Properties

Label 224.2.a.c.1.1
Level $224$
Weight $2$
Character 224.1
Self dual yes
Analytic conductor $1.789$
Analytic rank $0$
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] = [224,2,Mod(1,224)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 224.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: \(1.78864900528\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 224.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.23607 q^{3} -1.23607 q^{5} +1.00000 q^{7} +7.47214 q^{9} +O(q^{10})\) \(q-3.23607 q^{3} -1.23607 q^{5} +1.00000 q^{7} +7.47214 q^{9} +2.47214 q^{11} +5.23607 q^{13} +4.00000 q^{15} -4.47214 q^{17} +3.23607 q^{19} -3.23607 q^{21} +4.00000 q^{23} -3.47214 q^{25} -14.4721 q^{27} +4.47214 q^{29} +6.47214 q^{31} -8.00000 q^{33} -1.23607 q^{35} +4.47214 q^{37} -16.9443 q^{39} +0.472136 q^{41} -2.47214 q^{43} -9.23607 q^{45} +1.52786 q^{47} +1.00000 q^{49} +14.4721 q^{51} -10.0000 q^{53} -3.05573 q^{55} -10.4721 q^{57} -4.76393 q^{59} +6.76393 q^{61} +7.47214 q^{63} -6.47214 q^{65} +4.00000 q^{67} -12.9443 q^{69} +12.9443 q^{71} +14.9443 q^{73} +11.2361 q^{75} +2.47214 q^{77} -4.94427 q^{79} +24.4164 q^{81} -4.76393 q^{83} +5.52786 q^{85} -14.4721 q^{87} -6.00000 q^{89} +5.23607 q^{91} -20.9443 q^{93} -4.00000 q^{95} +3.52786 q^{97} +18.4721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 6 q^{9} - 4 q^{11} + 6 q^{13} + 8 q^{15} + 2 q^{19} - 2 q^{21} + 8 q^{23} + 2 q^{25} - 20 q^{27} + 4 q^{31} - 16 q^{33} + 2 q^{35} - 16 q^{39} - 8 q^{41} + 4 q^{43} - 14 q^{45} + 12 q^{47} + 2 q^{49} + 20 q^{51} - 20 q^{53} - 24 q^{55} - 12 q^{57} - 14 q^{59} + 18 q^{61} + 6 q^{63} - 4 q^{65} + 8 q^{67} - 8 q^{69} + 8 q^{71} + 12 q^{73} + 18 q^{75} - 4 q^{77} + 8 q^{79} + 22 q^{81} - 14 q^{83} + 20 q^{85} - 20 q^{87} - 12 q^{89} + 6 q^{91} - 24 q^{93} - 8 q^{95} + 16 q^{97} + 28 q^{99}+O(q^{100}) \) 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
\(3\) −3.23607 −1.86834 −0.934172 0.356822i \(-0.883860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 0 0
\(5\) −1.23607 −0.552786 −0.276393 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 7.47214 2.49071
\(10\) 0 0
\(11\) 2.47214 0.745377 0.372689 0.927957i \(-0.378436\pi\)
0.372689 + 0.927957i \(0.378436\pi\)
\(12\) 0 0
\(13\) 5.23607 1.45222 0.726112 0.687576i \(-0.241325\pi\)
0.726112 + 0.687576i \(0.241325\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) 3.23607 0.742405 0.371202 0.928552i \(-0.378946\pi\)
0.371202 + 0.928552i \(0.378946\pi\)
\(20\) 0 0
\(21\) −3.23607 −0.706168
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) 0 0
\(27\) −14.4721 −2.78516
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) 6.47214 1.16243 0.581215 0.813750i \(-0.302578\pi\)
0.581215 + 0.813750i \(0.302578\pi\)
\(32\) 0 0
\(33\) −8.00000 −1.39262
\(34\) 0 0
\(35\) −1.23607 −0.208934
\(36\) 0 0
\(37\) 4.47214 0.735215 0.367607 0.929981i \(-0.380177\pi\)
0.367607 + 0.929981i \(0.380177\pi\)
\(38\) 0 0
\(39\) −16.9443 −2.71325
\(40\) 0 0
\(41\) 0.472136 0.0737352 0.0368676 0.999320i \(-0.488262\pi\)
0.0368676 + 0.999320i \(0.488262\pi\)
\(42\) 0 0
\(43\) −2.47214 −0.376997 −0.188499 0.982073i \(-0.560362\pi\)
−0.188499 + 0.982073i \(0.560362\pi\)
\(44\) 0 0
\(45\) −9.23607 −1.37683
\(46\) 0 0
\(47\) 1.52786 0.222862 0.111431 0.993772i \(-0.464457\pi\)
0.111431 + 0.993772i \(0.464457\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 14.4721 2.02650
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) −3.05573 −0.412034
\(56\) 0 0
\(57\) −10.4721 −1.38707
\(58\) 0 0
\(59\) −4.76393 −0.620211 −0.310106 0.950702i \(-0.600364\pi\)
−0.310106 + 0.950702i \(0.600364\pi\)
\(60\) 0 0
\(61\) 6.76393 0.866033 0.433016 0.901386i \(-0.357449\pi\)
0.433016 + 0.901386i \(0.357449\pi\)
\(62\) 0 0
\(63\) 7.47214 0.941401
\(64\) 0 0
\(65\) −6.47214 −0.802770
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −12.9443 −1.55831
\(70\) 0 0
\(71\) 12.9443 1.53620 0.768101 0.640328i \(-0.221202\pi\)
0.768101 + 0.640328i \(0.221202\pi\)
\(72\) 0 0
\(73\) 14.9443 1.74909 0.874547 0.484940i \(-0.161159\pi\)
0.874547 + 0.484940i \(0.161159\pi\)
\(74\) 0 0
\(75\) 11.2361 1.29743
\(76\) 0 0
\(77\) 2.47214 0.281726
