Properties

Label 1200.4.f.r.49.2
Level $1200$
Weight $4$
Character 1200.49
Analytic conductor $70.802$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,4,Mod(49,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.f (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: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.49
Dual form 1200.4.f.r.49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} +4.00000i q^{7} -9.00000 q^{9} +48.0000 q^{11} -2.00000i q^{13} -114.000i q^{17} +140.000 q^{19} -12.0000 q^{21} +72.0000i q^{23} -27.0000i q^{27} -210.000 q^{29} -272.000 q^{31} +144.000i q^{33} -334.000i q^{37} +6.00000 q^{39} -198.000 q^{41} -268.000i q^{43} -216.000i q^{47} +327.000 q^{49} +342.000 q^{51} +78.0000i q^{53} +420.000i q^{57} +240.000 q^{59} +302.000 q^{61} -36.0000i q^{63} -596.000i q^{67} -216.000 q^{69} +768.000 q^{71} +478.000i q^{73} +192.000i q^{77} -640.000 q^{79} +81.0000 q^{81} -348.000i q^{83} -630.000i q^{87} -210.000 q^{89} +8.00000 q^{91} -816.000i q^{93} -1534.00i q^{97} -432.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} + 96 q^{11} + 280 q^{19} - 24 q^{21} - 420 q^{29} - 544 q^{31} + 12 q^{39} - 396 q^{41} + 654 q^{49} + 684 q^{51} + 480 q^{59} + 604 q^{61} - 432 q^{69} + 1536 q^{71} - 1280 q^{79} + 162 q^{81}+ \cdots - 864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000i 0.215980i 0.994152 + 0.107990i \(0.0344414\pi\)
−0.994152 + 0.107990i \(0.965559\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 48.0000 1.31569 0.657843 0.753155i \(-0.271469\pi\)
0.657843 + 0.753155i \(0.271469\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.0426692i −0.999772 0.0213346i \(-0.993208\pi\)
0.999772 0.0213346i \(-0.00679154\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 114.000i − 1.62642i −0.581974 0.813208i \(-0.697719\pi\)
0.581974 0.813208i \(-0.302281\pi\)
\(18\) 0 0
\(19\) 140.000 1.69043 0.845216 0.534425i \(-0.179472\pi\)
0.845216 + 0.534425i \(0.179472\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.124696
\(22\) 0 0
\(23\) 72.0000i 0.652741i 0.945242 + 0.326370i \(0.105826\pi\)
−0.945242 + 0.326370i \(0.894174\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) −210.000 −1.34469 −0.672345 0.740238i \(-0.734713\pi\)
−0.672345 + 0.740238i \(0.734713\pi\)
\(30\) 0 0
\(31\) −272.000 −1.57589 −0.787946 0.615745i \(-0.788855\pi\)
−0.787946 + 0.615745i \(0.788855\pi\)
\(32\) 0 0
\(33\) 144.000i 0.759612i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 334.000i − 1.48403i −0.670381 0.742017i \(-0.733869\pi\)
0.670381 0.742017i \(-0.266131\pi\)
\(38\) 0 0
\(39\) 6.00000 0.0246351
\(40\) 0 0
\(41\) −198.000 −0.754205 −0.377102 0.926172i \(-0.623080\pi\)
−0.377102 + 0.926172i \(0.623080\pi\)
\(42\) 0 0
\(43\) − 268.000i − 0.950456i −0.879863 0.475228i \(-0.842366\pi\)
0.879863 0.475228i \(-0.157634\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 216.000i − 0.670358i −0.942154 0.335179i \(-0.891203\pi\)
0.942154 0.335179i \(-0.108797\pi\)
\(48\) 0 0
\(49\) 327.000 0.953353
\(50\) 0 0
\(51\) 342.000 0.939011
\(52\) 0 0
\(53\) 78.0000i 0.202153i 0.994879 + 0.101077i \(0.0322287\pi\)
−0.994879 + 0.101077i \(0.967771\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 420.000i 0.975971i
\(58\) 0 0
\(59\) 240.000 0.529582 0.264791 0.964306i \(-0.414697\pi\)
0.264791 + 0.964306i \(0.414697\pi\)
\(60\) 0 0
\(61\) 302.000 0.633888 0.316944 0.948444i \(-0.397343\pi\)
0.316944 + 0.948444i \(0.397343\pi\)
\(62\) 0 0
\(63\) − 36.0000i − 0.0719932i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 596.000i − 1.08676i −0.839487 0.543381i \(-0.817144\pi\)
0.839487 0.543381i \(-0.182856\pi\)
\(68\) 0 0
\(69\) −216.000 −0.376860
\(70\) 0 0
\(71\) 768.000 1.28373 0.641865 0.766818i \(-0.278161\pi\)
0.641865 + 0.766818i \(0.278161\pi\)
\(72\) 0 0
\(73\) 478.000i 0.766379i 0.923670 + 0.383190i \(0.125174\pi\)
−0.923670 + 0.383190i \(0.874826\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 192.000i 0.284161i
\(78\) 0 0
\(79\) −640.000 −0.911464 −0.455732 0.890117i \(-0.650622\pi\)
−0.455732 + 0.890117i \(0.650622\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 348.000i − 0.460216i −0.973165 0.230108i \(-0.926092\pi\)
