Properties

Label 1296.3.e.e.161.4
Level $1296$
Weight $3$
Character 1296.161
Analytic conductor $35.313$
Analytic rank $0$
Dimension $4$
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] = [1296,3,Mod(161,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1296.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1296.e (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: \(35.3134422611\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.4
Root \(-1.18614 - 1.26217i\) of defining polynomial
Character \(\chi\) \(=\) 1296.161
Dual form 1296.3.e.e.161.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.57301i q^{5} -9.11684 q^{7} +O(q^{10})\) \(q+7.57301i q^{5} -9.11684 q^{7} -0.442430i q^{11} -11.1168 q^{13} +8.01544i q^{17} +8.11684 q^{19} -23.6039i q^{23} -32.3505 q^{25} +53.0111i q^{29} -29.3505 q^{31} -69.0420i q^{35} +18.4674 q^{37} -44.9956i q^{41} -23.0000 q^{43} -8.45787i q^{47} +34.1168 q^{49} -60.5841i q^{53} +3.35053 q^{55} -76.1726i q^{59} +5.35053 q^{61} -84.1880i q^{65} +109.701 q^{67} +16.0309i q^{71} -4.35053 q^{73} +4.03357i q^{77} -1.58422 q^{79} -8.45787i q^{83} -60.7011 q^{85} -64.1236i q^{89} +101.351 q^{91} +61.4690i q^{95} +115.234 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{7} - 10 q^{13} - 2 q^{19} - 26 q^{25} - 14 q^{31} - 64 q^{37} - 92 q^{43} + 102 q^{49} - 90 q^{55} - 82 q^{61} + 232 q^{67} + 86 q^{73} + 166 q^{79} - 36 q^{85} + 302 q^{91} + 392 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(5\) 7.57301i 1.51460i 0.653066 + 0.757301i \(0.273483\pi\)
−0.653066 + 0.757301i \(0.726517\pi\)
\(6\) 0 0
\(7\) −9.11684 −1.30241 −0.651203 0.758903i \(-0.725735\pi\)
−0.651203 + 0.758903i \(0.725735\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 0.442430i − 0.0402210i −0.999798 0.0201105i \(-0.993598\pi\)
0.999798 0.0201105i \(-0.00640179\pi\)
\(12\) 0 0
\(13\) −11.1168 −0.855142 −0.427571 0.903982i \(-0.640631\pi\)
−0.427571 + 0.903982i \(0.640631\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.01544i 0.471497i 0.971814 + 0.235748i \(0.0757541\pi\)
−0.971814 + 0.235748i \(0.924246\pi\)
\(18\) 0 0
\(19\) 8.11684 0.427202 0.213601 0.976921i \(-0.431481\pi\)
0.213601 + 0.976921i \(0.431481\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 23.6039i − 1.02626i −0.858312 0.513128i \(-0.828487\pi\)
0.858312 0.513128i \(-0.171513\pi\)
\(24\) 0 0
\(25\) −32.3505 −1.29402
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 53.0111i 1.82797i 0.405749 + 0.913984i \(0.367011\pi\)
−0.405749 + 0.913984i \(0.632989\pi\)
\(30\) 0 0
\(31\) −29.3505 −0.946791 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 69.0420i − 1.97263i
\(36\) 0 0
\(37\) 18.4674 0.499118 0.249559 0.968360i \(-0.419714\pi\)
0.249559 + 0.968360i \(0.419714\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 44.9956i − 1.09745i −0.836001 0.548727i \(-0.815112\pi\)
0.836001 0.548727i \(-0.184888\pi\)
\(42\) 0 0
\(43\) −23.0000 −0.534884 −0.267442 0.963574i \(-0.586178\pi\)
−0.267442 + 0.963574i \(0.586178\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 8.45787i − 0.179955i −0.995944 0.0899774i \(-0.971321\pi\)
0.995944 0.0899774i \(-0.0286795\pi\)
\(48\) 0 0
\(49\) 34.1168 0.696262
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 60.5841i − 1.14310i −0.820569 0.571548i \(-0.806343\pi\)
0.820569 0.571548i \(-0.193657\pi\)
\(54\) 0 0
\(55\) 3.35053 0.0609188
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 76.1726i − 1.29106i −0.763735 0.645530i \(-0.776636\pi\)
0.763735 0.645530i \(-0.223364\pi\)
\(60\) 0 0
\(61\) 5.35053 0.0877136 0.0438568 0.999038i \(-0.486035\pi\)
0.0438568 + 0.999038i \(0.486035\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 84.1880i − 1.29520i
\(66\) 0 0
\(67\) 109.701 1.63733 0.818665 0.574272i \(-0.194715\pi\)
0.818665 + 0.574272i \(0.194715\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 16.0309i 0.225787i 0.993607 + 0.112894i \(0.0360119\pi\)
−0.993607 + 0.112894i \(0.963988\pi\)
\(72\) 0 0
\(73\) −4.35053 −0.0595963 −0.0297982 0.999556i \(-0.509486\pi\)
−0.0297982 + 0.999556i \(0.509486\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.03357i 0.0523840i
\(78\) 0 0
\(79\) −1.58422 −0.0200534 −0.0100267 0.999950i \(-0.503192\pi\)
