Properties

Label 224.3.c.a.97.2
Level $224$
Weight $3$
Character 224.97
Analytic conductor $6.104$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,3,Mod(97,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, 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("224.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.c (of order \(2\), degree \(1\), 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: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4694952902656.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 20x^{6} + 56x^{4} + 20x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 97.2
Root \(-0.244539i\) of defining polynomial
Character \(\chi\) \(=\) 224.97
Dual form 224.3.c.a.97.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.43734i q^{3} +6.12900i q^{5} +(-6.21005 - 3.23038i) q^{7} -20.5647 q^{9} +O(q^{10})\) \(q-5.43734i q^{3} +6.12900i q^{5} +(-6.21005 - 3.23038i) q^{7} -20.5647 q^{9} -15.2485 q^{11} +3.00789i q^{13} +33.3255 q^{15} -21.6213i q^{17} +11.8981i q^{19} +(-17.5647 + 33.7661i) q^{21} -1.72205 q^{23} -12.5647 q^{25} +62.8810i q^{27} -41.1293 q^{29} -8.50759i q^{31} +82.9114i q^{33} +(19.7990 - 38.0614i) q^{35} +53.1293 q^{37} +16.3549 q^{39} -43.0162i q^{41} -36.6446 q^{43} -126.041i q^{45} -30.2569i q^{47} +(28.1293 + 40.1216i) q^{49} -117.563 q^{51} -30.0000 q^{53} -93.4582i q^{55} +64.6940 q^{57} -73.0525i q^{59} -5.67609i q^{61} +(127.707 + 66.4316i) q^{63} -18.4353 q^{65} +28.7750 q^{67} +9.36335i q^{69} -5.53191 q^{71} -94.9429i q^{73} +68.3183i q^{75} +(94.6940 + 49.2585i) q^{77} -81.5337 q^{79} +156.823 q^{81} +86.2943i q^{83} +132.517 q^{85} +223.634i q^{87} +27.6371i q^{89} +(9.71661 - 18.6791i) q^{91} -46.2586 q^{93} -72.9234 q^{95} -100.959i q^{97} +313.581 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 56 q^{9} - 32 q^{21} + 8 q^{25} - 112 q^{29} + 208 q^{37} + 8 q^{49} - 240 q^{53} + 192 q^{57} - 256 q^{65} + 432 q^{77} + 712 q^{81} + 192 q^{85} + 64 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 5.43734i 1.81245i −0.422800 0.906223i \(-0.638953\pi\)
0.422800 0.906223i \(-0.361047\pi\)
\(4\) 0 0
\(5\) 6.12900i 1.22580i 0.790160 + 0.612900i \(0.209997\pi\)
−0.790160 + 0.612900i \(0.790003\pi\)
\(6\) 0 0
\(7\) −6.21005 3.23038i −0.887149 0.461483i
\(8\) 0 0
\(9\) −20.5647 −2.28496
\(10\) 0 0
\(11\) −15.2485 −1.38623 −0.693114 0.720828i \(-0.743762\pi\)
−0.693114 + 0.720828i \(0.743762\pi\)
\(12\) 0 0
\(13\) 3.00789i 0.231376i 0.993286 + 0.115688i \(0.0369073\pi\)
−0.993286 + 0.115688i \(0.963093\pi\)
\(14\) 0 0
\(15\) 33.3255 2.22170
\(16\) 0 0
\(17\) 21.6213i 1.27184i −0.771753 0.635922i \(-0.780620\pi\)
0.771753 0.635922i \(-0.219380\pi\)
\(18\) 0 0
\(19\) 11.8981i 0.626216i 0.949718 + 0.313108i \(0.101370\pi\)
−0.949718 + 0.313108i \(0.898630\pi\)
\(20\) 0 0
\(21\) −17.5647 + 33.7661i −0.836412 + 1.60791i
\(22\) 0 0
\(23\) −1.72205 −0.0748715 −0.0374358 0.999299i \(-0.511919\pi\)
−0.0374358 + 0.999299i \(0.511919\pi\)
\(24\) 0 0
\(25\) −12.5647 −0.502586
\(26\) 0 0
\(27\) 62.8810i 2.32893i
\(28\) 0 0
\(29\) −41.1293 −1.41825 −0.709126 0.705082i \(-0.750910\pi\)
−0.709126 + 0.705082i \(0.750910\pi\)
\(30\) 0 0
\(31\) 8.50759i 0.274438i −0.990541 0.137219i \(-0.956184\pi\)
0.990541 0.137219i \(-0.0438165\pi\)
\(32\) 0 0
\(33\) 82.9114i 2.51247i
\(34\) 0 0
\(35\) 19.7990 38.0614i 0.565685 1.08747i
\(36\) 0 0
\(37\) 53.1293 1.43593 0.717964 0.696080i \(-0.245074\pi\)
0.717964 + 0.696080i \(0.245074\pi\)
\(38\) 0 0
\(39\) 16.3549 0.419356
\(40\) 0 0
\(41\) 43.0162i 1.04918i −0.851356 0.524588i \(-0.824219\pi\)
0.851356 0.524588i \(-0.175781\pi\)
\(42\) 0 0
\(43\) −36.6446 −0.852200 −0.426100 0.904676i \(-0.640113\pi\)
−0.426100 + 0.904676i \(0.640113\pi\)
\(44\) 0 0
\(45\) 126.041i 2.80091i
\(46\) 0 0
\(47\) 30.2569i 0.643765i −0.946780 0.321882i \(-0.895684\pi\)
0.946780 0.321882i \(-0.104316\pi\)
\(48\) 0 0
\(49\) 28.1293 + 40.1216i 0.574068 + 0.818808i
\(50\) 0 0
\(51\) −117.563 −2.30515
\(52\) 0 0
\(53\) −30.0000 −0.566038 −0.283019 0.959114i \(-0.591336\pi\)
−0.283019 + 0.959114i \(0.591336\pi\)
\(54\) 0 0
\(55\) 93.4582i 1.69924i
\(56\) 0 0
\(57\) 64.6940 1.13498
\(58\) 0 0
\(59\) 73.0525i 1.23818i −0.785321 0.619089i \(-0.787502\pi\)
0.785321 0.619089i \(-0.212498\pi\)
\(60\) 0 0
\(61\) 5.67609i 0.0930506i −0.998917 0.0465253i \(-0.985185\pi\)
0.998917 0.0465253i \(-0.0148148\pi\)
\(62\) 0 0
\(63\) 127.707 + 66.4316i 2.02710 + 1.05447i
\(64\) 0 0
\(65\) −18.4353 −0.283621
\(66\) 0 0