\(78\) 0 0
\(79\) −4.94427 −0.556274 −0.278137 0.960541i \(-0.589717\pi\)
−0.278137 + 0.960541i \(0.589717\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) 0 0
\(83\) −4.76393 −0.522909 −0.261455 0.965216i \(-0.584202\pi\)
−0.261455 + 0.965216i \(0.584202\pi\)
\(84\) 0 0
\(85\) 5.52786 0.599581
\(86\) 0 0
\(87\) −14.4721 −1.55158
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 5.23607 0.548889
\(92\) 0 0
\(93\) −20.9443 −2.17182
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 3.52786 0.358200 0.179100 0.983831i \(-0.442681\pi\)
0.179100 + 0.983831i \(0.442681\pi\)
\(98\) 0 0
\(99\) 18.4721 1.85652
\(100\) 0 0
\(101\) 11.7082 1.16501 0.582505 0.812827i \(-0.302073\pi\)
0.582505 + 0.812827i \(0.302073\pi\)
\(102\) 0 0
\(103\) −14.4721 −1.42598 −0.712991 0.701173i \(-0.752660\pi\)
−0.712991 + 0.701173i \(0.752660\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) −8.94427 −0.864675 −0.432338 0.901712i \(-0.642311\pi\)
−0.432338 + 0.901712i \(0.642311\pi\)
\(108\) 0 0
\(109\) −0.472136 −0.0452224 −0.0226112 0.999744i \(-0.507198\pi\)
−0.0226112 + 0.999744i \(0.507198\pi\)
\(110\) 0 0
\(111\) −14.4721 −1.37363
\(112\) 0 0
\(113\) −3.52786 −0.331874 −0.165937 0.986136i \(-0.553065\pi\)
−0.165937 + 0.986136i \(0.553065\pi\)
\(114\) 0 0
\(115\) −4.94427 −0.461056
\(116\) 0 0
\(117\) 39.1246 3.61707
\(118\) 0 0
\(119\) −4.47214 −0.409960
\(120\) 0 0
\(121\) −4.88854 −0.444413
\(122\) 0 0
\(123\) −1.52786 −0.137763
\(124\) 0 0
\(125\) 10.4721 0.936656
\(126\) 0 0
\(127\) −8.94427 −0.793676 −0.396838 0.917889i \(-0.629892\pi\)
−0.396838 + 0.917889i \(0.629892\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −1.70820 −0.149246 −0.0746232 0.997212i \(-0.523775\pi\)
−0.0746232 + 0.997212i \(0.523775\pi\)
\(132\) 0 0
\(133\) 3.23607 0.280603
\(134\) 0 0
\(135\) 17.8885 1.53960
\(136\) 0 0
\(137\) −2.94427 −0.251546 −0.125773 0.992059i \(-0.540141\pi\)
−0.125773 + 0.992059i \(0.540141\pi\)
\(138\) 0 0
\(139\) −3.23607 −0.274480 −0.137240 0.990538i \(-0.543823\pi\)
−0.137240 + 0.990538i \(0.543823\pi\)
\(140\) 0 0
\(141\) −4.94427 −0.416383
\(142\) 0 0
\(143\) 12.9443 1.08245
\(144\) 0 0
\(145\) −5.52786 −0.459064
\(146\) 0 0
\(147\) −3.23607 −0.266906
\(148\) 0 0
\(149\) −14.9443 −1.22428 −0.612141 0.790748i \(-0.709692\pi\)
−0.612141 + 0.790748i \(0.709692\pi\)
\(150\) 0 0
\(151\) 8.94427 0.727875 0.363937 0.931423i \(-0.381432\pi\)
0.363937 + 0.931423i \(0.381432\pi\)
\(152\) 0 0
\(153\) −33.4164 −2.70156
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) 5.23607 0.417884 0.208942 0.977928i \(-0.432998\pi\)
0.208942 + 0.977928i \(0.432998\pi\)
\(158\) 0 0
\(159\) 32.3607 2.56637
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 23.4164 1.83411 0.917057 0.398755i \(-0.130558\pi\)
0.917057 + 0.398755i \(0.130558\pi\)
\(164\) 0 0
\(165\) 9.88854 0.769822
\(166\) 0 0
\(167\) −3.41641 −0.264370 −0.132185 0.991225i \(-0.542199\pi\)
−0.132185 + 0.991225i \(0.542199\pi\)
\(168\) 0 0
\(169\) 14.4164 1.10895
\(170\) 0 0
\(171\) 24.1803 1.84912
\(172\) 0 0
\(173\) −7.70820 −0.586044 −0.293022 0.956106i \(-0.594661\pi\)
−0.293022 + 0.956106i \(0.594661\pi\)
\(174\) 0 0
\(175\) −3.47214 −0.262469
\(176\) 0 0
\(177\) 15.4164 1.15877
\(178\) 0 0
\(179\) −24.9443 −1.86442 −0.932211 0.361915i \(-0.882123\pi\)
−0.932211 + 0.361915i \(0.882123\pi\)
\(180\) 0 0
\(181\) 10.1803 0.756699 0.378349 0.925663i \(-0.376492\pi\)
0.378349 + 0.925663i \(0.376492\pi\)
\(182\) 0 0
\(183\) −21.8885 −1.61805
\(184\) 0 0
\(185\) −5.52786 −0.406417
\(186\) 0 0
\(187\) −11.0557 −0.808475
\(188\) 0 0
\(189\) −14.4721 −1.05269
\(190\) 0 0
\(191\) −12.9443 −0.936615 −0.468307 0.883566i \(-0.655136\pi\)
−0.468307 + 0.883566i \(0.655136\pi\)
\(192\) 0 0
\(193\) −8.47214 −0.609838 −0.304919 0.952378i \(-0.598629\pi\)
−0.304919 + 0.952378i \(0.598629\pi\)
\(194\) 0 0
\(195\) 20.9443 1.49985
\(196\) 0 0
\(197\) −6.94427 −0.494759 −0.247379 0.968919i \(-0.579569\pi\)
−0.247379 + 0.968919i \(0.579569\pi\)
\(198\) 0 0
\(199\) −11.4164 −0.809288 −0.404644 0.914474i \(-0.632605\pi\)
−0.404644 + 0.914474i \(0.632605\pi\)
\(200\) 0 0
\(201\) −12.9443 −0.913019
\(202\) 0 0
\(203\) 4.47214 0.313882
\(204\) 0 0