−0.0100267 + 0.999950i \(0.503192\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 8.45787i − 0.101902i −0.998701 0.0509511i \(-0.983775\pi\)
0.998701 0.0509511i \(-0.0162252\pi\)
\(84\) 0 0
\(85\) −60.7011 −0.714130
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 64.1236i − 0.720489i −0.932858 0.360245i \(-0.882693\pi\)
0.932858 0.360245i \(-0.117307\pi\)
\(90\) 0 0
\(91\) 101.351 1.11374
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 61.4690i 0.647042i
\(96\) 0 0
\(97\) 115.234 1.18798 0.593988 0.804474i \(-0.297553\pi\)
0.593988 + 0.804474i \(0.297553\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 132.281i − 1.30971i −0.755755 0.654855i \(-0.772730\pi\)
0.755755 0.654855i \(-0.227270\pi\)
\(102\) 0 0
\(103\) −125.351 −1.21700 −0.608498 0.793556i \(-0.708228\pi\)
−0.608498 + 0.793556i \(0.708228\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 36.5378i 0.341475i 0.985317 + 0.170737i \(0.0546149\pi\)
−0.985317 + 0.170737i \(0.945385\pi\)
\(108\) 0 0
\(109\) 134.701 1.23579 0.617895 0.786261i \(-0.287986\pi\)
0.617895 + 0.786261i \(0.287986\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 190.210i − 1.68328i −0.540042 0.841638i \(-0.681592\pi\)
0.540042 0.841638i \(-0.318408\pi\)
\(114\) 0 0
\(115\) 178.753 1.55437
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 73.0756i − 0.614080i
\(120\) 0 0
\(121\) 120.804 0.998382
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 55.6657i − 0.445325i
\(126\) 0 0
\(127\) −184.103 −1.44963 −0.724816 0.688943i \(-0.758075\pi\)
−0.724816 + 0.688943i \(0.758075\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 126.087i 0.962493i 0.876585 + 0.481247i \(0.159816\pi\)
−0.876585 + 0.481247i \(0.840184\pi\)
\(132\) 0 0
\(133\) −74.0000 −0.556391
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 124.265i − 0.907045i −0.891245 0.453523i \(-0.850167\pi\)
0.891245 0.453523i \(-0.149833\pi\)
\(138\) 0 0
\(139\) −26.7663 −0.192563 −0.0962817 0.995354i \(-0.530695\pi\)
−0.0962817 + 0.995354i \(0.530695\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.91843i 0.0343946i
\(144\) 0 0
\(145\) −401.454 −2.76865
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 80.6486i 0.541266i 0.962683 + 0.270633i \(0.0872329\pi\)
−0.962683 + 0.270633i \(0.912767\pi\)
\(150\) 0 0
\(151\) −99.9484 −0.661910 −0.330955 0.943647i \(-0.607371\pi\)
−0.330955 + 0.943647i \(0.607371\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 222.272i − 1.43401i
\(156\) 0 0
\(157\) −69.4537 −0.442380 −0.221190 0.975231i \(-0.570994\pi\)
−0.221190 + 0.975231i \(0.570994\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 215.193i 1.33660i
\(162\) 0 0
\(163\) −162.467 −0.996732 −0.498366 0.866967i \(-0.666066\pi\)
−0.498366 + 0.866967i \(0.666066\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 96.6795i − 0.578919i −0.957190 0.289459i \(-0.906524\pi\)
0.957190 0.289459i \(-0.0934755\pi\)
\(168\) 0 0
\(169\) −45.4158 −0.268732
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 28.0282i 0.162013i 0.996714 + 0.0810064i \(0.0258134\pi\)
−0.996714 + 0.0810064i \(0.974187\pi\)
\(174\) 0 0
\(175\) 294.935 1.68534
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 35.6012i − 0.198889i −0.995043 0.0994447i \(-0.968293\pi\)
0.995043 0.0994447i \(-0.0317067\pi\)
\(180\) 0 0
\(181\) −19.6358 −0.108485 −0.0542426 0.998528i \(-0.517274\pi\)
−0.0542426 + 0.998528i \(0.517274\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 139.854i 0.755966i
\(186\) 0 0
\(187\) 3.54628 0.0189640
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 217.848i 1.14056i 0.821449 + 0.570282i \(0.193166\pi\)
−0.821449 + 0.570282i \(0.806834\pi\)
\(192\) 0 0
\(193\) −49.0000 −0.253886 −0.126943 0.991910i \(-0.540517\pi\)
−0.126943 + 0.991910i \(0.540517\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 359.965i − 1.82723i −0.406575 0.913617i \(-0.633277\pi\)
0.406575 0.913617i \(-0.366723\pi\)
\(198\) 0 0
\(199\) 61.0652 0.306861 0.153430 0.988159i \(-0.450968\pi\)
0.153430 + 0.988159i \(0.450968\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 483.294i − 2.38076i
\(204\) 0 0
\(205\) 340.753 1.66221
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 3.59114i − 0.0171825i
\(210\) 0 0
\(211\) −386.986 −1.83406 −0.917029 0.398820i \(-0.869420\pi\)
