Properties

Label 4416.2.a.bl.1.1
Level $4416$
Weight $2$
Character 4416.1
Self dual yes
Analytic conductor $35.262$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4416,2,Mod(1,4416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4416, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4416.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4416 = 2^{6} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4416.a (trivial)

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: yes
Analytic conductor: \(35.2619375326\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 4416.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.23607 q^{5} +4.47214 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.23607 q^{5} +4.47214 q^{7} +1.00000 q^{9} -5.23607 q^{11} -4.47214 q^{13} -1.23607 q^{15} -4.00000 q^{17} +5.70820 q^{19} +4.47214 q^{21} -1.00000 q^{23} -3.47214 q^{25} +1.00000 q^{27} +4.47214 q^{29} +2.47214 q^{31} -5.23607 q^{33} -5.52786 q^{35} -11.2361 q^{37} -4.47214 q^{39} -2.00000 q^{41} -4.76393 q^{43} -1.23607 q^{45} -4.00000 q^{47} +13.0000 q^{49} -4.00000 q^{51} -5.23607 q^{53} +6.47214 q^{55} +5.70820 q^{57} -8.94427 q^{59} -0.763932 q^{61} +4.47214 q^{63} +5.52786 q^{65} +9.70820 q^{67} -1.00000 q^{69} -8.94427 q^{71} -4.47214 q^{73} -3.47214 q^{75} -23.4164 q^{77} -4.47214 q^{79} +1.00000 q^{81} +13.2361 q^{83} +4.94427 q^{85} +4.47214 q^{87} -10.4721 q^{89} -20.0000 q^{91} +2.47214 q^{93} -7.05573 q^{95} +0.472136 q^{97} -5.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9} - 6 q^{11} + 2 q^{15} - 8 q^{17} - 2 q^{19} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{31} - 6 q^{33} - 20 q^{35} - 18 q^{37} - 4 q^{41} - 14 q^{43} + 2 q^{45} - 8 q^{47} + 26 q^{49} - 8 q^{51} - 6 q^{53} + 4 q^{55} - 2 q^{57} - 6 q^{61} + 20 q^{65} + 6 q^{67} - 2 q^{69} + 2 q^{75} - 20 q^{77} + 2 q^{81} + 22 q^{83} - 8 q^{85} - 12 q^{89} - 40 q^{91} - 4 q^{93} - 32 q^{95} - 8 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.23607 −0.552786 −0.276393 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(6\) 0 0
\(7\) 4.47214 1.69031 0.845154 0.534522i \(-0.179509\pi\)
0.845154 + 0.534522i \(0.179509\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.23607 −1.57873 −0.789367 0.613922i \(-0.789591\pi\)
−0.789367 + 0.613922i \(0.789591\pi\)
\(12\) 0 0
\(13\) −4.47214 −1.24035 −0.620174 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) −1.23607 −0.319151
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 5.70820 1.30955 0.654776 0.755823i \(-0.272763\pi\)
0.654776 + 0.755823i \(0.272763\pi\)
\(20\) 0 0
\(21\) 4.47214 0.975900
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) 2.47214 0.444009 0.222004 0.975046i \(-0.428740\pi\)
0.222004 + 0.975046i \(0.428740\pi\)
\(32\) 0 0
\(33\) −5.23607 −0.911482
\(34\) 0 0
\(35\) −5.52786 −0.934380
\(36\) 0 0
\(37\) −11.2361 −1.84720 −0.923599 0.383360i \(-0.874767\pi\)
−0.923599 + 0.383360i \(0.874767\pi\)
\(38\) 0 0
\(39\) −4.47214 −0.716115
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −4.76393 −0.726493 −0.363246 0.931693i \(-0.618332\pi\)
−0.363246 + 0.931693i \(0.618332\pi\)
\(44\) 0 0
\(45\) −1.23607 −0.184262
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 13.0000 1.85714
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −5.23607 −0.719229 −0.359615 0.933101i \(-0.617092\pi\)
−0.359615 + 0.933101i \(0.617092\pi\)
\(54\) 0 0
\(55\) 6.47214 0.872703
\(56\) 0 0
\(57\) 5.70820 0.756070
\(58\) 0 0
\(59\) −8.94427 −1.16445 −0.582223 0.813029i \(-0.697817\pi\)
−0.582223 + 0.813029i \(0.697817\pi\)
\(60\) 0 0
\(61\) −0.763932 −0.0978115 −0.0489057 0.998803i \(-0.515573\pi\)
−0.0489057 + 0.998803i \(0.515573\pi\)
\(62\) 0 0
\(63\) 4.47214 0.563436
\(64\) 0 0
\(65\) 5.52786 0.685647
\(66\) 0 0
\(67\) 9.70820 1.18605 0.593023 0.805186i \(-0.297934\pi\)
0.593023 + 0.805186i \(0.297934\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −8.94427 −1.06149 −0.530745 0.847532i \(-0.678088\pi\)
−0.530745 + 0.847532i \(0.678088\pi\)
\(72\) 0 0
\(73\) −4.47214 −0.523424 −0.261712 0.965146i \(-0.584287\pi\)
