Properties

Label 56.3.h.c.13.2
Level $56$
Weight $3$
Character 56.13
Self dual yes
Analytic conductor $1.526$
Analytic rank $0$
Dimension $2$
CM discriminant -56
Inner twists $4$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [56,3,Mod(13,56)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(56, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("56.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 56.h (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.52588948042\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 13.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 56.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +2.82843 q^{3} +4.00000 q^{4} -8.48528 q^{5} +5.65685 q^{6} -7.00000 q^{7} +8.00000 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} +2.82843 q^{3} +4.00000 q^{4} -8.48528 q^{5} +5.65685 q^{6} -7.00000 q^{7} +8.00000 q^{8} -1.00000 q^{9} -16.9706 q^{10} +11.3137 q^{12} +25.4558 q^{13} -14.0000 q^{14} -24.0000 q^{15} +16.0000 q^{16} -2.00000 q^{18} -8.48528 q^{19} -33.9411 q^{20} -19.7990 q^{21} -10.0000 q^{23} +22.6274 q^{24} +47.0000 q^{25} +50.9117 q^{26} -28.2843 q^{27} -28.0000 q^{28} -48.0000 q^{30} +32.0000 q^{32} +59.3970 q^{35} -4.00000 q^{36} -16.9706 q^{38} +72.0000 q^{39} -67.8823 q^{40} -39.5980 q^{42} +8.48528 q^{45} -20.0000 q^{46} +45.2548 q^{48} +49.0000 q^{49} +94.0000 q^{50} +101.823 q^{52} -56.5685 q^{54} -56.0000 q^{56} -24.0000 q^{57} -76.3675 q^{59} -96.0000 q^{60} -8.48528 q^{61} +7.00000 q^{63} +64.0000 q^{64} -216.000 q^{65} -28.2843 q^{69} +118.794 q^{70} -110.000 q^{71} -8.00000 q^{72} +132.936 q^{75} -33.9411 q^{76} +144.000 q^{78} +130.000 q^{79} -135.765 q^{80} -71.0000 q^{81} +25.4558 q^{83} -79.1960 q^{84} +16.9706 q^{90} -178.191 q^{91} -40.0000 q^{92} +72.0000 q^{95} +90.5097 q^{96} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} - 14 q^{7} + 16 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 8 q^{4} - 14 q^{7} + 16 q^{8} - 2 q^{9} - 28 q^{14} - 48 q^{15} + 32 q^{16} - 4 q^{18} - 20 q^{23} + 94 q^{25} - 56 q^{28} - 96 q^{30} + 64 q^{32} - 8 q^{36} + 144 q^{39} - 40 q^{46} + 98 q^{49} + 188 q^{50} - 112 q^{56} - 48 q^{57} - 192 q^{60} + 14 q^{63} + 128 q^{64} - 432 q^{65} - 220 q^{71} - 16 q^{72} + 288 q^{78} + 260 q^{79} - 142 q^{81} - 80 q^{92} + 144 q^{95} + 196 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.00000
\(3\) 2.82843 0.942809 0.471405 0.881917i \(-0.343747\pi\)
0.471405 + 0.881917i \(0.343747\pi\)
\(4\) 4.00000 1.00000
\(5\) −8.48528 −1.69706 −0.848528 0.529150i \(-0.822511\pi\)
−0.848528 + 0.529150i \(0.822511\pi\)
\(6\) 5.65685 0.942809
\(7\) −7.00000 −1.00000
\(8\) 8.00000 1.00000
\(9\) −1.00000 −0.111111
\(10\) −16.9706 −1.69706
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 11.3137 0.942809
\(13\) 25.4558 1.95814 0.979071 0.203519i \(-0.0652380\pi\)
0.979071 + 0.203519i \(0.0652380\pi\)
\(14\) −14.0000 −1.00000
\(15\) −24.0000 −1.60000
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −2.00000 −0.111111
\(19\) −8.48528 −0.446594 −0.223297 0.974750i \(-0.571682\pi\)
−0.223297 + 0.974750i \(0.571682\pi\)
\(20\) −33.9411 −1.69706
\(21\) −19.7990 −0.942809
\(22\) 0 0
\(23\) −10.0000 −0.434783 −0.217391 0.976085i \(-0.569755\pi\)
−0.217391 + 0.976085i \(0.569755\pi\)
\(24\) 22.6274 0.942809
\(25\) 47.0000 1.88000
\(26\) 50.9117 1.95814
\(27\) −28.2843 −1.04757
\(28\) −28.0000 −1.00000
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) −48.0000 −1.60000
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 32.0000 1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 59.3970 1.69706
\(36\) −4.00000 −0.111111
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −16.9706 −0.446594
\(39\) 72.0000 1.84615
\(40\) −67.8823 −1.69706
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) −39.5980 −0.942809
