Properties

Label 3696.2.a.bc.1.2
Level $3696$
Weight $2$
Character 3696.1
Self dual yes
Analytic conductor $29.513$
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] = [3696,2,Mod(1,3696)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3696, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3696.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3696.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: \(29.5127085871\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3696.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +2.00000 q^{13} -3.46410 q^{15} -3.46410 q^{17} +1.46410 q^{19} +1.00000 q^{21} +6.92820 q^{23} +7.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +1.46410 q^{31} +1.00000 q^{33} -3.46410 q^{35} +8.92820 q^{37} -2.00000 q^{39} -3.46410 q^{41} -2.92820 q^{43} +3.46410 q^{45} -2.53590 q^{47} +1.00000 q^{49} +3.46410 q^{51} +12.9282 q^{53} -3.46410 q^{55} -1.46410 q^{57} +6.92820 q^{59} +2.00000 q^{61} -1.00000 q^{63} +6.92820 q^{65} -1.07180 q^{67} -6.92820 q^{69} +12.0000 q^{71} -7.46410 q^{73} -7.00000 q^{75} +1.00000 q^{77} -2.92820 q^{79} +1.00000 q^{81} +16.3923 q^{83} -12.0000 q^{85} +6.00000 q^{87} +12.9282 q^{89} -2.00000 q^{91} -1.46410 q^{93} +5.07180 q^{95} +2.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} - 4 q^{19} + 2 q^{21} + 14 q^{25} - 2 q^{27} - 12 q^{29} - 4 q^{31} + 2 q^{33} + 4 q^{37} - 4 q^{39} + 8 q^{43} - 12 q^{47} + 2 q^{49} + 12 q^{53} + 4 q^{57} + 4 q^{61} - 2 q^{63} - 16 q^{67} + 24 q^{71} - 8 q^{73} - 14 q^{75} + 2 q^{77} + 8 q^{79} + 2 q^{81} + 12 q^{83} - 24 q^{85} + 12 q^{87} + 12 q^{89} - 4 q^{91} + 4 q^{93} + 24 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 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
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −3.46410 −0.894427
\(16\) 0 0
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) 1.46410 0.335888 0.167944 0.985797i \(-0.446287\pi\)
0.167944 + 0.985797i \(0.446287\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 6.92820 1.44463 0.722315 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 1.46410 0.262960 0.131480 0.991319i \(-0.458027\pi\)
0.131480 + 0.991319i \(0.458027\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) −3.46410 −0.585540
\(36\) 0 0
\(37\) 8.92820 1.46779 0.733894 0.679264i \(-0.237701\pi\)
0.733894 + 0.679264i \(0.237701\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) 0 0
\(43\) −2.92820 −0.446547 −0.223273 0.974756i \(-0.571674\pi\)
−0.223273 + 0.974756i \(0.571674\pi\)
\(44\) 0 0
\(45\) 3.46410 0.516398
\(46\) 0 0
\(47\) −2.53590 −0.369899 −0.184949 0.982748i \(-0.559212\pi\)
−0.184949 + 0.982748i \(0.559212\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.46410 0.485071
\(52\) 0 0
\(53\) 12.9282 1.77583 0.887913 0.460012i \(-0.152155\pi\)
0.887913 + 0.460012i \(0.152155\pi\)
\(54\) 0 0
\(55\) −3.46410 −0.467099
\(56\) 0 0
\(57\) −1.46410 −0.193925
\(58\) 0 0
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 6.92820 0.859338
\(66\) 0 0
\(67\) −1.07180 −0.130941 −0.0654704 0.997855i \(-0.520855\pi\)
−0.0654704 + 0.997855i \(0.520855\pi\)
\(68\) 0 0
\(69\) −6.92820 −0.834058
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −7.46410 −0.873607 −0.436804 0.899557i \(-0.643889\pi\)
−0.436804 + 0.899557i \(0.643889\pi\)
\(74\) 0 0
\(75\) −7.00000 −0.808290
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −2.92820 −0.329449 −0.164724 0.986340i \(-0.552673\pi\)
−0.164724 + 0.986340i \(0.552673\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 16.3923 1.79929 0.899645 0.436623i \(-0.143826\pi\)
0.899645 + 0.436623i \(0.143826\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 12.9282 1.37039 0.685193 0.728361i \(-0.259718\pi\)
