Properties

Label 1472.2.a.t.1.2
Level $1472$
Weight $2$
Character 1472.1
Self dual yes
Analytic conductor $11.754$
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] = [1472,2,Mod(1,1472)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1472, 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("1472.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1472 = 2^{6} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1472.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: \(11.7539791775\)
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: no (minimal twist has level 23)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1472.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.23607 q^{3} -1.23607 q^{5} +3.23607 q^{7} +2.00000 q^{9} +5.23607 q^{11} -3.00000 q^{13} -2.76393 q^{15} +0.763932 q^{17} +2.00000 q^{19} +7.23607 q^{21} +1.00000 q^{23} -3.47214 q^{25} -2.23607 q^{27} +3.00000 q^{29} +6.70820 q^{31} +11.7082 q^{33} -4.00000 q^{35} +1.23607 q^{37} -6.70820 q^{39} -3.47214 q^{41} -2.47214 q^{45} -2.23607 q^{47} +3.47214 q^{49} +1.70820 q^{51} -0.472136 q^{53} -6.47214 q^{55} +4.47214 q^{57} -6.47214 q^{59} +6.94427 q^{61} +6.47214 q^{63} +3.70820 q^{65} +2.76393 q^{67} +2.23607 q^{69} +12.2361 q^{71} +6.52786 q^{73} -7.76393 q^{75} +16.9443 q^{77} -10.9443 q^{79} -11.0000 q^{81} +8.76393 q^{83} -0.944272 q^{85} +6.70820 q^{87} -10.4721 q^{89} -9.70820 q^{91} +15.0000 q^{93} -2.47214 q^{95} +17.7082 q^{97} +10.4721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7} + 4 q^{9} + 6 q^{11} - 6 q^{13} - 10 q^{15} + 6 q^{17} + 4 q^{19} + 10 q^{21} + 2 q^{23} + 2 q^{25} + 6 q^{29} + 10 q^{33} - 8 q^{35} - 2 q^{37} + 2 q^{41} + 4 q^{45} - 2 q^{49}+ \cdots + 12 q^{99}+O(q^{100}) \) 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
\(3\) 2.23607 1.29099 0.645497 0.763763i \(-0.276650\pi\)
0.645497 + 0.763763i \(0.276650\pi\)
\(4\) 0 0
\(5\) −1.23607 −0.552786 −0.276393 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(6\) 0 0
\(7\) 3.23607 1.22312 0.611559 0.791199i \(-0.290543\pi\)
0.611559 + 0.791199i \(0.290543\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 5.23607 1.57873 0.789367 0.613922i \(-0.210409\pi\)
0.789367 + 0.613922i \(0.210409\pi\)
\(12\) 0 0
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 0 0
\(15\) −2.76393 −0.713644
\(16\) 0 0
\(17\) 0.763932 0.185281 0.0926404 0.995700i \(-0.470469\pi\)
0.0926404 + 0.995700i \(0.470469\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 7.23607 1.57904
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) 0 0
\(27\) −2.23607 −0.430331
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 6.70820 1.20483 0.602414 0.798183i \(-0.294205\pi\)
0.602414 + 0.798183i \(0.294205\pi\)
\(32\) 0 0
\(33\) 11.7082 2.03814
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 1.23607 0.203208 0.101604 0.994825i \(-0.467602\pi\)
0.101604 + 0.994825i \(0.467602\pi\)
\(38\) 0 0
\(39\) −6.70820 −1.07417
\(40\) 0 0
\(41\) −3.47214 −0.542257 −0.271128 0.962543i \(-0.587397\pi\)
−0.271128 + 0.962543i \(0.587397\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −2.47214 −0.368524
\(46\) 0 0
\(47\) −2.23607 −0.326164 −0.163082 0.986613i \(-0.552144\pi\)
−0.163082 + 0.986613i \(0.552144\pi\)
\(48\) 0 0
\(49\) 3.47214 0.496019
\(50\) 0 0
\(51\) 1.70820 0.239196
\(52\) 0 0
\(53\) −0.472136 −0.0648529 −0.0324264 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) 0 0
\(55\) −6.47214 −0.872703
\(56\) 0 0
\(57\) 4.47214 0.592349
\(58\) 0 0
\(59\) −6.47214 −0.842600 −0.421300 0.906921i \(-0.638426\pi\)
−0.421300 + 0.906921i \(0.638426\pi\)
\(60\) 0 0
\(61\) 6.94427 0.889123 0.444561 0.895748i \(-0.353360\pi\)
0.444561 + 0.895748i \(0.353360\pi\)
\(62\) 0 0
\(63\) 6.47214 0.815412
\(64\) 0 0
\(65\) 3.70820 0.459946
\(66\) 0 0
\(67\) 2.76393 0.337668 0.168834 0.985644i \(-0.446000\pi\)
0.168834 + 0.985644i \(0.446000\pi\)
\(68\) 0 0
\(69\) 2.23607 0.269191
\(70\) 0 0
\(71\) 12.2361 1.45215 0.726077 0.687613i \(-0.241342\pi\)
0.726077 + 0.687613i \(0.241342\pi\)
\(72\) 0 0
\(73\) 6.52786 0.764029 0.382014 0.924156i \(-0.375230\pi\)
0.382014 + 0.924156i \(0.375230\pi\)
\(74\) 0 0
\(75\) −7.76393 −0.896502
\(76\) 0 0
\(77\) 16.9443 1.93098
\(78\) 0 0
\(79\) −10.9443 −1.23133 −0.615663 0.788009i \(-0.711112\pi\)