0.973165 0.230108i \(-0.0739080\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 630.000i − 0.776357i
\(88\) 0 0
\(89\) −210.000 −0.250112 −0.125056 0.992150i \(-0.539911\pi\)
−0.125056 + 0.992150i \(0.539911\pi\)
\(90\) 0 0
\(91\) 8.00000 0.00921569
\(92\) 0 0
\(93\) − 816.000i − 0.909841i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 1534.00i − 1.60571i −0.596173 0.802856i \(-0.703313\pi\)
0.596173 0.802856i \(-0.296687\pi\)
\(98\) 0 0
\(99\) −432.000 −0.438562
\(100\) 0 0
\(101\) 1722.00 1.69649 0.848245 0.529605i \(-0.177660\pi\)
0.848245 + 0.529605i \(0.177660\pi\)
\(102\) 0 0
\(103\) 1052.00i 1.00638i 0.864177 + 0.503188i \(0.167840\pi\)
−0.864177 + 0.503188i \(0.832160\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 564.000i 0.509570i 0.966998 + 0.254785i \(0.0820046\pi\)
−0.966998 + 0.254785i \(0.917995\pi\)
\(108\) 0 0
\(109\) 610.000 0.536031 0.268016 0.963415i \(-0.413632\pi\)
0.268016 + 0.963415i \(0.413632\pi\)
\(110\) 0 0
\(111\) 1002.00 0.856807
\(112\) 0 0
\(113\) − 1302.00i − 1.08391i −0.840407 0.541955i \(-0.817684\pi\)
0.840407 0.541955i \(-0.182316\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 18.0000i 0.0142231i
\(118\) 0 0
\(119\) 456.000 0.351273
\(120\) 0 0
\(121\) 973.000 0.731029
\(122\) 0 0
\(123\) − 594.000i − 0.435440i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 124.000i 0.0866395i 0.999061 + 0.0433198i \(0.0137934\pi\)
−0.999061 + 0.0433198i \(0.986207\pi\)
\(128\) 0 0
\(129\) 804.000 0.548746
\(130\) 0 0
\(131\) −192.000 −0.128054 −0.0640272 0.997948i \(-0.520394\pi\)
−0.0640272 + 0.997948i \(0.520394\pi\)
\(132\) 0 0
\(133\) 560.000i 0.365099i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 2514.00i − 1.56778i −0.620901 0.783889i \(-0.713233\pi\)
0.620901 0.783889i \(-0.286767\pi\)
\(138\) 0 0
\(139\) 1340.00 0.817679 0.408839 0.912606i \(-0.365934\pi\)
0.408839 + 0.912606i \(0.365934\pi\)
\(140\) 0 0
\(141\) 648.000 0.387032
\(142\) 0 0
\(143\) − 96.0000i − 0.0561393i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 981.000i 0.550418i
\(148\) 0 0
\(149\) −1410.00 −0.775246 −0.387623 0.921818i \(-0.626704\pi\)
−0.387623 + 0.921818i \(0.626704\pi\)
\(150\) 0 0
\(151\) 2128.00 1.14685 0.573424 0.819258i \(-0.305615\pi\)
0.573424 + 0.819258i \(0.305615\pi\)
\(152\) 0 0
\(153\) 1026.00i 0.542138i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3026.00i 1.53822i 0.639114 + 0.769112i \(0.279301\pi\)
−0.639114 + 0.769112i \(0.720699\pi\)
\(158\) 0 0
\(159\) −234.000 −0.116713
\(160\) 0 0
\(161\) −288.000 −0.140979
\(162\) 0 0
\(163\) 2612.00i 1.25514i 0.778561 + 0.627569i \(0.215950\pi\)
−0.778561 + 0.627569i \(0.784050\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 24.0000i 0.0111208i 0.999985 + 0.00556041i \(0.00176994\pi\)
−0.999985 + 0.00556041i \(0.998230\pi\)
\(168\) 0 0
\(169\) 2193.00 0.998179
\(170\) 0 0
\(171\) −1260.00 −0.563477
\(172\) 0 0
\(173\) − 1962.00i − 0.862243i −0.902294 0.431122i \(-0.858118\pi\)
0.902294 0.431122i \(-0.141882\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 720.000i 0.305754i
\(178\) 0 0
\(179\) −120.000 −0.0501074 −0.0250537 0.999686i \(-0.507976\pi\)
−0.0250537 + 0.999686i \(0.507976\pi\)
\(180\) 0 0
\(181\) 902.000 0.370415 0.185208 0.982699i \(-0.440704\pi\)
0.185208 + 0.982699i \(0.440704\pi\)
\(182\) 0 0
\(183\) 906.000i 0.365975i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 5472.00i − 2.13985i
\(188\) 0 0
\(189\) 108.000 0.0415653
\(190\) 0 0
\(191\) 168.000 0.0636443 0.0318221 0.999494i \(-0.489869\pi\)
0.0318221 + 0.999494i \(0.489869\pi\)
\(192\) 0 0
\(193\) 1318.00i 0.491563i 0.969325 + 0.245782i \(0.0790446\pi\)
−0.969325 + 0.245782i \(0.920955\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 4014.00i − 1.45170i −0.687851 0.725852i \(-0.741446\pi\)
0.687851 0.725852i \(-0.258554\pi\)
\(198\) 0 0
\(199\) 2000.00 0.712443 0.356222 0.934401i \(-0.384065\pi\)
0.356222 + 0.934401i \(0.384065\pi\)
\(200\) 0 0
\(201\) 1788.00 0.627442
\(202\) 0 0
\(203\) − 840.000i − 0.290426i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 648.000i − 0.217580i
\(208\) 0 0
\(209\) 6720.00 2.22408
\(210\) 0 0
\(211\) 3868.00 1.26201 0.631005 0.775779i \(-0.282643\pi\)
0.631005 + 0.775779i \(0.282643\pi\)