−0.917029 + 0.398820i \(0.869420\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 174.179i − 0.810136i
\(216\) 0 0
\(217\) 267.584 1.23311
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 89.1064i − 0.403197i
\(222\) 0 0
\(223\) −102.753 −0.460774 −0.230387 0.973099i \(-0.573999\pi\)
−0.230387 + 0.973099i \(0.573999\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 338.964i − 1.49323i −0.665254 0.746617i \(-0.731677\pi\)
0.665254 0.746617i \(-0.268323\pi\)
\(228\) 0 0
\(229\) −296.753 −1.29586 −0.647932 0.761699i \(-0.724366\pi\)
−0.647932 + 0.761699i \(0.724366\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 346.537i 1.48728i 0.668578 + 0.743642i \(0.266903\pi\)
−0.668578 + 0.743642i \(0.733097\pi\)
\(234\) 0 0
\(235\) 64.0516 0.272560
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 161.688i 0.676518i 0.941053 + 0.338259i \(0.109838\pi\)
−0.941053 + 0.338259i \(0.890162\pi\)
\(240\) 0 0
\(241\) −324.739 −1.34746 −0.673732 0.738975i \(-0.735310\pi\)
−0.673732 + 0.738975i \(0.735310\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 258.367i 1.05456i
\(246\) 0 0
\(247\) −90.2337 −0.365319
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 384.012i − 1.52993i −0.644074 0.764963i \(-0.722757\pi\)
0.644074 0.764963i \(-0.277243\pi\)
\(252\) 0 0
\(253\) −10.4431 −0.0412770
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 12.9339i − 0.0503264i −0.999683 0.0251632i \(-0.991989\pi\)
0.999683 0.0251632i \(-0.00801053\pi\)
\(258\) 0 0
\(259\) −168.364 −0.650055
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 49.4716i − 0.188105i −0.995567 0.0940526i \(-0.970018\pi\)
0.995567 0.0940526i \(-0.0299822\pi\)
\(264\) 0 0
\(265\) 458.804 1.73134
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 21.4434i − 0.0797154i −0.999205 0.0398577i \(-0.987310\pi\)
0.999205 0.0398577i \(-0.0126905\pi\)
\(270\) 0 0
\(271\) 326.907 1.20630 0.603150 0.797628i \(-0.293912\pi\)
0.603150 + 0.797628i \(0.293912\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 14.3129i 0.0520468i
\(276\) 0 0
\(277\) −477.454 −1.72366 −0.861830 0.507198i \(-0.830681\pi\)
−0.861830 + 0.507198i \(0.830681\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 119.008i 0.423515i 0.977322 + 0.211758i \(0.0679187\pi\)
−0.977322 + 0.211758i \(0.932081\pi\)
\(282\) 0 0
\(283\) 391.351 1.38286 0.691432 0.722442i \(-0.256980\pi\)
0.691432 + 0.722442i \(0.256980\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 410.218i 1.42933i
\(288\) 0 0
\(289\) 224.753 0.777691
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 71.6966i 0.244698i 0.992487 + 0.122349i \(0.0390427\pi\)
−0.992487 + 0.122349i \(0.960957\pi\)
\(294\) 0 0
\(295\) 576.856 1.95544
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 262.401i 0.877595i
\(300\) 0 0
\(301\) 209.687 0.696636
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 40.5196i 0.132851i
\(306\) 0 0
\(307\) 172.351 0.561402 0.280701 0.959795i \(-0.409433\pi\)
0.280701 + 0.959795i \(0.409433\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 605.347i 1.94645i 0.229848 + 0.973227i \(0.426177\pi\)
−0.229848 + 0.973227i \(0.573823\pi\)
\(312\) 0 0
\(313\) 327.467 1.04622 0.523111 0.852265i \(-0.324771\pi\)
0.523111 + 0.852265i \(0.324771\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 69.0420i 0.217798i 0.994053 + 0.108899i \(0.0347325\pi\)
−0.994053 + 0.108899i \(0.965267\pi\)
\(318\) 0 0
\(319\) 23.4537 0.0735226
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 65.0601i 0.201424i
\(324\) 0 0
\(325\) 359.636 1.10657
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 77.1091i 0.234374i
\(330\) 0 0
\(331\) −509.791 −1.54015 −0.770076 0.637952i \(-0.779782\pi\)
−0.770076 + 0.637952i \(0.779782\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 830.768i 2.47990i
\(336\) 0 0
\(337\) −337.440 −1.00131 −0.500653 0.865648i \(-0.666907\pi\)
−0.500653 + 0.865648i \(0.666907\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 12.9856i 0.0380808i
\(342\) 0 0
\(343\) 135.687 0.395590
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 215.245i − 0.620302i −0.950687 0.310151i \(-0.899620\pi\)
0.950687 0.310151i \(-0.100380\pi\)
\(348\) 0 0
\(349\) 362.024 1.03732 0.518659 0.854981i \(-0.326431\pi\)