\(67\) 28.7750 0.429477 0.214739 0.976672i \(-0.431110\pi\)
0.214739 + 0.976672i \(0.431110\pi\)
\(68\) 0 0
\(69\) 9.36335i 0.135701i
\(70\) 0 0
\(71\) −5.53191 −0.0779142 −0.0389571 0.999241i \(-0.512404\pi\)
−0.0389571 + 0.999241i \(0.512404\pi\)
\(72\) 0 0
\(73\) 94.9429i 1.30059i −0.759683 0.650294i \(-0.774646\pi\)
0.759683 0.650294i \(-0.225354\pi\)
\(74\) 0 0
\(75\) 68.3183i 0.910911i
\(76\) 0 0
\(77\) 94.6940 + 49.2585i 1.22979 + 0.639720i
\(78\) 0 0
\(79\) −81.5337 −1.03207 −0.516036 0.856567i \(-0.672593\pi\)
−0.516036 + 0.856567i \(0.672593\pi\)
\(80\) 0 0
\(81\) 156.823 1.93609
\(82\) 0 0
\(83\) 86.2943i 1.03969i 0.854261 + 0.519845i \(0.174010\pi\)
−0.854261 + 0.519845i \(0.825990\pi\)
\(84\) 0 0
\(85\) 132.517 1.55903
\(86\) 0 0
\(87\) 223.634i 2.57051i
\(88\) 0 0
\(89\) 27.6371i 0.310529i 0.987873 + 0.155265i \(0.0496231\pi\)
−0.987873 + 0.155265i \(0.950377\pi\)
\(90\) 0 0
\(91\) 9.71661 18.6791i 0.106776 0.205265i
\(92\) 0 0
\(93\) −46.2586 −0.497405
\(94\) 0 0
\(95\) −72.9234 −0.767615
\(96\) 0 0
\(97\) 100.959i 1.04081i −0.853919 0.520406i \(-0.825781\pi\)
0.853919 0.520406i \(-0.174219\pi\)
\(98\) 0 0
\(99\) 313.581 3.16748
\(100\) 0 0
\(101\) 107.541i 1.06476i 0.846506 + 0.532379i \(0.178702\pi\)
−0.846506 + 0.532379i \(0.821298\pi\)
\(102\) 0 0
\(103\) 59.1076i 0.573860i −0.957952 0.286930i \(-0.907365\pi\)
0.957952 0.286930i \(-0.0926347\pi\)
\(104\) 0 0
\(105\) −206.953 107.654i −1.97098 1.02527i
\(106\) 0 0
\(107\) −45.9954 −0.429864 −0.214932 0.976629i \(-0.568953\pi\)
−0.214932 + 0.976629i \(0.568953\pi\)
\(108\) 0 0
\(109\) −23.7414 −0.217811 −0.108905 0.994052i \(-0.534735\pi\)
−0.108905 + 0.994052i \(0.534735\pi\)
\(110\) 0 0
\(111\) 288.882i 2.60254i
\(112\) 0 0
\(113\) −187.211 −1.65674 −0.828368 0.560184i \(-0.810731\pi\)
−0.828368 + 0.560184i \(0.810731\pi\)
\(114\) 0 0
\(115\) 10.5544i 0.0917776i
\(116\) 0 0
\(117\) 61.8562i 0.528685i
\(118\) 0 0
\(119\) −69.8451 + 134.270i −0.586934 + 1.12832i
\(120\) 0 0
\(121\) 111.517 0.921630
\(122\) 0 0
\(123\) −233.894 −1.90158
\(124\) 0 0
\(125\) 76.2162i 0.609730i
\(126\) 0 0
\(127\) 206.350 1.62480 0.812402 0.583097i \(-0.198159\pi\)
0.812402 + 0.583097i \(0.198159\pi\)
\(128\) 0 0
\(129\) 199.249i 1.54457i
\(130\) 0 0
\(131\) 186.979i 1.42732i −0.700492 0.713660i \(-0.747036\pi\)
0.700492 0.713660i \(-0.252964\pi\)
\(132\) 0 0
\(133\) 38.4353 73.8877i 0.288988 0.555547i
\(134\) 0 0
\(135\) −385.398 −2.85480
\(136\) 0 0
\(137\) −2.51728 −0.0183743 −0.00918715 0.999958i \(-0.502924\pi\)
−0.00918715 + 0.999958i \(0.502924\pi\)
\(138\) 0 0
\(139\) 18.6791i 0.134382i 0.997740 + 0.0671910i \(0.0214037\pi\)
−0.997740 + 0.0671910i \(0.978596\pi\)
\(140\) 0 0
\(141\) −164.517 −1.16679
\(142\) 0 0
\(143\) 45.8658i 0.320740i
\(144\) 0 0
\(145\) 252.082i 1.73849i
\(146\) 0 0
\(147\) 218.155 152.949i 1.48405 1.04047i
\(148\) 0 0
\(149\) −2.51728 −0.0168945 −0.00844725 0.999964i \(-0.502689\pi\)
−0.00844725 + 0.999964i \(0.502689\pi\)
\(150\) 0 0
\(151\) −78.4553 −0.519572 −0.259786 0.965666i \(-0.583652\pi\)
−0.259786 + 0.965666i \(0.583652\pi\)
\(152\) 0 0
\(153\) 444.636i 2.90612i
\(154\) 0 0
\(155\) 52.1430 0.336407
\(156\) 0 0
\(157\) 298.559i 1.90165i 0.309732 + 0.950824i \(0.399761\pi\)
−0.309732 + 0.950824i \(0.600239\pi\)
\(158\) 0 0
\(159\) 163.120i 1.02591i
\(160\) 0 0
\(161\) 10.6940 + 5.56286i 0.0664222 + 0.0345519i
\(162\) 0 0
\(163\) −8.86012 −0.0543566 −0.0271783 0.999631i \(-0.508652\pi\)
−0.0271783 + 0.999631i \(0.508652\pi\)
\(164\) 0 0
\(165\) −508.164 −3.07978
\(166\) 0 0
\(167\) 252.930i 1.51455i 0.653095 + 0.757276i \(0.273470\pi\)
−0.653095 + 0.757276i \(0.726530\pi\)
\(168\) 0 0
\(169\) 159.953 0.946465
\(170\) 0 0
\(171\) 244.680i 1.43088i
\(172\) 0 0
\(173\) 7.89137i 0.0456149i 0.999740 + 0.0228074i \(0.00726046\pi\)
−0.999740 + 0.0228074i \(0.992740\pi\)
\(174\) 0 0
\(175\) 78.0271 + 40.5886i 0.445869 + 0.231935i
\(176\) 0 0
\(177\) −397.211 −2.24413
\(178\) 0 0
\(179\) 298.573 1.66800 0.834002 0.551761i \(-0.186044\pi\)
0.834002 + 0.551761i \(0.186044\pi\)
\(180\) 0 0
\(181\) 281.821i 1.55702i −0.627631 0.778511i \(-0.715975\pi\)
0.627631 0.778511i \(-0.284025\pi\)
\(182\) 0 0
\(183\) −30.8628 −0.168649
\(184\) 0 0
\(185\) 325.630i 1.76016i
\(186\) 0 0
\(187\) 329.694i 1.76307i
\(188\) 0 0
\(189\) 203.129 390.494i 1.07476 2.06610i
\(190\) 0 0
\(191\) −20.5396 −0.107537 −0.0537686 0.998553i \(-0.517123\pi\)
−0.0537686 + 0.998553i \(0.517123\pi\)
\(192\) 0 0