\(205\) −0.583592 −0.0407598
\(206\) 0 0
\(207\) 29.8885 2.07740
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) −41.8885 −2.87016
\(214\) 0 0
\(215\) 3.05573 0.208399
\(216\) 0 0
\(217\) 6.47214 0.439357
\(218\) 0 0
\(219\) −48.3607 −3.26791
\(220\) 0 0
\(221\) −23.4164 −1.57516
\(222\) 0 0
\(223\) −4.94427 −0.331093 −0.165546 0.986202i \(-0.552939\pi\)
−0.165546 + 0.986202i \(0.552939\pi\)
\(224\) 0 0
\(225\) −25.9443 −1.72962
\(226\) 0 0
\(227\) −12.7639 −0.847172 −0.423586 0.905856i \(-0.639229\pi\)
−0.423586 + 0.905856i \(0.639229\pi\)
\(228\) 0 0
\(229\) −25.5967 −1.69148 −0.845740 0.533595i \(-0.820841\pi\)
−0.845740 + 0.533595i \(0.820841\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) 19.8885 1.30294 0.651471 0.758674i \(-0.274152\pi\)
0.651471 + 0.758674i \(0.274152\pi\)
\(234\) 0 0
\(235\) −1.88854 −0.123195
\(236\) 0 0
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) 21.8885 1.41585 0.707926 0.706287i \(-0.249631\pi\)
0.707926 + 0.706287i \(0.249631\pi\)
\(240\) 0 0
\(241\) 3.52786 0.227250 0.113625 0.993524i \(-0.463754\pi\)
0.113625 + 0.993524i \(0.463754\pi\)
\(242\) 0 0
\(243\) −35.5967 −2.28353
\(244\) 0 0
\(245\) −1.23607 −0.0789695
\(246\) 0 0
\(247\) 16.9443 1.07814
\(248\) 0 0
\(249\) 15.4164 0.976975
\(250\) 0 0
\(251\) 17.7082 1.11773 0.558866 0.829258i \(-0.311237\pi\)
0.558866 + 0.829258i \(0.311237\pi\)
\(252\) 0 0
\(253\) 9.88854 0.621687
\(254\) 0 0
\(255\) −17.8885 −1.12022
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 4.47214 0.277885
\(260\) 0 0
\(261\) 33.4164 2.06842
\(262\) 0 0
\(263\) 28.9443 1.78478 0.892390 0.451265i \(-0.149027\pi\)
0.892390 + 0.451265i \(0.149027\pi\)
\(264\) 0 0
\(265\) 12.3607 0.759311
\(266\) 0 0
\(267\) 19.4164 1.18826
\(268\) 0 0
\(269\) 18.1803 1.10847 0.554237 0.832359i \(-0.313010\pi\)
0.554237 + 0.832359i \(0.313010\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) −16.9443 −1.02551
\(274\) 0 0
\(275\) −8.58359 −0.517610
\(276\) 0 0
\(277\) −27.8885 −1.67566 −0.837830 0.545931i \(-0.816176\pi\)
−0.837830 + 0.545931i \(0.816176\pi\)
\(278\) 0 0
\(279\) 48.3607 2.89528
\(280\) 0 0
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) 16.1803 0.961821 0.480911 0.876770i \(-0.340306\pi\)
0.480911 + 0.876770i \(0.340306\pi\)
\(284\) 0 0
\(285\) 12.9443 0.766752
\(286\) 0 0
\(287\) 0.472136 0.0278693
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) −11.4164 −0.669242
\(292\) 0 0
\(293\) −17.2361 −1.00694 −0.503471 0.864012i \(-0.667944\pi\)
−0.503471 + 0.864012i \(0.667944\pi\)
\(294\) 0 0
\(295\) 5.88854 0.342844
\(296\) 0 0
\(297\) −35.7771 −2.07600
\(298\) 0 0
\(299\) 20.9443 1.21124
\(300\) 0 0
\(301\) −2.47214 −0.142492
\(302\) 0 0
\(303\) −37.8885 −2.17664
\(304\) 0 0
\(305\) −8.36068 −0.478731
\(306\) 0 0
\(307\) 24.1803 1.38004 0.690022 0.723788i \(-0.257601\pi\)
0.690022 + 0.723788i \(0.257601\pi\)
\(308\) 0 0
\(309\) 46.8328 2.66423
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 0.472136 0.0266867 0.0133434 0.999911i \(-0.495753\pi\)
0.0133434 + 0.999911i \(0.495753\pi\)
\(314\) 0 0
\(315\) −9.23607 −0.520393
\(316\) 0 0
\(317\) 26.9443 1.51334 0.756671 0.653796i \(-0.226825\pi\)
0.756671 + 0.653796i \(0.226825\pi\)
\(318\) 0 0
\(319\) 11.0557 0.619002
\(320\) 0 0
\(321\) 28.9443 1.61551
\(322\) 0 0
\(323\) −14.4721 −0.805251
\(324\) 0 0
\(325\) −18.1803 −1.00846
\(326\) 0 0
\(327\) 1.52786 0.0844911
\(328\) 0 0
\(329\) 1.52786 0.0842339
\(330\) 0 0
\(331\) 13.5279 0.743559 0.371779 0.928321i \(-0.378748\pi\)
0.371779 + 0.928321i \(0.378748\pi\)
\(332\) 0 0
\(333\) 33.4164 1.83121
\(334\) 0 0
\(335\) −4.94427 −0.270134
\(336\) 0 0
\(337\) −34.3607 −1.87175 −0.935873 0.352338i \(-0.885387\pi\)
−0.935873 + 0.352338i \(0.885387\pi\)
\(338\) 0 0
\(339\) 11.4164 0.620054
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 16.0000 0.861411
\(346\) 0 0
\(347\) −2.47214 −0.132711 −0.0663556 0.997796i \(-0.521137\pi\)
−0.0663556 + 0.997796i \(0.521137\pi\)
\(348\) 0 0
\(349\) −4.65248 −0.249041 −0.124521 0.992217i \(-0.539739\pi\)
−0.124521 + 0.992217i \(0.539739\pi\)
\(350\) 0 0
\(351\) −75.7771 −4.04468
\(352\) 0 0
\(353\) 19.8885 1.05856 0.529280 0.848447i \(-0.322462\pi\)