0.518659 + 0.854981i \(0.326431\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 584.840i 1.65677i 0.560159 + 0.828385i \(0.310740\pi\)
−0.560159 + 0.828385i \(0.689260\pi\)
\(354\) 0 0
\(355\) −121.402 −0.341978
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 393.693i 1.09664i 0.836269 + 0.548319i \(0.184732\pi\)
−0.836269 + 0.548319i \(0.815268\pi\)
\(360\) 0 0
\(361\) −295.117 −0.817498
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 32.9466i − 0.0902648i
\(366\) 0 0
\(367\) −380.856 −1.03775 −0.518877 0.854849i \(-0.673650\pi\)
−0.518877 + 0.854849i \(0.673650\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 552.336i 1.48878i
\(372\) 0 0
\(373\) 132.883 0.356255 0.178128 0.984007i \(-0.442996\pi\)
0.178128 + 0.984007i \(0.442996\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 589.316i − 1.56317i
\(378\) 0 0
\(379\) −507.622 −1.33937 −0.669686 0.742644i \(-0.733571\pi\)
−0.669686 + 0.742644i \(0.733571\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 331.937i − 0.866676i −0.901231 0.433338i \(-0.857336\pi\)
0.901231 0.433338i \(-0.142664\pi\)
\(384\) 0 0
\(385\) −30.5463 −0.0793410
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 342.555i − 0.880605i −0.897850 0.440302i \(-0.854871\pi\)
0.897850 0.440302i \(-0.145129\pi\)
\(390\) 0 0
\(391\) 189.196 0.483877
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 11.9973i − 0.0303730i
\(396\) 0 0
\(397\) −24.8043 −0.0624792 −0.0312396 0.999512i \(-0.509945\pi\)
−0.0312396 + 0.999512i \(0.509945\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 60.1417i 0.149979i 0.997184 + 0.0749896i \(0.0238924\pi\)
−0.997184 + 0.0749896i \(0.976108\pi\)
\(402\) 0 0
\(403\) 326.285 0.809641
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 8.17053i − 0.0200750i
\(408\) 0 0
\(409\) −481.440 −1.17712 −0.588558 0.808455i \(-0.700304\pi\)
−0.588558 + 0.808455i \(0.700304\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 694.453i 1.68149i
\(414\) 0 0
\(415\) 64.0516 0.154341
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 553.715i 1.32152i 0.750599 + 0.660758i \(0.229765\pi\)
−0.750599 + 0.660758i \(0.770235\pi\)
\(420\) 0 0
\(421\) −381.894 −0.907111 −0.453556 0.891228i \(-0.649845\pi\)
−0.453556 + 0.891228i \(0.649845\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 259.304i − 0.610127i
\(426\) 0 0
\(427\) −48.7800 −0.114239
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 821.321i 1.90562i 0.303570 + 0.952809i \(0.401821\pi\)
−0.303570 + 0.952809i \(0.598179\pi\)
\(432\) 0 0
\(433\) −199.155 −0.459942 −0.229971 0.973198i \(-0.573863\pi\)
−0.229971 + 0.973198i \(0.573863\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 191.589i − 0.438419i
\(438\) 0 0
\(439\) 481.660 1.09718 0.548588 0.836093i \(-0.315166\pi\)
0.548588 + 0.836093i \(0.315166\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 540.287i − 1.21961i −0.792552 0.609805i \(-0.791248\pi\)
0.792552 0.609805i \(-0.208752\pi\)
\(444\) 0 0
\(445\) 485.609 1.09126
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 300.318i − 0.668859i −0.942421 0.334429i \(-0.891456\pi\)
0.942421 0.334429i \(-0.108544\pi\)
\(450\) 0 0
\(451\) −19.9074 −0.0441407
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 767.529i 1.68688i
\(456\) 0 0
\(457\) 155.701 0.340703 0.170351 0.985383i \(-0.445510\pi\)
0.170351 + 0.985383i \(0.445510\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 301.542i − 0.654103i −0.945006 0.327052i \(-0.893945\pi\)
0.945006 0.327052i \(-0.106055\pi\)
\(462\) 0 0
\(463\) −238.780 −0.515723 −0.257862 0.966182i \(-0.583018\pi\)
−0.257862 + 0.966182i \(0.583018\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 423.152i − 0.906107i −0.891483 0.453054i \(-0.850335\pi\)
0.891483 0.453054i \(-0.149665\pi\)
\(468\) 0 0
\(469\) −1000.13 −2.13247
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 10.1759i 0.0215135i
\(474\) 0 0
\(475\) −262.584 −0.552809
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 437.959i − 0.914320i −0.889385 0.457160i \(-0.848867\pi\)
0.889385 0.457160i \(-0.151133\pi\)
\(480\) 0 0
\(481\) −205.299 −0.426817
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 872.666i 1.79931i
\(486\) 0 0
\(487\) −401.945 −0.825350 −0.412675 0.910878i \(-0.635405\pi\)