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 8.48528 0.188562
\(46\) −20.0000 −0.434783
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 45.2548 0.942809
\(49\) 49.0000 1.00000
\(50\) 94.0000 1.88000
\(51\) 0 0
\(52\) 101.823 1.95814
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −56.5685 −1.04757
\(55\) 0 0
\(56\) −56.0000 −1.00000
\(57\) −24.0000 −0.421053
\(58\) 0 0
\(59\) −76.3675 −1.29436 −0.647182 0.762335i \(-0.724053\pi\)
−0.647182 + 0.762335i \(0.724053\pi\)
\(60\) −96.0000 −1.60000
\(61\) −8.48528 −0.139103 −0.0695515 0.997578i \(-0.522157\pi\)
−0.0695515 + 0.997578i \(0.522157\pi\)
\(62\) 0 0
\(63\) 7.00000 0.111111
\(64\) 64.0000 1.00000
\(65\) −216.000 −3.32308
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −28.2843 −0.409917
\(70\) 118.794 1.69706
\(71\) −110.000 −1.54930 −0.774648 0.632393i \(-0.782073\pi\)
−0.774648 + 0.632393i \(0.782073\pi\)
\(72\) −8.00000 −0.111111
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 132.936 1.77248
\(76\) −33.9411 −0.446594
\(77\) 0 0
\(78\) 144.000 1.84615
\(79\) 130.000 1.64557 0.822785 0.568353i \(-0.192419\pi\)
0.822785 + 0.568353i \(0.192419\pi\)
\(80\) −135.765 −1.69706
\(81\) −71.0000 −0.876543
\(82\) 0 0
\(83\) 25.4558 0.306697 0.153348 0.988172i \(-0.450994\pi\)
0.153348 + 0.988172i \(0.450994\pi\)
\(84\) −79.1960 −0.942809
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 16.9706 0.188562
\(91\) −178.191 −1.95814
\(92\) −40.0000 −0.434783
\(93\) 0 0
\(94\) 0 0
\(95\) 72.0000 0.757895
\(96\) 90.5097 0.942809
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 98.0000 1.00000
\(99\) 0 0
\(100\) 188.000 1.88000
\(101\) 161.220 1.59624 0.798121 0.602498i \(-0.205828\pi\)
0.798121 + 0.602498i \(0.205828\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 203.647 1.95814
\(105\) 168.000 1.60000
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −113.137 −1.04757
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −112.000 −1.00000
\(113\) −26.0000 −0.230088 −0.115044 0.993360i \(-0.536701\pi\)
−0.115044 + 0.993360i \(0.536701\pi\)
\(114\) −48.0000 −0.421053
\(115\) 84.8528 0.737851
\(116\) 0 0
\(117\) −25.4558 −0.217571
\(118\) −152.735 −1.29436
\(119\) 0 0
\(120\) −192.000 −1.60000
\(121\) 121.000 1.00000
\(122\) −16.9706 −0.139103
\(123\) 0 0
\(124\) 0 0
\(125\) −186.676 −1.49341
\(126\) 14.0000 0.111111
\(127\) −250.000 −1.96850 −0.984252 0.176771i \(-0.943435\pi\)
−0.984252 + 0.176771i \(0.943435\pi\)
\(128\) 128.000 1.00000
\(129\) 0 0
\(130\) −432.000 −3.32308
\(131\) −246.073 −1.87842 −0.939211 0.343342i \(-0.888441\pi\)
−0.939211 + 0.343342i \(0.888441\pi\)
\(132\) 0 0
\(133\) 59.3970 0.446594
\(134\) 0 0
\(135\) 240.000 1.77778
\(136\) 0 0
\(137\) 50.0000 0.364964 0.182482 0.983209i \(-0.441587\pi\)
0.182482 + 0.983209i \(0.441587\pi\)
\(138\) −56.5685 −0.409917
\(139\) 263.044 1.89240 0.946200 0.323581i \(-0.104887\pi\)
0.946200 + 0.323581i \(0.104887\pi\)
\(140\) 237.588 1.69706
\(141\) 0 0
\(142\) −220.000 −1.54930
\(143\) 0 0
\(144\) −16.0000 −0.111111
\(145\) 0 0
\(146\) 0 0
\(147\) 138.593 0.942809
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 265.872 1.77248
\(151\) −202.000 −1.33775 −0.668874 0.743376i \(-0.733224\pi\)
−0.668874 + 0.743376i \(0.733224\pi\)
\(152\) −67.8823 −0.446594
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 288.000 1.84615
\(157\) −313.955 −1.99972 −0.999858 0.0168519i \(-0.994636\pi\)
−0.999858 + 0.0168519i \(0.994636\pi\)
\(158\) 260.000 1.64557
\(159\) 0 0
\(160\) −271.529 −1.69706
\(161\) 70.0000 0.434783
\(162\) −142.000 −0.876543
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 50.9117 0.306697
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) −158.392 −0.942809
\(169\) 479.000 2.83432
\(170\) 0 0