−0.615663 + 0.788009i \(0.711112\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 8.76393 0.961967 0.480983 0.876730i \(-0.340280\pi\)
0.480983 + 0.876730i \(0.340280\pi\)
\(84\) 0 0
\(85\) −0.944272 −0.102421
\(86\) 0 0
\(87\) 6.70820 0.719195
\(88\) 0 0
\(89\) −10.4721 −1.11004 −0.555022 0.831836i \(-0.687290\pi\)
−0.555022 + 0.831836i \(0.687290\pi\)
\(90\) 0 0
\(91\) −9.70820 −1.01770
\(92\) 0 0
\(93\) 15.0000 1.55543
\(94\) 0 0
\(95\) −2.47214 −0.253636
\(96\) 0 0
\(97\) 17.7082 1.79800 0.898998 0.437953i \(-0.144296\pi\)
0.898998 + 0.437953i \(0.144296\pi\)
\(98\) 0 0
\(99\) 10.4721 1.05249
\(100\) 0 0
\(101\) −4.47214 −0.444994 −0.222497 0.974933i \(-0.571421\pi\)
−0.222497 + 0.974933i \(0.571421\pi\)
\(102\) 0 0
\(103\) −4.18034 −0.411901 −0.205951 0.978562i \(-0.566029\pi\)
−0.205951 + 0.978562i \(0.566029\pi\)
\(104\) 0 0
\(105\) −8.94427 −0.872872
\(106\) 0 0
\(107\) −13.4164 −1.29701 −0.648507 0.761209i \(-0.724606\pi\)
−0.648507 + 0.761209i \(0.724606\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 2.76393 0.262341
\(112\) 0 0
\(113\) 8.76393 0.824441 0.412221 0.911084i \(-0.364753\pi\)
0.412221 + 0.911084i \(0.364753\pi\)
\(114\) 0 0
\(115\) −1.23607 −0.115264
\(116\) 0 0
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) 2.47214 0.226620
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) 0 0
\(123\) −7.76393 −0.700050
\(124\) 0 0
\(125\) 10.4721 0.936656
\(126\) 0 0
\(127\) −7.29180 −0.647042 −0.323521 0.946221i \(-0.604867\pi\)
−0.323521 + 0.946221i \(0.604867\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −18.7082 −1.63454 −0.817272 0.576253i \(-0.804514\pi\)
−0.817272 + 0.576253i \(0.804514\pi\)
\(132\) 0 0
\(133\) 6.47214 0.561205
\(134\) 0 0
\(135\) 2.76393 0.237881
\(136\) 0 0
\(137\) −21.8885 −1.87006 −0.935032 0.354563i \(-0.884630\pi\)
−0.935032 + 0.354563i \(0.884630\pi\)
\(138\) 0 0
\(139\) 10.7082 0.908258 0.454129 0.890936i \(-0.349951\pi\)
0.454129 + 0.890936i \(0.349951\pi\)
\(140\) 0 0
\(141\) −5.00000 −0.421076
\(142\) 0 0
\(143\) −15.7082 −1.31359
\(144\) 0 0
\(145\) −3.70820 −0.307950
\(146\) 0 0
\(147\) 7.76393 0.640358
\(148\) 0 0
\(149\) −23.8885 −1.95703 −0.978513 0.206186i \(-0.933895\pi\)
−0.978513 + 0.206186i \(0.933895\pi\)
\(150\) 0 0
\(151\) 4.23607 0.344726 0.172363 0.985033i \(-0.444860\pi\)
0.172363 + 0.985033i \(0.444860\pi\)
\(152\) 0 0
\(153\) 1.52786 0.123520
\(154\) 0 0
\(155\) −8.29180 −0.666013
\(156\) 0 0
\(157\) 11.4164 0.911129 0.455564 0.890203i \(-0.349438\pi\)
0.455564 + 0.890203i \(0.349438\pi\)
\(158\) 0 0
\(159\) −1.05573 −0.0837247
\(160\) 0 0
\(161\) 3.23607 0.255038
\(162\) 0 0
\(163\) 5.76393 0.451466 0.225733 0.974189i \(-0.427522\pi\)
0.225733 + 0.974189i \(0.427522\pi\)
\(164\) 0 0
\(165\) −14.4721 −1.12665
\(166\) 0 0
\(167\) 1.52786 0.118230 0.0591148 0.998251i \(-0.481172\pi\)
0.0591148 + 0.998251i \(0.481172\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) −22.9443 −1.74442 −0.872210 0.489131i \(-0.837314\pi\)
−0.872210 + 0.489131i \(0.837314\pi\)
\(174\) 0 0
\(175\) −11.2361 −0.849367
\(176\) 0 0
\(177\) −14.4721 −1.08779
\(178\) 0 0
\(179\) −0.708204 −0.0529336 −0.0264668 0.999650i \(-0.508426\pi\)
−0.0264668 + 0.999650i \(0.508426\pi\)
\(180\) 0 0
\(181\) −16.6525 −1.23777 −0.618884 0.785482i \(-0.712415\pi\)
−0.618884 + 0.785482i \(0.712415\pi\)
\(182\) 0 0
\(183\) 15.5279 1.14785
\(184\) 0 0
\(185\) −1.52786 −0.112331
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) −7.23607 −0.526346
\(190\) 0 0
\(191\) −26.1803 −1.89434 −0.947171 0.320728i \(-0.896073\pi\)
−0.947171 + 0.320728i \(0.896073\pi\)
\(192\) 0 0
\(193\) 9.94427 0.715804 0.357902 0.933759i \(-0.383492\pi\)
0.357902 + 0.933759i \(0.383492\pi\)
\(194\) 0 0
\(195\) 8.29180 0.593788
\(196\) 0 0
\(197\) 1.47214 0.104885 0.0524427 0.998624i \(-0.483299\pi\)
0.0524427 + 0.998624i \(0.483299\pi\)
\(198\) 0 0
\(199\) −12.2918 −0.871342 −0.435671 0.900106i \(-0.643489\pi\)
−0.435671 + 0.900106i \(0.643489\pi\)
\(200\) 0 0
\(201\) 6.18034 0.435928
\(202\) 0 0
\(203\) 9.70820 0.681382
\(204\) 0 0
\(205\) 4.29180 0.299752
\(206\) 0 0
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) 10.4721 0.724373
\(210\) 0 0
\(211\) 23.4164 1.61205 0.806026 0.591880i \(-0.201614\pi\)
0.806026 + 0.591880i \(0.201614\pi\)
\(212\) 0 0
\(213\) 27.3607 1.87472
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 21.7082 1.47365