0.529280 + 0.848447i \(0.322462\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) 0 0
\(357\) 14.4721 0.765947
\(358\) 0 0
\(359\) 0.944272 0.0498368 0.0249184 0.999689i \(-0.492067\pi\)
0.0249184 + 0.999689i \(0.492067\pi\)
\(360\) 0 0
\(361\) −8.52786 −0.448835
\(362\) 0 0
\(363\) 15.8197 0.830317
\(364\) 0 0
\(365\) −18.4721 −0.966876
\(366\) 0 0
\(367\) 30.8328 1.60946 0.804730 0.593641i \(-0.202310\pi\)
0.804730 + 0.593641i \(0.202310\pi\)
\(368\) 0 0
\(369\) 3.52786 0.183653
\(370\) 0 0
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) −14.9443 −0.773785 −0.386893 0.922125i \(-0.626452\pi\)
−0.386893 + 0.922125i \(0.626452\pi\)
\(374\) 0 0
\(375\) −33.8885 −1.75000
\(376\) 0 0
\(377\) 23.4164 1.20601
\(378\) 0 0
\(379\) −31.4164 −1.61375 −0.806876 0.590721i \(-0.798844\pi\)
−0.806876 + 0.590721i \(0.798844\pi\)
\(380\) 0 0
\(381\) 28.9443 1.48286
\(382\) 0 0
\(383\) 11.4164 0.583351 0.291676 0.956517i \(-0.405787\pi\)
0.291676 + 0.956517i \(0.405787\pi\)
\(384\) 0 0
\(385\) −3.05573 −0.155734
\(386\) 0 0
\(387\) −18.4721 −0.938991
\(388\) 0 0
\(389\) 4.47214 0.226746 0.113373 0.993552i \(-0.463834\pi\)
0.113373 + 0.993552i \(0.463834\pi\)
\(390\) 0 0
\(391\) −17.8885 −0.904663
\(392\) 0 0
\(393\) 5.52786 0.278844
\(394\) 0 0
\(395\) 6.11146 0.307501
\(396\) 0 0
\(397\) −10.7639 −0.540226 −0.270113 0.962829i \(-0.587061\pi\)
−0.270113 + 0.962829i \(0.587061\pi\)
\(398\) 0 0
\(399\) −10.4721 −0.524263
\(400\) 0 0
\(401\) −32.4721 −1.62158 −0.810791 0.585336i \(-0.800962\pi\)
−0.810791 + 0.585336i \(0.800962\pi\)
\(402\) 0 0
\(403\) 33.8885 1.68811
\(404\) 0 0
\(405\) −30.1803 −1.49967
\(406\) 0 0
\(407\) 11.0557 0.548012
\(408\) 0 0
\(409\) 5.41641 0.267824 0.133912 0.990993i \(-0.457246\pi\)
0.133912 + 0.990993i \(0.457246\pi\)
\(410\) 0 0
\(411\) 9.52786 0.469975
\(412\) 0 0
\(413\) −4.76393 −0.234418
\(414\) 0 0
\(415\) 5.88854 0.289057
\(416\) 0 0
\(417\) 10.4721 0.512823
\(418\) 0 0
\(419\) −0.180340 −0.00881018 −0.00440509 0.999990i \(-0.501402\pi\)
−0.00440509 + 0.999990i \(0.501402\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 11.4164 0.555085
\(424\) 0 0
\(425\) 15.5279 0.753212
\(426\) 0 0
\(427\) 6.76393 0.327330
\(428\) 0 0
\(429\) −41.8885 −2.02240
\(430\) 0 0
\(431\) −28.0000 −1.34871 −0.674356 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) 0 0
\(433\) −17.4164 −0.836979 −0.418490 0.908222i \(-0.637440\pi\)
−0.418490 + 0.908222i \(0.637440\pi\)
\(434\) 0 0
\(435\) 17.8885 0.857690
\(436\) 0 0
\(437\) 12.9443 0.619208
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 7.47214 0.355816
\(442\) 0 0
\(443\) 21.8885 1.03996 0.519978 0.854180i \(-0.325940\pi\)
0.519978 + 0.854180i \(0.325940\pi\)
\(444\) 0 0
\(445\) 7.41641 0.351571
\(446\) 0 0
\(447\) 48.3607 2.28738
\(448\) 0 0
\(449\) 27.8885 1.31614 0.658071 0.752956i \(-0.271373\pi\)
0.658071 + 0.752956i \(0.271373\pi\)
\(450\) 0 0
\(451\) 1.16718 0.0549606
\(452\) 0 0
\(453\) −28.9443 −1.35992
\(454\) 0 0
\(455\) −6.47214 −0.303418
\(456\) 0 0
\(457\) −16.4721 −0.770534 −0.385267 0.922805i \(-0.625891\pi\)
−0.385267 + 0.922805i \(0.625891\pi\)
\(458\) 0 0
\(459\) 64.7214 3.02093
\(460\) 0 0
\(461\) 8.29180 0.386187 0.193094 0.981180i \(-0.438148\pi\)
0.193094 + 0.981180i \(0.438148\pi\)
\(462\) 0 0
\(463\) 35.7771 1.66270 0.831351 0.555748i \(-0.187568\pi\)
0.831351 + 0.555748i \(0.187568\pi\)
\(464\) 0 0
\(465\) 25.8885 1.20055
\(466\) 0 0
\(467\) 26.0689 1.20632 0.603162 0.797619i \(-0.293907\pi\)
0.603162 + 0.797619i \(0.293907\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −16.9443 −0.780751
\(472\) 0 0
\(473\) −6.11146 −0.281005
\(474\) 0 0
\(475\) −11.2361 −0.515546
\(476\) 0 0
\(477\) −74.7214 −3.42126
\(478\) 0 0
\(479\) −35.4164 −1.61822 −0.809108 0.587659i \(-0.800050\pi\)
−0.809108 + 0.587659i \(0.800050\pi\)
\(480\) 0 0
\(481\) 23.4164 1.06770
\(482\) 0 0
\(483\) −12.9443 −0.588985
\(484\) 0 0
\(485\) −4.36068 −0.198008
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 0 0
\(489\) −75.7771 −3.42676
\(490\) 0 0
\(491\) 2.11146 0.0952887 0.0476443 0.998864i \(-0.484829\pi\)
0.0476443 + 0.998864i \(0.484829\pi\)
\(492\) 0 0
\(493\) −20.0000 −0.900755