\(171\) 8.48528 0.0496215
\(172\) 0 0
\(173\) 229.103 1.32429 0.662146 0.749375i \(-0.269646\pi\)
0.662146 + 0.749375i \(0.269646\pi\)
\(174\) 0 0
\(175\) −329.000 −1.88000
\(176\) 0 0
\(177\) −216.000 −1.22034
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 33.9411 0.188562
\(181\) 263.044 1.45328 0.726640 0.687018i \(-0.241081\pi\)
0.726640 + 0.687018i \(0.241081\pi\)
\(182\) −356.382 −1.95814
\(183\) −24.0000 −0.131148
\(184\) −80.0000 −0.434783
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 197.990 1.04757
\(190\) 144.000 0.757895
\(191\) 130.000 0.680628 0.340314 0.940312i \(-0.389467\pi\)
0.340314 + 0.940312i \(0.389467\pi\)
\(192\) 181.019 0.942809
\(193\) −314.000 −1.62694 −0.813472 0.581605i \(-0.802425\pi\)
−0.813472 + 0.581605i \(0.802425\pi\)
\(194\) 0 0
\(195\) −610.940 −3.13303
\(196\) 196.000 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 376.000 1.88000
\(201\) 0 0
\(202\) 322.441 1.59624
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 10.0000 0.0483092
\(208\) 407.294 1.95814
\(209\) 0 0
\(210\) 336.000 1.60000
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) −311.127 −1.46069
\(214\) 0 0
\(215\) 0 0
\(216\) −226.274 −1.04757
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) −224.000 −1.00000
\(225\) −47.0000 −0.208889
\(226\) −52.0000 −0.230088
\(227\) 398.808 1.75686 0.878432 0.477867i \(-0.158590\pi\)
0.878432 + 0.477867i \(0.158590\pi\)
\(228\) −96.0000 −0.421053
\(229\) −246.073 −1.07456 −0.537278 0.843405i \(-0.680547\pi\)
−0.537278 + 0.843405i \(0.680547\pi\)
\(230\) 169.706 0.737851
\(231\) 0 0
\(232\) 0 0
\(233\) −430.000 −1.84549 −0.922747 0.385407i \(-0.874061\pi\)
−0.922747 + 0.385407i \(0.874061\pi\)
\(234\) −50.9117 −0.217571
\(235\) 0 0
\(236\) −305.470 −1.29436
\(237\) 367.696 1.55146
\(238\) 0 0
\(239\) 422.000 1.76569 0.882845 0.469664i \(-0.155625\pi\)
0.882845 + 0.469664i \(0.155625\pi\)
\(240\) −384.000 −1.60000
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 242.000 1.00000
\(243\) 53.7401 0.221153
\(244\) −33.9411 −0.139103
\(245\) −415.779 −1.69706
\(246\) 0 0
\(247\) −216.000 −0.874494
\(248\) 0 0
\(249\) 72.0000 0.289157
\(250\) −373.352 −1.49341
\(251\) 500.632 1.99455 0.997274 0.0737859i \(-0.0235081\pi\)
0.997274 + 0.0737859i \(0.0235081\pi\)
\(252\) 28.0000 0.111111
\(253\) 0 0
\(254\) −500.000 −1.96850
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −864.000 −3.32308
\(261\) 0 0
\(262\) −492.146 −1.87842
\(263\) 274.000 1.04183 0.520913 0.853610i \(-0.325592\pi\)
0.520913 + 0.853610i \(0.325592\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 118.794 0.446594
\(267\) 0 0
\(268\) 0 0
\(269\) 161.220 0.599332 0.299666 0.954044i \(-0.403125\pi\)
0.299666 + 0.954044i \(0.403125\pi\)
\(270\) 480.000 1.77778
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) −504.000 −1.84615
\(274\) 100.000 0.364964
\(275\) 0 0
\(276\) −113.137 −0.409917
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 526.087 1.89240
\(279\) 0 0
\(280\) 475.176 1.69706
\(281\) 338.000 1.20285 0.601423 0.798930i \(-0.294600\pi\)
0.601423 + 0.798930i \(0.294600\pi\)
\(282\) 0 0
\(283\) −313.955 −1.10938 −0.554692 0.832056i \(-0.687164\pi\)
−0.554692 + 0.832056i \(0.687164\pi\)
\(284\) −440.000 −1.54930
\(285\) 203.647 0.714550
\(286\) 0 0
\(287\) 0 0
\(288\) −32.0000 −0.111111
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −449.720 −1.53488 −0.767440 0.641121i \(-0.778470\pi\)
−0.767440 + 0.641121i \(0.778470\pi\)
\(294\) 277.186 0.942809
\(295\) 648.000 2.19661
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −254.558 −0.851366
\(300\) 531.744 1.77248
\(301\) 0 0
\(302\) −404.000 −1.33775
\(303\) 456.000 1.50495
\(304\) −135.765 −0.446594
\(305\) 72.0000 0.236066
\(306\) 0 0
\(307\) −449.720 −1.46489 −0.732443 0.680829i \(-0.761620\pi\)