−0.261712 + 0.965146i \(0.584287\pi\)
\(74\) 0 0
\(75\) −3.47214 −0.400928
\(76\) 0 0
\(77\) −23.4164 −2.66855
\(78\) 0 0
\(79\) −4.47214 −0.503155 −0.251577 0.967837i \(-0.580949\pi\)
−0.251577 + 0.967837i \(0.580949\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.2361 1.45285 0.726424 0.687247i \(-0.241181\pi\)
0.726424 + 0.687247i \(0.241181\pi\)
\(84\) 0 0
\(85\) 4.94427 0.536282
\(86\) 0 0
\(87\) 4.47214 0.479463
\(88\) 0 0
\(89\) −10.4721 −1.11004 −0.555022 0.831836i \(-0.687290\pi\)
−0.555022 + 0.831836i \(0.687290\pi\)
\(90\) 0 0
\(91\) −20.0000 −2.09657
\(92\) 0 0
\(93\) 2.47214 0.256349
\(94\) 0 0
\(95\) −7.05573 −0.723902
\(96\) 0 0
\(97\) 0.472136 0.0479381 0.0239691 0.999713i \(-0.492370\pi\)
0.0239691 + 0.999713i \(0.492370\pi\)
\(98\) 0 0
\(99\) −5.23607 −0.526245
\(100\) 0 0
\(101\) −4.47214 −0.444994 −0.222497 0.974933i \(-0.571421\pi\)
−0.222497 + 0.974933i \(0.571421\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) −5.52786 −0.539464
\(106\) 0 0
\(107\) −12.6525 −1.22316 −0.611581 0.791182i \(-0.709466\pi\)
−0.611581 + 0.791182i \(0.709466\pi\)
\(108\) 0 0
\(109\) −4.76393 −0.456302 −0.228151 0.973626i \(-0.573268\pi\)
−0.228151 + 0.973626i \(0.573268\pi\)
\(110\) 0 0
\(111\) −11.2361 −1.06648
\(112\) 0 0
\(113\) −5.52786 −0.520018 −0.260009 0.965606i \(-0.583725\pi\)
−0.260009 + 0.965606i \(0.583725\pi\)
\(114\) 0 0
\(115\) 1.23607 0.115264
\(116\) 0 0
\(117\) −4.47214 −0.413449
\(118\) 0 0
\(119\) −17.8885 −1.63984
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) 0 0
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) 10.4721 0.936656
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) −4.76393 −0.419441
\(130\) 0 0
\(131\) −9.52786 −0.832453 −0.416227 0.909261i \(-0.636648\pi\)
−0.416227 + 0.909261i \(0.636648\pi\)
\(132\) 0 0
\(133\) 25.5279 2.21355
\(134\) 0 0
\(135\) −1.23607 −0.106384
\(136\) 0 0
\(137\) 3.05573 0.261068 0.130534 0.991444i \(-0.458331\pi\)
0.130534 + 0.991444i \(0.458331\pi\)
\(138\) 0 0
\(139\) −16.9443 −1.43719 −0.718597 0.695427i \(-0.755215\pi\)
−0.718597 + 0.695427i \(0.755215\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) 23.4164 1.95818
\(144\) 0 0
\(145\) −5.52786 −0.459064
\(146\) 0 0
\(147\) 13.0000 1.07222
\(148\) 0 0
\(149\) 11.7082 0.959173 0.479587 0.877494i \(-0.340787\pi\)
0.479587 + 0.877494i \(0.340787\pi\)
\(150\) 0 0
\(151\) 14.4721 1.17773 0.588863 0.808233i \(-0.299576\pi\)
0.588863 + 0.808233i \(0.299576\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) −3.05573 −0.245442
\(156\) 0 0
\(157\) 6.65248 0.530925 0.265463 0.964121i \(-0.414475\pi\)
0.265463 + 0.964121i \(0.414475\pi\)
\(158\) 0 0
\(159\) −5.23607 −0.415247
\(160\) 0 0
\(161\) −4.47214 −0.352454
\(162\) 0 0
\(163\) −2.47214 −0.193633 −0.0968163 0.995302i \(-0.530866\pi\)
−0.0968163 + 0.995302i \(0.530866\pi\)
\(164\) 0 0
\(165\) 6.47214 0.503855
\(166\) 0 0
\(167\) −16.9443 −1.31119 −0.655594 0.755114i \(-0.727582\pi\)
−0.655594 + 0.755114i \(0.727582\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 5.70820 0.436517
\(172\) 0 0
\(173\) 17.4164 1.32414 0.662072 0.749440i \(-0.269677\pi\)
0.662072 + 0.749440i \(0.269677\pi\)
\(174\) 0 0
\(175\) −15.5279 −1.17380
\(176\) 0 0
\(177\) −8.94427 −0.672293
\(178\) 0 0
\(179\) 19.4164 1.45125 0.725625 0.688090i \(-0.241551\pi\)
0.725625 + 0.688090i \(0.241551\pi\)
\(180\) 0 0
\(181\) 11.2361 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(182\) 0 0
\(183\) −0.763932 −0.0564715
\(184\) 0 0
\(185\) 13.8885 1.02111
\(186\) 0 0
\(187\) 20.9443 1.53160
\(188\) 0 0
\(189\) 4.47214 0.325300
\(190\) 0 0
\(191\) −6.47214 −0.468307 −0.234154 0.972200i \(-0.575232\pi\)
−0.234154 + 0.972200i \(0.575232\pi\)
\(192\) 0 0
\(193\) 23.8885 1.71954 0.859768 0.510686i \(-0.170608\pi\)
0.859768 + 0.510686i \(0.170608\pi\)
\(194\) 0 0
\(195\) 5.52786 0.395859
\(196\) 0 0
\(197\) −2.94427 −0.209771 −0.104885 0.994484i \(-0.533448\pi\)
−0.104885 + 0.994484i \(0.533448\pi\)
\(198\) 0 0
\(199\) −17.4164 −1.23462 −0.617308 0.786721i \(-0.711777\pi\)
−0.617308 + 0.786721i \(0.711777\pi\)