\(494\) 0 0
\(495\) −22.8328 −1.02626
\(496\) 0 0
\(497\) 12.9443 0.580630
\(498\) 0 0
\(499\) −13.8885 −0.621737 −0.310868 0.950453i \(-0.600620\pi\)
−0.310868 + 0.950453i \(0.600620\pi\)
\(500\) 0 0
\(501\) 11.0557 0.493934
\(502\) 0 0
\(503\) 12.9443 0.577157 0.288578 0.957456i \(-0.406817\pi\)
0.288578 + 0.957456i \(0.406817\pi\)
\(504\) 0 0
\(505\) −14.4721 −0.644002
\(506\) 0 0
\(507\) −46.6525 −2.07191
\(508\) 0 0
\(509\) −0.875388 −0.0388009 −0.0194004 0.999812i \(-0.506176\pi\)
−0.0194004 + 0.999812i \(0.506176\pi\)
\(510\) 0 0
\(511\) 14.9443 0.661096
\(512\) 0 0
\(513\) −46.8328 −2.06772
\(514\) 0 0
\(515\) 17.8885 0.788263
\(516\) 0 0
\(517\) 3.77709 0.166116
\(518\) 0 0
\(519\) 24.9443 1.09493
\(520\) 0 0
\(521\) −33.4164 −1.46400 −0.732000 0.681305i \(-0.761413\pi\)
−0.732000 + 0.681305i \(0.761413\pi\)
\(522\) 0 0
\(523\) −17.7082 −0.774326 −0.387163 0.922011i \(-0.626545\pi\)
−0.387163 + 0.922011i \(0.626545\pi\)
\(524\) 0 0
\(525\) 11.2361 0.490382
\(526\) 0 0
\(527\) −28.9443 −1.26083
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −35.5967 −1.54477
\(532\) 0 0
\(533\) 2.47214 0.107080
\(534\) 0 0
\(535\) 11.0557 0.477981
\(536\) 0 0
\(537\) 80.7214 3.48338
\(538\) 0 0
\(539\) 2.47214 0.106482
\(540\) 0 0
\(541\) −22.9443 −0.986451 −0.493226 0.869901i \(-0.664182\pi\)
−0.493226 + 0.869901i \(0.664182\pi\)
\(542\) 0 0
\(543\) −32.9443 −1.41377
\(544\) 0 0
\(545\) 0.583592 0.0249983
\(546\) 0 0
\(547\) −31.4164 −1.34327 −0.671634 0.740883i \(-0.734407\pi\)
−0.671634 + 0.740883i \(0.734407\pi\)
\(548\) 0 0
\(549\) 50.5410 2.15704
\(550\) 0 0
\(551\) 14.4721 0.616534
\(552\) 0 0
\(553\) −4.94427 −0.210252
\(554\) 0 0
\(555\) 17.8885 0.759326
\(556\) 0 0
\(557\) 26.9443 1.14167 0.570833 0.821066i \(-0.306620\pi\)
0.570833 + 0.821066i \(0.306620\pi\)
\(558\) 0 0
\(559\) −12.9443 −0.547484
\(560\) 0 0
\(561\) 35.7771 1.51051
\(562\) 0 0
\(563\) −40.1803 −1.69340 −0.846700 0.532071i \(-0.821414\pi\)
−0.846700 + 0.532071i \(0.821414\pi\)
\(564\) 0 0
\(565\) 4.36068 0.183455
\(566\) 0 0
\(567\) 24.4164 1.02539
\(568\) 0 0
\(569\) −18.3607 −0.769720 −0.384860 0.922975i \(-0.625750\pi\)
−0.384860 + 0.922975i \(0.625750\pi\)
\(570\) 0 0
\(571\) −5.52786 −0.231334 −0.115667 0.993288i \(-0.536901\pi\)
−0.115667 + 0.993288i \(0.536901\pi\)
\(572\) 0 0
\(573\) 41.8885 1.74992
\(574\) 0 0
\(575\) −13.8885 −0.579192
\(576\) 0 0
\(577\) −6.00000 −0.249783 −0.124892 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 0 0
\(579\) 27.4164 1.13939
\(580\) 0 0
\(581\) −4.76393 −0.197641
\(582\) 0 0
\(583\) −24.7214 −1.02385
\(584\) 0 0
\(585\) −48.3607 −1.99947
\(586\) 0 0
\(587\) 9.70820 0.400700 0.200350 0.979724i \(-0.435792\pi\)
0.200350 + 0.979724i \(0.435792\pi\)
\(588\) 0 0
\(589\) 20.9443 0.862994
\(590\) 0 0
\(591\) 22.4721 0.924380
\(592\) 0 0
\(593\) −20.8328 −0.855501 −0.427751 0.903897i \(-0.640694\pi\)
−0.427751 + 0.903897i \(0.640694\pi\)
\(594\) 0 0
\(595\) 5.52786 0.226620
\(596\) 0 0
\(597\) 36.9443 1.51203
\(598\) 0 0
\(599\) −17.8885 −0.730906 −0.365453 0.930830i \(-0.619086\pi\)
−0.365453 + 0.930830i \(0.619086\pi\)
\(600\) 0 0
\(601\) −41.7771 −1.70412 −0.852061 0.523442i \(-0.824648\pi\)
−0.852061 + 0.523442i \(0.824648\pi\)
\(602\) 0 0
\(603\) 29.8885 1.21716
\(604\) 0 0
\(605\) 6.04257 0.245666
\(606\) 0 0
\(607\) −25.8885 −1.05078 −0.525392 0.850860i \(-0.676081\pi\)
−0.525392 + 0.850860i \(0.676081\pi\)
\(608\) 0 0
\(609\) −14.4721 −0.586441
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 2.58359 0.104350 0.0521752 0.998638i \(-0.483385\pi\)
0.0521752 + 0.998638i \(0.483385\pi\)
\(614\) 0 0
\(615\) 1.88854 0.0761534
\(616\) 0 0
\(617\) −10.3607 −0.417105 −0.208553 0.978011i \(-0.566875\pi\)
−0.208553 + 0.978011i \(0.566875\pi\)
\(618\) 0 0
\(619\) 10.0689 0.404703 0.202351 0.979313i \(-0.435142\pi\)
0.202351 + 0.979313i \(0.435142\pi\)
\(620\) 0 0
\(621\) −57.8885 −2.32299
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 0 0
\(627\) −25.8885 −1.03389
\(628\) 0 0
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) −27.0557 −1.07707 −0.538536 0.842603i \(-0.681022\pi\)
−0.538536 + 0.842603i \(0.681022\pi\)
\(632\) 0 0
\(633\) 38.8328 1.54347