−0.732443 + 0.680829i \(0.761620\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 576.000 1.84615
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) −627.911 −1.99972
\(315\) −59.3970 −0.188562
\(316\) 520.000 1.64557
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −543.058 −1.69706
\(321\) 0 0
\(322\) 140.000 0.434783
\(323\) 0 0
\(324\) −284.000 −0.876543
\(325\) 1196.42 3.68131
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 101.823 0.306697
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −316.784 −0.942809
\(337\) −26.0000 −0.0771513 −0.0385757 0.999256i \(-0.512282\pi\)
−0.0385757 + 0.999256i \(0.512282\pi\)
\(338\) 958.000 2.83432
\(339\) −73.5391 −0.216930
\(340\) 0 0
\(341\) 0 0
\(342\) 16.9706 0.0496215
\(343\) −343.000 −1.00000
\(344\) 0 0
\(345\) 240.000 0.695652
\(346\) 458.205 1.32429
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −687.308 −1.96936 −0.984682 0.174362i \(-0.944214\pi\)
−0.984682 + 0.174362i \(0.944214\pi\)
\(350\) −658.000 −1.88000
\(351\) −720.000 −2.05128
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) −432.000 −1.22034
\(355\) 933.381 2.62924
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −682.000 −1.89972 −0.949861 0.312673i \(-0.898775\pi\)
−0.949861 + 0.312673i \(0.898775\pi\)
\(360\) 67.8823 0.188562
\(361\) −289.000 −0.800554
\(362\) 526.087 1.45328
\(363\) 342.240 0.942809
\(364\) −712.764 −1.95814
\(365\) 0 0
\(366\) −48.0000 −0.131148
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −160.000 −0.434783
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −528.000 −1.40800
\(376\) 0 0
\(377\) 0 0
\(378\) 395.980 1.04757
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 288.000 0.757895
\(381\) −707.107 −1.85592
\(382\) 260.000 0.680628
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 362.039 0.942809
\(385\) 0 0
\(386\) −628.000 −1.62694
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) −1221.88 −3.13303
\(391\) 0 0
\(392\) 392.000 1.00000
\(393\) −696.000 −1.77099
\(394\) 0 0
\(395\) −1103.09 −2.79262
\(396\) 0 0
\(397\) −483.661 −1.21829 −0.609145 0.793059i \(-0.708487\pi\)
−0.609145 + 0.793059i \(0.708487\pi\)
\(398\) 0 0
\(399\) 168.000 0.421053
\(400\) 752.000 1.88000
\(401\) 550.000 1.37157 0.685786 0.727804i \(-0.259459\pi\)
0.685786 + 0.727804i \(0.259459\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 644.881 1.59624
\(405\) 602.455 1.48754
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 141.421 0.344091
\(412\) 0 0
\(413\) 534.573 1.29436
\(414\) 20.0000 0.0483092
\(415\) −216.000 −0.520482
\(416\) 814.587 1.95814
\(417\) 744.000 1.78417
\(418\) 0 0
\(419\) 500.632 1.19482 0.597412 0.801934i \(-0.296196\pi\)
0.597412 + 0.801934i \(0.296196\pi\)
\(420\) 672.000 1.60000
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) −622.254 −1.46069
\(427\) 59.3970 0.139103
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −538.000 −1.24826 −0.624130 0.781321i \(-0.714546\pi\)
−0.624130 + 0.781321i \(0.714546\pi\)
\(432\) −452.548 −1.04757
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 84.8528 0.194171
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −49.0000 −0.111111
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −448.000 −1.00000
\(449\) 2.00000 0.00445434 0.00222717 0.999998i \(-0.499291\pi\)
0.00222717 + 0.999998i \(0.499291\pi\)
\(450\) −94.0000 −0.208889
\(451\) 0 0
\(452\) −104.000 −0.230088
\(453\) −571.342 −1.26124
\(454\) 797.616 1.75686
\(455\) 1512.00 3.32308
\(456\) −192.000 −0.421053
\(457\) 886.000 1.93873 0.969365 0.245623i \(-0.0789925\pi\)
0.969365 + 0.245623i \(0.0789925\pi\)
\(458\) −492.146 −1.07456
\(459\) 0 0
\(460\) 339.411 0.737851
\(461\) 263.044 0.570594 0.285297 0.958439i \(-0.407908\pi\)
0.285297 + 0.958439i \(0.407908\pi\)