\(193\) 191.565 0.992563 0.496282 0.868162i \(-0.334698\pi\)
0.496282 + 0.868162i \(0.334698\pi\)
\(194\) 0 0
\(195\) 100.239i 0.514047i
\(196\) 0 0
\(197\) −113.129 −0.574261 −0.287130 0.957892i \(-0.592701\pi\)
−0.287130 + 0.957892i \(0.592701\pi\)
\(198\) 0 0
\(199\) 132.543i 0.666045i 0.942919 + 0.333023i \(0.108069\pi\)
−0.942919 + 0.333023i \(0.891931\pi\)
\(200\) 0 0
\(201\) 156.459i 0.778405i
\(202\) 0 0
\(203\) 255.415 + 132.863i 1.25820 + 0.654499i
\(204\) 0 0
\(205\) 263.647 1.28608
\(206\) 0 0
\(207\) 35.4133 0.171079
\(208\) 0 0
\(209\) 181.428i 0.868078i
\(210\) 0 0
\(211\) −235.366 −1.11548 −0.557739 0.830016i \(-0.688331\pi\)
−0.557739 + 0.830016i \(0.688331\pi\)
\(212\) 0 0
\(213\) 30.0789i 0.141215i
\(214\) 0 0
\(215\) 224.595i 1.04463i
\(216\) 0 0
\(217\) −27.4827 + 52.8325i −0.126648 + 0.243468i
\(218\) 0 0
\(219\) −516.237 −2.35725
\(220\) 0 0
\(221\) 65.0346 0.294274
\(222\) 0 0
\(223\) 209.306i 0.938593i 0.883041 + 0.469297i \(0.155492\pi\)
−0.883041 + 0.469297i \(0.844508\pi\)
\(224\) 0 0
\(225\) 258.388 1.14839
\(226\) 0 0
\(227\) 217.751i 0.959256i 0.877472 + 0.479628i \(0.159228\pi\)
−0.877472 + 0.479628i \(0.840772\pi\)
\(228\) 0 0
\(229\) 51.1341i 0.223293i −0.993748 0.111646i \(-0.964388\pi\)
0.993748 0.111646i \(-0.0356124\pi\)
\(230\) 0 0
\(231\) 267.835 514.883i 1.15946 2.22893i
\(232\) 0 0
\(233\) −242.517 −1.04085 −0.520423 0.853908i \(-0.674226\pi\)
−0.520423 + 0.853908i \(0.674226\pi\)
\(234\) 0 0
\(235\) 185.445 0.789127
\(236\) 0 0
\(237\) 443.326i 1.87057i
\(238\) 0 0
\(239\) −17.4613 −0.0730598 −0.0365299 0.999333i \(-0.511630\pi\)
−0.0365299 + 0.999333i \(0.511630\pi\)
\(240\) 0 0
\(241\) 68.8417i 0.285650i −0.989748 0.142825i \(-0.954381\pi\)
0.989748 0.142825i \(-0.0456186\pi\)
\(242\) 0 0
\(243\) 286.773i 1.18013i
\(244\) 0 0
\(245\) −245.905 + 172.405i −1.00369 + 0.703692i
\(246\) 0 0
\(247\) −35.7881 −0.144891
\(248\) 0 0
\(249\) 469.211 1.88438
\(250\) 0 0
\(251\) 122.246i 0.487037i −0.969896 0.243518i \(-0.921698\pi\)
0.969896 0.243518i \(-0.0783016\pi\)
\(252\) 0 0
\(253\) 26.2586 0.103789
\(254\) 0 0
\(255\) 720.541i 2.82565i
\(256\) 0 0
\(257\) 312.466i 1.21582i −0.794006 0.607910i \(-0.792008\pi\)
0.794006 0.607910i \(-0.207992\pi\)
\(258\) 0 0
\(259\) −329.935 171.628i −1.27388 0.662655i
\(260\) 0 0
\(261\) 845.810 3.24065
\(262\) 0 0
\(263\) −373.959 −1.42190 −0.710949 0.703244i \(-0.751734\pi\)
−0.710949 + 0.703244i \(0.751734\pi\)
\(264\) 0 0
\(265\) 183.870i 0.693849i
\(266\) 0 0
\(267\) 150.272 0.562818
\(268\) 0 0
\(269\) 180.636i 0.671508i −0.941950 0.335754i \(-0.891009\pi\)
0.941950 0.335754i \(-0.108991\pi\)
\(270\) 0 0
\(271\) 275.195i 1.01548i 0.861511 + 0.507740i \(0.169519\pi\)
−0.861511 + 0.507740i \(0.830481\pi\)
\(272\) 0 0
\(273\) −101.565 52.8325i −0.372032 0.193526i
\(274\) 0 0
\(275\) 191.592 0.696700
\(276\) 0 0
\(277\) 376.940 1.36079 0.680397 0.732844i \(-0.261807\pi\)
0.680397 + 0.732844i \(0.261807\pi\)
\(278\) 0 0
\(279\) 174.956i 0.627081i
\(280\) 0 0
\(281\) 62.5173 0.222481 0.111241 0.993793i \(-0.464518\pi\)
0.111241 + 0.993793i \(0.464518\pi\)
\(282\) 0 0
\(283\) 103.435i 0.365494i 0.983160 + 0.182747i \(0.0584989\pi\)
−0.983160 + 0.182747i \(0.941501\pi\)
\(284\) 0 0
\(285\) 396.510i 1.39126i
\(286\) 0 0
\(287\) −138.959 + 267.133i −0.484177 + 0.930776i
\(288\) 0 0
\(289\) −178.483 −0.617587
\(290\) 0 0
\(291\) −548.947 −1.88641
\(292\) 0 0
\(293\) 507.624i 1.73251i −0.499606 0.866253i \(-0.666522\pi\)
0.499606 0.866253i \(-0.333478\pi\)
\(294\) 0 0
\(295\) 447.739 1.51776
\(296\) 0 0
\(297\) 958.842i 3.22842i
\(298\) 0 0
\(299\) 5.17972i 0.0173235i
\(300\) 0 0
\(301\) 227.565 + 118.376i 0.756029 + 0.393275i
\(302\) 0 0
\(303\) 584.735 1.92982
\(304\) 0 0
\(305\) 34.7887 0.114061
\(306\) 0 0
\(307\) 419.316i 1.36585i −0.730489 0.682925i \(-0.760708\pi\)
0.730489 0.682925i \(-0.239292\pi\)
\(308\) 0 0
\(309\) −321.388 −1.04009
\(310\) 0 0
\(311\) 171.628i 0.551858i 0.961178 + 0.275929i \(0.0889854\pi\)
−0.961178 + 0.275929i \(0.911015\pi\)
\(312\) 0 0
\(313\) 503.484i 1.60857i 0.594240 + 0.804287i \(0.297453\pi\)
−0.594240 + 0.804287i \(0.702547\pi\)
\(314\) 0 0
\(315\) −407.159 + 782.719i −1.29257 + 2.48482i
\(316\) 0 0
\(317\) −205.647 −0.648727 −0.324364 0.945932i \(-0.605150\pi\)
−0.324364 + 0.945932i \(0.605150\pi\)
\(318\) 0 0
\(319\) 627.161 1.96602
\(320\) 0 0
\(321\) 250.093i 0.779105i
\(322\) 0 0
\(323\) 257.253 0.796448
\(324\) 0 0
\(325\) 37.7931i 0.116286i
\(326\) 0 0
\(327\) 129.090i 0.394770i