\(212\) 0 0
\(213\) 2304.00i 0.741162i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 1088.00i − 0.340361i
\(218\) 0 0
\(219\) −1434.00 −0.442469
\(220\) 0 0
\(221\) −228.000 −0.0693979
\(222\) 0 0
\(223\) − 3148.00i − 0.945317i −0.881246 0.472658i \(-0.843294\pi\)
0.881246 0.472658i \(-0.156706\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 2556.00i − 0.747347i −0.927560 0.373673i \(-0.878098\pi\)
0.927560 0.373673i \(-0.121902\pi\)
\(228\) 0 0
\(229\) 610.000 0.176026 0.0880130 0.996119i \(-0.471948\pi\)
0.0880130 + 0.996119i \(0.471948\pi\)
\(230\) 0 0
\(231\) −576.000 −0.164061
\(232\) 0 0
\(233\) 2058.00i 0.578644i 0.957232 + 0.289322i \(0.0934298\pi\)
−0.957232 + 0.289322i \(0.906570\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 1920.00i − 0.526234i
\(238\) 0 0
\(239\) 4920.00 1.33158 0.665792 0.746138i \(-0.268094\pi\)
0.665792 + 0.746138i \(0.268094\pi\)
\(240\) 0 0
\(241\) −1438.00 −0.384356 −0.192178 0.981360i \(-0.561555\pi\)
−0.192178 + 0.981360i \(0.561555\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 280.000i − 0.0721294i
\(248\) 0 0
\(249\) 1044.00 0.265706
\(250\) 0 0
\(251\) −792.000 −0.199166 −0.0995829 0.995029i \(-0.531751\pi\)
−0.0995829 + 0.995029i \(0.531751\pi\)
\(252\) 0 0
\(253\) 3456.00i 0.858802i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2166.00i 0.525725i 0.964833 + 0.262863i \(0.0846666\pi\)
−0.964833 + 0.262863i \(0.915333\pi\)
\(258\) 0 0
\(259\) 1336.00 0.320521
\(260\) 0 0
\(261\) 1890.00 0.448230
\(262\) 0 0
\(263\) 3192.00i 0.748392i 0.927350 + 0.374196i \(0.122081\pi\)
−0.927350 + 0.374196i \(0.877919\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 630.000i − 0.144402i
\(268\) 0 0
\(269\) −5490.00 −1.24435 −0.622177 0.782877i \(-0.713752\pi\)
−0.622177 + 0.782877i \(0.713752\pi\)
\(270\) 0 0
\(271\) 6328.00 1.41845 0.709223 0.704985i \(-0.249046\pi\)
0.709223 + 0.704985i \(0.249046\pi\)
\(272\) 0 0
\(273\) 24.0000i 0.00532068i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 574.000i − 0.124507i −0.998060 0.0622533i \(-0.980171\pi\)
0.998060 0.0622533i \(-0.0198287\pi\)
\(278\) 0 0
\(279\) 2448.00 0.525297
\(280\) 0 0
\(281\) 4242.00 0.900557 0.450278 0.892888i \(-0.351325\pi\)
0.450278 + 0.892888i \(0.351325\pi\)
\(282\) 0 0
\(283\) − 628.000i − 0.131911i −0.997823 0.0659553i \(-0.978991\pi\)
0.997823 0.0659553i \(-0.0210095\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 792.000i − 0.162893i
\(288\) 0 0
\(289\) −8083.00 −1.64523
\(290\) 0 0
\(291\) 4602.00 0.927058
\(292\) 0 0
\(293\) 558.000i 0.111258i 0.998451 + 0.0556292i \(0.0177165\pi\)
−0.998451 + 0.0556292i \(0.982284\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 1296.00i − 0.253204i
\(298\) 0 0
\(299\) 144.000 0.0278520
\(300\) 0 0
\(301\) 1072.00 0.205279
\(302\) 0 0
\(303\) 5166.00i 0.979468i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 6964.00i 1.29465i 0.762216 + 0.647323i \(0.224112\pi\)
−0.762216 + 0.647323i \(0.775888\pi\)
\(308\) 0 0
\(309\) −3156.00 −0.581031
\(310\) 0 0
\(311\) −2832.00 −0.516360 −0.258180 0.966097i \(-0.583123\pi\)
−0.258180 + 0.966097i \(0.583123\pi\)
\(312\) 0 0
\(313\) − 8642.00i − 1.56062i −0.625392 0.780311i \(-0.715061\pi\)
0.625392 0.780311i \(-0.284939\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2214.00i − 0.392273i −0.980577 0.196137i \(-0.937160\pi\)
0.980577 0.196137i \(-0.0628396\pi\)
\(318\) 0 0
\(319\) −10080.0 −1.76919
\(320\) 0 0
\(321\) −1692.00 −0.294200
\(322\) 0 0
\(323\) − 15960.0i − 2.74934i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1830.00i 0.309478i
\(328\) 0 0
\(329\) 864.000 0.144784
\(330\) 0 0
\(331\) −10772.0 −1.78877 −0.894385 0.447299i \(-0.852386\pi\)
−0.894385 + 0.447299i \(0.852386\pi\)
\(332\) 0 0
\(333\) 3006.00i 0.494678i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 1654.00i − 0.267356i −0.991025 0.133678i \(-0.957321\pi\)
0.991025 0.133678i \(-0.0426789\pi\)
\(338\) 0 0
\(339\) 3906.00 0.625796
\(340\) 0 0
\(341\) −13056.0 −2.07338
\(342\) 0 0
\(343\) 2680.00i 0.421885i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2196.00i − 0.339733i −0.985467 0.169867i \(-0.945666\pi\)
0.985467 0.169867i \(-0.0543337\pi\)
\(348\) 0 0