−0.412675 + 0.910878i \(0.635405\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 278.380i 0.566966i 0.958977 + 0.283483i \(0.0914899\pi\)
−0.958977 + 0.283483i \(0.908510\pi\)
\(492\) 0 0
\(493\) −424.907 −0.861881
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 146.151i − 0.294067i
\(498\) 0 0
\(499\) −545.310 −1.09280 −0.546402 0.837523i \(-0.684003\pi\)
−0.546402 + 0.837523i \(0.684003\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 306.460i − 0.609264i −0.952470 0.304632i \(-0.901466\pi\)
0.952470 0.304632i \(-0.0985335\pi\)
\(504\) 0 0
\(505\) 1001.76 1.98369
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 554.496i 1.08938i 0.838636 + 0.544692i \(0.183353\pi\)
−0.838636 + 0.544692i \(0.816647\pi\)
\(510\) 0 0
\(511\) 39.6631 0.0776186
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 949.281i − 1.84326i
\(516\) 0 0
\(517\) −3.74202 −0.00723795
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 154.167i − 0.295905i −0.988994 0.147953i \(-0.952732\pi\)
0.988994 0.147953i \(-0.0472683\pi\)
\(522\) 0 0
\(523\) −480.598 −0.918925 −0.459463 0.888197i \(-0.651958\pi\)
−0.459463 + 0.888197i \(0.651958\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 235.258i − 0.446409i
\(528\) 0 0
\(529\) −28.1441 −0.0532026
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 500.210i 0.938480i
\(534\) 0 0
\(535\) −276.701 −0.517198
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 15.0943i − 0.0280043i
\(540\) 0 0
\(541\) −300.543 −0.555533 −0.277766 0.960649i \(-0.589594\pi\)
−0.277766 + 0.960649i \(0.589594\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1020.09i 1.87173i
\(546\) 0 0
\(547\) 101.206 0.185021 0.0925104 0.995712i \(-0.470511\pi\)
0.0925104 + 0.995712i \(0.470511\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 430.283i 0.780912i
\(552\) 0 0
\(553\) 14.4431 0.0261177
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 433.041i 0.777452i 0.921353 + 0.388726i \(0.127085\pi\)
−0.921353 + 0.388726i \(0.872915\pi\)
\(558\) 0 0
\(559\) 255.687 0.457401
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 1041.77i − 1.85039i −0.379487 0.925197i \(-0.623899\pi\)
0.379487 0.925197i \(-0.376101\pi\)
\(564\) 0 0
\(565\) 1440.46 2.54949
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 297.272i − 0.522447i −0.965278 0.261223i \(-0.915874\pi\)
0.965278 0.261223i \(-0.0841260\pi\)
\(570\) 0 0
\(571\) 679.049 1.18923 0.594613 0.804012i \(-0.297305\pi\)
0.594613 + 0.804012i \(0.297305\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 763.599i 1.32800i
\(576\) 0 0
\(577\) −148.351 −0.257107 −0.128553 0.991703i \(-0.541033\pi\)
−0.128553 + 0.991703i \(0.541033\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 77.1091i 0.132718i
\(582\) 0 0
\(583\) −26.8043 −0.0459764
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 527.117i 0.897985i 0.893535 + 0.448993i \(0.148217\pi\)
−0.893535 + 0.448993i \(0.851783\pi\)
\(588\) 0 0
\(589\) −238.234 −0.404471
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 473.848i 0.799069i 0.916718 + 0.399534i \(0.130828\pi\)
−0.916718 + 0.399534i \(0.869172\pi\)
\(594\) 0 0
\(595\) 553.402 0.930088
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 694.453i 1.15935i 0.814846 + 0.579677i \(0.196821\pi\)
−0.814846 + 0.579677i \(0.803179\pi\)
\(600\) 0 0
\(601\) 186.712 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 914.852i 1.51215i
\(606\) 0 0
\(607\) 822.388 1.35484 0.677420 0.735596i \(-0.263098\pi\)
0.677420 + 0.735596i \(0.263098\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 94.0249i 0.153887i
\(612\) 0 0
\(613\) −482.206 −0.786634 −0.393317 0.919403i \(-0.628672\pi\)
−0.393317 + 0.919403i \(0.628672\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 688.104i − 1.11524i −0.830096 0.557621i \(-0.811714\pi\)
0.830096 0.557621i \(-0.188286\pi\)
\(618\) 0 0
\(619\) 760.505 1.22860 0.614302 0.789071i \(-0.289438\pi\)
0.614302 + 0.789071i \(0.289438\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 584.604i 0.938370i
\(624\) 0 0
\(625\) −387.206 −0.619530
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 148.024i 0.235333i
\(630\) 0 0
\(631\) −1008.08 −1.59758 −0.798792 0.601607i \(-0.794527\pi\)
−0.798792 + 0.601607i \(0.794527\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 1394.22i − 2.19562i