0.685193 + 0.728361i \(0.259718\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) −1.46410 −0.151820
\(94\) 0 0
\(95\) 5.07180 0.520355
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 7.85641 0.781742 0.390871 0.920446i \(-0.372174\pi\)
0.390871 + 0.920446i \(0.372174\pi\)
\(102\) 0 0
\(103\) 6.53590 0.644001 0.322001 0.946739i \(-0.395645\pi\)
0.322001 + 0.946739i \(0.395645\pi\)
\(104\) 0 0
\(105\) 3.46410 0.338062
\(106\) 0 0
\(107\) −6.92820 −0.669775 −0.334887 0.942258i \(-0.608698\pi\)
−0.334887 + 0.942258i \(0.608698\pi\)
\(108\) 0 0
\(109\) −11.8564 −1.13564 −0.567819 0.823154i \(-0.692213\pi\)
−0.567819 + 0.823154i \(0.692213\pi\)
\(110\) 0 0
\(111\) −8.92820 −0.847428
\(112\) 0 0
\(113\) −19.8564 −1.86793 −0.933967 0.357360i \(-0.883677\pi\)
−0.933967 + 0.357360i \(0.883677\pi\)
\(114\) 0 0
\(115\) 24.0000 2.23801
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 3.46410 0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 3.46410 0.312348
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) −2.92820 −0.259836 −0.129918 0.991525i \(-0.541471\pi\)
−0.129918 + 0.991525i \(0.541471\pi\)
\(128\) 0 0
\(129\) 2.92820 0.257814
\(130\) 0 0
\(131\) −16.3923 −1.43220 −0.716101 0.697997i \(-0.754075\pi\)
−0.716101 + 0.697997i \(0.754075\pi\)
\(132\) 0 0
\(133\) −1.46410 −0.126954
\(134\) 0 0
\(135\) −3.46410 −0.298142
\(136\) 0 0
\(137\) −19.8564 −1.69645 −0.848224 0.529638i \(-0.822328\pi\)
−0.848224 + 0.529638i \(0.822328\pi\)
\(138\) 0 0
\(139\) 6.53590 0.554368 0.277184 0.960817i \(-0.410599\pi\)
0.277184 + 0.960817i \(0.410599\pi\)
\(140\) 0 0
\(141\) 2.53590 0.213561
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) −20.7846 −1.72607
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) −3.46410 −0.280056
\(154\) 0 0
\(155\) 5.07180 0.407377
\(156\) 0 0
\(157\) 18.3923 1.46787 0.733933 0.679222i \(-0.237683\pi\)
0.733933 + 0.679222i \(0.237683\pi\)
\(158\) 0 0
\(159\) −12.9282 −1.02527
\(160\) 0 0
\(161\) −6.92820 −0.546019
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 3.46410 0.269680
\(166\) 0 0
\(167\) −5.07180 −0.392467 −0.196234 0.980557i \(-0.562871\pi\)
−0.196234 + 0.980557i \(0.562871\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 1.46410 0.111963
\(172\) 0 0
\(173\) 12.9282 0.982913 0.491457 0.870902i \(-0.336465\pi\)
0.491457 + 0.870902i \(0.336465\pi\)
\(174\) 0 0
\(175\) −7.00000 −0.529150
\(176\) 0 0
\(177\) −6.92820 −0.520756
\(178\) 0 0
\(179\) −20.7846 −1.55351 −0.776757 0.629800i \(-0.783137\pi\)
−0.776757 + 0.629800i \(0.783137\pi\)
\(180\) 0 0
\(181\) −14.3923 −1.06977 −0.534886 0.844924i \(-0.679645\pi\)
−0.534886 + 0.844924i \(0.679645\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) 30.9282 2.27389
\(186\) 0 0
\(187\) 3.46410 0.253320
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 20.7846 1.50392 0.751961 0.659208i \(-0.229108\pi\)
0.751961 + 0.659208i \(0.229108\pi\)
\(192\) 0 0
\(193\) 26.0000 1.87152 0.935760 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) 0 0
\(195\) −6.92820 −0.496139
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 20.3923 1.44557 0.722786 0.691072i \(-0.242861\pi\)
0.722786 + 0.691072i \(0.242861\pi\)
\(200\) 0 0
\(201\) 1.07180 0.0755987
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) 6.92820 0.481543
\(208\) 0 0
\(209\) −1.46410 −0.101274
\(210\) 0 0
\(211\) 24.7846 1.70624 0.853121 0.521712i \(-0.174707\pi\)
0.853121 + 0.521712i \(0.174707\pi\)
\(212\) 0 0
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) −10.1436 −0.691787
\(216\) 0 0
\(217\) −1.46410 −0.0993897
\(218\) 0 0
\(219\) 7.46410 0.504377
\(220\) 0 0
\(221\) −6.92820 −0.466041
\(222\) 0 0
\(223\) −12.3923 −0.829850 −0.414925 0.909856i \(-0.636192\pi\)
−0.414925 + 0.909856i \(0.636192\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) −16.3923 −1.08800 −0.543998 0.839087i \(-0.683090\pi\)
