Properties

Label 2240.2.a.ba.1.1
Level $2240$
Weight $2$
Character 2240.1
Self dual yes
Analytic conductor $17.886$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(1,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2240.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.00000 q^{15} -3.00000 q^{17} +8.00000 q^{19} -3.00000 q^{21} -4.00000 q^{23} +1.00000 q^{25} +9.00000 q^{27} -3.00000 q^{29} +6.00000 q^{31} +3.00000 q^{33} -1.00000 q^{35} +8.00000 q^{37} +3.00000 q^{39} +10.0000 q^{41} -12.0000 q^{43} +6.00000 q^{45} -3.00000 q^{47} +1.00000 q^{49} -9.00000 q^{51} -12.0000 q^{53} +1.00000 q^{55} +24.0000 q^{57} -2.00000 q^{61} -6.00000 q^{63} +1.00000 q^{65} +4.00000 q^{67} -12.0000 q^{69} +12.0000 q^{71} -10.0000 q^{73} +3.00000 q^{75} -1.00000 q^{77} +13.0000 q^{79} +9.00000 q^{81} -3.00000 q^{85} -9.00000 q^{87} -6.00000 q^{89} -1.00000 q^{91} +18.0000 q^{93} +8.00000 q^{95} +5.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −9.00000 −1.26025
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 24.0000 3.17888
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −6.00000 −0.755929
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 3.00000 0.346410
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 13.0000 1.46261 0.731307 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) −9.00000 −0.964901
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 18.0000 1.86651
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 5.00000 0.507673 0.253837 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) −4.00000 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 0 0
\(103\) 13.0000 1.28093 0.640464 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) 24.0000 2.27798
\(112\) 0 0
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 6.00000 0.554700
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 30.0000 2.70501
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) −36.0000 −3.16962
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) 9.00000 0.774597
\(136\) 0 0
\(137\) 22.0000 1.87959 0.939793 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) −9.00000 −0.757937
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) −18.0000 −1.45521
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) −36.0000 −2.85499
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) 0 0
\(165\) 3.00000 0.233550
\(166\) 0 0
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 48.0000 3.67065
\(172\) 0 0
\(173\) 11.0000 0.836315 0.418157 0.908375i \(-0.362676\pi\)
0.418157 + 0.908375i \(0.362676\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) −3.00000 −0.219382
\(188\) 0 0
\(189\) −9.00000 −0.654654
\(190\) 0 0
\(191\) −1.00000 −0.0723575 −0.0361787 0.999345i \(-0.511519\pi\)
−0.0361787 + 0.999345i \(0.511519\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 0 0
\(195\) 3.00000 0.214834
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) 3.00000 0.210559
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) 0 0
\(207\) −24.0000 −1.66812
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 0 0
\(213\) 36.0000 2.46668
\(214\) 0 0
\(215\) −12.0000 −0.818393
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 0 0
\(219\) −30.0000 −2.02721
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) 0 0
\(223\) 13.0000 0.870544 0.435272 0.900299i \(-0.356652\pi\)
0.435272 + 0.900299i \(0.356652\pi\)
\(224\) 0 0
\(225\) 6.00000 0.400000
\(226\) 0 0
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) −8.00000 −0.524097 −0.262049 0.965055i \(-0.584398\pi\)
−0.262049 + 0.965055i \(0.584398\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) 0 0
\(237\) 39.0000 2.53332
\(238\) 0 0
\(239\) −27.0000 −1.74648 −0.873242 0.487286i \(-0.837987\pi\)
−0.873242 + 0.487286i \(0.837987\pi\)
\(240\) 0 0
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 10.0000 0.631194 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) −9.00000 −0.563602
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 0 0
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 0 0
\(273\) −3.00000 −0.181568
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 0 0
\(279\) 36.0000 2.15526
\(280\) 0 0
\(281\) 15.0000 0.894825 0.447412 0.894328i \(-0.352346\pi\)
0.447412 + 0.894328i \(0.352346\pi\)
\(282\) 0 0