\(218\) 0 0
\(219\) 14.5967 0.986357
\(220\) 0 0
\(221\) −2.29180 −0.154163
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) −6.94427 −0.462951
\(226\) 0 0
\(227\) 12.1803 0.808438 0.404219 0.914662i \(-0.367543\pi\)
0.404219 + 0.914662i \(0.367543\pi\)
\(228\) 0 0
\(229\) 12.0000 0.792982 0.396491 0.918039i \(-0.370228\pi\)
0.396491 + 0.918039i \(0.370228\pi\)
\(230\) 0 0
\(231\) 37.8885 2.49288
\(232\) 0 0
\(233\) −6.52786 −0.427655 −0.213827 0.976871i \(-0.568593\pi\)
−0.213827 + 0.976871i \(0.568593\pi\)
\(234\) 0 0
\(235\) 2.76393 0.180299
\(236\) 0 0
\(237\) −24.4721 −1.58964
\(238\) 0 0
\(239\) 13.7639 0.890315 0.445157 0.895452i \(-0.353148\pi\)
0.445157 + 0.895452i \(0.353148\pi\)
\(240\) 0 0
\(241\) −23.1246 −1.48959 −0.744794 0.667295i \(-0.767452\pi\)
−0.744794 + 0.667295i \(0.767452\pi\)
\(242\) 0 0
\(243\) −17.8885 −1.14755
\(244\) 0 0
\(245\) −4.29180 −0.274193
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) 19.5967 1.24189
\(250\) 0 0
\(251\) −2.29180 −0.144657 −0.0723284 0.997381i \(-0.523043\pi\)
−0.0723284 + 0.997381i \(0.523043\pi\)
\(252\) 0 0
\(253\) 5.23607 0.329189
\(254\) 0 0
\(255\) −2.11146 −0.132225
\(256\) 0 0
\(257\) −7.47214 −0.466099 −0.233050 0.972465i \(-0.574870\pi\)
−0.233050 + 0.972465i \(0.574870\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 2.94427 0.181552 0.0907758 0.995871i \(-0.471065\pi\)
0.0907758 + 0.995871i \(0.471065\pi\)
\(264\) 0 0
\(265\) 0.583592 0.0358498
\(266\) 0 0
\(267\) −23.4164 −1.43306
\(268\) 0 0
\(269\) 7.94427 0.484371 0.242185 0.970230i \(-0.422136\pi\)
0.242185 + 0.970230i \(0.422136\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) −21.7082 −1.31384
\(274\) 0 0
\(275\) −18.1803 −1.09632
\(276\) 0 0
\(277\) −15.4721 −0.929631 −0.464815 0.885408i \(-0.653879\pi\)
−0.464815 + 0.885408i \(0.653879\pi\)
\(278\) 0 0
\(279\) 13.4164 0.803219
\(280\) 0 0
\(281\) −8.76393 −0.522812 −0.261406 0.965229i \(-0.584186\pi\)
−0.261406 + 0.965229i \(0.584186\pi\)
\(282\) 0 0
\(283\) −27.7082 −1.64708 −0.823541 0.567257i \(-0.808005\pi\)
−0.823541 + 0.567257i \(0.808005\pi\)
\(284\) 0 0
\(285\) −5.52786 −0.327442
\(286\) 0 0
\(287\) −11.2361 −0.663244
\(288\) 0 0
\(289\) −16.4164 −0.965671
\(290\) 0 0
\(291\) 39.5967 2.32120
\(292\) 0 0
\(293\) 1.52786 0.0892588 0.0446294 0.999004i \(-0.485789\pi\)
0.0446294 + 0.999004i \(0.485789\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) −11.7082 −0.679379
\(298\) 0 0
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −10.0000 −0.574485
\(304\) 0 0
\(305\) −8.58359 −0.491495
\(306\) 0 0
\(307\) −9.52786 −0.543784 −0.271892 0.962328i \(-0.587649\pi\)
−0.271892 + 0.962328i \(0.587649\pi\)
\(308\) 0 0
\(309\) −9.34752 −0.531762
\(310\) 0 0
\(311\) 13.1803 0.747389 0.373694 0.927552i \(-0.378091\pi\)
0.373694 + 0.927552i \(0.378091\pi\)
\(312\) 0 0
\(313\) 24.3607 1.37695 0.688474 0.725261i \(-0.258281\pi\)
0.688474 + 0.725261i \(0.258281\pi\)
\(314\) 0 0
\(315\) −8.00000 −0.450749
\(316\) 0 0
\(317\) −25.4164 −1.42753 −0.713764 0.700386i \(-0.753011\pi\)
−0.713764 + 0.700386i \(0.753011\pi\)
\(318\) 0 0
\(319\) 15.7082 0.879491
\(320\) 0 0
\(321\) −30.0000 −1.67444
\(322\) 0 0
\(323\) 1.52786 0.0850126
\(324\) 0 0
\(325\) 10.4164 0.577798
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −7.23607 −0.398937
\(330\) 0 0
\(331\) 19.6525 1.08020 0.540099 0.841602i \(-0.318387\pi\)
0.540099 + 0.841602i \(0.318387\pi\)
\(332\) 0 0
\(333\) 2.47214 0.135472
\(334\) 0 0
\(335\) −3.41641 −0.186658
\(336\) 0 0
\(337\) 23.4164 1.27557 0.637787 0.770213i \(-0.279850\pi\)
0.637787 + 0.770213i \(0.279850\pi\)
\(338\) 0 0
\(339\) 19.5967 1.06435
\(340\) 0 0
\(341\) 35.1246 1.90210
\(342\) 0 0
\(343\) −11.4164 −0.616428
\(344\) 0 0
\(345\) −2.76393 −0.148805
\(346\) 0 0
\(347\) 9.88854 0.530845 0.265422 0.964132i \(-0.414489\pi\)
0.265422 + 0.964132i \(0.414489\pi\)
\(348\) 0 0
\(349\) −24.4164 −1.30698 −0.653490 0.756935i \(-0.726696\pi\)
−0.653490 + 0.756935i \(0.726696\pi\)
\(350\) 0 0
\(351\) 6.70820 0.358057
\(352\) 0 0
\(353\) 9.36068 0.498219 0.249109 0.968475i \(-0.419862\pi\)
0.249109 + 0.968475i \(0.419862\pi\)
\(354\) 0 0
\(355\) −15.1246 −0.802731
\(356\) 0 0
\(357\) 5.52786 0.292566
\(358\) 0 0
\(359\) −19.8885 −1.04968 −0.524839 0.851202i \(-0.675874\pi\)