\(200\) 0 0
\(201\) 9.70820 0.684764
\(202\) 0 0
\(203\) 20.0000 1.40372
\(204\) 0 0
\(205\) 2.47214 0.172661
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −29.8885 −2.06743
\(210\) 0 0
\(211\) −23.4164 −1.61205 −0.806026 0.591880i \(-0.798386\pi\)
−0.806026 + 0.591880i \(0.798386\pi\)
\(212\) 0 0
\(213\) −8.94427 −0.612851
\(214\) 0 0
\(215\) 5.88854 0.401595
\(216\) 0 0
\(217\) 11.0557 0.750512
\(218\) 0 0
\(219\) −4.47214 −0.302199
\(220\) 0 0
\(221\) 17.8885 1.20331
\(222\) 0 0
\(223\) −19.4164 −1.30022 −0.650109 0.759841i \(-0.725277\pi\)
−0.650109 + 0.759841i \(0.725277\pi\)
\(224\) 0 0
\(225\) −3.47214 −0.231476
\(226\) 0 0
\(227\) −9.23607 −0.613019 −0.306510 0.951868i \(-0.599161\pi\)
−0.306510 + 0.951868i \(0.599161\pi\)
\(228\) 0 0
\(229\) −17.7082 −1.17019 −0.585096 0.810964i \(-0.698943\pi\)
−0.585096 + 0.810964i \(0.698943\pi\)
\(230\) 0 0
\(231\) −23.4164 −1.54069
\(232\) 0 0
\(233\) 19.8885 1.30294 0.651471 0.758674i \(-0.274152\pi\)
0.651471 + 0.758674i \(0.274152\pi\)
\(234\) 0 0
\(235\) 4.94427 0.322529
\(236\) 0 0
\(237\) −4.47214 −0.290496
\(238\) 0 0
\(239\) 4.94427 0.319818 0.159909 0.987132i \(-0.448880\pi\)
0.159909 + 0.987132i \(0.448880\pi\)
\(240\) 0 0
\(241\) −12.4721 −0.803401 −0.401700 0.915771i \(-0.631581\pi\)
−0.401700 + 0.915771i \(0.631581\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −16.0689 −1.02660
\(246\) 0 0
\(247\) −25.5279 −1.62430
\(248\) 0 0
\(249\) 13.2361 0.838802
\(250\) 0 0
\(251\) −19.7082 −1.24397 −0.621985 0.783029i \(-0.713674\pi\)
−0.621985 + 0.783029i \(0.713674\pi\)
\(252\) 0 0
\(253\) 5.23607 0.329189
\(254\) 0 0
\(255\) 4.94427 0.309622
\(256\) 0 0
\(257\) −11.8885 −0.741587 −0.370793 0.928715i \(-0.620914\pi\)
−0.370793 + 0.928715i \(0.620914\pi\)
\(258\) 0 0
\(259\) −50.2492 −3.12233
\(260\) 0 0
\(261\) 4.47214 0.276818
\(262\) 0 0
\(263\) −24.9443 −1.53813 −0.769065 0.639171i \(-0.779278\pi\)
−0.769065 + 0.639171i \(0.779278\pi\)
\(264\) 0 0
\(265\) 6.47214 0.397580
\(266\) 0 0
\(267\) −10.4721 −0.640884
\(268\) 0 0
\(269\) 13.0557 0.796022 0.398011 0.917381i \(-0.369701\pi\)
0.398011 + 0.917381i \(0.369701\pi\)
\(270\) 0 0
\(271\) 16.9443 1.02929 0.514646 0.857403i \(-0.327924\pi\)
0.514646 + 0.857403i \(0.327924\pi\)
\(272\) 0 0
\(273\) −20.0000 −1.21046
\(274\) 0 0
\(275\) 18.1803 1.09632
\(276\) 0 0
\(277\) 20.4721 1.23005 0.615026 0.788507i \(-0.289146\pi\)
0.615026 + 0.788507i \(0.289146\pi\)
\(278\) 0 0
\(279\) 2.47214 0.148003
\(280\) 0 0
\(281\) −13.5279 −0.807005 −0.403502 0.914979i \(-0.632207\pi\)
−0.403502 + 0.914979i \(0.632207\pi\)
\(282\) 0 0
\(283\) −3.81966 −0.227055 −0.113528 0.993535i \(-0.536215\pi\)
−0.113528 + 0.993535i \(0.536215\pi\)
\(284\) 0 0
\(285\) −7.05573 −0.417945
\(286\) 0 0
\(287\) −8.94427 −0.527964
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0.472136 0.0276771
\(292\) 0 0
\(293\) −0.291796 −0.0170469 −0.00852345 0.999964i \(-0.502713\pi\)
−0.00852345 + 0.999964i \(0.502713\pi\)
\(294\) 0 0
\(295\) 11.0557 0.643689
\(296\) 0 0
\(297\) −5.23607 −0.303827
\(298\) 0 0
\(299\) 4.47214 0.258630
\(300\) 0 0
\(301\) −21.3050 −1.22800
\(302\) 0 0
\(303\) −4.47214 −0.256917
\(304\) 0 0
\(305\) 0.944272 0.0540689
\(306\) 0 0
\(307\) 15.4164 0.879861 0.439930 0.898032i \(-0.355003\pi\)
0.439930 + 0.898032i \(0.355003\pi\)
\(308\) 0 0
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) 20.9443 1.18764 0.593820 0.804598i \(-0.297619\pi\)
0.593820 + 0.804598i \(0.297619\pi\)
\(312\) 0 0
\(313\) 15.5279 0.877687 0.438843 0.898564i \(-0.355388\pi\)
0.438843 + 0.898564i \(0.355388\pi\)
\(314\) 0 0
\(315\) −5.52786 −0.311460
\(316\) 0 0
\(317\) −19.5279 −1.09679 −0.548397 0.836218i \(-0.684762\pi\)
−0.548397 + 0.836218i \(0.684762\pi\)
\(318\) 0 0
\(319\) −23.4164 −1.31107
\(320\) 0 0
\(321\) −12.6525 −0.706192
\(322\) 0 0
\(323\) −22.8328 −1.27045
\(324\) 0 0
\(325\) 15.5279 0.861331
\(326\) 0 0
\(327\) −4.76393 −0.263446
\(328\) 0 0
\(329\) −17.8885 −0.986227
\(330\) 0 0
\(331\) −10.4721 −0.575601 −0.287800 0.957690i \(-0.592924\pi\)
−0.287800 + 0.957690i \(0.592924\pi\)