\(283\) 29.0000 1.72387 0.861936 0.507018i \(-0.169252\pi\)
0.861936 + 0.507018i \(0.169252\pi\)
\(284\) 0 0
\(285\) 24.0000 1.42164
\(286\) 0 0
\(287\) −10.0000 −0.590281
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 15.0000 0.879316
\(292\) 0 0
\(293\) −33.0000 −1.92788 −0.963940 0.266119i \(-0.914259\pi\)
−0.963940 + 0.266119i \(0.914259\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 9.00000 0.522233
\(298\) 0 0
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) −12.0000 −0.689382
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 0 0
\(309\) 39.0000 2.21863
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 0 0
\(315\) −6.00000 −0.338062
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) −3.00000 −0.167968
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) −3.00000 −0.165900
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) 48.0000 2.63038
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 0 0
\(339\) −48.0000 −2.60700
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −12.0000 −0.646058
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) 9.00000 0.480384
\(352\) 0 0
\(353\) 25.0000 1.33062 0.665308 0.746569i \(-0.268300\pi\)
0.665308 + 0.746569i \(0.268300\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 0 0
\(357\) 9.00000 0.476331
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) −30.0000 −1.57459
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 33.0000 1.72259 0.861293 0.508109i \(-0.169655\pi\)
0.861293 + 0.508109i \(0.169655\pi\)
\(368\) 0 0
\(369\) 60.0000 3.12348
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) −48.0000 −2.45911
\(382\) 0 0
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −72.0000 −3.65997
\(388\) 0 0
\(389\) 19.0000 0.963338 0.481669 0.876353i \(-0.340031\pi\)
0.481669 + 0.876353i \(0.340031\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) −48.0000 −2.42128
\(394\) 0 0
\(395\) 13.0000 0.654101
\(396\) 0 0
\(397\) 39.0000 1.95735 0.978677 0.205406i \(-0.0658513\pi\)
0.978677 + 0.205406i \(0.0658513\pi\)
\(398\) 0 0
\(399\) −24.0000 −1.20150
\(400\) 0 0
\(401\) 13.0000 0.649189 0.324595 0.945853i \(-0.394772\pi\)
0.324595 + 0.945853i \(0.394772\pi\)
\(402\) 0 0
\(403\) 6.00000 0.298881
\(404\) 0 0
\(405\) 9.00000 0.447214
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 34.0000 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 0 0
\(411\) 66.0000 3.25554
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 42.0000 2.05675
\(418\) 0 0
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) 35.0000 1.70580 0.852898 0.522078i \(-0.174843\pi\)
0.852898 + 0.522078i \(0.174843\pi\)
\(422\) 0 0
\(423\) −18.0000 −0.875190
\(424\) 0 0
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) 3.00000 0.144841
\(430\) 0 0
\(431\) −5.00000 −0.240842 −0.120421 0.992723i \(-0.538424\pi\)
−0.120421 + 0.992723i \(0.538424\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) 0 0
\(437\) −32.0000 −1.53077
\(438\) 0 0
\(439\) −34.0000 −1.62273 −0.811366 0.584539i \(-0.801275\pi\)
−0.811366 + 0.584539i \(0.801275\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) −28.0000 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) −30.0000 −1.41895
\(448\) 0 0
\(449\) −33.0000 −1.55737 −0.778683 0.627417i \(-0.784112\pi\)
−0.778683 + 0.627417i \(0.784112\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 0 0
\(453\) 15.0000 0.704761
\(454\) 0 0
\(455\) −1.00000 −0.0468807
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 0 0
\(459\) −27.0000 −1.26025
\(460\) 0 0
\(461\) −16.0000 −0.745194 −0.372597 0.927993i \(-0.621533\pi\)
−0.372597 + 0.927993i \(0.621533\pi\)
\(462\) 0 0
\(463\) −2.00000 −0.0929479 −0.0464739 0.998920i \(-0.514798\pi\)
−0.0464739 + 0.998920i \(0.514798\pi\)
\(464\) 0 0
\(465\) 18.0000 0.834730
\(466\) 0 0
\(467\) −11.0000 −0.509019 −0.254510 0.967070i \(-0.581914\pi\)
−0.254510 + 0.967070i \(0.581914\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −30.0000 −1.38233
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) −72.0000 −3.29665
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) 12.0000 0.546019
\(484\) 0 0
\(485\) 5.00000 0.227038
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 0 0
\(489\) −42.0000 −1.89931
\(490\) 0 0
\(491\) −25.0000 −1.12823 −0.564117 0.825695i \(-0.690783\pi\)