\(349\) −8270.00 −1.26843 −0.634216 0.773156i \(-0.718677\pi\)
−0.634216 + 0.773156i \(0.718677\pi\)
\(350\) 0 0
\(351\) −54.0000 −0.00821170
\(352\) 0 0
\(353\) − 10302.0i − 1.55331i −0.629923 0.776657i \(-0.716914\pi\)
0.629923 0.776657i \(-0.283086\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1368.00i 0.202807i
\(358\) 0 0
\(359\) −2280.00 −0.335192 −0.167596 0.985856i \(-0.553600\pi\)
−0.167596 + 0.985856i \(0.553600\pi\)
\(360\) 0 0
\(361\) 12741.0 1.85756
\(362\) 0 0
\(363\) 2919.00i 0.422060i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8764.00i 1.24653i 0.782010 + 0.623266i \(0.214195\pi\)
−0.782010 + 0.623266i \(0.785805\pi\)
\(368\) 0 0
\(369\) 1782.00 0.251402
\(370\) 0 0
\(371\) −312.000 −0.0436610
\(372\) 0 0
\(373\) 1318.00i 0.182958i 0.995807 + 0.0914792i \(0.0291595\pi\)
−0.995807 + 0.0914792i \(0.970841\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 420.000i 0.0573769i
\(378\) 0 0
\(379\) 1100.00 0.149085 0.0745425 0.997218i \(-0.476250\pi\)
0.0745425 + 0.997218i \(0.476250\pi\)
\(380\) 0 0
\(381\) −372.000 −0.0500214
\(382\) 0 0
\(383\) − 3528.00i − 0.470685i −0.971912 0.235343i \(-0.924379\pi\)
0.971912 0.235343i \(-0.0756212\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2412.00i 0.316819i
\(388\) 0 0
\(389\) 9630.00 1.25517 0.627584 0.778549i \(-0.284044\pi\)
0.627584 + 0.778549i \(0.284044\pi\)
\(390\) 0 0
\(391\) 8208.00 1.06163
\(392\) 0 0
\(393\) − 576.000i − 0.0739322i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 3094.00i − 0.391142i −0.980690 0.195571i \(-0.937344\pi\)
0.980690 0.195571i \(-0.0626560\pi\)
\(398\) 0 0
\(399\) −1680.00 −0.210790
\(400\) 0 0
\(401\) −1638.00 −0.203985 −0.101992 0.994785i \(-0.532522\pi\)
−0.101992 + 0.994785i \(0.532522\pi\)
\(402\) 0 0
\(403\) 544.000i 0.0672421i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 16032.0i − 1.95252i
\(408\) 0 0
\(409\) 13750.0 1.66233 0.831166 0.556024i \(-0.187674\pi\)
0.831166 + 0.556024i \(0.187674\pi\)
\(410\) 0 0
\(411\) 7542.00 0.905157
\(412\) 0 0
\(413\) 960.000i 0.114379i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4020.00i 0.472087i
\(418\) 0 0
\(419\) −12480.0 −1.45510 −0.727551 0.686053i \(-0.759342\pi\)
−0.727551 + 0.686053i \(0.759342\pi\)
\(420\) 0 0
\(421\) 7262.00 0.840685 0.420342 0.907366i \(-0.361910\pi\)
0.420342 + 0.907366i \(0.361910\pi\)
\(422\) 0 0
\(423\) 1944.00i 0.223453i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1208.00i 0.136907i
\(428\) 0 0
\(429\) 288.000 0.0324121
\(430\) 0 0
\(431\) −9792.00 −1.09435 −0.547174 0.837019i \(-0.684296\pi\)
−0.547174 + 0.837019i \(0.684296\pi\)
\(432\) 0 0
\(433\) − 1802.00i − 0.199997i −0.994988 0.0999984i \(-0.968116\pi\)
0.994988 0.0999984i \(-0.0318838\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 10080.0i 1.10341i
\(438\) 0 0
\(439\) −2320.00 −0.252227 −0.126113 0.992016i \(-0.540250\pi\)
−0.126113 + 0.992016i \(0.540250\pi\)
\(440\) 0 0
\(441\) −2943.00 −0.317784
\(442\) 0 0
\(443\) 11172.0i 1.19819i 0.800678 + 0.599095i \(0.204473\pi\)
−0.800678 + 0.599095i \(0.795527\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 4230.00i − 0.447589i
\(448\) 0 0
\(449\) −6810.00 −0.715777 −0.357888 0.933764i \(-0.616503\pi\)
−0.357888 + 0.933764i \(0.616503\pi\)
\(450\) 0 0
\(451\) −9504.00 −0.992297
\(452\) 0 0
\(453\) 6384.00i 0.662134i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 17066.0i 1.74686i 0.486952 + 0.873429i \(0.338109\pi\)
−0.486952 + 0.873429i \(0.661891\pi\)
\(458\) 0 0
\(459\) −3078.00 −0.313004
\(460\) 0 0
\(461\) −18918.0 −1.91128 −0.955639 0.294541i \(-0.904833\pi\)
−0.955639 + 0.294541i \(0.904833\pi\)
\(462\) 0 0
\(463\) 1052.00i 0.105595i 0.998605 + 0.0527976i \(0.0168138\pi\)
−0.998605 + 0.0527976i \(0.983186\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 11076.0i − 1.09751i −0.835984 0.548754i \(-0.815102\pi\)
0.835984 0.548754i \(-0.184898\pi\)
\(468\) 0 0
\(469\) 2384.00 0.234718
\(470\) 0 0
\(471\) −9078.00 −0.888094
\(472\) 0 0
\(473\) − 12864.0i − 1.25050i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 702.000i − 0.0673844i
\(478\) 0 0
\(479\) −9000.00 −0.858498 −0.429249 0.903186i \(-0.641222\pi\)
−0.429249 + 0.903186i \(0.641222\pi\)