\(332\) 0 0
\(333\) −11.2361 −0.615733
\(334\) 0 0
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −19.8885 −1.08340 −0.541699 0.840573i \(-0.682219\pi\)
−0.541699 + 0.840573i \(0.682219\pi\)
\(338\) 0 0
\(339\) −5.52786 −0.300232
\(340\) 0 0
\(341\) −12.9443 −0.700972
\(342\) 0 0
\(343\) 26.8328 1.44884
\(344\) 0 0
\(345\) 1.23607 0.0665477
\(346\) 0 0
\(347\) 30.4721 1.63583 0.817915 0.575339i \(-0.195130\pi\)
0.817915 + 0.575339i \(0.195130\pi\)
\(348\) 0 0
\(349\) −3.88854 −0.208149 −0.104074 0.994570i \(-0.533188\pi\)
−0.104074 + 0.994570i \(0.533188\pi\)
\(350\) 0 0
\(351\) −4.47214 −0.238705
\(352\) 0 0
\(353\) −3.88854 −0.206966 −0.103483 0.994631i \(-0.532999\pi\)
−0.103483 + 0.994631i \(0.532999\pi\)
\(354\) 0 0
\(355\) 11.0557 0.586777
\(356\) 0 0
\(357\) −17.8885 −0.946762
\(358\) 0 0
\(359\) −29.3050 −1.54666 −0.773328 0.634006i \(-0.781409\pi\)
−0.773328 + 0.634006i \(0.781409\pi\)
\(360\) 0 0
\(361\) 13.5836 0.714926
\(362\) 0 0
\(363\) 16.4164 0.861638
\(364\) 0 0
\(365\) 5.52786 0.289342
\(366\) 0 0
\(367\) −9.41641 −0.491532 −0.245766 0.969329i \(-0.579040\pi\)
−0.245766 + 0.969329i \(0.579040\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −23.4164 −1.21572
\(372\) 0 0
\(373\) 35.5967 1.84313 0.921565 0.388224i \(-0.126911\pi\)
0.921565 + 0.388224i \(0.126911\pi\)
\(374\) 0 0
\(375\) 10.4721 0.540779
\(376\) 0 0
\(377\) −20.0000 −1.03005
\(378\) 0 0
\(379\) 13.7082 0.704143 0.352072 0.935973i \(-0.385477\pi\)
0.352072 + 0.935973i \(0.385477\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) −7.05573 −0.360531 −0.180265 0.983618i \(-0.557696\pi\)
−0.180265 + 0.983618i \(0.557696\pi\)
\(384\) 0 0
\(385\) 28.9443 1.47514
\(386\) 0 0
\(387\) −4.76393 −0.242164
\(388\) 0 0
\(389\) −23.7082 −1.20205 −0.601027 0.799229i \(-0.705242\pi\)
−0.601027 + 0.799229i \(0.705242\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) −9.52786 −0.480617
\(394\) 0 0
\(395\) 5.52786 0.278137
\(396\) 0 0
\(397\) −26.9443 −1.35229 −0.676147 0.736767i \(-0.736352\pi\)
−0.676147 + 0.736767i \(0.736352\pi\)
\(398\) 0 0
\(399\) 25.5279 1.27799
\(400\) 0 0
\(401\) 8.94427 0.446656 0.223328 0.974743i \(-0.428308\pi\)
0.223328 + 0.974743i \(0.428308\pi\)
\(402\) 0 0
\(403\) −11.0557 −0.550725
\(404\) 0 0
\(405\) −1.23607 −0.0614207
\(406\) 0 0
\(407\) 58.8328 2.91623
\(408\) 0 0
\(409\) −35.8885 −1.77457 −0.887287 0.461217i \(-0.847413\pi\)
−0.887287 + 0.461217i \(0.847413\pi\)
\(410\) 0 0
\(411\) 3.05573 0.150728
\(412\) 0 0
\(413\) −40.0000 −1.96827
\(414\) 0 0
\(415\) −16.3607 −0.803114
\(416\) 0 0
\(417\) −16.9443 −0.829765
\(418\) 0 0
\(419\) 28.0689 1.37125 0.685627 0.727953i \(-0.259528\pi\)
0.685627 + 0.727953i \(0.259528\pi\)
\(420\) 0 0
\(421\) 0.763932 0.0372318 0.0186159 0.999827i \(-0.494074\pi\)
0.0186159 + 0.999827i \(0.494074\pi\)
\(422\) 0 0
\(423\) −4.00000 −0.194487
\(424\) 0 0
\(425\) 13.8885 0.673693
\(426\) 0 0
\(427\) −3.41641 −0.165332
\(428\) 0 0
\(429\) 23.4164 1.13055
\(430\) 0 0
\(431\) 23.4164 1.12793 0.563964 0.825799i \(-0.309276\pi\)
0.563964 + 0.825799i \(0.309276\pi\)
\(432\) 0 0
\(433\) −21.4164 −1.02921 −0.514603 0.857428i \(-0.672061\pi\)
−0.514603 + 0.857428i \(0.672061\pi\)
\(434\) 0 0
\(435\) −5.52786 −0.265041
\(436\) 0 0
\(437\) −5.70820 −0.273060
\(438\) 0 0
\(439\) −19.0557 −0.909480 −0.454740 0.890624i \(-0.650268\pi\)
−0.454740 + 0.890624i \(0.650268\pi\)
\(440\) 0 0
\(441\) 13.0000 0.619048
\(442\) 0 0
\(443\) −15.0557 −0.715319 −0.357660 0.933852i \(-0.616425\pi\)
−0.357660 + 0.933852i \(0.616425\pi\)
\(444\) 0 0
\(445\) 12.9443 0.613617
\(446\) 0 0
\(447\) 11.7082 0.553779
\(448\) 0 0
\(449\) 15.8885 0.749827 0.374913 0.927060i \(-0.377672\pi\)
0.374913 + 0.927060i \(0.377672\pi\)
\(450\) 0 0
\(451\) 10.4721 0.493114
\(452\) 0 0
\(453\) 14.4721 0.679960
\(454\) 0 0
\(455\) 24.7214 1.15896
\(456\) 0 0
\(457\) 27.5279 1.28770 0.643850 0.765152i \(-0.277336\pi\)
0.643850 + 0.765152i \(0.277336\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −22.0000 −1.02464 −0.512321 0.858794i \(-0.671214\pi\)