−0.543998 + 0.839087i \(0.683090\pi\)
\(228\) 0 0
\(229\) 13.3205 0.880244 0.440122 0.897938i \(-0.354935\pi\)
0.440122 + 0.897938i \(0.354935\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) 19.8564 1.30084 0.650418 0.759576i \(-0.274594\pi\)
0.650418 + 0.759576i \(0.274594\pi\)
\(234\) 0 0
\(235\) −8.78461 −0.573045
\(236\) 0 0
\(237\) 2.92820 0.190207
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 30.3923 1.95774 0.978870 0.204482i \(-0.0655511\pi\)
0.978870 + 0.204482i \(0.0655511\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.46410 0.221313
\(246\) 0 0
\(247\) 2.92820 0.186317
\(248\) 0 0
\(249\) −16.3923 −1.03882
\(250\) 0 0
\(251\) −17.0718 −1.07756 −0.538781 0.842446i \(-0.681115\pi\)
−0.538781 + 0.842446i \(0.681115\pi\)
\(252\) 0 0
\(253\) −6.92820 −0.435572
\(254\) 0 0
\(255\) 12.0000 0.751469
\(256\) 0 0
\(257\) −0.928203 −0.0578997 −0.0289499 0.999581i \(-0.509216\pi\)
−0.0289499 + 0.999581i \(0.509216\pi\)
\(258\) 0 0
\(259\) −8.92820 −0.554772
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 18.9282 1.16716 0.583582 0.812055i \(-0.301651\pi\)
0.583582 + 0.812055i \(0.301651\pi\)
\(264\) 0 0
\(265\) 44.7846 2.75110
\(266\) 0 0
\(267\) −12.9282 −0.791193
\(268\) 0 0
\(269\) −24.2487 −1.47847 −0.739235 0.673448i \(-0.764813\pi\)
−0.739235 + 0.673448i \(0.764813\pi\)
\(270\) 0 0
\(271\) −16.7846 −1.01959 −0.509796 0.860295i \(-0.670279\pi\)
−0.509796 + 0.860295i \(0.670279\pi\)
\(272\) 0 0
\(273\) 2.00000 0.121046
\(274\) 0 0
\(275\) −7.00000 −0.422116
\(276\) 0 0
\(277\) 15.8564 0.952719 0.476360 0.879251i \(-0.341956\pi\)
0.476360 + 0.879251i \(0.341956\pi\)
\(278\) 0 0
\(279\) 1.46410 0.0876535
\(280\) 0 0
\(281\) 19.8564 1.18453 0.592267 0.805742i \(-0.298233\pi\)
0.592267 + 0.805742i \(0.298233\pi\)
\(282\) 0 0
\(283\) 6.53590 0.388519 0.194259 0.980950i \(-0.437770\pi\)
0.194259 + 0.980950i \(0.437770\pi\)
\(284\) 0 0
\(285\) −5.07180 −0.300427
\(286\) 0 0
\(287\) 3.46410 0.204479
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 0 0
\(293\) −11.0718 −0.646821 −0.323411 0.946259i \(-0.604830\pi\)
−0.323411 + 0.946259i \(0.604830\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 13.8564 0.801337
\(300\) 0 0
\(301\) 2.92820 0.168779
\(302\) 0 0
\(303\) −7.85641 −0.451339
\(304\) 0 0
\(305\) 6.92820 0.396708
\(306\) 0 0
\(307\) −17.4641 −0.996729 −0.498364 0.866968i \(-0.666066\pi\)
−0.498364 + 0.866968i \(0.666066\pi\)
\(308\) 0 0
\(309\) −6.53590 −0.371814
\(310\) 0 0
\(311\) −7.60770 −0.431393 −0.215696 0.976460i \(-0.569202\pi\)
−0.215696 + 0.976460i \(0.569202\pi\)
\(312\) 0 0
\(313\) −3.07180 −0.173628 −0.0868141 0.996225i \(-0.527669\pi\)
−0.0868141 + 0.996225i \(0.527669\pi\)
\(314\) 0 0
\(315\) −3.46410 −0.195180
\(316\) 0 0
\(317\) −0.928203 −0.0521331 −0.0260665 0.999660i \(-0.508298\pi\)
−0.0260665 + 0.999660i \(0.508298\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 6.92820 0.386695
\(322\) 0 0
\(323\) −5.07180 −0.282202
\(324\) 0 0
\(325\) 14.0000 0.776580
\(326\) 0 0
\(327\) 11.8564 0.655661
\(328\) 0 0
\(329\) 2.53590 0.139809
\(330\) 0 0
\(331\) 22.9282 1.26025 0.630124 0.776495i \(-0.283004\pi\)
0.630124 + 0.776495i \(0.283004\pi\)
\(332\) 0 0
\(333\) 8.92820 0.489263
\(334\) 0 0
\(335\) −3.71281 −0.202853
\(336\) 0 0
\(337\) 7.07180 0.385225 0.192613 0.981275i \(-0.438304\pi\)
0.192613 + 0.981275i \(0.438304\pi\)
\(338\) 0 0
\(339\) 19.8564 1.07845
\(340\) 0 0
\(341\) −1.46410 −0.0792855
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −24.0000 −1.29212
\(346\) 0 0
\(347\) −20.7846 −1.11578 −0.557888 0.829916i \(-0.688388\pi\)
−0.557888 + 0.829916i \(0.688388\pi\)
\(348\) 0 0
\(349\) −30.7846 −1.64786 −0.823931 0.566690i \(-0.808224\pi\)