\(636\) 0 0
\(637\) −379.272 −0.595403
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 563.603i 0.879256i 0.898180 + 0.439628i \(0.144890\pi\)
−0.898180 + 0.439628i \(0.855110\pi\)
\(642\) 0 0
\(643\) 577.000 0.897356 0.448678 0.893693i \(-0.351895\pi\)
0.448678 + 0.893693i \(0.351895\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 1024.52i − 1.58349i −0.610853 0.791744i \(-0.709173\pi\)
0.610853 0.791744i \(-0.290827\pi\)
\(648\) 0 0
\(649\) −33.7011 −0.0519277
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 399.393i 0.611628i 0.952091 + 0.305814i \(0.0989286\pi\)
−0.952091 + 0.305814i \(0.901071\pi\)
\(654\) 0 0
\(655\) −954.856 −1.45780
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 746.292i − 1.13246i −0.824247 0.566231i \(-0.808401\pi\)
0.824247 0.566231i \(-0.191599\pi\)
\(660\) 0 0
\(661\) 951.247 1.43910 0.719552 0.694439i \(-0.244347\pi\)
0.719552 + 0.694439i \(0.244347\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 560.403i − 0.842711i
\(666\) 0 0
\(667\) 1251.27 1.87596
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 2.36724i − 0.00352793i
\(672\) 0 0
\(673\) −344.231 −0.511487 −0.255743 0.966745i \(-0.582320\pi\)
−0.255743 + 0.966745i \(0.582320\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 985.664i − 1.45593i −0.685615 0.727965i \(-0.740467\pi\)
0.685615 0.727965i \(-0.259533\pi\)
\(678\) 0 0
\(679\) −1050.57 −1.54723
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 166.658i 0.244009i 0.992530 + 0.122004i \(0.0389322\pi\)
−0.992530 + 0.122004i \(0.961068\pi\)
\(684\) 0 0
\(685\) 941.062 1.37381
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 673.504i 0.977510i
\(690\) 0 0
\(691\) 898.155 1.29979 0.649895 0.760024i \(-0.274813\pi\)
0.649895 + 0.760024i \(0.274813\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 202.702i − 0.291657i
\(696\) 0 0
\(697\) 360.660 0.517446
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 730.549i − 1.04215i −0.853510 0.521076i \(-0.825531\pi\)
0.853510 0.521076i \(-0.174469\pi\)
\(702\) 0 0
\(703\) 149.897 0.213224
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1205.98i 1.70577i
\(708\) 0 0
\(709\) −229.921 −0.324289 −0.162145 0.986767i \(-0.551841\pi\)
−0.162145 + 0.986767i \(0.551841\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 692.787i 0.971651i
\(714\) 0 0
\(715\) −37.2473 −0.0520942
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 907.095i − 1.26161i −0.775943 0.630803i \(-0.782725\pi\)
0.775943 0.630803i \(-0.217275\pi\)
\(720\) 0 0
\(721\) 1142.80 1.58502
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 1714.94i − 2.36543i
\(726\) 0 0
\(727\) −215.742 −0.296757 −0.148378 0.988931i \(-0.547405\pi\)
−0.148378 + 0.988931i \(0.547405\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 184.355i − 0.252196i
\(732\) 0 0
\(733\) 629.269 0.858484 0.429242 0.903190i \(-0.358781\pi\)
0.429242 + 0.903190i \(0.358781\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 48.5351i − 0.0658549i
\(738\) 0 0
\(739\) −547.649 −0.741068 −0.370534 0.928819i \(-0.620825\pi\)
−0.370534 + 0.928819i \(0.620825\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 534.145i 0.718902i 0.933164 + 0.359451i \(0.117036\pi\)
−0.933164 + 0.359451i \(0.882964\pi\)
\(744\) 0 0
\(745\) −610.753 −0.819802
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 333.109i − 0.444739i
\(750\) 0 0
\(751\) 451.090 0.600652 0.300326 0.953837i \(-0.402905\pi\)
0.300326 + 0.953837i \(0.402905\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 756.911i − 1.00253i
\(756\) 0 0
\(757\) −352.391 −0.465511 −0.232755 0.972535i \(-0.574774\pi\)
−0.232755 + 0.972535i \(0.574774\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 1073.78i − 1.41101i −0.708703 0.705507i \(-0.750719\pi\)
0.708703 0.705507i \(-0.249281\pi\)
\(762\) 0 0
\(763\) −1228.05 −1.60950
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 846.799i 1.10404i
\(768\) 0 0
\(769\) 355.976 0.462907 0.231454 0.972846i \(-0.425652\pi\)
0.231454 + 0.972846i \(0.425652\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 370.790i − 0.479677i −0.970813 0.239838i \(-0.922906\pi\)
0.970813 0.239838i \(-0.0770945\pi\)
\(774\) 0 0
\(775\) 949.505 1.22517