−0.524839 + 0.851202i \(0.675874\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 36.7082 1.92668
\(364\) 0 0
\(365\) −8.06888 −0.422345
\(366\) 0 0
\(367\) −4.18034 −0.218212 −0.109106 0.994030i \(-0.534799\pi\)
−0.109106 + 0.994030i \(0.534799\pi\)
\(368\) 0 0
\(369\) −6.94427 −0.361504
\(370\) 0 0
\(371\) −1.52786 −0.0793227
\(372\) 0 0
\(373\) −7.70820 −0.399116 −0.199558 0.979886i \(-0.563951\pi\)
−0.199558 + 0.979886i \(0.563951\pi\)
\(374\) 0 0
\(375\) 23.4164 1.20922
\(376\) 0 0
\(377\) −9.00000 −0.463524
\(378\) 0 0
\(379\) −24.3607 −1.25132 −0.625662 0.780094i \(-0.715171\pi\)
−0.625662 + 0.780094i \(0.715171\pi\)
\(380\) 0 0
\(381\) −16.3050 −0.835328
\(382\) 0 0
\(383\) 7.05573 0.360531 0.180265 0.983618i \(-0.442304\pi\)
0.180265 + 0.983618i \(0.442304\pi\)
\(384\) 0 0
\(385\) −20.9443 −1.06742
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −25.5279 −1.29431 −0.647157 0.762357i \(-0.724042\pi\)
−0.647157 + 0.762357i \(0.724042\pi\)
\(390\) 0 0
\(391\) 0.763932 0.0386337
\(392\) 0 0
\(393\) −41.8328 −2.11019
\(394\) 0 0
\(395\) 13.5279 0.680661
\(396\) 0 0
\(397\) 24.4164 1.22542 0.612712 0.790306i \(-0.290078\pi\)
0.612712 + 0.790306i \(0.290078\pi\)
\(398\) 0 0
\(399\) 14.4721 0.724513
\(400\) 0 0
\(401\) −14.1803 −0.708132 −0.354066 0.935220i \(-0.615201\pi\)
−0.354066 + 0.935220i \(0.615201\pi\)
\(402\) 0 0
\(403\) −20.1246 −1.00248
\(404\) 0 0
\(405\) 13.5967 0.675628
\(406\) 0 0
\(407\) 6.47214 0.320812
\(408\) 0 0
\(409\) 21.3607 1.05622 0.528109 0.849177i \(-0.322901\pi\)
0.528109 + 0.849177i \(0.322901\pi\)
\(410\) 0 0
\(411\) −48.9443 −2.41424
\(412\) 0 0
\(413\) −20.9443 −1.03060
\(414\) 0 0
\(415\) −10.8328 −0.531762
\(416\) 0 0
\(417\) 23.9443 1.17256
\(418\) 0 0
\(419\) 4.58359 0.223923 0.111962 0.993713i \(-0.464287\pi\)
0.111962 + 0.993713i \(0.464287\pi\)
\(420\) 0 0
\(421\) 10.2918 0.501591 0.250796 0.968040i \(-0.419308\pi\)
0.250796 + 0.968040i \(0.419308\pi\)
\(422\) 0 0
\(423\) −4.47214 −0.217443
\(424\) 0 0
\(425\) −2.65248 −0.128664
\(426\) 0 0
\(427\) 22.4721 1.08750
\(428\) 0 0
\(429\) −35.1246 −1.69583
\(430\) 0 0
\(431\) −17.5279 −0.844288 −0.422144 0.906529i \(-0.638722\pi\)
−0.422144 + 0.906529i \(0.638722\pi\)
\(432\) 0 0
\(433\) 17.8197 0.856358 0.428179 0.903694i \(-0.359155\pi\)
0.428179 + 0.903694i \(0.359155\pi\)
\(434\) 0 0
\(435\) −8.29180 −0.397561
\(436\) 0 0
\(437\) 2.00000 0.0956730
\(438\) 0 0
\(439\) −18.7082 −0.892894 −0.446447 0.894810i \(-0.647311\pi\)
−0.446447 + 0.894810i \(0.647311\pi\)
\(440\) 0 0
\(441\) 6.94427 0.330680
\(442\) 0 0
\(443\) −38.1246 −1.81135 −0.905677 0.423967i \(-0.860637\pi\)
−0.905677 + 0.423967i \(0.860637\pi\)
\(444\) 0 0
\(445\) 12.9443 0.613617
\(446\) 0 0
\(447\) −53.4164 −2.52651
\(448\) 0 0
\(449\) −14.9443 −0.705264 −0.352632 0.935762i \(-0.614713\pi\)
−0.352632 + 0.935762i \(0.614713\pi\)
\(450\) 0 0
\(451\) −18.1803 −0.856079
\(452\) 0 0
\(453\) 9.47214 0.445040
\(454\) 0 0
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) −5.12461 −0.239719 −0.119860 0.992791i \(-0.538244\pi\)
−0.119860 + 0.992791i \(0.538244\pi\)
\(458\) 0 0
\(459\) −1.70820 −0.0797321
\(460\) 0 0
\(461\) 1.47214 0.0685642 0.0342821 0.999412i \(-0.489086\pi\)
0.0342821 + 0.999412i \(0.489086\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) −18.5410 −0.859819
\(466\) 0 0
\(467\) 13.0557 0.604147 0.302074 0.953285i \(-0.402321\pi\)
0.302074 + 0.953285i \(0.402321\pi\)
\(468\) 0 0
\(469\) 8.94427 0.413008
\(470\) 0 0
\(471\) 25.5279 1.17626
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −6.94427 −0.318625
\(476\) 0 0
\(477\) −0.944272 −0.0432352
\(478\) 0 0
\(479\) 31.5967 1.44369 0.721846 0.692054i \(-0.243294\pi\)
0.721846 + 0.692054i \(0.243294\pi\)
\(480\) 0 0
\(481\) −3.70820 −0.169080
\(482\) 0 0
\(483\) 7.23607 0.329252
\(484\) 0 0
\(485\) −21.8885 −0.993908
\(486\) 0 0
\(487\) −14.7082 −0.666492 −0.333246 0.942840i \(-0.608144\pi\)
−0.333246 + 0.942840i \(0.608144\pi\)
\(488\) 0 0
\(489\) 12.8885 0.582840
\(490\) 0 0
\(491\) −8.34752 −0.376718 −0.188359 0.982100i \(-0.560317\pi\)
−0.188359 + 0.982100i \(0.560317\pi\)
\(492\) 0 0
\(493\) 2.29180 0.103217
\(494\) 0 0
\(495\) −12.9443 −0.581802
\(496\) 0 0
\(497\) 39.5967 1.77616
\(498\) 0 0
\(499\) −19.2918 −0.863619 −0.431810 0.901965i \(-0.642125\pi\)
−0.431810 + 0.901965i \(0.642125\pi\)