−0.823931 + 0.566690i \(0.808224\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) −0.928203 −0.0494033 −0.0247016 0.999695i \(-0.507864\pi\)
−0.0247016 + 0.999695i \(0.507864\pi\)
\(354\) 0 0
\(355\) 41.5692 2.20627
\(356\) 0 0
\(357\) −3.46410 −0.183340
\(358\) 0 0
\(359\) 18.9282 0.998992 0.499496 0.866316i \(-0.333518\pi\)
0.499496 + 0.866316i \(0.333518\pi\)
\(360\) 0 0
\(361\) −16.8564 −0.887179
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −25.8564 −1.35339
\(366\) 0 0
\(367\) 15.3205 0.799724 0.399862 0.916575i \(-0.369058\pi\)
0.399862 + 0.916575i \(0.369058\pi\)
\(368\) 0 0
\(369\) −3.46410 −0.180334
\(370\) 0 0
\(371\) −12.9282 −0.671199
\(372\) 0 0
\(373\) −25.7128 −1.33136 −0.665679 0.746238i \(-0.731858\pi\)
−0.665679 + 0.746238i \(0.731858\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −9.85641 −0.506290 −0.253145 0.967428i \(-0.581465\pi\)
−0.253145 + 0.967428i \(0.581465\pi\)
\(380\) 0 0
\(381\) 2.92820 0.150016
\(382\) 0 0
\(383\) −21.4641 −1.09676 −0.548382 0.836228i \(-0.684756\pi\)
−0.548382 + 0.836228i \(0.684756\pi\)
\(384\) 0 0
\(385\) 3.46410 0.176547
\(386\) 0 0
\(387\) −2.92820 −0.148849
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 0 0
\(393\) 16.3923 0.826882
\(394\) 0 0
\(395\) −10.1436 −0.510380
\(396\) 0 0
\(397\) 18.3923 0.923083 0.461542 0.887119i \(-0.347296\pi\)
0.461542 + 0.887119i \(0.347296\pi\)
\(398\) 0 0
\(399\) 1.46410 0.0732968
\(400\) 0 0
\(401\) 31.8564 1.59083 0.795417 0.606063i \(-0.207252\pi\)
0.795417 + 0.606063i \(0.207252\pi\)
\(402\) 0 0
\(403\) 2.92820 0.145864
\(404\) 0 0
\(405\) 3.46410 0.172133
\(406\) 0 0
\(407\) −8.92820 −0.442555
\(408\) 0 0
\(409\) 25.3205 1.25202 0.626009 0.779816i \(-0.284687\pi\)
0.626009 + 0.779816i \(0.284687\pi\)
\(410\) 0 0
\(411\) 19.8564 0.979444
\(412\) 0 0
\(413\) −6.92820 −0.340915
\(414\) 0 0
\(415\) 56.7846 2.78745
\(416\) 0 0
\(417\) −6.53590 −0.320064
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) 17.7128 0.863270 0.431635 0.902048i \(-0.357937\pi\)
0.431635 + 0.902048i \(0.357937\pi\)
\(422\) 0 0
\(423\) −2.53590 −0.123300
\(424\) 0 0
\(425\) −24.2487 −1.17624
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 0 0
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) 5.07180 0.244300 0.122150 0.992512i \(-0.461021\pi\)
0.122150 + 0.992512i \(0.461021\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 20.7846 0.996546
\(436\) 0 0
\(437\) 10.1436 0.485234
\(438\) 0 0
\(439\) −16.7846 −0.801086 −0.400543 0.916278i \(-0.631178\pi\)
−0.400543 + 0.916278i \(0.631178\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −20.7846 −0.987507 −0.493753 0.869602i \(-0.664375\pi\)
−0.493753 + 0.869602i \(0.664375\pi\)
\(444\) 0 0
\(445\) 44.7846 2.12299
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 7.85641 0.370767 0.185383 0.982666i \(-0.440647\pi\)
0.185383 + 0.982666i \(0.440647\pi\)
\(450\) 0 0
\(451\) 3.46410 0.163118
\(452\) 0 0
\(453\) −16.0000 −0.751746
\(454\) 0 0
\(455\) −6.92820 −0.324799
\(456\) 0 0
\(457\) −11.8564 −0.554619 −0.277310 0.960781i \(-0.589443\pi\)
−0.277310 + 0.960781i \(0.589443\pi\)
\(458\) 0 0
\(459\) 3.46410 0.161690
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) −5.07180 −0.235199
\(466\) 0 0
\(467\) −39.7128 −1.83769 −0.918845 0.394619i \(-0.870877\pi\)
−0.918845 + 0.394619i \(0.870877\pi\)
\(468\) 0 0
\(469\) 1.07180 0.0494910
\(470\) 0 0
\(471\) −18.3923 −0.847473
\(472\) 0 0
\(473\) 2.92820 0.134639
\(474\) 0 0
\(475\) 10.2487 0.470243
\(476\) 0 0
\(477\) 12.9282 0.591942
\(478\) 0 0
\(479\) −13.8564 −0.633115 −0.316558 0.948573i \(-0.602527\pi\)
−0.316558 + 0.948573i \(0.602527\pi\)
\(480\) 0 0
\(481\) 17.8564 0.814182
\(482\) 0 0
\(483\) 6.92820 0.315244
\(484\) 0 0
\(485\) 6.92820 0.314594
\(486\) 0 0