−0.512321 + 0.858794i \(0.671214\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 0 0
\(465\) −3.05573 −0.141706
\(466\) 0 0
\(467\) −17.8197 −0.824596 −0.412298 0.911049i \(-0.635274\pi\)
−0.412298 + 0.911049i \(0.635274\pi\)
\(468\) 0 0
\(469\) 43.4164 2.00478
\(470\) 0 0
\(471\) 6.65248 0.306530
\(472\) 0 0
\(473\) 24.9443 1.14694
\(474\) 0 0
\(475\) −19.8197 −0.909388
\(476\) 0 0
\(477\) −5.23607 −0.239743
\(478\) 0 0
\(479\) 28.9443 1.32250 0.661249 0.750167i \(-0.270027\pi\)
0.661249 + 0.750167i \(0.270027\pi\)
\(480\) 0 0
\(481\) 50.2492 2.29117
\(482\) 0 0
\(483\) −4.47214 −0.203489
\(484\) 0 0
\(485\) −0.583592 −0.0264996
\(486\) 0 0
\(487\) −9.88854 −0.448093 −0.224046 0.974578i \(-0.571927\pi\)
−0.224046 + 0.974578i \(0.571927\pi\)
\(488\) 0 0
\(489\) −2.47214 −0.111794
\(490\) 0 0
\(491\) 29.3050 1.32251 0.661257 0.750159i \(-0.270023\pi\)
0.661257 + 0.750159i \(0.270023\pi\)
\(492\) 0 0
\(493\) −17.8885 −0.805659
\(494\) 0 0
\(495\) 6.47214 0.290901
\(496\) 0 0
\(497\) −40.0000 −1.79425
\(498\) 0 0
\(499\) 0.583592 0.0261252 0.0130626 0.999915i \(-0.495842\pi\)
0.0130626 + 0.999915i \(0.495842\pi\)
\(500\) 0 0
\(501\) −16.9443 −0.757014
\(502\) 0 0
\(503\) −10.4721 −0.466929 −0.233465 0.972365i \(-0.575006\pi\)
−0.233465 + 0.972365i \(0.575006\pi\)
\(504\) 0 0
\(505\) 5.52786 0.245987
\(506\) 0 0
\(507\) 7.00000 0.310881
\(508\) 0 0
\(509\) 33.4164 1.48116 0.740578 0.671970i \(-0.234552\pi\)
0.740578 + 0.671970i \(0.234552\pi\)
\(510\) 0 0
\(511\) −20.0000 −0.884748
\(512\) 0 0
\(513\) 5.70820 0.252023
\(514\) 0 0
\(515\) −7.41641 −0.326806
\(516\) 0 0
\(517\) 20.9443 0.921128
\(518\) 0 0
\(519\) 17.4164 0.764495
\(520\) 0 0
\(521\) 16.0000 0.700973 0.350486 0.936568i \(-0.386016\pi\)
0.350486 + 0.936568i \(0.386016\pi\)
\(522\) 0 0
\(523\) 25.7082 1.12414 0.562071 0.827089i \(-0.310005\pi\)
0.562071 + 0.827089i \(0.310005\pi\)
\(524\) 0 0
\(525\) −15.5279 −0.677692
\(526\) 0 0
\(527\) −9.88854 −0.430752
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −8.94427 −0.388148
\(532\) 0 0
\(533\) 8.94427 0.387419
\(534\) 0 0
\(535\) 15.6393 0.676147
\(536\) 0 0
\(537\) 19.4164 0.837880
\(538\) 0 0
\(539\) −68.0689 −2.93193
\(540\) 0 0
\(541\) −8.11146 −0.348739 −0.174369 0.984680i \(-0.555789\pi\)
−0.174369 + 0.984680i \(0.555789\pi\)
\(542\) 0 0
\(543\) 11.2361 0.482186
\(544\) 0 0
\(545\) 5.88854 0.252238
\(546\) 0 0
\(547\) 41.3050 1.76607 0.883036 0.469305i \(-0.155496\pi\)
0.883036 + 0.469305i \(0.155496\pi\)
\(548\) 0 0
\(549\) −0.763932 −0.0326038
\(550\) 0 0
\(551\) 25.5279 1.08752
\(552\) 0 0
\(553\) −20.0000 −0.850487
\(554\) 0 0
\(555\) 13.8885 0.589536
\(556\) 0 0
\(557\) 15.1246 0.640850 0.320425 0.947274i \(-0.396174\pi\)
0.320425 + 0.947274i \(0.396174\pi\)
\(558\) 0 0
\(559\) 21.3050 0.901103
\(560\) 0 0
\(561\) 20.9443 0.884268
\(562\) 0 0
\(563\) 24.6525 1.03898 0.519489 0.854477i \(-0.326122\pi\)
0.519489 + 0.854477i \(0.326122\pi\)
\(564\) 0 0
\(565\) 6.83282 0.287459
\(566\) 0 0
\(567\) 4.47214 0.187812
\(568\) 0 0
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) −14.2918 −0.598093 −0.299047 0.954239i \(-0.596669\pi\)
−0.299047 + 0.954239i \(0.596669\pi\)
\(572\) 0 0
\(573\) −6.47214 −0.270377
\(574\) 0 0
\(575\) 3.47214 0.144798
\(576\) 0 0
\(577\) −34.3607 −1.43045 −0.715227 0.698892i \(-0.753677\pi\)
−0.715227 + 0.698892i \(0.753677\pi\)
\(578\) 0 0
\(579\) 23.8885 0.992774
\(580\) 0 0
\(581\) 59.1935 2.45576
\(582\) 0 0
\(583\) 27.4164 1.13547
\(584\) 0 0
\(585\) 5.52786 0.228549
\(586\) 0 0
\(587\) 6.47214 0.267134 0.133567 0.991040i \(-0.457357\pi\)
0.133567 + 0.991040i \(0.457357\pi\)
\(588\) 0 0
\(589\) 14.1115 0.581452
\(590\) 0 0
\(591\) −2.94427 −0.121111
\(592\) 0 0
\(593\) 33.7771 1.38706 0.693529 0.720428i \(-0.256055\pi\)
0.693529 + 0.720428i \(0.256055\pi\)
\(594\) 0 0
\(595\) 22.1115 0.906481
\(596\) 0 0
\(597\) −17.4164 −0.712806
\(598\) 0 0
\(599\) −33.8885 −1.38465 −0.692324 0.721587i \(-0.743413\pi\)
−0.692324 + 0.721587i \(0.743413\pi\)
\(600\) 0 0
\(601\) −10.3607 −0.422621 −0.211310 0.977419i \(-0.567773\pi\)
−0.211310 + 0.977419i \(0.567773\pi\)