\(500\) 0 0
\(501\) 3.41641 0.152634
\(502\) 0 0
\(503\) −26.9443 −1.20139 −0.600693 0.799480i \(-0.705109\pi\)
−0.600693 + 0.799480i \(0.705109\pi\)
\(504\) 0 0
\(505\) 5.52786 0.245987
\(506\) 0 0
\(507\) −8.94427 −0.397229
\(508\) 0 0
\(509\) 28.3050 1.25459 0.627297 0.778780i \(-0.284161\pi\)
0.627297 + 0.778780i \(0.284161\pi\)
\(510\) 0 0
\(511\) 21.1246 0.934498
\(512\) 0 0
\(513\) −4.47214 −0.197450
\(514\) 0 0
\(515\) 5.16718 0.227693
\(516\) 0 0
\(517\) −11.7082 −0.514926
\(518\) 0 0
\(519\) −51.3050 −2.25204
\(520\) 0 0
\(521\) 31.4164 1.37638 0.688189 0.725532i \(-0.258406\pi\)
0.688189 + 0.725532i \(0.258406\pi\)
\(522\) 0 0
\(523\) −41.1246 −1.79825 −0.899127 0.437688i \(-0.855797\pi\)
−0.899127 + 0.437688i \(0.855797\pi\)
\(524\) 0 0
\(525\) −25.1246 −1.09653
\(526\) 0 0
\(527\) 5.12461 0.223232
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −12.9443 −0.561734
\(532\) 0 0
\(533\) 10.4164 0.451185
\(534\) 0 0
\(535\) 16.5836 0.716971
\(536\) 0 0
\(537\) −1.58359 −0.0683370
\(538\) 0 0
\(539\) 18.1803 0.783083
\(540\) 0 0
\(541\) 34.4164 1.47968 0.739838 0.672785i \(-0.234902\pi\)
0.739838 + 0.672785i \(0.234902\pi\)
\(542\) 0 0
\(543\) −37.2361 −1.59795
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 29.5410 1.26308 0.631541 0.775342i \(-0.282423\pi\)
0.631541 + 0.775342i \(0.282423\pi\)
\(548\) 0 0
\(549\) 13.8885 0.592749
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) −35.4164 −1.50606
\(554\) 0 0
\(555\) −3.41641 −0.145018
\(556\) 0 0
\(557\) 7.41641 0.314243 0.157122 0.987579i \(-0.449779\pi\)
0.157122 + 0.987579i \(0.449779\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 8.94427 0.377627
\(562\) 0 0
\(563\) 32.9443 1.38844 0.694218 0.719765i \(-0.255750\pi\)
0.694218 + 0.719765i \(0.255750\pi\)
\(564\) 0 0
\(565\) −10.8328 −0.455740
\(566\) 0 0
\(567\) −35.5967 −1.49492
\(568\) 0 0
\(569\) −22.1803 −0.929848 −0.464924 0.885351i \(-0.653918\pi\)
−0.464924 + 0.885351i \(0.653918\pi\)
\(570\) 0 0
\(571\) 14.2918 0.598093 0.299047 0.954239i \(-0.403331\pi\)
0.299047 + 0.954239i \(0.403331\pi\)
\(572\) 0 0
\(573\) −58.5410 −2.44559
\(574\) 0 0
\(575\) −3.47214 −0.144798
\(576\) 0 0
\(577\) 22.8885 0.952863 0.476431 0.879212i \(-0.341930\pi\)
0.476431 + 0.879212i \(0.341930\pi\)
\(578\) 0 0
\(579\) 22.2361 0.924099
\(580\) 0 0
\(581\) 28.3607 1.17660
\(582\) 0 0
\(583\) −2.47214 −0.102385
\(584\) 0 0
\(585\) 7.41641 0.306631
\(586\) 0 0
\(587\) 24.7082 1.01982 0.509908 0.860229i \(-0.329679\pi\)
0.509908 + 0.860229i \(0.329679\pi\)
\(588\) 0 0
\(589\) 13.4164 0.552813
\(590\) 0 0
\(591\) 3.29180 0.135406
\(592\) 0 0
\(593\) −2.94427 −0.120907 −0.0604534 0.998171i \(-0.519255\pi\)
−0.0604534 + 0.998171i \(0.519255\pi\)
\(594\) 0 0
\(595\) −3.05573 −0.125273
\(596\) 0 0
\(597\) −27.4853 −1.12490
\(598\) 0 0
\(599\) 33.8885 1.38465 0.692324 0.721587i \(-0.256587\pi\)
0.692324 + 0.721587i \(0.256587\pi\)
\(600\) 0 0
\(601\) 46.8885 1.91262 0.956312 0.292349i \(-0.0944368\pi\)
0.956312 + 0.292349i \(0.0944368\pi\)
\(602\) 0 0
\(603\) 5.52786 0.225112
\(604\) 0 0
\(605\) −20.2918 −0.824979
\(606\) 0 0
\(607\) 26.4721 1.07447 0.537235 0.843432i \(-0.319469\pi\)
0.537235 + 0.843432i \(0.319469\pi\)
\(608\) 0 0
\(609\) 21.7082 0.879661
\(610\) 0 0
\(611\) 6.70820 0.271385
\(612\) 0 0
\(613\) −5.70820 −0.230552 −0.115276 0.993333i \(-0.536775\pi\)
−0.115276 + 0.993333i \(0.536775\pi\)
\(614\) 0 0
\(615\) 9.59675 0.386978
\(616\) 0 0
\(617\) −7.52786 −0.303060 −0.151530 0.988453i \(-0.548420\pi\)
−0.151530 + 0.988453i \(0.548420\pi\)
\(618\) 0 0
\(619\) −19.4164 −0.780411 −0.390206 0.920728i \(-0.627596\pi\)
−0.390206 + 0.920728i \(0.627596\pi\)
\(620\) 0 0
\(621\) −2.23607 −0.0897303
\(622\) 0 0
\(623\) −33.8885 −1.35772
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 0 0
\(627\) 23.4164 0.935161
\(628\) 0 0
\(629\) 0.944272 0.0376506
\(630\) 0 0
\(631\) 12.3607 0.492071 0.246035 0.969261i \(-0.420872\pi\)
0.246035 + 0.969261i \(0.420872\pi\)
\(632\) 0 0
\(633\) 52.3607 2.08115
\(634\) 0 0
\(635\) 9.01316 0.357676
\(636\) 0 0
\(637\) −10.4164 −0.412713
\(638\) 0 0
\(639\) 24.4721 0.968103
\(640\) 0 0
\(641\) −17.3050 −0.683504 −0.341752 0.939790i \(-0.611020\pi\)
−0.341752 + 0.939790i \(0.611020\pi\)
\(642\) 0 0
\(643\) 29.5967 1.16718 0.583591 0.812048i \(-0.301647\pi\)