\(487\) 0.784610 0.0355541 0.0177770 0.999842i \(-0.494341\pi\)
0.0177770 + 0.999842i \(0.494341\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −25.8564 −1.16688 −0.583442 0.812155i \(-0.698294\pi\)
−0.583442 + 0.812155i \(0.698294\pi\)
\(492\) 0 0
\(493\) 20.7846 0.936092
\(494\) 0 0
\(495\) −3.46410 −0.155700
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 22.9282 1.02641 0.513204 0.858267i \(-0.328459\pi\)
0.513204 + 0.858267i \(0.328459\pi\)
\(500\) 0 0
\(501\) 5.07180 0.226591
\(502\) 0 0
\(503\) −5.07180 −0.226140 −0.113070 0.993587i \(-0.536068\pi\)
−0.113070 + 0.993587i \(0.536068\pi\)
\(504\) 0 0
\(505\) 27.2154 1.21107
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) −6.67949 −0.296063 −0.148032 0.988983i \(-0.547294\pi\)
−0.148032 + 0.988983i \(0.547294\pi\)
\(510\) 0 0
\(511\) 7.46410 0.330192
\(512\) 0 0
\(513\) −1.46410 −0.0646417
\(514\) 0 0
\(515\) 22.6410 0.997682
\(516\) 0 0
\(517\) 2.53590 0.111529
\(518\) 0 0
\(519\) −12.9282 −0.567485
\(520\) 0 0
\(521\) 23.0718 1.01079 0.505397 0.862887i \(-0.331346\pi\)
0.505397 + 0.862887i \(0.331346\pi\)
\(522\) 0 0
\(523\) −3.60770 −0.157753 −0.0788767 0.996884i \(-0.525133\pi\)
−0.0788767 + 0.996884i \(0.525133\pi\)
\(524\) 0 0
\(525\) 7.00000 0.305505
\(526\) 0 0
\(527\) −5.07180 −0.220931
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 6.92820 0.300658
\(532\) 0 0
\(533\) −6.92820 −0.300094
\(534\) 0 0
\(535\) −24.0000 −1.03761
\(536\) 0 0
\(537\) 20.7846 0.896922
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 29.7128 1.27745 0.638727 0.769434i \(-0.279461\pi\)
0.638727 + 0.769434i \(0.279461\pi\)
\(542\) 0 0
\(543\) 14.3923 0.617633
\(544\) 0 0
\(545\) −41.0718 −1.75932
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −8.78461 −0.374237
\(552\) 0 0
\(553\) 2.92820 0.124520
\(554\) 0 0
\(555\) −30.9282 −1.31283
\(556\) 0 0
\(557\) −33.7128 −1.42846 −0.714229 0.699912i \(-0.753222\pi\)
−0.714229 + 0.699912i \(0.753222\pi\)
\(558\) 0 0
\(559\) −5.85641 −0.247700
\(560\) 0 0
\(561\) −3.46410 −0.146254
\(562\) 0 0
\(563\) −30.2487 −1.27483 −0.637416 0.770520i \(-0.719997\pi\)
−0.637416 + 0.770520i \(0.719997\pi\)
\(564\) 0 0
\(565\) −68.7846 −2.89379
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −4.14359 −0.173708 −0.0868542 0.996221i \(-0.527681\pi\)
−0.0868542 + 0.996221i \(0.527681\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) −20.7846 −0.868290
\(574\) 0 0
\(575\) 48.4974 2.02248
\(576\) 0 0
\(577\) 10.7846 0.448969 0.224485 0.974478i \(-0.427930\pi\)
0.224485 + 0.974478i \(0.427930\pi\)
\(578\) 0 0
\(579\) −26.0000 −1.08052
\(580\) 0 0
\(581\) −16.3923 −0.680067
\(582\) 0 0
\(583\) −12.9282 −0.535431
\(584\) 0 0
\(585\) 6.92820 0.286446
\(586\) 0 0
\(587\) −20.7846 −0.857873 −0.428936 0.903335i \(-0.641112\pi\)
−0.428936 + 0.903335i \(0.641112\pi\)
\(588\) 0 0
\(589\) 2.14359 0.0883252
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) −27.4641 −1.12782 −0.563908 0.825838i \(-0.690703\pi\)
−0.563908 + 0.825838i \(0.690703\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) 0 0
\(597\) −20.3923 −0.834601
\(598\) 0 0
\(599\) 25.8564 1.05646 0.528232 0.849100i \(-0.322855\pi\)
0.528232 + 0.849100i \(0.322855\pi\)
\(600\) 0 0
\(601\) −2.39230 −0.0975842 −0.0487921 0.998809i \(-0.515537\pi\)
−0.0487921 + 0.998809i \(0.515537\pi\)
\(602\) 0 0
\(603\) −1.07180 −0.0436469
\(604\) 0 0
\(605\) 3.46410 0.140836
\(606\) 0 0
\(607\) −35.7128 −1.44954 −0.724769 0.688992i \(-0.758054\pi\)
−0.724769 + 0.688992i \(0.758054\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) −5.07180 −0.205183
\(612\) 0 0
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 0 0
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) 7.85641 0.316287 0.158144 0.987416i \(-0.449449\pi\)