\(462\) 0 0
\(463\) 226.000 0.488121 0.244060 0.969760i \(-0.421520\pi\)
0.244060 + 0.969760i \(0.421520\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −860.000 −1.84549
\(467\) −483.661 −1.03568 −0.517838 0.855478i \(-0.673263\pi\)
−0.517838 + 0.855478i \(0.673263\pi\)
\(468\) −101.823 −0.217571
\(469\) 0 0
\(470\) 0 0
\(471\) −888.000 −1.88535
\(472\) −610.940 −1.29436
\(473\) 0 0
\(474\) 735.391 1.55146
\(475\) −398.808 −0.839596
\(476\) 0 0
\(477\) 0 0
\(478\) 844.000 1.76569
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −768.000 −1.60000
\(481\) 0 0
\(482\) 0 0
\(483\) 197.990 0.409917
\(484\) 484.000 1.00000
\(485\) 0 0
\(486\) 107.480 0.221153
\(487\) 470.000 0.965092 0.482546 0.875871i \(-0.339712\pi\)
0.482546 + 0.875871i \(0.339712\pi\)
\(488\) −67.8823 −0.139103
\(489\) 0 0
\(490\) −831.558 −1.69706
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −432.000 −0.874494
\(495\) 0 0
\(496\) 0 0
\(497\) 770.000 1.54930
\(498\) 144.000 0.289157
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −746.705 −1.49341
\(501\) 0 0
\(502\) 1001.26 1.99455
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 56.0000 0.111111
\(505\) −1368.00 −2.70891
\(506\) 0 0
\(507\) 1354.82 2.67222
\(508\) −1000.00 −1.96850
\(509\) 941.866 1.85042 0.925212 0.379450i \(-0.123887\pi\)
0.925212 + 0.379450i \(0.123887\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 512.000 1.00000
\(513\) 240.000 0.467836
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 648.000 1.24855
\(520\) −1728.00 −3.32308
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −958.837 −1.83334 −0.916670 0.399645i \(-0.869133\pi\)
−0.916670 + 0.399645i \(0.869133\pi\)
\(524\) −984.293 −1.87842
\(525\) −930.553 −1.77248
\(526\) 548.000 1.04183
\(527\) 0 0
\(528\) 0 0
\(529\) −429.000 −0.810964
\(530\) 0 0
\(531\) 76.3675 0.143818
\(532\) 237.588 0.446594
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 322.441 0.599332
\(539\) 0 0
\(540\) 960.000 1.77778
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 744.000 1.37017
\(544\) 0 0
\(545\) 0 0
\(546\) −1008.00 −1.84615
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 200.000 0.364964
\(549\) 8.48528 0.0154559
\(550\) 0 0
\(551\) 0 0
\(552\) −226.274 −0.409917
\(553\) −910.000 −1.64557
\(554\) 0 0
\(555\) 0 0
\(556\) 1052.17 1.89240
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 950.352 1.69706
\(561\) 0 0
\(562\) 676.000 1.20285
\(563\) 398.808 0.708363 0.354181 0.935177i \(-0.384760\pi\)
0.354181 + 0.935177i \(0.384760\pi\)
\(564\) 0 0
\(565\) 220.617 0.390473
\(566\) −627.911 −1.10938
\(567\) 497.000 0.876543
\(568\) −880.000 −1.54930
\(569\) −1130.00 −1.98594 −0.992970 0.118365i \(-0.962235\pi\)
−0.992970 + 0.118365i \(0.962235\pi\)
\(570\) 407.294 0.714550
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 367.696 0.641702
\(574\) 0 0
\(575\) −470.000 −0.817391
\(576\) −64.0000 −0.111111
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 578.000 1.00000
\(579\) −888.126 −1.53390
\(580\) 0 0
\(581\) −178.191 −0.306697
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 216.000 0.369231
\(586\) −899.440 −1.53488
\(587\) −1162.48 −1.98038 −0.990190 0.139725i \(-0.955378\pi\)
−0.990190 + 0.139725i \(0.955378\pi\)
\(588\) 554.372 0.942809
\(589\) 0 0
\(590\) 1296.00 2.19661
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −509.117 −0.851366
\(599\) −1070.00 −1.78631 −0.893155 0.449748i \(-0.851514\pi\)
−0.893155 + 0.449748i \(0.851514\pi\)
\(600\) 1063.49 1.77248
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −808.000 −1.33775
\(605\) −1026.72 −1.69706
\(606\) 912.000 1.50495
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) −271.529 −0.446594
\(609\) 0 0
\(610\) 144.000 0.236066