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 365.223i − 0.468835i
\(780\) 0 0
\(781\) 7.09255 0.00908137
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 525.974i − 0.670031i
\(786\) 0 0
\(787\) −1153.06 −1.46514 −0.732568 0.680694i \(-0.761678\pi\)
−0.732568 + 0.680694i \(0.761678\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1734.12i 2.19231i
\(792\) 0 0
\(793\) −59.4810 −0.0750076
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 879.745i − 1.10382i −0.833903 0.551910i \(-0.813899\pi\)
0.833903 0.551910i \(-0.186101\pi\)
\(798\) 0 0
\(799\) 67.7936 0.0848481
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.92481i 0.00239702i
\(804\) 0 0
\(805\) −1629.66 −2.02442
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 884.508i − 1.09334i −0.837350 0.546668i \(-0.815896\pi\)
0.837350 0.546668i \(-0.184104\pi\)
\(810\) 0 0
\(811\) 961.464 1.18553 0.592765 0.805376i \(-0.298036\pi\)
0.592765 + 0.805376i \(0.298036\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1230.37i − 1.50965i
\(816\) 0 0
\(817\) −186.687 −0.228504
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 898.431i − 1.09431i −0.837030 0.547156i \(-0.815710\pi\)
0.837030 0.547156i \(-0.184290\pi\)
\(822\) 0 0
\(823\) −216.182 −0.262676 −0.131338 0.991338i \(-0.541927\pi\)
−0.131338 + 0.991338i \(0.541927\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 113.883i 0.137706i 0.997627 + 0.0688528i \(0.0219339\pi\)
−0.997627 + 0.0688528i \(0.978066\pi\)
\(828\) 0 0
\(829\) 101.326 0.122227 0.0611135 0.998131i \(-0.480535\pi\)
0.0611135 + 0.998131i \(0.480535\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 273.462i 0.328285i
\(834\) 0 0
\(835\) 732.155 0.876832
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 69.8235i 0.0832222i 0.999134 + 0.0416111i \(0.0132491\pi\)
−0.999134 + 0.0416111i \(0.986751\pi\)
\(840\) 0 0
\(841\) −1969.18 −2.34147
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 343.934i − 0.407023i
\(846\) 0 0
\(847\) −1101.35 −1.30030
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 435.902i − 0.512223i
\(852\) 0 0
\(853\) −1146.65 −1.34425 −0.672127 0.740435i \(-0.734619\pi\)
−0.672127 + 0.740435i \(0.734619\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 251.576i − 0.293554i −0.989170 0.146777i \(-0.953110\pi\)
0.989170 0.146777i \(-0.0468900\pi\)
\(858\) 0 0
\(859\) 488.533 0.568722 0.284361 0.958717i \(-0.408218\pi\)
0.284361 + 0.958717i \(0.408218\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 596.889i − 0.691644i −0.938300 0.345822i \(-0.887600\pi\)
0.938300 0.345822i \(-0.112400\pi\)
\(864\) 0 0
\(865\) −212.258 −0.245385
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0.700907i 0 0.000806567i
\(870\) 0 0
\(871\) −1219.53 −1.40015
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 507.495i 0.579995i
\(876\) 0 0
\(877\) 716.416 0.816894 0.408447 0.912782i \(-0.366071\pi\)
0.408447 + 0.912782i \(0.366071\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 1200.86i − 1.36306i −0.731790 0.681531i \(-0.761315\pi\)
0.731790 0.681531i \(-0.238685\pi\)
\(882\) 0 0
\(883\) −22.8938 −0.0259273 −0.0129636 0.999916i \(-0.504127\pi\)
−0.0129636 + 0.999916i \(0.504127\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 602.589i − 0.679356i −0.940542 0.339678i \(-0.889682\pi\)
0.940542 0.339678i \(-0.110318\pi\)
\(888\) 0 0
\(889\) 1678.44 1.88801
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 68.6512i − 0.0768771i
\(894\) 0 0
\(895\) 269.609 0.301239
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 1555.90i − 1.73071i
\(900\) 0 0
\(901\) 485.609 0.538966
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 148.702i − 0.164312i
\(906\) 0 0
\(907\) −422.945 −0.466312 −0.233156 0.972439i \(-0.574905\pi\)
−0.233156 + 0.972439i \(0.574905\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 144.772i 0.158916i 0.996838 + 0.0794578i \(0.0253189\pi\)
−0.996838 + 0.0794578i \(0.974681\pi\)
\(912\) 0 0
\(913\) −3.74202 −0.00409860
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1149.51i − 1.25356i
\(918\) 0 0
\(919\) 869.093 0.945694 0.472847 0.881145i \(-0.343226\pi\)
0.472847 + 0.881145i \(0.343226\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 178.213i − 0.193080i
\(924\) 0 0
\(925\) −597.429 −0.645870