\(480\) 0 0
\(481\) −668.000 −0.0633226
\(482\) 0 0
\(483\) − 864.000i − 0.0813941i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8764.00i 0.815472i 0.913100 + 0.407736i \(0.133682\pi\)
−0.913100 + 0.407736i \(0.866318\pi\)
\(488\) 0 0
\(489\) −7836.00 −0.724655
\(490\) 0 0
\(491\) −5592.00 −0.513978 −0.256989 0.966414i \(-0.582730\pi\)
−0.256989 + 0.966414i \(0.582730\pi\)
\(492\) 0 0
\(493\) 23940.0i 2.18703i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3072.00i 0.277260i
\(498\) 0 0
\(499\) 4700.00 0.421645 0.210823 0.977524i \(-0.432386\pi\)
0.210823 + 0.977524i \(0.432386\pi\)
\(500\) 0 0
\(501\) −72.0000 −0.00642060
\(502\) 0 0
\(503\) − 11808.0i − 1.04671i −0.852116 0.523353i \(-0.824681\pi\)
0.852116 0.523353i \(-0.175319\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6579.00i 0.576299i
\(508\) 0 0
\(509\) −1170.00 −0.101885 −0.0509424 0.998702i \(-0.516222\pi\)
−0.0509424 + 0.998702i \(0.516222\pi\)
\(510\) 0 0
\(511\) −1912.00 −0.165522
\(512\) 0 0
\(513\) − 3780.00i − 0.325324i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 10368.0i − 0.881981i
\(518\) 0 0
\(519\) 5886.00 0.497816
\(520\) 0 0
\(521\) −16638.0 −1.39909 −0.699543 0.714590i \(-0.746613\pi\)
−0.699543 + 0.714590i \(0.746613\pi\)
\(522\) 0 0
\(523\) 15692.0i 1.31198i 0.754771 + 0.655988i \(0.227748\pi\)
−0.754771 + 0.655988i \(0.772252\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 31008.0i 2.56305i
\(528\) 0 0
\(529\) 6983.00 0.573929
\(530\) 0 0
\(531\) −2160.00 −0.176527
\(532\) 0 0
\(533\) 396.000i 0.0321814i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 360.000i − 0.0289295i
\(538\) 0 0
\(539\) 15696.0 1.25431
\(540\) 0 0
\(541\) −22018.0 −1.74977 −0.874887 0.484327i \(-0.839064\pi\)
−0.874887 + 0.484327i \(0.839064\pi\)
\(542\) 0 0
\(543\) 2706.00i 0.213859i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 4564.00i 0.356751i 0.983963 + 0.178375i \(0.0570841\pi\)
−0.983963 + 0.178375i \(0.942916\pi\)
\(548\) 0 0
\(549\) −2718.00 −0.211296
\(550\) 0 0
\(551\) −29400.0 −2.27311
\(552\) 0 0
\(553\) − 2560.00i − 0.196858i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 7734.00i − 0.588331i −0.955755 0.294165i \(-0.904958\pi\)
0.955755 0.294165i \(-0.0950416\pi\)
\(558\) 0 0
\(559\) −536.000 −0.0405552
\(560\) 0 0
\(561\) 16416.0 1.23544
\(562\) 0 0
\(563\) − 20148.0i − 1.50824i −0.656739 0.754118i \(-0.728065\pi\)
0.656739 0.754118i \(-0.271935\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 324.000i 0.0239977i
\(568\) 0 0
\(569\) 24030.0 1.77046 0.885228 0.465156i \(-0.154002\pi\)
0.885228 + 0.465156i \(0.154002\pi\)
\(570\) 0 0
\(571\) −2372.00 −0.173844 −0.0869222 0.996215i \(-0.527703\pi\)
−0.0869222 + 0.996215i \(0.527703\pi\)
\(572\) 0 0
\(573\) 504.000i 0.0367450i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 8546.00i 0.616594i 0.951290 + 0.308297i \(0.0997590\pi\)
−0.951290 + 0.308297i \(0.900241\pi\)
\(578\) 0 0
\(579\) −3954.00 −0.283804
\(580\) 0 0
\(581\) 1392.00 0.0993974
\(582\) 0 0
\(583\) 3744.00i 0.265970i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15444.0i 1.08593i 0.839755 + 0.542966i \(0.182699\pi\)
−0.839755 + 0.542966i \(0.817301\pi\)
\(588\) 0 0
\(589\) −38080.0 −2.66394
\(590\) 0 0
\(591\) 12042.0 0.838142
\(592\) 0 0
\(593\) − 18342.0i − 1.27018i −0.772439 0.635089i \(-0.780963\pi\)
0.772439 0.635089i \(-0.219037\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6000.00i 0.411329i
\(598\) 0 0
\(599\) 24600.0 1.67801 0.839006 0.544123i \(-0.183137\pi\)
0.839006 + 0.544123i \(0.183137\pi\)
\(600\) 0 0
\(601\) −8998.00 −0.610709 −0.305354 0.952239i \(-0.598775\pi\)
−0.305354 + 0.952239i \(0.598775\pi\)
\(602\) 0 0
\(603\) 5364.00i 0.362254i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 4076.00i − 0.272553i −0.990671 0.136277i \(-0.956486\pi\)
0.990671 0.136277i \(-0.0435136\pi\)
\(608\) 0 0
\(609\) 2520.00 0.167677
\(610\) 0 0
\(611\) −432.000 −0.0286037
\(612\) 0 0
\(613\) 4078.00i 0.268693i 0.990934 + 0.134347i \(0.0428935\pi\)
−0.990934 + 0.134347i \(0.957106\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10086.0i 0.658099i 0.944313 + 0.329049i \(0.106728\pi\)
−0.944313 + 0.329049i \(0.893272\pi\)
\(618\) 0 0