\(602\) 0 0
\(603\) 9.70820 0.395349
\(604\) 0 0
\(605\) −20.2918 −0.824979
\(606\) 0 0
\(607\) 17.5279 0.711434 0.355717 0.934594i \(-0.384237\pi\)
0.355717 + 0.934594i \(0.384237\pi\)
\(608\) 0 0
\(609\) 20.0000 0.810441
\(610\) 0 0
\(611\) 17.8885 0.723693
\(612\) 0 0
\(613\) 25.1246 1.01477 0.507387 0.861718i \(-0.330612\pi\)
0.507387 + 0.861718i \(0.330612\pi\)
\(614\) 0 0
\(615\) 2.47214 0.0996861
\(616\) 0 0
\(617\) 20.3607 0.819690 0.409845 0.912155i \(-0.365583\pi\)
0.409845 + 0.912155i \(0.365583\pi\)
\(618\) 0 0
\(619\) 18.2918 0.735209 0.367605 0.929982i \(-0.380178\pi\)
0.367605 + 0.929982i \(0.380178\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −46.8328 −1.87632
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 0 0
\(627\) −29.8885 −1.19363
\(628\) 0 0
\(629\) 44.9443 1.79205
\(630\) 0 0
\(631\) 6.00000 0.238856 0.119428 0.992843i \(-0.461894\pi\)
0.119428 + 0.992843i \(0.461894\pi\)
\(632\) 0 0
\(633\) −23.4164 −0.930719
\(634\) 0 0
\(635\) −4.94427 −0.196207
\(636\) 0 0
\(637\) −58.1378 −2.30350
\(638\) 0 0
\(639\) −8.94427 −0.353830
\(640\) 0 0
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) 0 0
\(643\) −32.5410 −1.28329 −0.641646 0.767001i \(-0.721748\pi\)
−0.641646 + 0.767001i \(0.721748\pi\)
\(644\) 0 0
\(645\) 5.88854 0.231861
\(646\) 0 0
\(647\) −12.9443 −0.508892 −0.254446 0.967087i \(-0.581893\pi\)
−0.254446 + 0.967087i \(0.581893\pi\)
\(648\) 0 0
\(649\) 46.8328 1.83835
\(650\) 0 0
\(651\) 11.0557 0.433308
\(652\) 0 0
\(653\) −9.41641 −0.368493 −0.184246 0.982880i \(-0.558984\pi\)
−0.184246 + 0.982880i \(0.558984\pi\)
\(654\) 0 0
\(655\) 11.7771 0.460169
\(656\) 0 0
\(657\) −4.47214 −0.174475
\(658\) 0 0
\(659\) −32.6525 −1.27196 −0.635980 0.771706i \(-0.719404\pi\)
−0.635980 + 0.771706i \(0.719404\pi\)
\(660\) 0 0
\(661\) −3.81966 −0.148568 −0.0742838 0.997237i \(-0.523667\pi\)
−0.0742838 + 0.997237i \(0.523667\pi\)
\(662\) 0 0
\(663\) 17.8885 0.694733
\(664\) 0 0
\(665\) −31.5542 −1.22362
\(666\) 0 0
\(667\) −4.47214 −0.173162
\(668\) 0 0
\(669\) −19.4164 −0.750682
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 4.47214 0.172388 0.0861941 0.996278i \(-0.472529\pi\)
0.0861941 + 0.996278i \(0.472529\pi\)
\(674\) 0 0
\(675\) −3.47214 −0.133643
\(676\) 0 0
\(677\) −19.1246 −0.735019 −0.367509 0.930020i \(-0.619789\pi\)
−0.367509 + 0.930020i \(0.619789\pi\)
\(678\) 0 0
\(679\) 2.11146 0.0810303
\(680\) 0 0
\(681\) −9.23607 −0.353927
\(682\) 0 0
\(683\) 14.4721 0.553761 0.276880 0.960904i \(-0.410699\pi\)
0.276880 + 0.960904i \(0.410699\pi\)
\(684\) 0 0
\(685\) −3.77709 −0.144315
\(686\) 0 0
\(687\) −17.7082 −0.675610
\(688\) 0 0
\(689\) 23.4164 0.892094
\(690\) 0 0
\(691\) 16.5836 0.630870 0.315435 0.948947i \(-0.397850\pi\)
0.315435 + 0.948947i \(0.397850\pi\)
\(692\) 0 0
\(693\) −23.4164 −0.889516
\(694\) 0 0
\(695\) 20.9443 0.794462
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) 0 0
\(699\) 19.8885 0.752254
\(700\) 0 0
\(701\) 3.12461 0.118015 0.0590075 0.998258i \(-0.481206\pi\)
0.0590075 + 0.998258i \(0.481206\pi\)
\(702\) 0 0
\(703\) −64.1378 −2.41900
\(704\) 0 0
\(705\) 4.94427 0.186212
\(706\) 0 0
\(707\) −20.0000 −0.752177
\(708\) 0 0
\(709\) −35.0132 −1.31495 −0.657473 0.753478i \(-0.728375\pi\)
−0.657473 + 0.753478i \(0.728375\pi\)
\(710\) 0 0
\(711\) −4.47214 −0.167718
\(712\) 0 0
\(713\) −2.47214 −0.0925822
\(714\) 0 0
\(715\) −28.9443 −1.08245
\(716\) 0 0
\(717\) 4.94427 0.184647
\(718\) 0 0
\(719\) 3.05573 0.113959 0.0569797 0.998375i \(-0.481853\pi\)
0.0569797 + 0.998375i \(0.481853\pi\)
\(720\) 0 0
\(721\) 26.8328 0.999306
\(722\) 0 0
\(723\) −12.4721 −0.463844
\(724\) 0 0
\(725\) −15.5279 −0.576690
\(726\) 0 0
\(727\) 19.3050 0.715981 0.357991 0.933725i \(-0.383462\pi\)
0.357991 + 0.933725i \(0.383462\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 19.0557 0.704802
\(732\) 0 0
\(733\) 21.7082 0.801811 0.400905 0.916119i \(-0.368696\pi\)
0.400905 + 0.916119i \(0.368696\pi\)
\(734\) 0 0
\(735\) −16.0689 −0.592710
\(736\) 0 0
\(737\) −50.8328 −1.87245
\(738\) 0 0