\(634\) 0 0
\(635\) 11.0557 0.438733
\(636\) 0 0
\(637\) 5.23607 0.207461
\(638\) 0 0
\(639\) 96.7214 3.82624
\(640\) 0 0
\(641\) 41.4164 1.63585 0.817925 0.575325i \(-0.195124\pi\)
0.817925 + 0.575325i \(0.195124\pi\)
\(642\) 0 0
\(643\) −30.2918 −1.19459 −0.597296 0.802021i \(-0.703758\pi\)
−0.597296 + 0.802021i \(0.703758\pi\)
\(644\) 0 0
\(645\) −9.88854 −0.389361
\(646\) 0 0
\(647\) 24.3607 0.957717 0.478859 0.877892i \(-0.341051\pi\)
0.478859 + 0.877892i \(0.341051\pi\)
\(648\) 0 0
\(649\) −11.7771 −0.462291
\(650\) 0 0
\(651\) −20.9443 −0.820871
\(652\) 0 0
\(653\) 17.4164 0.681557 0.340778 0.940144i \(-0.389309\pi\)
0.340778 + 0.940144i \(0.389309\pi\)
\(654\) 0 0
\(655\) 2.11146 0.0825014
\(656\) 0 0
\(657\) 111.666 4.35649
\(658\) 0 0
\(659\) −25.3050 −0.985741 −0.492870 0.870103i \(-0.664052\pi\)
−0.492870 + 0.870103i \(0.664052\pi\)
\(660\) 0 0
\(661\) 8.65248 0.336542 0.168271 0.985741i \(-0.446182\pi\)
0.168271 + 0.985741i \(0.446182\pi\)
\(662\) 0 0
\(663\) 75.7771 2.94294
\(664\) 0 0
\(665\) −4.00000 −0.155113
\(666\) 0 0
\(667\) 17.8885 0.692647
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 16.7214 0.645521
\(672\) 0 0
\(673\) 14.9443 0.576059 0.288030 0.957621i \(-0.407000\pi\)
0.288030 + 0.957621i \(0.407000\pi\)
\(674\) 0 0
\(675\) 50.2492 1.93409
\(676\) 0 0
\(677\) 42.1803 1.62112 0.810561 0.585654i \(-0.199162\pi\)
0.810561 + 0.585654i \(0.199162\pi\)
\(678\) 0 0
\(679\) 3.52786 0.135387
\(680\) 0 0
\(681\) 41.3050 1.58281
\(682\) 0 0
\(683\) −47.7771 −1.82814 −0.914070 0.405557i \(-0.867078\pi\)
−0.914070 + 0.405557i \(0.867078\pi\)
\(684\) 0 0
\(685\) 3.63932 0.139051
\(686\) 0 0
\(687\) 82.8328 3.16027
\(688\) 0 0
\(689\) −52.3607 −1.99478
\(690\) 0 0
\(691\) 8.18034 0.311195 0.155597 0.987821i \(-0.450270\pi\)
0.155597 + 0.987821i \(0.450270\pi\)
\(692\) 0 0
\(693\) 18.4721 0.701698
\(694\) 0 0
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) −2.11146 −0.0799771
\(698\) 0 0
\(699\) −64.3607 −2.43434
\(700\) 0 0
\(701\) −14.5836 −0.550815 −0.275407 0.961328i \(-0.588813\pi\)
−0.275407 + 0.961328i \(0.588813\pi\)
\(702\) 0 0
\(703\) 14.4721 0.545827
\(704\) 0 0
\(705\) 6.11146 0.230171
\(706\) 0 0
\(707\) 11.7082 0.440332
\(708\) 0 0
\(709\) 46.3607 1.74111 0.870556 0.492069i \(-0.163759\pi\)
0.870556 + 0.492069i \(0.163759\pi\)
\(710\) 0 0
\(711\) −36.9443 −1.38552
\(712\) 0 0
\(713\) 25.8885 0.969534
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 0 0
\(717\) −70.8328 −2.64530
\(718\) 0 0
\(719\) −47.1935 −1.76002 −0.880010 0.474955i \(-0.842464\pi\)
−0.880010 + 0.474955i \(0.842464\pi\)
\(720\) 0 0
\(721\) −14.4721 −0.538971
\(722\) 0 0
\(723\) −11.4164 −0.424581
\(724\) 0 0
\(725\) −15.5279 −0.576690
\(726\) 0 0
\(727\) −32.3607 −1.20019 −0.600096 0.799928i \(-0.704871\pi\)
−0.600096 + 0.799928i \(0.704871\pi\)
\(728\) 0 0
\(729\) 41.9443 1.55349
\(730\) 0 0
\(731\) 11.0557 0.408911
\(732\) 0 0
\(733\) −9.23607 −0.341142 −0.170571 0.985345i \(-0.554561\pi\)
−0.170571 + 0.985345i \(0.554561\pi\)
\(734\) 0 0
\(735\) 4.00000 0.147542
\(736\) 0 0
\(737\) 9.88854 0.364249
\(738\) 0 0
\(739\) 23.4164 0.861386 0.430693 0.902498i \(-0.358269\pi\)
0.430693 + 0.902498i \(0.358269\pi\)
\(740\) 0 0
\(741\) −54.8328 −2.01433
\(742\) 0 0
\(743\) 7.05573 0.258850 0.129425 0.991589i \(-0.458687\pi\)
0.129425 + 0.991589i \(0.458687\pi\)
\(744\) 0 0
\(745\) 18.4721 0.676767
\(746\) 0 0
\(747\) −35.5967 −1.30242
\(748\) 0 0
\(749\) −8.94427 −0.326817
\(750\) 0 0
\(751\) 36.0000 1.31366 0.656829 0.754039i \(-0.271897\pi\)
0.656829 + 0.754039i \(0.271897\pi\)
\(752\) 0 0
\(753\) −57.3050 −2.08831
\(754\) 0 0
\(755\) −11.0557 −0.402359
\(756\) 0 0
\(757\) −23.3050 −0.847033 −0.423516 0.905888i \(-0.639204\pi\)
−0.423516 + 0.905888i \(0.639204\pi\)
\(758\) 0 0
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) −12.4721 −0.452115 −0.226057 0.974114i \(-0.572584\pi\)
−0.226057 + 0.974114i \(0.572584\pi\)
\(762\) 0 0
\(763\) −0.472136 −0.0170925
\(764\) 0 0
\(765\) 41.3050 1.49338
\(766\) 0 0
\(767\) −24.9443 −0.900685
\(768\) 0 0
\(769\) 26.3607 0.950590 0.475295 0.879826i \(-0.342341\pi\)
0.475295 + 0.879826i \(0.342341\pi\)
\(770\) 0 0