\(328\) 0 0
\(329\) −97.7414 + 187.897i −0.297086 + 0.571115i
\(330\) 0 0
\(331\) −19.6740 −0.0594382 −0.0297191 0.999558i \(-0.509461\pi\)
−0.0297191 + 0.999558i \(0.509461\pi\)
\(332\) 0 0
\(333\) −1092.59 −3.28104
\(334\) 0 0
\(335\) 176.362i 0.526454i
\(336\) 0 0
\(337\) 317.823 0.943096 0.471548 0.881840i \(-0.343695\pi\)
0.471548 + 0.881840i \(0.343695\pi\)
\(338\) 0 0
\(339\) 1017.93i 3.00275i
\(340\) 0 0
\(341\) 129.728i 0.380434i
\(342\) 0 0
\(343\) −45.0765 340.025i −0.131418 0.991327i
\(344\) 0 0
\(345\) −57.3880 −0.166342
\(346\) 0 0
\(347\) −416.385 −1.19996 −0.599979 0.800016i \(-0.704824\pi\)
−0.599979 + 0.800016i \(0.704824\pi\)
\(348\) 0 0
\(349\) 333.344i 0.955140i −0.878594 0.477570i \(-0.841518\pi\)
0.878594 0.477570i \(-0.158482\pi\)
\(350\) 0 0
\(351\) −189.139 −0.538857
\(352\) 0 0
\(353\) 230.864i 0.654005i 0.945023 + 0.327003i \(0.106039\pi\)
−0.945023 + 0.327003i \(0.893961\pi\)
\(354\) 0 0
\(355\) 33.9051i 0.0955073i
\(356\) 0 0
\(357\) 730.069 + 379.772i 2.04501 + 1.06379i
\(358\) 0 0
\(359\) 16.9796 0.0472970 0.0236485 0.999720i \(-0.492472\pi\)
0.0236485 + 0.999720i \(0.492472\pi\)
\(360\) 0 0
\(361\) 219.435 0.607854
\(362\) 0 0
\(363\) 606.357i 1.67041i
\(364\) 0 0
\(365\) 581.905 1.59426
\(366\) 0 0
\(367\) 457.377i 1.24626i −0.782119 0.623129i \(-0.785861\pi\)
0.782119 0.623129i \(-0.214139\pi\)
\(368\) 0 0
\(369\) 884.614i 2.39733i
\(370\) 0 0
\(371\) 186.301 + 96.9113i 0.502160 + 0.261217i
\(372\) 0 0
\(373\) −541.974 −1.45301 −0.726507 0.687159i \(-0.758858\pi\)
−0.726507 + 0.687159i \(0.758858\pi\)
\(374\) 0 0
\(375\) 414.413 1.10510
\(376\) 0 0
\(377\) 123.712i 0.328149i
\(378\) 0 0
\(379\) −254.799 −0.672294 −0.336147 0.941810i \(-0.609124\pi\)
−0.336147 + 0.941810i \(0.609124\pi\)
\(380\) 0 0
\(381\) 1122.00i 2.94487i
\(382\) 0 0
\(383\) 295.983i 0.772803i −0.922331 0.386401i \(-0.873718\pi\)
0.922331 0.386401i \(-0.126282\pi\)
\(384\) 0 0
\(385\) −301.905 + 580.380i −0.784169 + 1.50748i
\(386\) 0 0
\(387\) 753.584 1.94725
\(388\) 0 0
\(389\) −538.164 −1.38345 −0.691727 0.722159i \(-0.743150\pi\)
−0.691727 + 0.722159i \(0.743150\pi\)
\(390\) 0 0
\(391\) 37.2329i 0.0952249i
\(392\) 0 0
\(393\) −1016.67 −2.58694
\(394\) 0 0
\(395\) 499.720i 1.26511i
\(396\) 0 0
\(397\) 547.293i 1.37857i −0.724490 0.689286i \(-0.757925\pi\)
0.724490 0.689286i \(-0.242075\pi\)
\(398\) 0 0
\(399\) −401.753 208.986i −1.00690 0.523774i
\(400\) 0 0
\(401\) 203.375 0.507170 0.253585 0.967313i \(-0.418390\pi\)
0.253585 + 0.967313i \(0.418390\pi\)
\(402\) 0 0
\(403\) 25.5899 0.0634984
\(404\) 0 0
\(405\) 961.170i 2.37326i
\(406\) 0 0
\(407\) −810.143 −1.99052
\(408\) 0 0
\(409\) 103.627i 0.253366i 0.991943 + 0.126683i \(0.0404332\pi\)
−0.991943 + 0.126683i \(0.959567\pi\)
\(410\) 0 0
\(411\) 13.6873i 0.0333024i
\(412\) 0 0
\(413\) −235.987 + 453.659i −0.571398 + 1.09845i
\(414\) 0 0
\(415\) −528.898 −1.27445
\(416\) 0 0
\(417\) 101.565 0.243560
\(418\) 0 0
\(419\) 438.503i 1.04655i −0.852165 0.523273i \(-0.824711\pi\)
0.852165 0.523273i \(-0.175289\pi\)
\(420\) 0 0
\(421\) −741.293 −1.76079 −0.880396 0.474240i \(-0.842723\pi\)
−0.880396 + 0.474240i \(0.842723\pi\)
\(422\) 0 0
\(423\) 622.224i 1.47098i
\(424\) 0 0
\(425\) 271.665i 0.639212i
\(426\) 0 0
\(427\) −18.3359 + 35.2488i −0.0429412 + 0.0825498i
\(428\) 0 0
\(429\) −249.388 −0.581324
\(430\) 0 0
\(431\) −82.8809 −0.192299 −0.0961495 0.995367i \(-0.530653\pi\)
−0.0961495 + 0.995367i \(0.530653\pi\)
\(432\) 0 0
\(433\) 607.840i 1.40379i −0.712282 0.701893i \(-0.752338\pi\)
0.712282 0.701893i \(-0.247662\pi\)
\(434\) 0 0
\(435\) −1370.65 −3.15093
\(436\) 0 0
\(437\) 20.4891i 0.0468857i
\(438\) 0 0
\(439\) 654.207i 1.49022i −0.666941 0.745111i \(-0.732397\pi\)
0.666941 0.745111i \(-0.267603\pi\)
\(440\) 0 0
\(441\) −578.470 825.087i −1.31172 1.87094i
\(442\) 0 0
\(443\) 566.176 1.27805 0.639025 0.769186i \(-0.279338\pi\)
0.639025 + 0.769186i \(0.279338\pi\)
\(444\) 0 0
\(445\) −169.388 −0.380647
\(446\) 0 0
\(447\) 13.6873i 0.0306204i
\(448\) 0 0
\(449\) 348.776 0.776784 0.388392 0.921494i \(-0.373031\pi\)
0.388392 + 0.921494i \(0.373031\pi\)
\(450\) 0 0
\(451\) 655.934i 1.45440i
\(452\) 0 0
\(453\) 426.588i 0.941696i
\(454\) 0 0
\(455\) 114.484 + 59.5531i 0.251614 + 0.130886i
\(456\) 0 0
\(457\) 354.694 0.776136 0.388068 0.921631i \(-0.373143\pi\)
0.388068 + 0.921631i \(0.373143\pi\)
\(458\) 0 0
\(459\) 1359.57 2.96203
\(460\) 0 0
\(461\) 220.531i 0.478375i −0.970973 0.239187i \(-0.923119\pi\)