0.158144 + 0.987416i \(0.449449\pi\)
\(618\) 0 0
\(619\) −23.7128 −0.953098 −0.476549 0.879148i \(-0.658113\pi\)
−0.476549 + 0.879148i \(0.658113\pi\)
\(620\) 0 0
\(621\) −6.92820 −0.278019
\(622\) 0 0
\(623\) −12.9282 −0.517958
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 1.46410 0.0584706
\(628\) 0 0
\(629\) −30.9282 −1.23319
\(630\) 0 0
\(631\) −40.7846 −1.62361 −0.811805 0.583929i \(-0.801515\pi\)
−0.811805 + 0.583929i \(0.801515\pi\)
\(632\) 0 0
\(633\) −24.7846 −0.985100
\(634\) 0 0
\(635\) −10.1436 −0.402536
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 4.14359 0.163662 0.0818311 0.996646i \(-0.473923\pi\)
0.0818311 + 0.996646i \(0.473923\pi\)
\(642\) 0 0
\(643\) −1.07180 −0.0422675 −0.0211338 0.999777i \(-0.506728\pi\)
−0.0211338 + 0.999777i \(0.506728\pi\)
\(644\) 0 0
\(645\) 10.1436 0.399404
\(646\) 0 0
\(647\) −35.3205 −1.38859 −0.694296 0.719689i \(-0.744284\pi\)
−0.694296 + 0.719689i \(0.744284\pi\)
\(648\) 0 0
\(649\) −6.92820 −0.271956
\(650\) 0 0
\(651\) 1.46410 0.0573827
\(652\) 0 0
\(653\) 21.7128 0.849688 0.424844 0.905267i \(-0.360329\pi\)
0.424844 + 0.905267i \(0.360329\pi\)
\(654\) 0 0
\(655\) −56.7846 −2.21876
\(656\) 0 0
\(657\) −7.46410 −0.291202
\(658\) 0 0
\(659\) 30.9282 1.20479 0.602396 0.798197i \(-0.294213\pi\)
0.602396 + 0.798197i \(0.294213\pi\)
\(660\) 0 0
\(661\) −4.24871 −0.165256 −0.0826279 0.996580i \(-0.526331\pi\)
−0.0826279 + 0.996580i \(0.526331\pi\)
\(662\) 0 0
\(663\) 6.92820 0.269069
\(664\) 0 0
\(665\) −5.07180 −0.196676
\(666\) 0 0
\(667\) −41.5692 −1.60957
\(668\) 0 0
\(669\) 12.3923 0.479114
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 7.07180 0.272598 0.136299 0.990668i \(-0.456479\pi\)
0.136299 + 0.990668i \(0.456479\pi\)
\(674\) 0 0
\(675\) −7.00000 −0.269430
\(676\) 0 0
\(677\) −38.7846 −1.49061 −0.745307 0.666722i \(-0.767697\pi\)
−0.745307 + 0.666722i \(0.767697\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 16.3923 0.628154
\(682\) 0 0
\(683\) −44.7846 −1.71364 −0.856818 0.515619i \(-0.827562\pi\)
−0.856818 + 0.515619i \(0.827562\pi\)
\(684\) 0 0
\(685\) −68.7846 −2.62812
\(686\) 0 0
\(687\) −13.3205 −0.508209
\(688\) 0 0
\(689\) 25.8564 0.985051
\(690\) 0 0
\(691\) 46.9282 1.78523 0.892616 0.450817i \(-0.148867\pi\)
0.892616 + 0.450817i \(0.148867\pi\)
\(692\) 0 0
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) 22.6410 0.858823
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0 0
\(699\) −19.8564 −0.751038
\(700\) 0 0
\(701\) −9.71281 −0.366848 −0.183424 0.983034i \(-0.558718\pi\)
−0.183424 + 0.983034i \(0.558718\pi\)
\(702\) 0 0
\(703\) 13.0718 0.493012
\(704\) 0 0
\(705\) 8.78461 0.330848
\(706\) 0 0
\(707\) −7.85641 −0.295471
\(708\) 0 0
\(709\) 5.21539 0.195868 0.0979340 0.995193i \(-0.468777\pi\)
0.0979340 + 0.995193i \(0.468777\pi\)
\(710\) 0 0
\(711\) −2.92820 −0.109816
\(712\) 0 0
\(713\) 10.1436 0.379881
\(714\) 0 0
\(715\) −6.92820 −0.259100
\(716\) 0 0
\(717\) −24.0000 −0.896296
\(718\) 0 0
\(719\) −25.1769 −0.938940 −0.469470 0.882948i \(-0.655555\pi\)
−0.469470 + 0.882948i \(0.655555\pi\)
\(720\) 0 0
\(721\) −6.53590 −0.243410
\(722\) 0 0
\(723\) −30.3923 −1.13030
\(724\) 0 0
\(725\) −42.0000 −1.55984
\(726\) 0 0
\(727\) 29.1769 1.08211 0.541056 0.840987i \(-0.318025\pi\)
0.541056 + 0.840987i \(0.318025\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 10.1436 0.375174
\(732\) 0 0
\(733\) 39.8564 1.47213 0.736065 0.676911i \(-0.236682\pi\)
0.736065 + 0.676911i \(0.236682\pi\)
\(734\) 0 0
\(735\) −3.46410 −0.127775
\(736\) 0 0
\(737\) 1.07180 0.0394801
\(738\) 0 0
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 0 0
\(741\) −2.92820 −0.107570
\(742\) 0 0
\(743\) −41.5692 −1.52503 −0.762513 0.646972i \(-0.776035\pi\)
−0.762513 + 0.646972i \(0.776035\pi\)
\(744\) 0 0