\(771\) 45.3050 1.63162
\(772\) 0 0
\(773\) 17.8197 0.640929 0.320464 0.947261i \(-0.396161\pi\)
0.320464 + 0.947261i \(0.396161\pi\)
\(774\) 0 0
\(775\) −22.4721 −0.807223
\(776\) 0 0
\(777\) −14.4721 −0.519185
\(778\) 0 0
\(779\) 1.52786 0.0547414
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) −64.7214 −2.31295
\(784\) 0 0
\(785\) −6.47214 −0.231000
\(786\) 0 0
\(787\) −25.7082 −0.916399 −0.458199 0.888850i \(-0.651505\pi\)
−0.458199 + 0.888850i \(0.651505\pi\)
\(788\) 0 0
\(789\) −93.6656 −3.33458
\(790\) 0 0
\(791\) −3.52786 −0.125436
\(792\) 0 0
\(793\) 35.4164 1.25767
\(794\) 0 0
\(795\) −40.0000 −1.41865
\(796\) 0 0
\(797\) −29.8197 −1.05627 −0.528133 0.849161i \(-0.677108\pi\)
−0.528133 + 0.849161i \(0.677108\pi\)
\(798\) 0 0
\(799\) −6.83282 −0.241728
\(800\) 0 0
\(801\) −44.8328 −1.58409
\(802\) 0 0
\(803\) 36.9443 1.30374
\(804\) 0 0
\(805\) −4.94427 −0.174263
\(806\) 0 0
\(807\) −58.8328 −2.07101
\(808\) 0 0
\(809\) 9.41641 0.331063 0.165532 0.986204i \(-0.447066\pi\)
0.165532 + 0.986204i \(0.447066\pi\)
\(810\) 0 0
\(811\) 23.0132 0.808101 0.404051 0.914737i \(-0.367602\pi\)
0.404051 + 0.914737i \(0.367602\pi\)
\(812\) 0 0
\(813\) 77.6656 2.72385
\(814\) 0 0
\(815\) −28.9443 −1.01387
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) 0 0
\(819\) 39.1246 1.36712
\(820\) 0 0
\(821\) 39.8885 1.39212 0.696060 0.717984i \(-0.254935\pi\)
0.696060 + 0.717984i \(0.254935\pi\)
\(822\) 0 0
\(823\) 51.7771 1.80484 0.902418 0.430862i \(-0.141790\pi\)
0.902418 + 0.430862i \(0.141790\pi\)
\(824\) 0 0
\(825\) 27.7771 0.967074
\(826\) 0 0
\(827\) 32.9443 1.14558 0.572792 0.819701i \(-0.305860\pi\)
0.572792 + 0.819701i \(0.305860\pi\)
\(828\) 0 0
\(829\) 6.76393 0.234921 0.117461 0.993078i \(-0.462525\pi\)
0.117461 + 0.993078i \(0.462525\pi\)
\(830\) 0 0
\(831\) 90.2492 3.13071
\(832\) 0 0
\(833\) −4.47214 −0.154950
\(834\) 0 0
\(835\) 4.22291 0.146140
\(836\) 0 0
\(837\) −93.6656 −3.23756
\(838\) 0 0
\(839\) −30.4721 −1.05201 −0.526007 0.850480i \(-0.676312\pi\)
−0.526007 + 0.850480i \(0.676312\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) −84.1378 −2.89786
\(844\) 0 0
\(845\) −17.8197 −0.613015
\(846\) 0 0
\(847\) −4.88854 −0.167972
\(848\) 0 0
\(849\) −52.3607 −1.79701
\(850\) 0 0
\(851\) 17.8885 0.613211
\(852\) 0 0
\(853\) 0.291796 0.00999091 0.00499545 0.999988i \(-0.498410\pi\)
0.00499545 + 0.999988i \(0.498410\pi\)
\(854\) 0 0
\(855\) −29.8885 −1.02217
\(856\) 0 0
\(857\) −36.4721 −1.24586 −0.622932 0.782276i \(-0.714059\pi\)
−0.622932 + 0.782276i \(0.714059\pi\)
\(858\) 0 0
\(859\) −56.1803 −1.91685 −0.958424 0.285347i \(-0.907891\pi\)
−0.958424 + 0.285347i \(0.907891\pi\)
\(860\) 0 0
\(861\) −1.52786 −0.0520695
\(862\) 0 0
\(863\) −3.05573 −0.104018 −0.0520091 0.998647i \(-0.516562\pi\)
−0.0520091 + 0.998647i \(0.516562\pi\)
\(864\) 0 0
\(865\) 9.52786 0.323957
\(866\) 0 0
\(867\) −9.70820 −0.329708
\(868\) 0 0
\(869\) −12.2229 −0.414634
\(870\) 0 0
\(871\) 20.9443 0.709670
\(872\) 0 0
\(873\) 26.3607 0.892174
\(874\) 0 0
\(875\) 10.4721 0.354023
\(876\) 0 0
\(877\) −13.4164 −0.453040 −0.226520 0.974007i \(-0.572735\pi\)
−0.226520 + 0.974007i \(0.572735\pi\)
\(878\) 0 0
\(879\) 55.7771 1.88131
\(880\) 0 0
\(881\) −28.8328 −0.971402 −0.485701 0.874125i \(-0.661436\pi\)
−0.485701 + 0.874125i \(0.661436\pi\)
\(882\) 0 0
\(883\) 40.9443 1.37788 0.688942 0.724816i \(-0.258075\pi\)
0.688942 + 0.724816i \(0.258075\pi\)
\(884\) 0 0
\(885\) −19.0557 −0.640551
\(886\) 0 0
\(887\) 29.3050 0.983964 0.491982 0.870605i \(-0.336273\pi\)
0.491982 + 0.870605i \(0.336273\pi\)
\(888\) 0 0
\(889\) −8.94427 −0.299981
\(890\) 0 0
\(891\) 60.3607 2.02216
\(892\) 0 0
\(893\) 4.94427 0.165454
\(894\) 0 0
\(895\) 30.8328 1.03063
\(896\) 0 0
\(897\) −67.7771 −2.26301
\(898\) 0 0
\(899\) 28.9443 0.965346
\(900\) 0 0
\(901\) 44.7214 1.48988
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 0 0
\(905\) −12.5836 −0.418293
\(906\) 0 0
\(907\) −16.9443 −0.562625 −0.281313 0.959616i \(-0.590770\pi\)
−0.281313 + 0.959616i \(0.590770\pi\)
\(908\) 0 0
\(909\) 87.4853 2.90170
\(910\) 0 0
\(911\) 18.8328 0.623959 0.311980 0.950089i \(-0.399008\pi\)
0.311980 + 0.950089i \(0.399008\pi\)