0.583591 + 0.812048i \(0.301647\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.70820 0.263727 0.131863 0.991268i \(-0.457904\pi\)
0.131863 + 0.991268i \(0.457904\pi\)
\(648\) 0 0
\(649\) −33.8885 −1.33024
\(650\) 0 0
\(651\) 48.5410 1.90247
\(652\) 0 0
\(653\) 38.3050 1.49899 0.749494 0.662011i \(-0.230297\pi\)
0.749494 + 0.662011i \(0.230297\pi\)
\(654\) 0 0
\(655\) 23.1246 0.903553
\(656\) 0 0
\(657\) 13.0557 0.509352
\(658\) 0 0
\(659\) 10.6525 0.414962 0.207481 0.978239i \(-0.433474\pi\)
0.207481 + 0.978239i \(0.433474\pi\)
\(660\) 0 0
\(661\) 22.9443 0.892429 0.446214 0.894926i \(-0.352772\pi\)
0.446214 + 0.894926i \(0.352772\pi\)
\(662\) 0 0
\(663\) −5.12461 −0.199023
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 3.00000 0.116160
\(668\) 0 0
\(669\) 8.94427 0.345806
\(670\) 0 0
\(671\) 36.3607 1.40369
\(672\) 0 0
\(673\) 3.00000 0.115642 0.0578208 0.998327i \(-0.481585\pi\)
0.0578208 + 0.998327i \(0.481585\pi\)
\(674\) 0 0
\(675\) 7.76393 0.298834
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 57.3050 2.19916
\(680\) 0 0
\(681\) 27.2361 1.04369
\(682\) 0 0
\(683\) −26.5967 −1.01770 −0.508848 0.860856i \(-0.669929\pi\)
−0.508848 + 0.860856i \(0.669929\pi\)
\(684\) 0 0
\(685\) 27.0557 1.03375
\(686\) 0 0
\(687\) 26.8328 1.02374
\(688\) 0 0
\(689\) 1.41641 0.0539608
\(690\) 0 0
\(691\) −7.05573 −0.268413 −0.134206 0.990953i \(-0.542848\pi\)
−0.134206 + 0.990953i \(0.542848\pi\)
\(692\) 0 0
\(693\) 33.8885 1.28732
\(694\) 0 0
\(695\) −13.2361 −0.502073
\(696\) 0 0
\(697\) −2.65248 −0.100470
\(698\) 0 0
\(699\) −14.5967 −0.552100
\(700\) 0 0
\(701\) 3.81966 0.144267 0.0721333 0.997395i \(-0.477019\pi\)
0.0721333 + 0.997395i \(0.477019\pi\)
\(702\) 0 0
\(703\) 2.47214 0.0932384
\(704\) 0 0
\(705\) 6.18034 0.232765
\(706\) 0 0
\(707\) −14.4721 −0.544281
\(708\) 0 0
\(709\) 42.0689 1.57993 0.789965 0.613152i \(-0.210099\pi\)
0.789965 + 0.613152i \(0.210099\pi\)
\(710\) 0 0
\(711\) −21.8885 −0.820885
\(712\) 0 0
\(713\) 6.70820 0.251224
\(714\) 0 0
\(715\) 19.4164 0.726132
\(716\) 0 0
\(717\) 30.7771 1.14939
\(718\) 0 0
\(719\) −3.05573 −0.113959 −0.0569797 0.998375i \(-0.518147\pi\)
−0.0569797 + 0.998375i \(0.518147\pi\)
\(720\) 0 0
\(721\) −13.5279 −0.503804
\(722\) 0 0
\(723\) −51.7082 −1.92305
\(724\) 0 0
\(725\) −10.4164 −0.386856
\(726\) 0 0
\(727\) −27.7082 −1.02764 −0.513820 0.857898i \(-0.671770\pi\)
−0.513820 + 0.857898i \(0.671770\pi\)
\(728\) 0 0
\(729\) −7.00000 −0.259259
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 31.2361 1.15373 0.576865 0.816839i \(-0.304276\pi\)
0.576865 + 0.816839i \(0.304276\pi\)
\(734\) 0 0
\(735\) −9.59675 −0.353981
\(736\) 0 0
\(737\) 14.4721 0.533088
\(738\) 0 0
\(739\) −26.8197 −0.986577 −0.493289 0.869866i \(-0.664205\pi\)
−0.493289 + 0.869866i \(0.664205\pi\)
\(740\) 0 0
\(741\) −13.4164 −0.492864
\(742\) 0 0
\(743\) 41.1246 1.50872 0.754358 0.656463i \(-0.227948\pi\)
0.754358 + 0.656463i \(0.227948\pi\)
\(744\) 0 0
\(745\) 29.5279 1.08182
\(746\) 0 0
\(747\) 17.5279 0.641311
\(748\) 0 0
\(749\) −43.4164 −1.58640
\(750\) 0 0
\(751\) 0.360680 0.0131614 0.00658070 0.999978i \(-0.497905\pi\)
0.00658070 + 0.999978i \(0.497905\pi\)
\(752\) 0 0
\(753\) −5.12461 −0.186751
\(754\) 0 0
\(755\) −5.23607 −0.190560
\(756\) 0 0
\(757\) −1.59675 −0.0580348 −0.0290174 0.999579i \(-0.509238\pi\)
−0.0290174 + 0.999579i \(0.509238\pi\)
\(758\) 0 0
\(759\) 11.7082 0.424981
\(760\) 0 0
\(761\) 46.3050 1.67855 0.839277 0.543705i \(-0.182979\pi\)
0.839277 + 0.543705i \(0.182979\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.88854 −0.0682804
\(766\) 0 0
\(767\) 19.4164 0.701086
\(768\) 0 0
\(769\) −23.1246 −0.833895 −0.416947 0.908931i \(-0.636900\pi\)
−0.416947 + 0.908931i \(0.636900\pi\)
\(770\) 0 0
\(771\) −16.7082 −0.601731
\(772\) 0 0
\(773\) 5.52786 0.198823 0.0994117 0.995046i \(-0.468304\pi\)
0.0994117 + 0.995046i \(0.468304\pi\)
\(774\) 0 0
\(775\) −23.2918 −0.836666
\(776\) 0 0
\(777\) 8.94427 0.320874
\(778\) 0 0
\(779\) −6.94427 −0.248804
\(780\) 0 0
\(781\) 64.0689 2.29256
\(782\) 0 0
\(783\) −6.70820 −0.239732
\(784\) 0 0
\(785\) −14.1115 −0.503659
\(786\) 0 0
\(787\) −24.5836 −0.876310 −0.438155 0.898899i \(-0.644368\pi\)
−0.438155 + 0.898899i \(0.644368\pi\)
\(788\) 0 0
\(789\) 6.58359 0.234382
\(790\) 0 0
\(791\) 28.3607 1.00839
\(792\) 0 0
