
\(714\) 0 0
\(715\) − 6021.82i − 0.314970i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 17141.2 0.889094 0.444547 0.895756i \(-0.353365\pi\)
0.444547 + 0.895756i \(0.353365\pi\)
\(720\) 0 0
\(721\) 18710.8 0.966470
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1691.23i 0.0866352i
\(726\) 0 0
\(727\) 15946.4 0.813508 0.406754 0.913538i \(-0.366661\pi\)
0.406754 + 0.913538i \(0.366661\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 9990.26i − 0.505476i
\(732\) 0 0
\(733\) − 15914.2i − 0.801917i −0.916096 0.400958i \(-0.868677\pi\)
0.916096 0.400958i \(-0.131323\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7787.94 −0.389243
\(738\) 0 0
\(739\) 13555.4i 0.674755i 0.941369 + 0.337377i \(0.109540\pi\)
−0.941369 + 0.337377i \(0.890460\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1772.73 −0.0875303 −0.0437652 0.999042i \(-0.513935\pi\)
−0.0437652 + 0.999042i \(0.513935\pi\)
\(744\) 0 0
\(745\) 11814.3 0.580997
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 12536.7i 0.611589i
\(750\) 0 0
\(751\) 1006.65 0.0489124 0.0244562 0.999701i \(-0.492215\pi\)
0.0244562 + 0.999701i \(0.492215\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 10761.6i − 0.518750i
\(756\) 0 0
\(757\) 28774.0i 1.38152i 0.723086 + 0.690758i \(0.242723\pi\)
−0.723086 + 0.690758i \(0.757277\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18393.5 −0.876166 −0.438083 0.898934i \(-0.644342\pi\)
−0.438083 + 0.898934i \(0.644342\pi\)
\(762\) 0 0
\(763\) 17158.2i 0.814112i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6940.72 0.326747
\(768\) 0 0
\(769\) −14672.2 −0.688027 −0.344014 0.938965i \(-0.611787\pi\)
−0.344014 + 0.938965i \(0.611787\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 16256.4i 0.756405i 0.925723 + 0.378202i \(0.123458\pi\)
−0.925723 + 0.378202i \(0.876542\pi\)
\(774\) 0 0
\(775\) 7727.21 0.358154
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 36712.2i 1.68851i
\(780\) 0 0
\(781\) − 6200.24i − 0.284074i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9997.92 −0.454575
\(786\) 0 0
\(787\) 23988.3i 1.08652i 0.839565 + 0.543259i \(0.182810\pi\)
−0.839565 + 0.543259i \(0.817190\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 26900.7 1.20920
\(792\) 0 0
\(793\) −24170.6 −1.08238
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 32966.9i 1.46518i 0.680672 + 0.732589i \(0.261688\pi\)
−0.680672 + 0.732589i \(0.738312\pi\)
\(798\) 0 0
\(799\) −6107.50 −0.270423
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10352.8i 0.454973i
\(804\) 0 0
\(805\) − 2300.47i − 0.100721i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 41700.3 1.81224 0.906122 0.423017i \(-0.139029\pi\)
0.906122 + 0.423017i \(0.139029\pi\)
\(810\) 0 0
\(811\) − 5981.80i − 0.259000i −0.991579 0.129500i \(-0.958663\pi\)
0.991579 0.129500i \(-0.0413373\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −33204.4 −1.42712
\(816\) 0 0
\(817\) −11599.2 −0.496702
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 9846.06i − 0.418550i −0.977857 0.209275i \(-0.932890\pi\)
0.977857 0.209275i \(-0.0671104\pi\)
\(822\) 0 0
\(823\) −47001.9 −1.99074 −0.995372 0.0960935i \(-0.969365\pi\)
−0.995372 + 0.0960935i \(0.969365\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 21727.4i 0.913587i 0.889573 + 0.456794i \(0.151002\pi\)
−0.889573 + 0.456794i \(0.848998\pi\)
\(828\) 0 0
\(829\) 22772.3i 0.954058i 0.878888 + 0.477029i \(0.158286\pi\)
−0.878888 + 0.477029i \(0.841714\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 45679.8 1.90002
\(834\) 0 0
\(835\) 419.304i 0.0173780i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 11010.4 0.453064 0.226532 0.974004i \(-0.427261\pi\)
0.226532 + 0.974004i \(0.427261\pi\)
\(840\) 0 0
\(841\) 22703.0 0.930872
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 10711.0i 0.436059i
\(846\) 0 0
\(847\) −24942.1 −1.01183
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 1045.99i − 0.0421340i
\(852\) 0 0
\(853\) 38177.4i 1.53244i 0.642579 + 0.766219i \(0.277864\pi\)
−0.642579 + 0.766219i \(0.722136\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −8848.01 −0.352675 −0.176337 0.984330i \(-0.556425\pi\)
−0.176337 + 0.984330i \(0.556425\pi\)
\(858\) 0 0
\(859\) − 4347.66i − 0.172690i −0.996265 0.0863448i \(-0.972481\pi\)
0.996265 0.0863448i \(-0.0275187\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −33669.9 −1.32808 −0.664042 0.747695i \(-0.731161\pi\)
−0.664042 + 0.747695i \(0.731161\pi\)
\(864\) 0 0
\(865\) −22475.3 −0.883450
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6194.10i 0.241796i
\(870\) 0 0
\(871\) −12159.7 −0.473039
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 41713.8i − 1.61164i