\(912\) 0 0
\(913\) −11.7771 −0.389765
\(914\) 0 0
\(915\) 27.0557 0.894435
\(916\) 0 0
\(917\) −1.70820 −0.0564099
\(918\) 0 0
\(919\) 35.7771 1.18018 0.590089 0.807338i \(-0.299093\pi\)
0.590089 + 0.807338i \(0.299093\pi\)
\(920\) 0 0
\(921\) −78.2492 −2.57840
\(922\) 0 0
\(923\) 67.7771 2.23091
\(924\) 0 0
\(925\) −15.5279 −0.510553
\(926\) 0 0
\(927\) −108.138 −3.55171
\(928\) 0 0
\(929\) 15.3050 0.502139 0.251070 0.967969i \(-0.419218\pi\)
0.251070 + 0.967969i \(0.419218\pi\)
\(930\) 0 0
\(931\) 3.23607 0.106058
\(932\) 0 0
\(933\) −25.8885 −0.847553
\(934\) 0 0
\(935\) 13.6656 0.446914
\(936\) 0 0
\(937\) −26.9443 −0.880231 −0.440115 0.897941i \(-0.645063\pi\)
−0.440115 + 0.897941i \(0.645063\pi\)
\(938\) 0 0
\(939\) −1.52786 −0.0498600
\(940\) 0 0
\(941\) 13.5967 0.443241 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(942\) 0 0
\(943\) 1.88854 0.0614994
\(944\) 0 0
\(945\) 17.8885 0.581914
\(946\) 0 0
\(947\) −31.4164 −1.02090 −0.510448 0.859909i \(-0.670520\pi\)
−0.510448 + 0.859909i \(0.670520\pi\)
\(948\) 0 0
\(949\) 78.2492 2.54008
\(950\) 0 0
\(951\) −87.1935 −2.82744
\(952\) 0 0
\(953\) 16.1115 0.521901 0.260951 0.965352i \(-0.415964\pi\)
0.260951 + 0.965352i \(0.415964\pi\)
\(954\) 0 0
\(955\) 16.0000 0.517748
\(956\) 0 0
\(957\) −35.7771 −1.15651
\(958\) 0 0
\(959\) −2.94427 −0.0950755
\(960\) 0 0
\(961\) 10.8885 0.351243
\(962\) 0 0
\(963\) −66.8328 −2.15366
\(964\) 0 0
\(965\) 10.4721 0.337110
\(966\) 0 0
\(967\) −5.88854 −0.189363 −0.0946814 0.995508i \(-0.530183\pi\)
−0.0946814 + 0.995508i \(0.530183\pi\)
\(968\) 0 0
\(969\) 46.8328 1.50449
\(970\) 0 0
\(971\) 27.2361 0.874047 0.437024 0.899450i \(-0.356033\pi\)
0.437024 + 0.899450i \(0.356033\pi\)
\(972\) 0 0
\(973\) −3.23607 −0.103744
\(974\) 0 0
\(975\) 58.8328 1.88416
\(976\) 0 0
\(977\) 40.8328 1.30636 0.653179 0.757204i \(-0.273435\pi\)
0.653179 + 0.757204i \(0.273435\pi\)
\(978\) 0 0
\(979\) −14.8328 −0.474059
\(980\) 0 0
\(981\) −3.52786 −0.112636
\(982\) 0 0
\(983\) −14.4721 −0.461589 −0.230795 0.973002i \(-0.574133\pi\)
−0.230795 + 0.973002i \(0.574133\pi\)
\(984\) 0 0
\(985\) 8.58359 0.273496
\(986\) 0 0
\(987\) −4.94427 −0.157378
\(988\) 0 0
\(989\) −9.88854 −0.314437
\(990\) 0 0
\(991\) −24.0000 −0.762385 −0.381193 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 0 0
\(993\) −43.7771 −1.38922
\(994\) 0 0
\(995\) 14.1115 0.447363
\(996\) 0 0
\(997\) −40.0689 −1.26899 −0.634497 0.772925i \(-0.718793\pi\)
−0.634497 + 0.772925i \(0.718793\pi\)
\(998\) 0 0
\(999\) −64.7214 −2.04769
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 224.2.a.c.1.1 2
3.2 odd 2 2016.2.a.r.1.2 2
4.3 odd 2 224.2.a.d.1.2 yes 2
5.4 even 2 5600.2.a.bk.1.2 2
7.2 even 3 1568.2.i.v.1537.2 4
7.3 odd 6 1568.2.i.n.961.1 4
7.4 even 3 1568.2.i.v.961.2 4
7.5 odd 6 1568.2.i.n.1537.1 4
7.6 odd 2 1568.2.a.v.1.2 2
8.3 odd 2 448.2.a.i.1.1 2
8.5 even 2 448.2.a.j.1.2 2
12.11 even 2 2016.2.a.o.1.2 2
16.3 odd 4 1792.2.b.m.897.4 4
16.5 even 4 1792.2.b.k.897.4 4
16.11 odd 4 1792.2.b.m.897.1 4
16.13 even 4 1792.2.b.k.897.1 4
20.19 odd 2 5600.2.a.z.1.1 2
24.5 odd 2 4032.2.a.bw.1.1 2
24.11 even 2 4032.2.a.bv.1.1 2
28.3 even 6 1568.2.i.w.961.2 4
28.11 odd 6 1568.2.i.m.961.1 4
28.19 even 6 1568.2.i.w.1537.2 4
28.23 odd 6 1568.2.i.m.1537.1 4
28.27 even 2 1568.2.a.k.1.1 2
56.13 odd 2 3136.2.a.bf.1.1 2
56.27 even 2 3136.2.a.by.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.a.c.1.1 2 1.1 even 1 trivial
224.2.a.d.1.2 yes 2 4.3 odd 2
448.2.a.i.1.1 2 8.3 odd 2
448.2.a.j.1.2 2 8.5 even 2
1568.2.a.k.1.1 2 28.27 even 2
1568.2.a.v.1.2 2 7.6 odd 2
1568.2.i.m.961.1 4 28.11 odd 6
1568.2.i.m.1537.1 4 28.23 odd 6
1568.2.i.n.961.1 4 7.3 odd 6
1568.2.i.n.1537.1 4 7.5 odd 6
1568.2.i.v.961.2 4 7.4 even 3
1568.2.i.v.1537.2 4 7.2 even 3
1568.2.i.w.961.2 4 28.3 even 6
1568.2.i.w.1537.2 4 28.19 even 6
1792.2.b.k.897.1 4 16.13 even 4
1792.2.b.k.897.4 4 16.5 even 4
1792.2.b.m.897.1 4 16.11 odd 4
1792.2.b.m.897.4 4 16.3 odd 4
2016.2.a.o.1.2 2 12.11 even 2
2016.2.a.r.1.2 2 3.2 odd 2
3136.2.a.bf.1.1 2 56.13 odd 2
3136.2.a.by.1.2 2 56.27 even 2
4032.2.a.bv.1.1 2 24.11 even 2
4032.2.a.bw.1.1 2 24.5 odd 2
5600.2.a.z.1.1 2 20.19 odd 2
5600.2.a.bk.1.2 2 5.4 even 2