\(745\) −20.7846 −0.761489
\(746\) 0 0
\(747\) 16.3923 0.599763
\(748\) 0 0
\(749\) 6.92820 0.253151
\(750\) 0 0
\(751\) −16.7846 −0.612479 −0.306240 0.951954i \(-0.599071\pi\)
−0.306240 + 0.951954i \(0.599071\pi\)
\(752\) 0 0
\(753\) 17.0718 0.622131
\(754\) 0 0
\(755\) 55.4256 2.01715
\(756\) 0 0
\(757\) −18.7846 −0.682738 −0.341369 0.939929i \(-0.610891\pi\)
−0.341369 + 0.939929i \(0.610891\pi\)
\(758\) 0 0
\(759\) 6.92820 0.251478
\(760\) 0 0
\(761\) −46.3923 −1.68172 −0.840860 0.541253i \(-0.817950\pi\)
−0.840860 + 0.541253i \(0.817950\pi\)
\(762\) 0 0
\(763\) 11.8564 0.429231
\(764\) 0 0
\(765\) −12.0000 −0.433861
\(766\) 0 0
\(767\) 13.8564 0.500326
\(768\) 0 0
\(769\) 35.4641 1.27887 0.639434 0.768846i \(-0.279169\pi\)
0.639434 + 0.768846i \(0.279169\pi\)
\(770\) 0 0
\(771\) 0.928203 0.0334284
\(772\) 0 0
\(773\) −39.4641 −1.41943 −0.709713 0.704491i \(-0.751175\pi\)
−0.709713 + 0.704491i \(0.751175\pi\)
\(774\) 0 0
\(775\) 10.2487 0.368145
\(776\) 0 0
\(777\) 8.92820 0.320298
\(778\) 0 0
\(779\) −5.07180 −0.181716
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 63.7128 2.27401
\(786\) 0 0
\(787\) 29.1769 1.04004 0.520022 0.854153i \(-0.325924\pi\)
0.520022 + 0.854153i \(0.325924\pi\)
\(788\) 0 0
\(789\) −18.9282 −0.673862
\(790\) 0 0
\(791\) 19.8564 0.706013
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 0 0
\(795\) −44.7846 −1.58835
\(796\) 0 0
\(797\) −39.4641 −1.39789 −0.698945 0.715175i \(-0.746347\pi\)
−0.698945 + 0.715175i \(0.746347\pi\)
\(798\) 0 0
\(799\) 8.78461 0.310777
\(800\) 0 0
\(801\) 12.9282 0.456796
\(802\) 0 0
\(803\) 7.46410 0.263402
\(804\) 0 0
\(805\) −24.0000 −0.845889
\(806\) 0 0
\(807\) 24.2487 0.853595
\(808\) 0 0
\(809\) 24.9282 0.876429 0.438214 0.898870i \(-0.355611\pi\)
0.438214 + 0.898870i \(0.355611\pi\)
\(810\) 0 0
\(811\) −45.1769 −1.58638 −0.793188 0.608977i \(-0.791580\pi\)
−0.793188 + 0.608977i \(0.791580\pi\)
\(812\) 0 0
\(813\) 16.7846 0.588662
\(814\) 0 0
\(815\) 13.8564 0.485369
\(816\) 0 0
\(817\) −4.28719 −0.149990
\(818\) 0 0
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) 0.784610 0.0273498 0.0136749 0.999906i \(-0.495647\pi\)
0.0136749 + 0.999906i \(0.495647\pi\)
\(824\) 0 0
\(825\) 7.00000 0.243709
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −4.24871 −0.147564 −0.0737819 0.997274i \(-0.523507\pi\)
−0.0737819 + 0.997274i \(0.523507\pi\)
\(830\) 0 0
\(831\) −15.8564 −0.550053
\(832\) 0 0
\(833\) −3.46410 −0.120024
\(834\) 0 0
\(835\) −17.5692 −0.608008
\(836\) 0 0
\(837\) −1.46410 −0.0506068
\(838\) 0 0
\(839\) 26.5359 0.916121 0.458060 0.888921i \(-0.348544\pi\)
0.458060 + 0.888921i \(0.348544\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −19.8564 −0.683891
\(844\) 0 0
\(845\) −31.1769 −1.07252
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) −6.53590 −0.224311
\(850\) 0 0
\(851\) 61.8564 2.12041
\(852\) 0 0
\(853\) 5.71281 0.195603 0.0978015 0.995206i \(-0.468819\pi\)
0.0978015 + 0.995206i \(0.468819\pi\)
\(854\) 0 0
\(855\) 5.07180 0.173452
\(856\) 0 0
\(857\) 29.3205 1.00157 0.500785 0.865572i \(-0.333045\pi\)
0.500785 + 0.865572i \(0.333045\pi\)
\(858\) 0 0
\(859\) −11.2154 −0.382664 −0.191332 0.981525i \(-0.561281\pi\)
−0.191332 + 0.981525i \(0.561281\pi\)
\(860\) 0 0
\(861\) −3.46410 −0.118056
\(862\) 0 0
\(863\) −3.21539 −0.109453 −0.0547266 0.998501i \(-0.517429\pi\)
−0.0547266 + 0.998501i \(0.517429\pi\)
\(864\) 0 0
\(865\) 44.7846 1.52272
\(866\) 0 0
\(867\) 5.00000 0.169809
\(868\) 0 0
\(869\) 2.92820 0.0993325
\(870\) 0 0
\(871\) −2.14359 −0.0726329
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) −6.92820 −0.234216
\(876\) 0 0
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) 0 0
\(879\) 11.0718 0.373442
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 14.1436 0.475970 0.237985 0.971269i \(-0.423513\pi\)