\(619\) 8780.00 0.570110 0.285055 0.958511i \(-0.407988\pi\)
0.285055 + 0.958511i \(0.407988\pi\)
\(620\) 0 0
\(621\) 1944.00 0.125620
\(622\) 0 0
\(623\) − 840.000i − 0.0540191i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 20160.0i 1.28407i
\(628\) 0 0
\(629\) −38076.0 −2.41366
\(630\) 0 0
\(631\) −2792.00 −0.176145 −0.0880727 0.996114i \(-0.528071\pi\)
−0.0880727 + 0.996114i \(0.528071\pi\)
\(632\) 0 0
\(633\) 11604.0i 0.728622i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 654.000i − 0.0406788i
\(638\) 0 0
\(639\) −6912.00 −0.427910
\(640\) 0 0
\(641\) 7602.00 0.468426 0.234213 0.972185i \(-0.424749\pi\)
0.234213 + 0.972185i \(0.424749\pi\)
\(642\) 0 0
\(643\) 24212.0i 1.48496i 0.669869 + 0.742479i \(0.266350\pi\)
−0.669869 + 0.742479i \(0.733650\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 9456.00i − 0.574581i −0.957844 0.287290i \(-0.907246\pi\)
0.957844 0.287290i \(-0.0927545\pi\)
\(648\) 0 0
\(649\) 11520.0 0.696764
\(650\) 0 0
\(651\) 3264.00 0.196507
\(652\) 0 0
\(653\) 9558.00i 0.572792i 0.958111 + 0.286396i \(0.0924574\pi\)
−0.958111 + 0.286396i \(0.907543\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 4302.00i − 0.255460i
\(658\) 0 0
\(659\) −29280.0 −1.73078 −0.865392 0.501095i \(-0.832931\pi\)
−0.865392 + 0.501095i \(0.832931\pi\)
\(660\) 0 0
\(661\) −29098.0 −1.71223 −0.856113 0.516789i \(-0.827127\pi\)
−0.856113 + 0.516789i \(0.827127\pi\)
\(662\) 0 0
\(663\) − 684.000i − 0.0400669i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 15120.0i − 0.877734i
\(668\) 0 0
\(669\) 9444.00 0.545779
\(670\) 0 0
\(671\) 14496.0 0.833997
\(672\) 0 0
\(673\) 11638.0i 0.666585i 0.942823 + 0.333293i \(0.108160\pi\)
−0.942823 + 0.333293i \(0.891840\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3426.00i 0.194493i 0.995260 + 0.0972466i \(0.0310035\pi\)
−0.995260 + 0.0972466i \(0.968996\pi\)
\(678\) 0 0
\(679\) 6136.00 0.346801
\(680\) 0 0
\(681\) 7668.00 0.431481
\(682\) 0 0
\(683\) − 20148.0i − 1.12876i −0.825516 0.564379i \(-0.809116\pi\)
0.825516 0.564379i \(-0.190884\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1830.00i 0.101629i
\(688\) 0 0
\(689\) 156.000 0.00862573
\(690\) 0 0
\(691\) 29428.0 1.62011 0.810053 0.586356i \(-0.199438\pi\)
0.810053 + 0.586356i \(0.199438\pi\)
\(692\) 0 0
\(693\) − 1728.00i − 0.0947205i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 22572.0i 1.22665i
\(698\) 0 0
\(699\) −6174.00 −0.334080
\(700\) 0 0
\(701\) 16242.0 0.875110 0.437555 0.899192i \(-0.355845\pi\)
0.437555 + 0.899192i \(0.355845\pi\)
\(702\) 0 0
\(703\) − 46760.0i − 2.50866i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6888.00i 0.366407i
\(708\) 0 0
\(709\) −2030.00 −0.107529 −0.0537646 0.998554i \(-0.517122\pi\)
−0.0537646 + 0.998554i \(0.517122\pi\)
\(710\) 0 0
\(711\) 5760.00 0.303821
\(712\) 0 0
\(713\) − 19584.0i − 1.02865i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 14760.0i 0.768790i
\(718\) 0 0
\(719\) 6960.00 0.361007 0.180504 0.983574i \(-0.442227\pi\)
0.180504 + 0.983574i \(0.442227\pi\)
\(720\) 0 0
\(721\) −4208.00 −0.217357
\(722\) 0 0
\(723\) − 4314.00i − 0.221908i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 18596.0i − 0.948676i −0.880343 0.474338i \(-0.842687\pi\)
0.880343 0.474338i \(-0.157313\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −30552.0 −1.54584
\(732\) 0 0
\(733\) − 21242.0i − 1.07038i −0.844731 0.535192i \(-0.820239\pi\)
0.844731 0.535192i \(-0.179761\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 28608.0i − 1.42984i
\(738\) 0 0
\(739\) −340.000 −0.0169244 −0.00846218 0.999964i \(-0.502694\pi\)
−0.00846218 + 0.999964i \(0.502694\pi\)
\(740\) 0 0
\(741\) 840.000 0.0416440
\(742\) 0 0
\(743\) − 21888.0i − 1.08074i −0.841426 0.540372i \(-0.818284\pi\)
0.841426 0.540372i \(-0.181716\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3132.00i 0.153405i
\(748\) 0 0
\(749\) −2256.00 −0.110057
\(750\) 0 0
\(751\) −17792.0 −0.864500 −0.432250 0.901754i \(-0.642280\pi\)
−0.432250 + 0.901754i \(0.642280\pi\)
\(752\) 0 0
\(753\) − 2376.00i − 0.114988i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 37346.0i 1.79308i 0.442960 + 0.896541i \(0.353928\pi\)
−0.442960 + 0.896541i \(0.646072\pi\)
\(758\) 0 0