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 95.0131i − 0.102275i −0.998692 0.0511373i \(-0.983715\pi\)
0.998692 0.0511373i \(-0.0162846\pi\)
\(930\) 0 0
\(931\) 276.921 0.297445
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 26.8560i 0.0287230i
\(936\) 0 0
\(937\) −1555.55 −1.66014 −0.830071 0.557657i \(-0.811700\pi\)
−0.830071 + 0.557657i \(0.811700\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 387.109i 0.411380i 0.978617 + 0.205690i \(0.0659438\pi\)
−0.978617 + 0.205690i \(0.934056\pi\)
\(942\) 0 0
\(943\) −1062.07 −1.12627
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 969.788i − 1.02406i −0.858967 0.512032i \(-0.828893\pi\)
0.858967 0.512032i \(-0.171107\pi\)
\(948\) 0 0
\(949\) 48.3642 0.0509633
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1294.65i 1.35850i 0.733909 + 0.679248i \(0.237694\pi\)
−0.733909 + 0.679248i \(0.762306\pi\)
\(954\) 0 0
\(955\) −1649.76 −1.72750
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1132.91i 1.18134i
\(960\) 0 0
\(961\) −99.5463 −0.103586
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 371.078i − 0.384536i
\(966\) 0 0
\(967\) −824.073 −0.852195 −0.426098 0.904677i \(-0.640112\pi\)
−0.426098 + 0.904677i \(0.640112\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1518.35i 1.56370i 0.623469 + 0.781848i \(0.285723\pi\)
−0.623469 + 0.781848i \(0.714277\pi\)
\(972\) 0 0
\(973\) 244.024 0.250796
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 371.233i − 0.379972i −0.981787 0.189986i \(-0.939156\pi\)
0.981787 0.189986i \(-0.0608443\pi\)
\(978\) 0 0
\(979\) −28.3702 −0.0289788
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 933.044i − 0.949180i −0.880207 0.474590i \(-0.842596\pi\)
0.880207 0.474590i \(-0.157404\pi\)
\(984\) 0 0
\(985\) 2726.02 2.76753
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 542.890i 0.548928i
\(990\) 0 0
\(991\) −1615.53 −1.63020 −0.815099 0.579321i \(-0.803318\pi\)
−0.815099 + 0.579321i \(0.803318\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 462.448i 0.464772i
\(996\) 0 0
\(997\) 1038.75 1.04188 0.520939 0.853594i \(-0.325582\pi\)
0.520939 + 0.853594i \(0.325582\pi\)
\(998\) 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 1296.3.e.e.161.4 4
3.2 odd 2 inner 1296.3.e.e.161.1 4
4.3 odd 2 324.3.c.b.161.4 4
9.2 odd 6 144.3.q.b.113.2 4
9.4 even 3 144.3.q.b.65.2 4
9.5 odd 6 432.3.q.b.305.1 4
9.7 even 3 432.3.q.b.17.1 4
12.11 even 2 324.3.c.b.161.1 4
36.7 odd 6 108.3.g.a.17.1 4
36.11 even 6 36.3.g.a.5.1 4
36.23 even 6 108.3.g.a.89.1 4
36.31 odd 6 36.3.g.a.29.1 yes 4
72.5 odd 6 1728.3.q.h.1601.2 4
72.11 even 6 576.3.q.d.257.2 4
72.13 even 6 576.3.q.g.65.1 4
72.29 odd 6 576.3.q.g.257.1 4
72.43 odd 6 1728.3.q.g.449.2 4
72.59 even 6 1728.3.q.g.1601.2 4
72.61 even 6 1728.3.q.h.449.2 4
72.67 odd 6 576.3.q.d.65.2 4
180.7 even 12 2700.3.u.b.449.4 8
180.23 odd 12 2700.3.u.b.2249.4 8
180.43 even 12 2700.3.u.b.449.1 8
180.47 odd 12 900.3.u.a.149.4 8
180.59 even 6 2700.3.p.b.1601.2 4
180.67 even 12 900.3.u.a.749.1 8
180.79 odd 6 2700.3.p.b.2501.2 4
180.83 odd 12 900.3.u.a.149.1 8
180.103 even 12 900.3.u.a.749.4 8
180.119 even 6 900.3.p.a.401.2 4
180.139 odd 6 900.3.p.a.101.2 4
180.167 odd 12 2700.3.u.b.2249.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.g.a.5.1 4 36.11 even 6
36.3.g.a.29.1 yes 4 36.31 odd 6
108.3.g.a.17.1 4 36.7 odd 6
108.3.g.a.89.1 4 36.23 even 6
144.3.q.b.65.2 4 9.4 even 3
144.3.q.b.113.2 4 9.2 odd 6
324.3.c.b.161.1 4 12.11 even 2
324.3.c.b.161.4 4 4.3 odd 2
432.3.q.b.17.1 4 9.7 even 3
432.3.q.b.305.1 4 9.5 odd 6
576.3.q.d.65.2 4 72.67 odd 6
576.3.q.d.257.2 4 72.11 even 6
576.3.q.g.65.1 4 72.13 even 6
576.3.q.g.257.1 4 72.29 odd 6
900.3.p.a.101.2 4 180.139 odd 6
900.3.p.a.401.2 4 180.119 even 6
900.3.u.a.149.1 8 180.83 odd 12
900.3.u.a.149.4 8 180.47 odd 12
900.3.u.a.749.1 8 180.67 even 12
900.3.u.a.749.4 8 180.103 even 12
1296.3.e.e.161.1 4 3.2 odd 2 inner
1296.3.e.e.161.4 4 1.1 even 1 trivial
1728.3.q.g.449.2 4 72.43 odd 6
1728.3.q.g.1601.2 4 72.59 even 6
1728.3.q.h.449.2 4 72.61 even 6
1728.3.q.h.1601.2 4 72.5 odd 6
2700.3.p.b.1601.2 4 180.59 even 6
2700.3.p.b.2501.2 4 180.79 odd 6
2700.3.u.b.449.1 8 180.43 even 12
2700.3.u.b.449.4 8 180.7 even 12
2700.3.u.b.2249.1 8 180.167 odd 12
2700.3.u.b.2249.4 8 180.23 odd 12