\(739\) −32.9443 −1.21187 −0.605937 0.795512i \(-0.707202\pi\)
−0.605937 + 0.795512i \(0.707202\pi\)
\(740\) 0 0
\(741\) −25.5279 −0.937790
\(742\) 0 0
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) −14.4721 −0.530218
\(746\) 0 0
\(747\) 13.2361 0.484282
\(748\) 0 0
\(749\) −56.5836 −2.06752
\(750\) 0 0
\(751\) 18.0000 0.656829 0.328415 0.944534i \(-0.393486\pi\)
0.328415 + 0.944534i \(0.393486\pi\)
\(752\) 0 0
\(753\) −19.7082 −0.718207
\(754\) 0 0
\(755\) −17.8885 −0.651031
\(756\) 0 0
\(757\) 1.34752 0.0489766 0.0244883 0.999700i \(-0.492204\pi\)
0.0244883 + 0.999700i \(0.492204\pi\)
\(758\) 0 0
\(759\) 5.23607 0.190057
\(760\) 0 0
\(761\) −5.05573 −0.183270 −0.0916350 0.995793i \(-0.529209\pi\)
−0.0916350 + 0.995793i \(0.529209\pi\)
\(762\) 0 0
\(763\) −21.3050 −0.771291
\(764\) 0 0
\(765\) 4.94427 0.178761
\(766\) 0 0
\(767\) 40.0000 1.44432
\(768\) 0 0
\(769\) 12.4721 0.449757 0.224878 0.974387i \(-0.427802\pi\)
0.224878 + 0.974387i \(0.427802\pi\)
\(770\) 0 0
\(771\) −11.8885 −0.428155
\(772\) 0 0
\(773\) 38.1803 1.37325 0.686626 0.727011i \(-0.259091\pi\)
0.686626 + 0.727011i \(0.259091\pi\)
\(774\) 0 0
\(775\) −8.58359 −0.308332
\(776\) 0 0
\(777\) −50.2492 −1.80268
\(778\) 0 0
\(779\) −11.4164 −0.409035
\(780\) 0 0
\(781\) 46.8328 1.67581
\(782\) 0 0
\(783\) 4.47214 0.159821
\(784\) 0 0
\(785\) −8.22291 −0.293488
\(786\) 0 0
\(787\) 35.2361 1.25603 0.628015 0.778201i \(-0.283868\pi\)
0.628015 + 0.778201i \(0.283868\pi\)
\(788\) 0 0
\(789\) −24.9443 −0.888040
\(790\) 0 0
\(791\) −24.7214 −0.878990
\(792\) 0 0
\(793\) 3.41641 0.121320
\(794\) 0 0
\(795\) 6.47214 0.229543
\(796\) 0 0
\(797\) −41.5967 −1.47343 −0.736716 0.676202i \(-0.763625\pi\)
−0.736716 + 0.676202i \(0.763625\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −10.4721 −0.370015
\(802\) 0 0
\(803\) 23.4164 0.826347
\(804\) 0 0
\(805\) 5.52786 0.194832
\(806\) 0 0
\(807\) 13.0557 0.459583
\(808\) 0 0
\(809\) 42.9443 1.50984 0.754920 0.655817i \(-0.227676\pi\)
0.754920 + 0.655817i \(0.227676\pi\)
\(810\) 0 0
\(811\) 41.3050 1.45041 0.725207 0.688531i \(-0.241744\pi\)
0.725207 + 0.688531i \(0.241744\pi\)
\(812\) 0 0
\(813\) 16.9443 0.594262
\(814\) 0 0
\(815\) 3.05573 0.107037
\(816\) 0 0
\(817\) −27.1935 −0.951380
\(818\) 0 0
\(819\) −20.0000 −0.698857
\(820\) 0 0
\(821\) −39.5279 −1.37953 −0.689766 0.724032i \(-0.742287\pi\)
−0.689766 + 0.724032i \(0.742287\pi\)
\(822\) 0 0
\(823\) 52.3607 1.82518 0.912589 0.408877i \(-0.134080\pi\)
0.912589 + 0.408877i \(0.134080\pi\)
\(824\) 0 0
\(825\) 18.1803 0.632958
\(826\) 0 0
\(827\) −21.5967 −0.750993 −0.375496 0.926824i \(-0.622528\pi\)
−0.375496 + 0.926824i \(0.622528\pi\)
\(828\) 0 0
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) 20.4721 0.710171
\(832\) 0 0
\(833\) −52.0000 −1.80169
\(834\) 0 0
\(835\) 20.9443 0.724806
\(836\) 0 0
\(837\) 2.47214 0.0854495
\(838\) 0 0
\(839\) 45.8885 1.58425 0.792124 0.610360i \(-0.208975\pi\)
0.792124 + 0.610360i \(0.208975\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) −13.5279 −0.465924
\(844\) 0 0
\(845\) −8.65248 −0.297654
\(846\) 0 0
\(847\) 73.4164 2.52262
\(848\) 0 0
\(849\) −3.81966 −0.131090
\(850\) 0 0
\(851\) 11.2361 0.385167
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) −7.05573 −0.241301
\(856\) 0 0
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) −8.94427 −0.304820
\(862\) 0 0
\(863\) −14.8328 −0.504915 −0.252457 0.967608i \(-0.581239\pi\)
−0.252457 + 0.967608i \(0.581239\pi\)
\(864\) 0 0
\(865\) −21.5279 −0.731969
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 23.4164 0.794347
\(870\) 0 0
\(871\) −43.4164 −1.47111
\(872\) 0 0
\(873\) 0.472136 0.0159794
\(874\) 0 0
\(875\) 46.8328 1.58324
\(876\) 0 0
\(877\) 48.2492 1.62926 0.814630 0.579981i \(-0.196940\pi\)
0.814630 + 0.579981i \(0.196940\pi\)
\(878\) 0 0
\(879\) −0.291796 −0.00984204
\(880\) 0 0
\(881\) 39.4164 1.32797 0.663986 0.747745i \(-0.268863\pi\)
0.663986 + 0.747745i \(0.268863\pi\)
\(882\) 0 0
\(883\) 13.5279 0.455249 0.227624 0.973749i \(-0.426904\pi\)
0.227624 + 0.973749i \(0.426904\pi\)
\(884\) 0 0
\(885\) 11.0557 0.371634