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −899.440 −1.46489
\(615\) 0 0
\(616\) 0 0
\(617\) −1034.00 −1.67585 −0.837925 0.545785i \(-0.816232\pi\)
−0.837925 + 0.545785i \(0.816232\pi\)
\(618\) 0 0
\(619\) 1111.57 1.79575 0.897877 0.440246i \(-0.145109\pi\)
0.897877 + 0.440246i \(0.145109\pi\)
\(620\) 0 0
\(621\) 282.843 0.455463
\(622\) 0 0
\(623\) 0 0
\(624\) 1152.00 1.84615
\(625\) 409.000 0.654400
\(626\) 0 0
\(627\) 0 0
\(628\) −1255.82 −1.99972
\(629\) 0 0
\(630\) −118.794 −0.188562
\(631\) −110.000 −0.174326 −0.0871632 0.996194i \(-0.527780\pi\)
−0.0871632 + 0.996194i \(0.527780\pi\)
\(632\) 1040.00 1.64557
\(633\) 0 0
\(634\) 0 0
\(635\) 2121.32 3.34066
\(636\) 0 0
\(637\) 1247.34 1.95814
\(638\) 0 0
\(639\) 110.000 0.172144
\(640\) −1086.12 −1.69706
\(641\) 1030.00 1.60686 0.803432 0.595396i \(-0.203005\pi\)
0.803432 + 0.595396i \(0.203005\pi\)
\(642\) 0 0
\(643\) 738.219 1.14809 0.574043 0.818825i \(-0.305374\pi\)
0.574043 + 0.818825i \(0.305374\pi\)
\(644\) 280.000 0.434783
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −568.000 −0.876543
\(649\) 0 0
\(650\) 2392.85 3.68131
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 2088.00 3.18779
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1264.31 −1.91272 −0.956359 0.292193i \(-0.905615\pi\)
−0.956359 + 0.292193i \(0.905615\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 203.647 0.306697
\(665\) −504.000 −0.757895
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) −633.568 −0.942809
\(673\) −670.000 −0.995542 −0.497771 0.867308i \(-0.665848\pi\)
−0.497771 + 0.867308i \(0.665848\pi\)
\(674\) −52.0000 −0.0771513
\(675\) −1329.36 −1.96942
\(676\) 1916.00 2.83432
\(677\) −483.661 −0.714418 −0.357209 0.934024i \(-0.616272\pi\)
−0.357209 + 0.934024i \(0.616272\pi\)
\(678\) −147.078 −0.216930
\(679\) 0 0
\(680\) 0 0
\(681\) 1128.00 1.65639
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 33.9411 0.0496215
\(685\) −424.264 −0.619364
\(686\) −686.000 −1.00000
\(687\) −696.000 −1.01310
\(688\) 0 0
\(689\) 0 0
\(690\) 480.000 0.695652
\(691\) −246.073 −0.356112 −0.178056 0.984020i \(-0.556981\pi\)
−0.178056 + 0.984020i \(0.556981\pi\)
\(692\) 916.410 1.32429
\(693\) 0 0
\(694\) 0 0
\(695\) −2232.00 −3.21151
\(696\) 0 0
\(697\) 0 0
\(698\) −1374.62 −1.96936
\(699\) −1216.22 −1.73995
\(700\) −1316.00 −1.88000
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) −1440.00 −2.05128
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1128.54 −1.59624
\(708\) −864.000 −1.22034
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 1866.76 2.62924
\(711\) −130.000 −0.182841
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1193.60 1.66471
\(718\) −1364.00 −1.89972
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 135.765 0.188562
\(721\) 0 0
\(722\) −578.000 −0.800554
\(723\) 0 0
\(724\) 1052.17 1.45328
\(725\) 0 0
\(726\) 684.479 0.942809
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) −1425.53 −1.95814
\(729\) 791.000 1.08505
\(730\) 0 0
\(731\) 0 0
\(732\) −96.0000 −0.131148
\(733\) 1417.04 1.93321 0.966604 0.256273i \(-0.0824947\pi\)
0.966604 + 0.256273i \(0.0824947\pi\)
\(734\) 0 0
\(735\) −1176.00 −1.60000
\(736\) −320.000 −0.434783
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) −610.940 −0.824481
\(742\) 0 0
\(743\) 1430.00 1.92463 0.962315 0.271937i \(-0.0876644\pi\)
0.962315 + 0.271937i \(0.0876644\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −25.4558 −0.0340774
\(748\) 0 0
\(749\) 0 0
\(750\) −1056.00 −1.40800
\(751\) 998.000 1.32889 0.664447 0.747335i \(-0.268667\pi\)
0.664447 + 0.747335i \(0.268667\pi\)
\(752\) 0 0
\(753\) 1416.00 1.88048
\(754\) 0 0
\(755\) 1714.03 2.27023
\(756\) 791.960 1.04757
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 576.000 0.757895