0.970973 0.239187i \(-0.0768810\pi\)
\(462\) 0 0
\(463\) −497.178 −1.07382 −0.536910 0.843640i \(-0.680408\pi\)
−0.536910 + 0.843640i \(0.680408\pi\)
\(464\) 0 0
\(465\) 283.519i 0.609719i
\(466\) 0 0
\(467\) 299.499i 0.641326i −0.947193 0.320663i \(-0.896094\pi\)
0.947193 0.320663i \(-0.103906\pi\)
\(468\) 0 0
\(469\) −178.694 92.9541i −0.381011 0.198196i
\(470\) 0 0
\(471\) 1623.37 3.44663
\(472\) 0 0
\(473\) 558.776 1.18134
\(474\) 0 0
\(475\) 149.496i 0.314727i
\(476\) 0 0
\(477\) 616.940 1.29337
\(478\) 0 0
\(479\) 524.867i 1.09576i −0.836558 0.547878i \(-0.815436\pi\)
0.836558 0.547878i \(-0.184564\pi\)
\(480\) 0 0
\(481\) 159.807i 0.332239i
\(482\) 0 0
\(483\) 30.2471 58.1468i 0.0626235 0.120387i
\(484\) 0 0
\(485\) 618.776 1.27583
\(486\) 0 0
\(487\) 62.2345 0.127792 0.0638958 0.997957i \(-0.479647\pi\)
0.0638958 + 0.997957i \(0.479647\pi\)
\(488\) 0 0
\(489\) 48.1755i 0.0985184i
\(490\) 0 0
\(491\) 127.136 0.258933 0.129467 0.991584i \(-0.458674\pi\)
0.129467 + 0.991584i \(0.458674\pi\)
\(492\) 0 0
\(493\) 889.271i 1.80380i
\(494\) 0 0
\(495\) 1921.94i 3.88270i
\(496\) 0 0
\(497\) 34.3534 + 17.8702i 0.0691215 + 0.0359560i
\(498\) 0 0
\(499\) −699.460 −1.40172 −0.700862 0.713297i \(-0.747201\pi\)
−0.700862 + 0.713297i \(0.747201\pi\)
\(500\) 0 0
\(501\) 1375.27 2.74505
\(502\) 0 0
\(503\) 856.217i 1.70222i 0.524986 + 0.851111i \(0.324070\pi\)
−0.524986 + 0.851111i \(0.675930\pi\)
\(504\) 0 0
\(505\) −659.116 −1.30518
\(506\) 0 0
\(507\) 869.717i 1.71542i
\(508\) 0 0
\(509\) 83.6547i 0.164351i −0.996618 0.0821755i \(-0.973813\pi\)
0.996618 0.0821755i \(-0.0261868\pi\)
\(510\) 0 0
\(511\) −306.701 + 589.600i −0.600199 + 1.15382i
\(512\) 0 0
\(513\) −748.164 −1.45841
\(514\) 0 0
\(515\) 362.270 0.703438
\(516\) 0 0
\(517\) 461.374i 0.892405i
\(518\) 0 0
\(519\) 42.9081 0.0826745
\(520\) 0 0
\(521\) 563.287i 1.08117i 0.841291 + 0.540583i \(0.181796\pi\)
−0.841291 + 0.540583i \(0.818204\pi\)
\(522\) 0 0
\(523\) 414.011i 0.791607i 0.918335 + 0.395804i \(0.129534\pi\)
−0.918335 + 0.395804i \(0.870466\pi\)
\(524\) 0 0
\(525\) 220.694 424.260i 0.420369 0.808114i
\(526\) 0 0
\(527\) −183.946 −0.349043
\(528\) 0 0
\(529\) −526.035 −0.994394
\(530\) 0 0
\(531\) 1502.30i 2.82919i
\(532\) 0 0
\(533\) 129.388 0.242754
\(534\) 0 0
\(535\) 281.906i 0.526927i
\(536\) 0 0
\(537\) 1623.44i 3.02317i
\(538\) 0 0
\(539\) −428.930 611.795i −0.795789 1.13505i
\(540\) 0 0
\(541\) −86.7068 −0.160271 −0.0801357 0.996784i \(-0.525535\pi\)
−0.0801357 + 0.996784i \(0.525535\pi\)
\(542\) 0 0
\(543\) −1532.36 −2.82202
\(544\) 0 0
\(545\) 145.511i 0.266992i
\(546\) 0 0
\(547\) 827.355 1.51253 0.756266 0.654265i \(-0.227022\pi\)
0.756266 + 0.654265i \(0.227022\pi\)
\(548\) 0 0
\(549\) 116.727i 0.212617i
\(550\) 0 0
\(551\) 489.361i 0.888132i
\(552\) 0 0
\(553\) 506.328 + 263.385i 0.915602 + 0.476283i
\(554\) 0 0
\(555\) 1770.56 3.19020
\(556\) 0 0
\(557\) −343.741 −0.617130 −0.308565 0.951203i \(-0.599849\pi\)
−0.308565 + 0.951203i \(0.599849\pi\)
\(558\) 0 0
\(559\) 110.223i 0.197179i
\(560\) 0 0
\(561\) 1792.66 3.19546
\(562\) 0 0
\(563\) 308.968i 0.548788i −0.961617 0.274394i \(-0.911523\pi\)
0.961617 0.274394i \(-0.0884772\pi\)
\(564\) 0 0
\(565\) 1147.42i 2.03083i
\(566\) 0 0
\(567\) −973.880 506.598i −1.71760 0.893472i
\(568\) 0 0
\(569\) 182.177 0.320170 0.160085 0.987103i \(-0.448823\pi\)
0.160085 + 0.987103i \(0.448823\pi\)
\(570\) 0 0
\(571\) 378.037 0.662061 0.331031 0.943620i \(-0.392604\pi\)
0.331031 + 0.943620i \(0.392604\pi\)
\(572\) 0 0
\(573\) 111.681i 0.194905i
\(574\) 0 0
\(575\) 21.6369 0.0376294
\(576\) 0 0
\(577\) 81.6019i 0.141424i 0.997497 + 0.0707122i \(0.0225272\pi\)
−0.997497 + 0.0707122i \(0.977473\pi\)
\(578\) 0 0
\(579\) 1041.60i 1.79897i
\(580\) 0 0
\(581\) 278.763 535.891i 0.479799 0.922360i
\(582\) 0 0
\(583\) 457.456 0.784658
\(584\) 0 0
\(585\) 379.116 0.648062
\(586\) 0 0
\(587\) 37.9917i 0.0647217i −0.999476 0.0323609i \(-0.989697\pi\)
0.999476 0.0323609i \(-0.0103026\pi\)
\(588\) 0 0
\(589\) 101.224 0.171858
\(590\) 0 0
\(591\) 615.123i 1.04082i
\(592\) 0 0
\(593\) 406.099i 0.684822i −0.939550 0.342411i \(-0.888757\pi\)
0.939550 0.342411i \(-0.111243\pi\)
\(594\) 0 0
\(595\) −822.938 428.081i −1.38309 0.719464i
\(596\) 0 0
\(597\) 720.681 1.20717
\(598\) 0 0
\(599\) −292.800 −0.488815 −0.244407 0.969673i \(-0.578593\pi\)
−0.244407 + 0.969673i \(0.578593\pi\)
\(600\) 0 0
\(601\) 14.6997i 0.0244588i −0.999925 0.0122294i \(-0.996107\pi\)
0.999925 0.0122294i \(-0.00389284\pi\)