\(876\) 0 0
\(877\) 50102.0i 1.92910i 0.263892 + 0.964552i \(0.414994\pi\)
−0.263892 + 0.964552i \(0.585006\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −18716.9 −0.715766 −0.357883 0.933766i \(-0.616501\pi\)
−0.357883 + 0.933766i \(0.616501\pi\)
\(882\) 0 0
\(883\) 7514.19i 0.286379i 0.989695 + 0.143189i \(0.0457358\pi\)
−0.989695 + 0.143189i \(0.954264\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −15544.6 −0.588429 −0.294215 0.955739i \(-0.595058\pi\)
−0.294215 + 0.955739i \(0.595058\pi\)
\(888\) 0 0
\(889\) −22154.8 −0.835825
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7091.14i 0.265729i
\(894\) 0 0
\(895\) −9395.42 −0.350898
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 7703.21i 0.285780i
\(900\) 0 0
\(901\) − 81238.8i − 3.00384i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 33864.3 1.24385
\(906\) 0 0
\(907\) 8713.10i 0.318979i 0.987200 + 0.159489i \(0.0509848\pi\)
−0.987200 + 0.159489i \(0.949015\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1975.97 0.0718627 0.0359313 0.999354i \(-0.488560\pi\)
0.0359313 + 0.999354i \(0.488560\pi\)
\(912\) 0 0
\(913\) 12294.6 0.445666
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 30456.8i 1.09681i
\(918\) 0 0
\(919\) 18430.5 0.661552 0.330776 0.943709i \(-0.392689\pi\)
0.330776 + 0.943709i \(0.392689\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 9680.79i − 0.345230i
\(924\) 0 0
\(925\) − 4700.74i − 0.167091i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −12506.8 −0.441697 −0.220848 0.975308i \(-0.570883\pi\)
−0.220848 + 0.975308i \(0.570883\pi\)
\(930\) 0 0
\(931\) − 53036.7i − 1.86703i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 21001.0 0.734554
\(936\) 0 0
\(937\) 39267.5 1.36906 0.684532 0.728982i \(-0.260006\pi\)
0.684532 + 0.728982i \(0.260006\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 23727.2i − 0.821981i −0.911640 0.410991i \(-0.865183\pi\)
0.911640 0.410991i \(-0.134817\pi\)
\(942\) 0 0
\(943\) −2592.94 −0.0895418
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 23399.8i − 0.802948i −0.915870 0.401474i \(-0.868498\pi\)
0.915870 0.401474i \(-0.131502\pi\)
\(948\) 0 0
\(949\) 16164.4i 0.552919i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −41497.1 −1.41052 −0.705258 0.708950i \(-0.749169\pi\)
−0.705258 + 0.708950i \(0.749169\pi\)
\(954\) 0 0
\(955\) 46771.6i 1.58481i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12797.5 −0.430921
\(960\) 0 0
\(961\) 5405.00 0.181431
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 12946.2i − 0.431868i
\(966\) 0 0
\(967\) 49123.6 1.63362 0.816808 0.576909i \(-0.195741\pi\)
0.816808 + 0.576909i \(0.195741\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 20346.8i 0.672462i 0.941780 + 0.336231i \(0.109152\pi\)
−0.941780 + 0.336231i \(0.890848\pi\)
\(972\) 0 0
\(973\) − 9644.71i − 0.317775i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −40602.1 −1.32955 −0.664777 0.747042i \(-0.731474\pi\)
−0.664777 + 0.747042i \(0.731474\pi\)
\(978\) 0 0
\(979\) − 5696.32i − 0.185960i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 50425.9 1.63615 0.818075 0.575112i \(-0.195041\pi\)
0.818075 + 0.575112i \(0.195041\pi\)
\(984\) 0 0
\(985\) 25786.2 0.834127
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 819.241i − 0.0263401i
\(990\) 0 0
\(991\) 8511.62 0.272836 0.136418 0.990651i \(-0.456441\pi\)
0.136418 + 0.990651i \(0.456441\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8683.90i 0.276681i
\(996\) 0 0
\(997\) 25302.1i 0.803738i 0.915697 + 0.401869i \(0.131639\pi\)
−0.915697 + 0.401869i \(0.868361\pi\)
\(998\) 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 288.4.d.d.145.5 6
3.2 odd 2 96.4.d.a.49.1 6
4.3 odd 2 72.4.d.d.37.3 6
8.3 odd 2 72.4.d.d.37.4 6
8.5 even 2 inner 288.4.d.d.145.2 6
12.11 even 2 24.4.d.a.13.4 yes 6
16.3 odd 4 2304.4.a.bt.1.3 3
16.5 even 4 2304.4.a.bw.1.1 3
16.11 odd 4 2304.4.a.bv.1.1 3
16.13 even 4 2304.4.a.bu.1.3 3
24.5 odd 2 96.4.d.a.49.6 6
24.11 even 2 24.4.d.a.13.3 6
48.5 odd 4 768.4.a.q.1.3 3
48.11 even 4 768.4.a.s.1.3 3
48.29 odd 4 768.4.a.t.1.1 3
48.35 even 4 768.4.a.r.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.4.d.a.13.3 6 24.11 even 2
24.4.d.a.13.4 yes 6 12.11 even 2
72.4.d.d.37.3 6 4.3 odd 2
72.4.d.d.37.4 6 8.3 odd 2
96.4.d.a.49.1 6 3.2 odd 2
96.4.d.a.49.6 6 24.5 odd 2
288.4.d.d.145.2 6 8.5 even 2 inner
288.4.d.d.145.5 6 1.1 even 1 trivial
768.4.a.q.1.3 3 48.5 odd 4
768.4.a.r.1.1 3 48.35 even 4
768.4.a.s.1.3 3 48.11 even 4
768.4.a.t.1.1 3 48.29 odd 4
2304.4.a.bt.1.3 3 16.3 odd 4
2304.4.a.bu.1.3 3 16.13 even 4
2304.4.a.bv.1.1 3 16.11 odd 4
2304.4.a.bw.1.1 3 16.5 even 4