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) −1414.21 −1.85592
\(763\) 0 0
\(764\) 520.000 0.680628
\(765\) 0 0
\(766\) 0 0
\(767\) −1944.00 −2.53455
\(768\) 724.077 0.942809
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1256.00 −1.62694
\(773\) 873.984 1.13064 0.565320 0.824872i \(-0.308753\pi\)
0.565320 + 0.824872i \(0.308753\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) −2443.76 −3.13303
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 784.000 1.00000
\(785\) 2664.00 3.39363
\(786\) −1392.00 −1.77099
\(787\) −1501.89 −1.90838 −0.954190 0.299202i \(-0.903280\pi\)
−0.954190 + 0.299202i \(0.903280\pi\)
\(788\) 0 0
\(789\) 774.989 0.982242
\(790\) −2206.17 −2.79262
\(791\) 182.000 0.230088
\(792\) 0 0
\(793\) −216.000 −0.272383
\(794\) −967.322 −1.21829
\(795\) 0 0
\(796\) 0 0
\(797\) −449.720 −0.564266 −0.282133 0.959375i \(-0.591042\pi\)
−0.282133 + 0.959375i \(0.591042\pi\)
\(798\) 336.000 0.421053
\(799\) 0 0
\(800\) 1504.00 1.88000
\(801\) 0 0
\(802\) 1100.00 1.37157
\(803\) 0 0
\(804\) 0 0
\(805\) −593.970 −0.737851
\(806\) 0 0
\(807\) 456.000 0.565056
\(808\) 1289.76 1.59624
\(809\) −650.000 −0.803461 −0.401731 0.915758i \(-0.631591\pi\)
−0.401731 + 0.915758i \(0.631591\pi\)
\(810\) 1204.91 1.48754
\(811\) 1450.98 1.78913 0.894564 0.446939i \(-0.147486\pi\)
0.894564 + 0.446939i \(0.147486\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 178.191 0.217571
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 282.843 0.344091
\(823\) 946.000 1.14945 0.574727 0.818345i \(-0.305108\pi\)
0.574727 + 0.818345i \(0.305108\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 1069.15 1.29436
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 40.0000 0.0483092
\(829\) −76.3675 −0.0921201 −0.0460600 0.998939i \(-0.514667\pi\)
−0.0460600 + 0.998939i \(0.514667\pi\)
\(830\) −432.000 −0.520482
\(831\) 0 0
\(832\) 1629.17 1.95814
\(833\) 0 0
\(834\) 1488.00 1.78417
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 1001.26 1.19482
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 1344.00 1.60000
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) 956.008 1.13406
\(844\) 0 0
\(845\) −4064.45 −4.81000
\(846\) 0 0
\(847\) −847.000 −1.00000
\(848\) 0 0
\(849\) −888.000 −1.04594
\(850\) 0 0
\(851\) 0 0
\(852\) −1244.51 −1.46069
\(853\) −449.720 −0.527221 −0.263611 0.964629i \(-0.584913\pi\)
−0.263611 + 0.964629i \(0.584913\pi\)
\(854\) 118.794 0.139103
\(855\) −72.0000 −0.0842105
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1196.42 −1.39281 −0.696406 0.717648i \(-0.745218\pi\)
−0.696406 + 0.717648i \(0.745218\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1076.00 −1.24826
\(863\) 1474.00 1.70800 0.853998 0.520277i \(-0.174171\pi\)
0.853998 + 0.520277i \(0.174171\pi\)
\(864\) −905.097 −1.04757
\(865\) −1944.00 −2.24740
\(866\) 0 0
\(867\) 817.415 0.942809
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 169.706 0.194171
\(875\) 1306.73 1.49341
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) −1272.00 −1.44710
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −98.0000 −0.111111
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 1832.82 2.07098
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1750.00 1.96850
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −896.000 −1.00000
\(897\) −720.000 −0.802676
\(898\) 4.00000 0.00445434
\(899\) 0 0
\(900\) −188.000 −0.208889
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −208.000 −0.230088
\(905\) −2232.00 −2.46630
\(906\) −1142.68 −1.26124
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 1595.23 1.75686
\(909\) −161.220 −0.177360
\(910\) 3024.00 3.32308
\(911\) −922.000 −1.01207 −0.506037 0.862512i \(-0.668890\pi\)
−0.506037 + 0.862512i \(0.668890\pi\)
\(912\) −384.000 −0.421053
\(913\) 0 0
\(914\) 1772.00 1.93873