\(602\) 0 0
\(603\) −591.748 −0.981340
\(604\) 0 0
\(605\) 683.490i 1.12973i
\(606\) 0 0
\(607\) 959.534i 1.58078i −0.612604 0.790390i \(-0.709878\pi\)
0.612604 0.790390i \(-0.290122\pi\)
\(608\) 0 0
\(609\) 722.423 1388.78i 1.18624 2.28042i
\(610\) 0 0
\(611\) 91.0095 0.148952
\(612\) 0 0
\(613\) −896.397 −1.46231 −0.731156 0.682211i \(-0.761019\pi\)
−0.731156 + 0.682211i \(0.761019\pi\)
\(614\) 0 0
\(615\) 1433.54i 2.33095i
\(616\) 0 0
\(617\) 137.280 0.222497 0.111248 0.993793i \(-0.464515\pi\)
0.111248 + 0.993793i \(0.464515\pi\)
\(618\) 0 0
\(619\) 703.714i 1.13686i −0.822733 0.568428i \(-0.807552\pi\)
0.822733 0.568428i \(-0.192448\pi\)
\(620\) 0 0
\(621\) 108.284i 0.174370i
\(622\) 0 0
\(623\) 89.2783 171.628i 0.143304 0.275486i
\(624\) 0 0
\(625\) −781.246 −1.24999
\(626\) 0 0
\(627\) −986.487 −1.57334
\(628\) 0 0
\(629\) 1148.73i 1.82628i
\(630\) 0 0
\(631\) −87.6903 −0.138970 −0.0694852 0.997583i \(-0.522136\pi\)
−0.0694852 + 0.997583i \(0.522136\pi\)
\(632\) 0 0
\(633\) 1279.77i 2.02175i
\(634\) 0 0
\(635\) 1264.72i 1.99169i
\(636\) 0 0
\(637\) −120.681 + 84.6098i −0.189452 + 0.132825i
\(638\) 0 0
\(639\) 113.762 0.178031
\(640\) 0 0
\(641\) −380.953 −0.594310 −0.297155 0.954829i \(-0.596038\pi\)
−0.297155 + 0.954829i \(0.596038\pi\)
\(642\) 0 0
\(643\) 441.511i 0.686642i 0.939218 + 0.343321i \(0.111552\pi\)
−0.939218 + 0.343321i \(0.888448\pi\)
\(644\) 0 0
\(645\) −1221.20 −1.89333
\(646\) 0 0
\(647\) 844.897i 1.30587i 0.757414 + 0.652934i \(0.226462\pi\)
−0.757414 + 0.652934i \(0.773538\pi\)
\(648\) 0 0
\(649\) 1113.94i 1.71640i
\(650\) 0 0
\(651\) 287.268 + 149.433i 0.441272 + 0.229544i
\(652\) 0 0
\(653\) 478.845 0.733300 0.366650 0.930359i \(-0.380505\pi\)
0.366650 + 0.930359i \(0.380505\pi\)
\(654\) 0 0
\(655\) 1145.99 1.74961
\(656\) 0 0
\(657\) 1952.47i 2.97179i
\(658\) 0 0
\(659\) 257.012 0.390003 0.195002 0.980803i \(-0.437529\pi\)
0.195002 + 0.980803i \(0.437529\pi\)
\(660\) 0 0
\(661\) 788.652i 1.19312i 0.802568 + 0.596560i \(0.203466\pi\)
−0.802568 + 0.596560i \(0.796534\pi\)
\(662\) 0 0
\(663\) 353.615i 0.533356i
\(664\) 0 0
\(665\) 452.858 + 235.570i 0.680989 + 0.354241i
\(666\) 0 0
\(667\) 70.8266 0.106187
\(668\) 0 0
\(669\) 1138.07 1.70115
\(670\) 0 0
\(671\) 86.5519i 0.128989i
\(672\) 0 0
\(673\) −511.362 −0.759825 −0.379913 0.925022i \(-0.624046\pi\)
−0.379913 + 0.925022i \(0.624046\pi\)
\(674\) 0 0
\(675\) 790.078i 1.17049i
\(676\) 0 0
\(677\) 543.542i 0.802868i −0.915888 0.401434i \(-0.868512\pi\)
0.915888 0.401434i \(-0.131488\pi\)
\(678\) 0 0
\(679\) −326.135 + 626.958i −0.480316 + 0.923355i
\(680\) 0 0
\(681\) 1183.99 1.73860
\(682\) 0 0
\(683\) −785.312 −1.14980 −0.574899 0.818224i \(-0.694959\pi\)
−0.574899 + 0.818224i \(0.694959\pi\)
\(684\) 0 0
\(685\) 15.4284i 0.0225232i
\(686\) 0 0
\(687\) −278.033 −0.404706
\(688\) 0 0
\(689\) 90.2366i 0.130967i
\(690\) 0 0
\(691\) 1063.98i 1.53978i 0.638179 + 0.769888i \(0.279688\pi\)
−0.638179 + 0.769888i \(0.720312\pi\)
\(692\) 0 0
\(693\) −1947.35 1012.98i −2.81003 1.46174i
\(694\) 0 0
\(695\) −114.484 −0.164726
\(696\) 0 0
\(697\) −930.069 −1.33439
\(698\) 0 0
\(699\) 1318.65i 1.88648i
\(700\) 0 0
\(701\) −113.810 −0.162354 −0.0811772 0.996700i \(-0.525868\pi\)
−0.0811772 + 0.996700i \(0.525868\pi\)
\(702\) 0 0
\(703\) 632.138i 0.899200i
\(704\) 0 0
\(705\) 1008.33i 1.43025i
\(706\) 0 0
\(707\) 347.397 667.832i 0.491367 0.944600i
\(708\) 0 0
\(709\) 418.000 0.589563 0.294781 0.955565i \(-0.404753\pi\)
0.294781 + 0.955565i \(0.404753\pi\)
\(710\) 0 0
\(711\) 1676.71 2.35824
\(712\) 0 0
\(713\) 14.6505i 0.0205476i
\(714\) 0 0
\(715\) 281.112 0.393163
\(716\) 0 0
\(717\) 94.9429i 0.132417i
\(718\) 0 0
\(719\) 752.720i 1.04690i 0.852057 + 0.523449i \(0.175355\pi\)
−0.852057 + 0.523449i \(0.824645\pi\)
\(720\) 0 0
\(721\) −190.940 + 367.061i −0.264826 + 0.509099i
\(722\) 0 0
\(723\) −374.316 −0.517726
\(724\) 0 0
\(725\) 516.776 0.712794
\(726\) 0 0
\(727\) 269.945i 0.371314i −0.982615 0.185657i \(-0.940559\pi\)
0.982615 0.185657i \(-0.0594414\pi\)
\(728\) 0 0
\(729\) −147.871 −0.202840
\(730\) 0 0
\(731\) 792.306i 1.08387i
\(732\) 0 0
\(733\) 805.971i 1.09955i 0.835312 + 0.549776i \(0.185287\pi\)
−0.835312 + 0.549776i \(0.814713\pi\)
\(734\) 0 0
\(735\) 937.423 + 1337.07i 1.27540 + 1.81914i
\(736\) 0 0
\(737\) −438.776 −0.595354
\(738\) 0 0
\(739\) 672.425 0.909912 0.454956 0.890514i \(-0.349655\pi\)
0.454956 + 0.890514i \(0.349655\pi\)
\(740\) 0 0
\(741\) 194.592i 0.262607i
\(742\) 0 0