\(793\) −20.8328 −0.739795
\(794\) 0 0
\(795\) 1.30495 0.0462819
\(796\) 0 0
\(797\) 34.3607 1.21712 0.608559 0.793509i \(-0.291748\pi\)
0.608559 + 0.793509i \(0.291748\pi\)
\(798\) 0 0
\(799\) −1.70820 −0.0604319
\(800\) 0 0
\(801\) −20.9443 −0.740029
\(802\) 0 0
\(803\) 34.1803 1.20620
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 17.7639 0.625320
\(808\) 0 0
\(809\) 12.1115 0.425816 0.212908 0.977072i \(-0.431707\pi\)
0.212908 + 0.977072i \(0.431707\pi\)
\(810\) 0 0
\(811\) 24.3475 0.854957 0.427479 0.904025i \(-0.359402\pi\)
0.427479 + 0.904025i \(0.359402\pi\)
\(812\) 0 0
\(813\) 17.8885 0.627379
\(814\) 0 0
\(815\) −7.12461 −0.249564
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −19.4164 −0.678464
\(820\) 0 0
\(821\) 38.9443 1.35916 0.679582 0.733599i \(-0.262161\pi\)
0.679582 + 0.733599i \(0.262161\pi\)
\(822\) 0 0
\(823\) −39.5410 −1.37831 −0.689157 0.724612i \(-0.742019\pi\)
−0.689157 + 0.724612i \(0.742019\pi\)
\(824\) 0 0
\(825\) −40.6525 −1.41534
\(826\) 0 0
\(827\) −1.52786 −0.0531290 −0.0265645 0.999647i \(-0.508457\pi\)
−0.0265645 + 0.999647i \(0.508457\pi\)
\(828\) 0 0
\(829\) −40.2492 −1.39791 −0.698957 0.715164i \(-0.746352\pi\)
−0.698957 + 0.715164i \(0.746352\pi\)
\(830\) 0 0
\(831\) −34.5967 −1.20015
\(832\) 0 0
\(833\) 2.65248 0.0919028
\(834\) 0 0
\(835\) −1.88854 −0.0653558
\(836\) 0 0
\(837\) −15.0000 −0.518476
\(838\) 0 0
\(839\) −41.1246 −1.41978 −0.709890 0.704313i \(-0.751255\pi\)
−0.709890 + 0.704313i \(0.751255\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −19.5967 −0.674948
\(844\) 0 0
\(845\) 4.94427 0.170088
\(846\) 0 0
\(847\) 53.1246 1.82538
\(848\) 0 0
\(849\) −61.9574 −2.12637
\(850\) 0 0
\(851\) 1.23607 0.0423719
\(852\) 0 0
\(853\) 10.5836 0.362375 0.181188 0.983449i \(-0.442006\pi\)
0.181188 + 0.983449i \(0.442006\pi\)
\(854\) 0 0
\(855\) −4.94427 −0.169091
\(856\) 0 0
\(857\) 1.47214 0.0502872 0.0251436 0.999684i \(-0.491996\pi\)
0.0251436 + 0.999684i \(0.491996\pi\)
\(858\) 0 0
\(859\) 16.7082 0.570077 0.285038 0.958516i \(-0.407994\pi\)
0.285038 + 0.958516i \(0.407994\pi\)
\(860\) 0 0
\(861\) −25.1246 −0.856244
\(862\) 0 0
\(863\) −21.5410 −0.733265 −0.366632 0.930366i \(-0.619489\pi\)
−0.366632 + 0.930366i \(0.619489\pi\)
\(864\) 0 0
\(865\) 28.3607 0.964292
\(866\) 0 0
\(867\) −36.7082 −1.24668
\(868\) 0 0
\(869\) −57.3050 −1.94394
\(870\) 0 0
\(871\) −8.29180 −0.280957
\(872\) 0 0
\(873\) 35.4164 1.19866
\(874\) 0 0
\(875\) 33.8885 1.14564
\(876\) 0 0
\(877\) 36.4721 1.23158 0.615788 0.787912i \(-0.288838\pi\)
0.615788 + 0.787912i \(0.288838\pi\)
\(878\) 0 0
\(879\) 3.41641 0.115233
\(880\) 0 0
\(881\) 44.1803 1.48847 0.744237 0.667916i \(-0.232813\pi\)
0.744237 + 0.667916i \(0.232813\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 17.8885 0.601317
\(886\) 0 0
\(887\) 23.0689 0.774577 0.387289 0.921959i \(-0.373412\pi\)
0.387289 + 0.921959i \(0.373412\pi\)
\(888\) 0 0
\(889\) −23.5967 −0.791410
\(890\) 0 0
\(891\) −57.5967 −1.92956
\(892\) 0 0
\(893\) −4.47214 −0.149654
\(894\) 0 0
\(895\) 0.875388 0.0292610
\(896\) 0 0
\(897\) −6.70820 −0.223980
\(898\) 0 0
\(899\) 20.1246 0.671193
\(900\) 0 0
\(901\) −0.360680 −0.0120160
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 20.5836 0.684222
\(906\) 0 0
\(907\) 40.2492 1.33645 0.668227 0.743958i \(-0.267054\pi\)
0.668227 + 0.743958i \(0.267054\pi\)
\(908\) 0 0
\(909\) −8.94427 −0.296663
\(910\) 0 0
\(911\) 31.3050 1.03718 0.518590 0.855023i \(-0.326457\pi\)
0.518590 + 0.855023i \(0.326457\pi\)
\(912\) 0 0
\(913\) 45.8885 1.51869
\(914\) 0 0
\(915\) −19.1935 −0.634517
\(916\) 0 0
\(917\) −60.5410 −1.99924
\(918\) 0 0
\(919\) 41.1246 1.35658 0.678288 0.734796i \(-0.262722\pi\)
0.678288 + 0.734796i \(0.262722\pi\)
\(920\) 0 0
\(921\) −21.3050 −0.702022
\(922\) 0 0
\(923\) −36.7082 −1.20827
\(924\) 0 0
\(925\) −4.29180 −0.141113
\(926\) 0 0
\(927\) −8.36068 −0.274601
\(928\) 0 0
\(929\) −24.0557 −0.789243 −0.394621 0.918844i \(-0.629124\pi\)
−0.394621 + 0.918844i \(0.629124\pi\)
\(930\) 0 0
\(931\) 6.94427 0.227589
\(932\) 0 0
\(933\) 29.4721 0.964874
\(934\) 0 0
\(935\) −4.94427 −0.161695
\(936\) 0 0
\(937\) 34.1803 1.11662 0.558312 0.829631i \(-0.311449\pi\)
0.558312 + 0.829631i \(0.311449\pi\)
\(938\) 0 0
\(939\) 54.4721 1.77763
\(940\) 0 0
\(941\) −6.65248 −0.216865 −0.108432 0.994104i \(-0.534583\pi\)