\(915\) 203.647 0.222565
\(916\) −984.293 −1.07456
\(917\) 1722.51 1.87842
\(918\) 0 0
\(919\) −1550.00 −1.68662 −0.843308 0.537431i \(-0.819395\pi\)
−0.843308 + 0.537431i \(0.819395\pi\)
\(920\) 678.823 0.737851
\(921\) −1272.00 −1.38111
\(922\) 526.087 0.570594
\(923\) −2800.14 −3.03374
\(924\) 0 0
\(925\) 0 0
\(926\) 452.000 0.488121
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −415.779 −0.446594
\(932\) −1720.00 −1.84549
\(933\) 0 0
\(934\) −967.322 −1.03568
\(935\) 0 0
\(936\) −203.647 −0.217571
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1111.57 1.18127 0.590633 0.806940i \(-0.298878\pi\)
0.590633 + 0.806940i \(0.298878\pi\)
\(942\) −1776.00 −1.88535
\(943\) 0 0
\(944\) −1221.88 −1.29436
\(945\) −1680.00 −1.77778
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 1470.78 1.55146
\(949\) 0 0
\(950\) −797.616 −0.839596
\(951\) 0 0
\(952\) 0 0
\(953\) 1010.00 1.05981 0.529906 0.848057i \(-0.322227\pi\)
0.529906 + 0.848057i \(0.322227\pi\)
\(954\) 0 0
\(955\) −1103.09 −1.15506
\(956\) 1688.00 1.76569
\(957\) 0 0
\(958\) 0 0
\(959\) −350.000 −0.364964
\(960\) −1536.00 −1.60000
\(961\) 961.000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2664.38 2.76101
\(966\) 395.980 0.409917
\(967\) 1430.00 1.47880 0.739400 0.673266i \(-0.235109\pi\)
0.739400 + 0.673266i \(0.235109\pi\)
\(968\) 968.000 1.00000
\(969\) 0 0
\(970\) 0 0
\(971\) −8.48528 −0.00873870 −0.00436935 0.999990i \(-0.501391\pi\)
−0.00436935 + 0.999990i \(0.501391\pi\)
\(972\) 214.960 0.221153
\(973\) −1841.31 −1.89240
\(974\) 940.000 0.965092
\(975\) 3384.00 3.47077
\(976\) −135.765 −0.139103
\(977\) −1630.00 −1.66837 −0.834186 0.551483i \(-0.814062\pi\)
−0.834186 + 0.551483i \(0.814062\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1663.12 −1.69706
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −864.000 −0.874494
\(989\) 0 0
\(990\) 0 0
\(991\) 610.000 0.615540 0.307770 0.951461i \(-0.400417\pi\)
0.307770 + 0.951461i \(0.400417\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 1540.00 1.54930
\(995\) 0 0
\(996\) 288.000 0.289157
\(997\) −1977.07 −1.98302 −0.991510 0.130032i \(-0.958492\pi\)
−0.991510 + 0.130032i \(0.958492\pi\)
\(998\) 0 0
\(999\) 0 0
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 56.3.h.c.13.2 yes 2
3.2 odd 2 504.3.l.a.181.2 2
4.3 odd 2 224.3.h.c.209.1 2
7.2 even 3 392.3.j.a.325.1 4
7.3 odd 6 392.3.j.a.117.2 4
7.4 even 3 392.3.j.a.117.1 4
7.5 odd 6 392.3.j.a.325.2 4
7.6 odd 2 inner 56.3.h.c.13.1 2
8.3 odd 2 224.3.h.c.209.2 2
8.5 even 2 inner 56.3.h.c.13.1 2
12.11 even 2 2016.3.l.c.433.2 2
21.20 even 2 504.3.l.a.181.1 2
24.5 odd 2 504.3.l.a.181.1 2
24.11 even 2 2016.3.l.c.433.1 2
28.27 even 2 224.3.h.c.209.2 2
56.5 odd 6 392.3.j.a.325.1 4
56.13 odd 2 CM 56.3.h.c.13.2 yes 2
56.27 even 2 224.3.h.c.209.1 2
56.37 even 6 392.3.j.a.325.2 4
56.45 odd 6 392.3.j.a.117.1 4
56.53 even 6 392.3.j.a.117.2 4
84.83 odd 2 2016.3.l.c.433.1 2
168.83 odd 2 2016.3.l.c.433.2 2
168.125 even 2 504.3.l.a.181.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.h.c.13.1 2 7.6 odd 2 inner
56.3.h.c.13.1 2 8.5 even 2 inner
56.3.h.c.13.2 yes 2 1.1 even 1 trivial
56.3.h.c.13.2 yes 2 56.13 odd 2 CM
224.3.h.c.209.1 2 4.3 odd 2
224.3.h.c.209.1 2 56.27 even 2
224.3.h.c.209.2 2 8.3 odd 2
224.3.h.c.209.2 2 28.27 even 2
392.3.j.a.117.1 4 7.4 even 3
392.3.j.a.117.1 4 56.45 odd 6
392.3.j.a.117.2 4 7.3 odd 6
392.3.j.a.117.2 4 56.53 even 6
392.3.j.a.325.1 4 7.2 even 3
392.3.j.a.325.1 4 56.5 odd 6
392.3.j.a.325.2 4 7.5 odd 6
392.3.j.a.325.2 4 56.37 even 6
504.3.l.a.181.1 2 21.20 even 2
504.3.l.a.181.1 2 24.5 odd 2
504.3.l.a.181.2 2 3.2 odd 2
504.3.l.a.181.2 2 168.125 even 2
2016.3.l.c.433.1 2 24.11 even 2
2016.3.l.c.433.1 2 84.83 odd 2
2016.3.l.c.433.2 2 12.11 even 2
2016.3.l.c.433.2 2 168.83 odd 2