\(743\) −367.686 −0.494867 −0.247434 0.968905i \(-0.579587\pi\)
−0.247434 + 0.968905i \(0.579587\pi\)
\(744\) 0 0
\(745\) 15.4284i 0.0207093i
\(746\) 0 0
\(747\) 1774.61i 2.37565i
\(748\) 0 0
\(749\) 285.634 + 148.583i 0.381354 + 0.198375i
\(750\) 0 0
\(751\) −181.778 −0.242048 −0.121024 0.992650i \(-0.538618\pi\)
−0.121024 + 0.992650i \(0.538618\pi\)
\(752\) 0 0
\(753\) −664.694 −0.882728
\(754\) 0 0
\(755\) 480.853i 0.636891i
\(756\) 0 0
\(757\) −63.7414 −0.0842026 −0.0421013 0.999113i \(-0.513405\pi\)
−0.0421013 + 0.999113i \(0.513405\pi\)
\(758\) 0 0
\(759\) 142.777i 0.188112i
\(760\) 0 0
\(761\) 1279.94i 1.68192i −0.541096 0.840961i \(-0.681990\pi\)
0.541096 0.840961i \(-0.318010\pi\)
\(762\) 0 0
\(763\) 147.435 + 76.6936i 0.193231 + 0.100516i
\(764\) 0 0
\(765\) −2725.17 −3.56232
\(766\) 0 0
\(767\) 219.734 0.286485
\(768\) 0 0
\(769\) 321.701i 0.418337i −0.977880 0.209169i \(-0.932924\pi\)
0.977880 0.209169i \(-0.0670758\pi\)
\(770\) 0 0
\(771\) −1698.98 −2.20361
\(772\) 0 0
\(773\) 697.333i 0.902112i −0.892496 0.451056i \(-0.851047\pi\)
0.892496 0.451056i \(-0.148953\pi\)
\(774\) 0 0
\(775\) 106.895i 0.137929i
\(776\) 0 0
\(777\) −933.198 + 1793.97i −1.20103 + 2.30884i
\(778\) 0 0
\(779\) 511.811 0.657011
\(780\) 0 0
\(781\) 84.3534 0.108007
\(782\) 0 0
\(783\) 2586.25i 3.30300i
\(784\) 0 0
\(785\) −1829.87 −2.33104
\(786\) 0 0
\(787\) 330.202i 0.419570i 0.977748 + 0.209785i \(0.0672764\pi\)
−0.977748 + 0.209785i \(0.932724\pi\)
\(788\) 0 0
\(789\) 2033.34i 2.57711i
\(790\) 0 0
\(791\) 1162.59 + 604.763i 1.46977 + 0.764555i
\(792\) 0 0
\(793\) 17.0730 0.0215297
\(794\) 0 0
\(795\) −999.764 −1.25756
\(796\) 0 0
\(797\) 456.879i 0.573248i 0.958043 + 0.286624i \(0.0925331\pi\)
−0.958043 + 0.286624i \(0.907467\pi\)
\(798\) 0 0
\(799\) −654.196 −0.818768
\(800\) 0 0
\(801\) 568.348i 0.709548i
\(802\) 0 0
\(803\) 1447.74i 1.80291i
\(804\) 0 0
\(805\) −34.0948 + 65.5434i −0.0423537 + 0.0814204i
\(806\) 0 0
\(807\) −982.178 −1.21707
\(808\) 0 0
\(809\) 175.211 0.216578 0.108289 0.994119i \(-0.465463\pi\)
0.108289 + 0.994119i \(0.465463\pi\)
\(810\) 0 0
\(811\) 149.621i 0.184489i −0.995736 0.0922446i \(-0.970596\pi\)
0.995736 0.0922446i \(-0.0294042\pi\)
\(812\) 0 0
\(813\) 1496.33 1.84050
\(814\) 0 0
\(815\) 54.3037i 0.0666303i
\(816\) 0 0
\(817\) 436.001i 0.533661i
\(818\) 0 0
\(819\) −199.819 + 384.130i −0.243979 + 0.469023i
\(820\) 0 0
\(821\) −880.750 −1.07278 −0.536389 0.843971i \(-0.680212\pi\)
−0.536389 + 0.843971i \(0.680212\pi\)
\(822\) 0 0
\(823\) 1149.93 1.39724 0.698621 0.715492i \(-0.253798\pi\)
0.698621 + 0.715492i \(0.253798\pi\)
\(824\) 0 0
\(825\) 1041.75i 1.26273i
\(826\) 0 0
\(827\) −633.041 −0.765466 −0.382733 0.923859i \(-0.625017\pi\)
−0.382733 + 0.923859i \(0.625017\pi\)
\(828\) 0 0
\(829\) 900.963i 1.08681i 0.839472 + 0.543404i \(0.182865\pi\)
−0.839472 + 0.543404i \(0.817135\pi\)
\(830\) 0 0
\(831\) 2049.55i 2.46637i
\(832\) 0 0
\(833\) 867.483 608.194i 1.04140 0.730125i
\(834\) 0 0
\(835\) −1550.21 −1.85654
\(836\) 0 0
\(837\) 534.965 0.639146
\(838\) 0 0
\(839\) 1587.19i 1.89176i −0.324514 0.945881i \(-0.605201\pi\)
0.324514 0.945881i \(-0.394799\pi\)
\(840\) 0 0
\(841\) 850.621 1.01144
\(842\) 0 0
\(843\) 339.928i 0.403236i
\(844\) 0 0
\(845\) 980.350i 1.16018i
\(846\) 0 0
\(847\) −692.527 360.243i −0.817624 0.425316i
\(848\) 0 0
\(849\) 562.410 0.662438
\(850\) 0 0
\(851\) −91.4911 −0.107510
\(852\) 0 0
\(853\) 614.776i 0.720722i 0.932813 + 0.360361i \(0.117346\pi\)
−0.932813 + 0.360361i \(0.882654\pi\)
\(854\) 0 0
\(855\) 1499.65 1.75397
\(856\) 0 0
\(857\) 741.142i 0.864809i −0.901680 0.432405i \(-0.857665\pi\)
0.901680 0.432405i \(-0.142335\pi\)
\(858\) 0 0
\(859\) 686.448i 0.799125i −0.916706 0.399563i \(-0.869162\pi\)
0.916706 0.399563i \(-0.130838\pi\)
\(860\) 0 0
\(861\) 1452.49 + 755.566i 1.68698 + 0.877544i
\(862\) 0 0
\(863\) 1375.44 1.59378 0.796892 0.604122i \(-0.206476\pi\)
0.796892 + 0.604122i \(0.206476\pi\)
\(864\) 0 0
\(865\) −48.3662 −0.0559147
\(866\) 0 0
\(867\) 970.471i 1.11934i
\(868\) 0 0
\(869\) 1243.27 1.43069
\(870\) 0 0
\(871\) 86.5519i 0.0993707i
\(872\) 0 0
\(873\) 2076.18i 2.37821i
\(874\) 0 0
\(875\) 246.207 473.306i 0.281380 0.540921i
\(876\) 0 0
\(877\) −1319.39 −1.50443 −0.752217 0.658916i \(-0.771015\pi\)
−0.752217 + 0.658916i \(0.771015\pi\)
\(878\) 0 0
\(879\) −2760.12 −3.14007
\(880\) 0 0
\(881\) 270.582i 0.307130i −0.988139 0.153565i \(-0.950924\pi\)
0.988139 0.153565i \(-0.0490755\pi\)
\(882\) 0 0