0.237985 + 0.971269i \(0.423513\pi\)
\(884\) 0 0
\(885\) −24.0000 −0.806751
\(886\) 0 0
\(887\) −37.8564 −1.27109 −0.635547 0.772062i \(-0.719225\pi\)
−0.635547 + 0.772062i \(0.719225\pi\)
\(888\) 0 0
\(889\) 2.92820 0.0982088
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) −3.71281 −0.124245
\(894\) 0 0
\(895\) −72.0000 −2.40669
\(896\) 0 0
\(897\) −13.8564 −0.462652
\(898\) 0 0
\(899\) −8.78461 −0.292983
\(900\) 0 0
\(901\) −44.7846 −1.49199
\(902\) 0 0
\(903\) −2.92820 −0.0974445
\(904\) 0 0
\(905\) −49.8564 −1.65728
\(906\) 0 0
\(907\) −1.07180 −0.0355884 −0.0177942 0.999842i \(-0.505664\pi\)
−0.0177942 + 0.999842i \(0.505664\pi\)
\(908\) 0 0
\(909\) 7.85641 0.260581
\(910\) 0 0
\(911\) −48.4974 −1.60679 −0.803396 0.595446i \(-0.796976\pi\)
−0.803396 + 0.595446i \(0.796976\pi\)
\(912\) 0 0
\(913\) −16.3923 −0.542506
\(914\) 0 0
\(915\) −6.92820 −0.229039
\(916\) 0 0
\(917\) 16.3923 0.541322
\(918\) 0 0
\(919\) 5.85641 0.193185 0.0965925 0.995324i \(-0.469206\pi\)
0.0965925 + 0.995324i \(0.469206\pi\)
\(920\) 0 0
\(921\) 17.4641 0.575462
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 62.4974 2.05490
\(926\) 0 0
\(927\) 6.53590 0.214667
\(928\) 0 0
\(929\) −28.6410 −0.939681 −0.469841 0.882751i \(-0.655689\pi\)
−0.469841 + 0.882751i \(0.655689\pi\)
\(930\) 0 0
\(931\) 1.46410 0.0479840
\(932\) 0 0
\(933\) 7.60770 0.249065
\(934\) 0 0
\(935\) 12.0000 0.392442
\(936\) 0 0
\(937\) 6.39230 0.208827 0.104414 0.994534i \(-0.466703\pi\)
0.104414 + 0.994534i \(0.466703\pi\)
\(938\) 0 0
\(939\) 3.07180 0.100244
\(940\) 0 0
\(941\) −24.9282 −0.812636 −0.406318 0.913732i \(-0.633188\pi\)
−0.406318 + 0.913732i \(0.633188\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) 3.46410 0.112687
\(946\) 0 0
\(947\) −22.1436 −0.719570 −0.359785 0.933035i \(-0.617150\pi\)
−0.359785 + 0.933035i \(0.617150\pi\)
\(948\) 0 0
\(949\) −14.9282 −0.484590
\(950\) 0 0
\(951\) 0.928203 0.0300991
\(952\) 0 0
\(953\) 14.7846 0.478920 0.239460 0.970906i \(-0.423030\pi\)
0.239460 + 0.970906i \(0.423030\pi\)
\(954\) 0 0
\(955\) 72.0000 2.32987
\(956\) 0 0
\(957\) −6.00000 −0.193952
\(958\) 0 0
\(959\) 19.8564 0.641197
\(960\) 0 0
\(961\) −28.8564 −0.930852
\(962\) 0 0
\(963\) −6.92820 −0.223258
\(964\) 0 0
\(965\) 90.0666 2.89935
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 5.07180 0.162930
\(970\) 0 0
\(971\) −39.7128 −1.27444 −0.637222 0.770680i \(-0.719917\pi\)
−0.637222 + 0.770680i \(0.719917\pi\)
\(972\) 0 0
\(973\) −6.53590 −0.209531
\(974\) 0 0
\(975\) −14.0000 −0.448359
\(976\) 0 0
\(977\) 35.5692 1.13796 0.568980 0.822351i \(-0.307338\pi\)
0.568980 + 0.822351i \(0.307338\pi\)
\(978\) 0 0
\(979\) −12.9282 −0.413187
\(980\) 0 0
\(981\) −11.8564 −0.378546
\(982\) 0 0
\(983\) 11.3205 0.361068 0.180534 0.983569i \(-0.442217\pi\)
0.180534 + 0.983569i \(0.442217\pi\)
\(984\) 0 0
\(985\) 62.3538 1.98676
\(986\) 0 0
\(987\) −2.53590 −0.0807185
\(988\) 0 0
\(989\) −20.2872 −0.645095
\(990\) 0 0
\(991\) 29.8564 0.948420 0.474210 0.880412i \(-0.342734\pi\)
0.474210 + 0.880412i \(0.342734\pi\)
\(992\) 0 0
\(993\) −22.9282 −0.727605
\(994\) 0 0
\(995\) 70.6410 2.23947
\(996\) 0 0
\(997\) −44.6410 −1.41380 −0.706898 0.707316i \(-0.749906\pi\)
−0.706898 + 0.707316i \(0.749906\pi\)
\(998\) 0 0
\(999\) −8.92820 −0.282476
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 3696.2.a.bc.1.2 2
4.3 odd 2 462.2.a.h.1.2 2
12.11 even 2 1386.2.a.p.1.1 2
28.27 even 2 3234.2.a.x.1.1 2
44.43 even 2 5082.2.a.bu.1.2 2
84.83 odd 2 9702.2.a.dd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.h.1.2 2 4.3 odd 2
1386.2.a.p.1.1 2 12.11 even 2
3234.2.a.x.1.1 2 28.27 even 2
3696.2.a.bc.1.2 2 1.1 even 1 trivial
5082.2.a.bu.1.2 2 44.43 even 2
9702.2.a.dd.1.2 2 84.83 odd 2