\(759\) −10368.0 −0.495829
\(760\) 0 0
\(761\) −11358.0 −0.541034 −0.270517 0.962715i \(-0.587195\pi\)
−0.270517 + 0.962715i \(0.587195\pi\)
\(762\) 0 0
\(763\) 2440.00i 0.115772i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 480.000i − 0.0225969i
\(768\) 0 0
\(769\) 34270.0 1.60703 0.803516 0.595283i \(-0.202960\pi\)
0.803516 + 0.595283i \(0.202960\pi\)
\(770\) 0 0
\(771\) −6498.00 −0.303528
\(772\) 0 0
\(773\) 13278.0i 0.617822i 0.951091 + 0.308911i \(0.0999645\pi\)
−0.951091 + 0.308911i \(0.900035\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 4008.00i 0.185053i
\(778\) 0 0
\(779\) −27720.0 −1.27493
\(780\) 0 0
\(781\) 36864.0 1.68899
\(782\) 0 0
\(783\) 5670.00i 0.258786i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 11164.0i 0.505659i 0.967511 + 0.252829i \(0.0813612\pi\)
−0.967511 + 0.252829i \(0.918639\pi\)
\(788\) 0 0
\(789\) −9576.00 −0.432084
\(790\) 0 0
\(791\) 5208.00 0.234103
\(792\) 0 0
\(793\) − 604.000i − 0.0270475i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 5094.00i − 0.226397i −0.993572 0.113199i \(-0.963890\pi\)
0.993572 0.113199i \(-0.0361097\pi\)
\(798\) 0 0
\(799\) −24624.0 −1.09028
\(800\) 0 0
\(801\) 1890.00 0.0833706
\(802\) 0 0
\(803\) 22944.0i 1.00831i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 16470.0i − 0.718428i
\(808\) 0 0
\(809\) 8790.00 0.382002 0.191001 0.981590i \(-0.438827\pi\)
0.191001 + 0.981590i \(0.438827\pi\)
\(810\) 0 0
\(811\) −5852.00 −0.253380 −0.126690 0.991942i \(-0.540435\pi\)
−0.126690 + 0.991942i \(0.540435\pi\)
\(812\) 0 0
\(813\) 18984.0i 0.818940i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 37520.0i − 1.60668i
\(818\) 0 0
\(819\) −72.0000 −0.00307190
\(820\) 0 0
\(821\) −29478.0 −1.25309 −0.626546 0.779384i \(-0.715532\pi\)
−0.626546 + 0.779384i \(0.715532\pi\)
\(822\) 0 0
\(823\) 39332.0i 1.66589i 0.553356 + 0.832945i \(0.313347\pi\)
−0.553356 + 0.832945i \(0.686653\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 6756.00i − 0.284074i −0.989861 0.142037i \(-0.954635\pi\)
0.989861 0.142037i \(-0.0453652\pi\)
\(828\) 0 0
\(829\) −3950.00 −0.165488 −0.0827438 0.996571i \(-0.526368\pi\)
−0.0827438 + 0.996571i \(0.526368\pi\)
\(830\) 0 0
\(831\) 1722.00 0.0718839
\(832\) 0 0
\(833\) − 37278.0i − 1.55055i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7344.00i 0.303280i
\(838\) 0 0
\(839\) 12360.0 0.508599 0.254300 0.967126i \(-0.418155\pi\)
0.254300 + 0.967126i \(0.418155\pi\)
\(840\) 0 0
\(841\) 19711.0 0.808192
\(842\) 0 0
\(843\) 12726.0i 0.519937i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3892.00i 0.157887i
\(848\) 0 0
\(849\) 1884.00 0.0761587
\(850\) 0 0
\(851\) 24048.0 0.968690
\(852\) 0 0
\(853\) 35998.0i 1.44496i 0.691394 + 0.722478i \(0.256997\pi\)
−0.691394 + 0.722478i \(0.743003\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 21594.0i − 0.860720i −0.902657 0.430360i \(-0.858387\pi\)
0.902657 0.430360i \(-0.141613\pi\)
\(858\) 0 0
\(859\) 9260.00 0.367808 0.183904 0.982944i \(-0.441126\pi\)
0.183904 + 0.982944i \(0.441126\pi\)
\(860\) 0 0
\(861\) 2376.00 0.0940463
\(862\) 0 0
\(863\) 31632.0i 1.24770i 0.781544 + 0.623850i \(0.214433\pi\)
−0.781544 + 0.623850i \(0.785567\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 24249.0i − 0.949872i
\(868\) 0 0
\(869\) −30720.0 −1.19920
\(870\) 0 0
\(871\) −1192.00 −0.0463713
\(872\) 0 0
\(873\) 13806.0i 0.535237i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 39694.0i − 1.52836i −0.645003 0.764180i \(-0.723144\pi\)
0.645003 0.764180i \(-0.276856\pi\)
\(878\) 0 0
\(879\) −1674.00 −0.0642351
\(880\) 0 0
\(881\) 1242.00 0.0474961 0.0237480 0.999718i \(-0.492440\pi\)
0.0237480 + 0.999718i \(0.492440\pi\)
\(882\) 0 0
\(883\) − 2668.00i − 0.101682i −0.998707 0.0508411i \(-0.983810\pi\)
0.998707 0.0508411i \(-0.0161902\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 4344.00i 0.164439i 0.996614 + 0.0822194i \(0.0262008\pi\)
−0.996614 + 0.0822194i \(0.973799\pi\)
\(888\) 0 0
\(889\) −496.000 −0.0187124
\(890\) 0 0
\(891\) 3888.00 0.146187
\(892\) 0 0
\(893\) − 30240.0i − 1.13319i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 432.000i 0.0160803i
\(898\) 0 0
\(899\) 57120.0 2.11909
\(900\) 0 0
\(901\) 8892.00 0.328785