\(883\) 1491.67 1.68932 0.844660 0.535303i \(-0.179803\pi\)
0.844660 + 0.535303i \(0.179803\pi\)
\(884\) 0 0
\(885\) 2434.51i 2.75086i
\(886\) 0 0
\(887\) 79.8264i 0.0899960i 0.998987 + 0.0449980i \(0.0143281\pi\)
−0.998987 + 0.0449980i \(0.985672\pi\)
\(888\) 0 0
\(889\) −1281.44 666.589i −1.44144 0.749819i
\(890\) 0 0
\(891\) −2391.32 −2.68386
\(892\) 0 0
\(893\) 360.000 0.403135
\(894\) 0 0
\(895\) 1829.95i 2.04464i
\(896\) 0 0
\(897\) −28.1639 −0.0313979
\(898\) 0 0
\(899\) 349.911i 0.389223i
\(900\) 0 0
\(901\) 648.640i 0.719912i
\(902\) 0 0
\(903\) 643.650 1237.35i 0.712791 1.37026i
\(904\) 0 0
\(905\) 1727.28 1.90860
\(906\) 0 0
\(907\) 845.557 0.932256 0.466128 0.884717i \(-0.345649\pi\)
0.466128 + 0.884717i \(0.345649\pi\)
\(908\) 0 0
\(909\) 2211.54i 2.43293i
\(910\) 0 0
\(911\) 1377.51 1.51209 0.756045 0.654519i \(-0.227129\pi\)
0.756045 + 0.654519i \(0.227129\pi\)
\(912\) 0 0
\(913\) 1315.86i 1.44125i
\(914\) 0 0
\(915\) 189.158i 0.206730i
\(916\) 0 0
\(917\) −604.013 + 1161.15i −0.658684 + 1.26625i
\(918\) 0 0
\(919\) −648.950 −0.706148 −0.353074 0.935595i \(-0.614864\pi\)
−0.353074 + 0.935595i \(0.614864\pi\)
\(920\) 0 0
\(921\) −2279.96 −2.47553
\(922\) 0 0
\(923\) 16.6394i 0.0180275i
\(924\) 0 0
\(925\) −667.552 −0.721678
\(926\) 0 0
\(927\) 1215.53i 1.31125i
\(928\) 0 0
\(929\) 1394.62i 1.50120i 0.660756 + 0.750601i \(0.270236\pi\)
−0.660756 + 0.750601i \(0.729764\pi\)
\(930\) 0 0
\(931\) −477.370 + 334.685i −0.512750 + 0.359490i
\(932\) 0 0
\(933\) 933.198 1.00021
\(934\) 0 0
\(935\) −2020.69 −2.16117
\(936\) 0 0
\(937\) 433.382i 0.462521i 0.972892 + 0.231260i \(0.0742850\pi\)
−0.972892 + 0.231260i \(0.925715\pi\)
\(938\) 0 0
\(939\) 2737.61 2.91546
\(940\) 0 0
\(941\) 891.147i 0.947021i −0.880788 0.473511i \(-0.842987\pi\)
0.880788 0.473511i \(-0.157013\pi\)
\(942\) 0 0
\(943\) 74.0759i 0.0785535i
\(944\) 0 0
\(945\) 2393.34 + 1244.98i 2.53263 + 1.31744i
\(946\) 0 0
\(947\) 1021.88 1.07907 0.539537 0.841962i \(-0.318599\pi\)
0.539537 + 0.841962i \(0.318599\pi\)
\(948\) 0 0
\(949\) 285.577 0.300925
\(950\) 0 0
\(951\) 1118.17i 1.17578i
\(952\) 0 0
\(953\) −1322.52 −1.38774 −0.693871 0.720100i \(-0.744096\pi\)
−0.693871 + 0.720100i \(0.744096\pi\)
\(954\) 0 0
\(955\) 125.887i 0.131819i
\(956\) 0 0
\(957\) 3410.09i 3.56331i
\(958\) 0 0
\(959\) 15.6324 + 8.13176i 0.0163008 + 0.00847942i
\(960\) 0 0
\(961\) 888.621 0.924684
\(962\) 0 0
\(963\) 945.881 0.982223
\(964\) 0 0
\(965\) 1174.10i 1.21668i
\(966\) 0 0
\(967\) 1681.56 1.73894 0.869471 0.493983i \(-0.164460\pi\)
0.869471 + 0.493983i \(0.164460\pi\)
\(968\) 0 0
\(969\) 1398.77i 1.44352i
\(970\) 0 0
\(971\) 1177.91i 1.21309i 0.795049 + 0.606546i \(0.207445\pi\)
−0.795049 + 0.606546i \(0.792555\pi\)
\(972\) 0 0
\(973\) 60.3406 115.998i 0.0620150 0.119217i
\(974\) 0 0
\(975\) −205.494 −0.210763
\(976\) 0 0
\(977\) 1440.21 1.47411 0.737056 0.675832i \(-0.236215\pi\)
0.737056 + 0.675832i \(0.236215\pi\)
\(978\) 0 0
\(979\) 421.425i 0.430465i
\(980\) 0 0
\(981\) 488.233 0.497689
\(982\) 0 0
\(983\) 283.257i 0.288156i −0.989566 0.144078i \(-0.953978\pi\)
0.989566 0.144078i \(-0.0460215\pi\)
\(984\) 0 0
\(985\) 693.370i 0.703929i
\(986\) 0 0
\(987\) 1021.66 + 531.453i 1.03512 + 0.538453i
\(988\) 0 0
\(989\) 63.1037 0.0638055
\(990\) 0 0
\(991\) 1079.85 1.08966 0.544830 0.838547i \(-0.316594\pi\)
0.544830 + 0.838547i \(0.316594\pi\)
\(992\) 0 0
\(993\) 106.974i 0.107729i
\(994\) 0 0
\(995\) −812.356 −0.816438
\(996\) 0 0
\(997\) 1572.31i 1.57704i −0.615010 0.788520i \(-0.710848\pi\)
0.615010 0.788520i \(-0.289152\pi\)
\(998\) 0 0
\(999\) 3340.82i 3.34417i
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 224.3.c.a.97.2 yes 8
3.2 odd 2 2016.3.f.c.1441.1 8
4.3 odd 2 inner 224.3.c.a.97.8 yes 8
7.6 odd 2 inner 224.3.c.a.97.7 yes 8
8.3 odd 2 448.3.c.g.321.1 8
8.5 even 2 448.3.c.g.321.7 8
12.11 even 2 2016.3.f.c.1441.2 8
21.20 even 2 2016.3.f.c.1441.7 8
28.27 even 2 inner 224.3.c.a.97.1 8
56.13 odd 2 448.3.c.g.321.2 8
56.27 even 2 448.3.c.g.321.8 8
84.83 odd 2 2016.3.f.c.1441.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.3.c.a.97.1 8 28.27 even 2 inner
224.3.c.a.97.2 yes 8 1.1 even 1 trivial
224.3.c.a.97.7 yes 8 7.6 odd 2 inner
224.3.c.a.97.8 yes 8 4.3 odd 2 inner
448.3.c.g.321.1 8 8.3 odd 2
448.3.c.g.321.2 8 56.13 odd 2
448.3.c.g.321.7 8 8.5 even 2
448.3.c.g.321.8 8 56.27 even 2
2016.3.f.c.1441.1 8 3.2 odd 2
2016.3.f.c.1441.2 8 12.11 even 2
2016.3.f.c.1441.7 8 21.20 even 2
2016.3.f.c.1441.8 8 84.83 odd 2