\(886\) 0 0
\(887\) 16.9443 0.568933 0.284466 0.958686i \(-0.408184\pi\)
0.284466 + 0.958686i \(0.408184\pi\)
\(888\) 0 0
\(889\) 17.8885 0.599963
\(890\) 0 0
\(891\) −5.23607 −0.175415
\(892\) 0 0
\(893\) −22.8328 −0.764071
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) 4.47214 0.149320
\(898\) 0 0
\(899\) 11.0557 0.368729
\(900\) 0 0
\(901\) 20.9443 0.697755
\(902\) 0 0
\(903\) −21.3050 −0.708984
\(904\) 0 0
\(905\) −13.8885 −0.461671
\(906\) 0 0
\(907\) 4.18034 0.138806 0.0694030 0.997589i \(-0.477891\pi\)
0.0694030 + 0.997589i \(0.477891\pi\)
\(908\) 0 0
\(909\) −4.47214 −0.148331
\(910\) 0 0
\(911\) −16.5836 −0.549439 −0.274719 0.961524i \(-0.588585\pi\)
−0.274719 + 0.961524i \(0.588585\pi\)
\(912\) 0 0
\(913\) −69.3050 −2.29366
\(914\) 0 0
\(915\) 0.944272 0.0312167
\(916\) 0 0
\(917\) −42.6099 −1.40710
\(918\) 0 0
\(919\) 16.4721 0.543366 0.271683 0.962387i \(-0.412420\pi\)
0.271683 + 0.962387i \(0.412420\pi\)
\(920\) 0 0
\(921\) 15.4164 0.507988
\(922\) 0 0
\(923\) 40.0000 1.31662
\(924\) 0 0
\(925\) 39.0132 1.28274
\(926\) 0 0
\(927\) 6.00000 0.197066
\(928\) 0 0
\(929\) −24.8328 −0.814738 −0.407369 0.913264i \(-0.633554\pi\)
−0.407369 + 0.913264i \(0.633554\pi\)
\(930\) 0 0
\(931\) 74.2067 2.43202
\(932\) 0 0
\(933\) 20.9443 0.685685
\(934\) 0 0
\(935\) −25.8885 −0.846646
\(936\) 0 0
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 0 0
\(939\) 15.5279 0.506733
\(940\) 0 0
\(941\) −38.1803 −1.24464 −0.622322 0.782762i \(-0.713810\pi\)
−0.622322 + 0.782762i \(0.713810\pi\)
\(942\) 0 0
\(943\) 2.00000 0.0651290
\(944\) 0 0
\(945\) −5.52786 −0.179821
\(946\) 0 0
\(947\) −47.1935 −1.53358 −0.766791 0.641897i \(-0.778148\pi\)
−0.766791 + 0.641897i \(0.778148\pi\)
\(948\) 0 0
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) −19.5279 −0.633234
\(952\) 0 0
\(953\) −47.7771 −1.54765 −0.773826 0.633398i \(-0.781659\pi\)
−0.773826 + 0.633398i \(0.781659\pi\)
\(954\) 0 0
\(955\) 8.00000 0.258874
\(956\) 0 0
\(957\) −23.4164 −0.756945
\(958\) 0 0
\(959\) 13.6656 0.441286
\(960\) 0 0
\(961\) −24.8885 −0.802856
\(962\) 0 0
\(963\) −12.6525 −0.407720
\(964\) 0 0
\(965\) −29.5279 −0.950536
\(966\) 0 0
\(967\) −15.4164 −0.495758 −0.247879 0.968791i \(-0.579734\pi\)
−0.247879 + 0.968791i \(0.579734\pi\)
\(968\) 0 0
\(969\) −22.8328 −0.733496
\(970\) 0 0
\(971\) −19.1246 −0.613738 −0.306869 0.951752i \(-0.599281\pi\)
−0.306869 + 0.951752i \(0.599281\pi\)
\(972\) 0 0
\(973\) −75.7771 −2.42930
\(974\) 0 0
\(975\) 15.5279 0.497290
\(976\) 0 0
\(977\) 1.16718 0.0373415 0.0186708 0.999826i \(-0.494057\pi\)
0.0186708 + 0.999826i \(0.494057\pi\)
\(978\) 0 0
\(979\) 54.8328 1.75246
\(980\) 0 0
\(981\) −4.76393 −0.152101
\(982\) 0 0
\(983\) −0.583592 −0.0186137 −0.00930685 0.999957i \(-0.502963\pi\)
−0.00930685 + 0.999957i \(0.502963\pi\)
\(984\) 0 0
\(985\) 3.63932 0.115958
\(986\) 0 0
\(987\) −17.8885 −0.569399
\(988\) 0 0
\(989\) 4.76393 0.151484
\(990\) 0 0
\(991\) 10.4721 0.332658 0.166329 0.986070i \(-0.446809\pi\)
0.166329 + 0.986070i \(0.446809\pi\)
\(992\) 0 0
\(993\) −10.4721 −0.332323
\(994\) 0 0
\(995\) 21.5279 0.682479
\(996\) 0 0
\(997\) −5.05573 −0.160117 −0.0800583 0.996790i \(-0.525511\pi\)
−0.0800583 + 0.996790i \(0.525511\pi\)
\(998\) 0 0
\(999\) −11.2361 −0.355493
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 4416.2.a.bl.1.1 2
4.3 odd 2 4416.2.a.bh.1.1 2
8.3 odd 2 138.2.a.d.1.2 2
8.5 even 2 1104.2.a.j.1.2 2
24.5 odd 2 3312.2.a.bc.1.1 2
24.11 even 2 414.2.a.f.1.1 2
40.3 even 4 3450.2.d.x.2899.2 4
40.19 odd 2 3450.2.a.be.1.2 2
40.27 even 4 3450.2.d.x.2899.3 4
56.27 even 2 6762.2.a.cb.1.1 2
184.91 even 2 3174.2.a.s.1.1 2
552.275 odd 2 9522.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.d.1.2 2 8.3 odd 2
414.2.a.f.1.1 2 24.11 even 2
1104.2.a.j.1.2 2 8.5 even 2
3174.2.a.s.1.1 2 184.91 even 2
3312.2.a.bc.1.1 2 24.5 odd 2
3450.2.a.be.1.2 2 40.19 odd 2
3450.2.d.x.2899.2 4 40.3 even 4
3450.2.d.x.2899.3 4 40.27 even 4
4416.2.a.bh.1.1 2 4.3 odd 2
4416.2.a.bl.1.1 2 1.1 even 1 trivial
6762.2.a.cb.1.1 2 56.27 even 2
9522.2.a.q.1.2 2 552.275 odd 2