−0.108432 + 0.994104i \(0.534583\pi\)
\(942\) 0 0
\(943\) −3.47214 −0.113068
\(944\) 0 0
\(945\) 8.94427 0.290957
\(946\) 0 0
\(947\) 10.8197 0.351592 0.175796 0.984427i \(-0.443750\pi\)
0.175796 + 0.984427i \(0.443750\pi\)
\(948\) 0 0
\(949\) −19.5836 −0.635710
\(950\) 0 0
\(951\) −56.8328 −1.84293
\(952\) 0 0
\(953\) 20.4721 0.663158 0.331579 0.943428i \(-0.392419\pi\)
0.331579 + 0.943428i \(0.392419\pi\)
\(954\) 0 0
\(955\) 32.3607 1.04717
\(956\) 0 0
\(957\) 35.1246 1.13542
\(958\) 0 0
\(959\) −70.8328 −2.28731
\(960\) 0 0
\(961\) 14.0000 0.451613
\(962\) 0 0
\(963\) −26.8328 −0.864675
\(964\) 0 0
\(965\) −12.2918 −0.395687
\(966\) 0 0
\(967\) 27.5410 0.885659 0.442830 0.896606i \(-0.353975\pi\)
0.442830 + 0.896606i \(0.353975\pi\)
\(968\) 0 0
\(969\) 3.41641 0.109751
\(970\) 0 0
\(971\) −16.4721 −0.528616 −0.264308 0.964438i \(-0.585144\pi\)
−0.264308 + 0.964438i \(0.585144\pi\)
\(972\) 0 0
\(973\) 34.6525 1.11091
\(974\) 0 0
\(975\) 23.2918 0.745934
\(976\) 0 0
\(977\) −23.3475 −0.746953 −0.373477 0.927640i \(-0.621834\pi\)
−0.373477 + 0.927640i \(0.621834\pi\)
\(978\) 0 0
\(979\) −54.8328 −1.75246
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −40.4721 −1.29086 −0.645430 0.763819i \(-0.723322\pi\)
−0.645430 + 0.763819i \(0.723322\pi\)
\(984\) 0 0
\(985\) −1.81966 −0.0579792
\(986\) 0 0
\(987\) −16.1803 −0.515026
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 24.0000 0.762385 0.381193 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(992\) 0 0
\(993\) 43.9443 1.39453
\(994\) 0 0
\(995\) 15.1935 0.481666
\(996\) 0 0
\(997\) −16.8328 −0.533101 −0.266550 0.963821i \(-0.585884\pi\)
−0.266550 + 0.963821i \(0.585884\pi\)
\(998\) 0 0
\(999\) −2.76393 −0.0874469
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 1472.2.a.t.1.2 2
4.3 odd 2 1472.2.a.s.1.1 2
8.3 odd 2 368.2.a.h.1.2 2
8.5 even 2 23.2.a.a.1.2 2
24.5 odd 2 207.2.a.d.1.1 2
24.11 even 2 3312.2.a.ba.1.1 2
40.13 odd 4 575.2.b.d.24.2 4
40.19 odd 2 9200.2.a.bt.1.1 2
40.29 even 2 575.2.a.f.1.1 2
40.37 odd 4 575.2.b.d.24.3 4
56.13 odd 2 1127.2.a.c.1.2 2
88.21 odd 2 2783.2.a.c.1.1 2
104.77 even 2 3887.2.a.i.1.1 2
120.29 odd 2 5175.2.a.be.1.2 2
136.101 even 2 6647.2.a.b.1.2 2
152.37 odd 2 8303.2.a.e.1.1 2
184.5 odd 22 529.2.c.n.255.2 20
184.13 even 22 529.2.c.o.399.1 20
184.21 odd 22 529.2.c.n.487.2 20
184.29 even 22 529.2.c.o.266.1 20
184.37 odd 22 529.2.c.n.334.2 20
184.45 odd 2 529.2.a.a.1.2 2
184.53 odd 22 529.2.c.n.118.1 20
184.61 odd 22 529.2.c.n.501.1 20
184.77 even 22 529.2.c.o.501.1 20
184.85 even 22 529.2.c.o.118.1 20
184.91 even 2 8464.2.a.bb.1.2 2
184.101 even 22 529.2.c.o.334.2 20
184.109 odd 22 529.2.c.n.266.1 20
184.117 even 22 529.2.c.o.487.2 20
184.125 odd 22 529.2.c.n.399.1 20
184.133 even 22 529.2.c.o.255.2 20
184.141 even 22 529.2.c.o.170.1 20
184.149 odd 22 529.2.c.n.466.2 20
184.157 odd 22 529.2.c.n.177.1 20
184.165 even 22 529.2.c.o.177.1 20
184.173 even 22 529.2.c.o.466.2 20
184.181 odd 22 529.2.c.n.170.1 20
552.413 even 2 4761.2.a.w.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.2.a.a.1.2 2 8.5 even 2
207.2.a.d.1.1 2 24.5 odd 2
368.2.a.h.1.2 2 8.3 odd 2
529.2.a.a.1.2 2 184.45 odd 2
529.2.c.n.118.1 20 184.53 odd 22
529.2.c.n.170.1 20 184.181 odd 22
529.2.c.n.177.1 20 184.157 odd 22
529.2.c.n.255.2 20 184.5 odd 22
529.2.c.n.266.1 20 184.109 odd 22
529.2.c.n.334.2 20 184.37 odd 22
529.2.c.n.399.1 20 184.125 odd 22
529.2.c.n.466.2 20 184.149 odd 22
529.2.c.n.487.2 20 184.21 odd 22
529.2.c.n.501.1 20 184.61 odd 22
529.2.c.o.118.1 20 184.85 even 22
529.2.c.o.170.1 20 184.141 even 22
529.2.c.o.177.1 20 184.165 even 22
529.2.c.o.255.2 20 184.133 even 22
529.2.c.o.266.1 20 184.29 even 22
529.2.c.o.334.2 20 184.101 even 22
529.2.c.o.399.1 20 184.13 even 22
529.2.c.o.466.2 20 184.173 even 22
529.2.c.o.487.2 20 184.117 even 22
529.2.c.o.501.1 20 184.77 even 22
575.2.a.f.1.1 2 40.29 even 2
575.2.b.d.24.2 4 40.13 odd 4
575.2.b.d.24.3 4 40.37 odd 4
1127.2.a.c.1.2 2 56.13 odd 2
1472.2.a.s.1.1 2 4.3 odd 2
1472.2.a.t.1.2 2 1.1 even 1 trivial
2783.2.a.c.1.1 2 88.21 odd 2
3312.2.a.ba.1.1 2 24.11 even 2
3887.2.a.i.1.1 2 104.77 even 2
4761.2.a.w.1.1 2 552.413 even 2
5175.2.a.be.1.2 2 120.29 odd 2
6647.2.a.b.1.2 2 136.101 even 2
8303.2.a.e.1.1 2 152.37 odd 2
8464.2.a.bb.1.2 2 184.91 even 2
9200.2.a.bt.1.1 2 40.19 odd 2