\(902\) 0 0
\(903\) 3216.00i 0.118518i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 4436.00i − 0.162398i −0.996698 0.0811990i \(-0.974125\pi\)
0.996698 0.0811990i \(-0.0258749\pi\)
\(908\) 0 0
\(909\) −15498.0 −0.565496
\(910\) 0 0
\(911\) −22752.0 −0.827450 −0.413725 0.910402i \(-0.635773\pi\)
−0.413725 + 0.910402i \(0.635773\pi\)
\(912\) 0 0
\(913\) − 16704.0i − 0.605500i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 768.000i − 0.0276571i
\(918\) 0 0
\(919\) −27160.0 −0.974892 −0.487446 0.873153i \(-0.662071\pi\)
−0.487446 + 0.873153i \(0.662071\pi\)
\(920\) 0 0
\(921\) −20892.0 −0.747465
\(922\) 0 0
\(923\) − 1536.00i − 0.0547758i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 9468.00i − 0.335458i
\(928\) 0 0
\(929\) 33030.0 1.16650 0.583250 0.812292i \(-0.301781\pi\)
0.583250 + 0.812292i \(0.301781\pi\)
\(930\) 0 0
\(931\) 45780.0 1.61158
\(932\) 0 0
\(933\) − 8496.00i − 0.298121i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 29974.0i − 1.04505i −0.852625 0.522523i \(-0.824991\pi\)
0.852625 0.522523i \(-0.175009\pi\)
\(938\) 0 0
\(939\) 25926.0 0.901026
\(940\) 0 0
\(941\) 13962.0 0.483686 0.241843 0.970315i \(-0.422248\pi\)
0.241843 + 0.970315i \(0.422248\pi\)
\(942\) 0 0
\(943\) − 14256.0i − 0.492300i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 35196.0i − 1.20773i −0.797088 0.603863i \(-0.793627\pi\)
0.797088 0.603863i \(-0.206373\pi\)
\(948\) 0 0
\(949\) 956.000 0.0327008
\(950\) 0 0
\(951\) 6642.00 0.226479
\(952\) 0 0
\(953\) 28338.0i 0.963230i 0.876383 + 0.481615i \(0.159950\pi\)
−0.876383 + 0.481615i \(0.840050\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 30240.0i − 1.02144i
\(958\) 0 0
\(959\) 10056.0 0.338608
\(960\) 0 0
\(961\) 44193.0 1.48343
\(962\) 0 0
\(963\) − 5076.00i − 0.169857i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 17524.0i 0.582765i 0.956607 + 0.291383i \(0.0941153\pi\)
−0.956607 + 0.291383i \(0.905885\pi\)
\(968\) 0 0
\(969\) 47880.0 1.58733
\(970\) 0 0
\(971\) 26808.0 0.886004 0.443002 0.896521i \(-0.353913\pi\)
0.443002 + 0.896521i \(0.353913\pi\)
\(972\) 0 0
\(973\) 5360.00i 0.176602i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 10914.0i − 0.357390i −0.983905 0.178695i \(-0.942813\pi\)
0.983905 0.178695i \(-0.0571875\pi\)
\(978\) 0 0
\(979\) −10080.0 −0.329069
\(980\) 0 0
\(981\) −5490.00 −0.178677
\(982\) 0 0
\(983\) 22272.0i 0.722652i 0.932440 + 0.361326i \(0.117676\pi\)
−0.932440 + 0.361326i \(0.882324\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 2592.00i 0.0835910i
\(988\) 0 0
\(989\) 19296.0 0.620402
\(990\) 0 0
\(991\) −14072.0 −0.451071 −0.225536 0.974235i \(-0.572413\pi\)
−0.225536 + 0.974235i \(0.572413\pi\)
\(992\) 0 0
\(993\) − 32316.0i − 1.03275i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 4826.00i 0.153301i 0.997058 + 0.0766504i \(0.0244225\pi\)
−0.997058 + 0.0766504i \(0.975577\pi\)
\(998\) 0 0
\(999\) −9018.00 −0.285602
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 1200.4.f.r.49.2 2
4.3 odd 2 150.4.c.c.49.2 2
5.2 odd 4 1200.4.a.ba.1.1 1
5.3 odd 4 240.4.a.b.1.1 1
5.4 even 2 inner 1200.4.f.r.49.1 2
12.11 even 2 450.4.c.j.199.1 2
15.8 even 4 720.4.a.y.1.1 1
20.3 even 4 30.4.a.b.1.1 1
20.7 even 4 150.4.a.b.1.1 1
20.19 odd 2 150.4.c.c.49.1 2
40.3 even 4 960.4.a.n.1.1 1
40.13 odd 4 960.4.a.bg.1.1 1
60.23 odd 4 90.4.a.c.1.1 1
60.47 odd 4 450.4.a.r.1.1 1
60.59 even 2 450.4.c.j.199.2 2
140.83 odd 4 1470.4.a.r.1.1 1
180.23 odd 12 810.4.e.p.541.1 2
180.43 even 12 810.4.e.i.271.1 2
180.83 odd 12 810.4.e.p.271.1 2
180.103 even 12 810.4.e.i.541.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.4.a.b.1.1 1 20.3 even 4
90.4.a.c.1.1 1 60.23 odd 4
150.4.a.b.1.1 1 20.7 even 4
150.4.c.c.49.1 2 20.19 odd 2
150.4.c.c.49.2 2 4.3 odd 2
240.4.a.b.1.1 1 5.3 odd 4
450.4.a.r.1.1 1 60.47 odd 4
450.4.c.j.199.1 2 12.11 even 2
450.4.c.j.199.2 2 60.59 even 2
720.4.a.y.1.1 1 15.8 even 4
810.4.e.i.271.1 2 180.43 even 12
810.4.e.i.541.1 2 180.103 even 12
810.4.e.p.271.1 2 180.83 odd 12
810.4.e.p.541.1 2 180.23 odd 12
960.4.a.n.1.1 1 40.3 even 4
960.4.a.bg.1.1 1 40.13 odd 4
1200.4.a.ba.1.1 1 5.2 odd 4
1200.4.f.r.49.1 2 5.4 even 2 inner
1200.4.f.r.49.2 2 1.1 even 1 trivial
1470.4.a.r.1.1 1 140.83 odd 4