−0.564117 + 0.825695i \(0.690783\pi\)
\(492\) 0 0
\(493\) 9.00000 0.405340
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 25.0000 1.11915 0.559577 0.828778i \(-0.310964\pi\)
0.559577 + 0.828778i \(0.310964\pi\)
\(500\) 0 0
\(501\) 63.0000 2.81463
\(502\) 0 0
\(503\) −21.0000 −0.936344 −0.468172 0.883637i \(-0.655087\pi\)
−0.468172 + 0.883637i \(0.655087\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) 0 0
\(507\) −36.0000 −1.59882
\(508\) 0 0
\(509\) −42.0000 −1.86162 −0.930809 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 0 0
\(513\) 72.0000 3.17888
\(514\) 0 0
\(515\) 13.0000 0.572848
\(516\) 0 0
\(517\) −3.00000 −0.131940
\(518\) 0 0
\(519\) 33.0000 1.44854
\(520\) 0 0
\(521\) 4.00000 0.175243 0.0876216 0.996154i \(-0.472073\pi\)
0.0876216 + 0.996154i \(0.472073\pi\)
\(522\) 0 0
\(523\) 12.0000 0.524723 0.262362 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) 0 0
\(525\) −3.00000 −0.130931
\(526\) 0 0
\(527\) −18.0000 −0.784092
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 10.0000 0.433148
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −35.0000 −1.50477 −0.752384 0.658725i \(-0.771096\pi\)
−0.752384 + 0.658725i \(0.771096\pi\)
\(542\) 0 0
\(543\) −36.0000 −1.54491
\(544\) 0 0
\(545\) −1.00000 −0.0428353
\(546\) 0 0
\(547\) 6.00000 0.256541 0.128271 0.991739i \(-0.459057\pi\)
0.128271 + 0.991739i \(0.459057\pi\)
\(548\) 0 0
\(549\) −12.0000 −0.512148
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) −13.0000 −0.552816
\(554\) 0 0
\(555\) 24.0000 1.01874
\(556\) 0 0
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 0 0
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) −16.0000 −0.673125
\(566\) 0 0
\(567\) −9.00000 −0.377964
\(568\) 0 0
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) −3.00000 −0.125327
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 27.0000 1.12402 0.562012 0.827129i \(-0.310027\pi\)
0.562012 + 0.827129i \(0.310027\pi\)
\(578\) 0 0
\(579\) −54.0000 −2.24416
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) 0 0
\(585\) 6.00000 0.248069
\(586\) 0 0
\(587\) −20.0000 −0.825488 −0.412744 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) 0 0
\(589\) 48.0000 1.97781
\(590\) 0 0
\(591\) −54.0000 −2.22126
\(592\) 0 0
\(593\) 43.0000 1.76580 0.882899 0.469563i \(-0.155588\pi\)
0.882899 + 0.469563i \(0.155588\pi\)
\(594\) 0 0
\(595\) 3.00000 0.122988
\(596\) 0 0
\(597\) −12.0000 −0.491127
\(598\) 0 0
\(599\) −21.0000 −0.858037 −0.429018 0.903296i \(-0.641140\pi\)
−0.429018 + 0.903296i \(0.641140\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) 24.0000 0.977356
\(604\) 0 0
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) −13.0000 −0.527654 −0.263827 0.964570i \(-0.584985\pi\)
−0.263827 + 0.964570i \(0.584985\pi\)
\(608\) 0 0
\(609\) 9.00000 0.364698
\(610\) 0 0
\(611\) −3.00000 −0.121367
\(612\) 0 0
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) 0 0
\(615\) 30.0000 1.20972
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) −36.0000 −1.44463
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 24.0000 0.958468
\(628\) 0 0
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) 19.0000 0.756378 0.378189 0.925728i \(-0.376547\pi\)
0.378189 + 0.925728i \(0.376547\pi\)
\(632\) 0 0
\(633\) −3.00000 −0.119239
\(634\) 0 0
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 72.0000 2.84828
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −5.00000 −0.197181 −0.0985904 0.995128i \(-0.531433\pi\)
−0.0985904 + 0.995128i \(0.531433\pi\)
\(644\) 0 0
\(645\) −36.0000 −1.41750
\(646\) 0 0
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −18.0000 −0.705476
\(652\) 0 0
\(653\) 20.0000 0.782660 0.391330 0.920250i \(-0.372015\pi\)
0.391330 + 0.920250i \(0.372015\pi\)
\(654\) 0 0
\(655\) −16.0000 −0.625172
\(656\) 0 0
\(657\) −60.0000 −2.34082
\(658\) 0 0
\(659\) 29.0000 1.12968 0.564840 0.825201i \(-0.308938\pi\)
0.564840 + 0.825201i \(0.308938\pi\)
\(660\) 0 0
\(661\) −20.0000 −0.777910 −0.388955 0.921257i \(-0.627164\pi\)
−0.388955 + 0.921257i \(0.627164\pi\)
\(662\) 0 0
\(663\) −9.00000 −0.349531
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 12.0000 0.464642
\(668\) 0 0
\(669\) 39.0000 1.50783
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 9.00000 0.346410
\(676\) 0 0
\(677\) −11.0000 −0.422764 −0.211382 0.977403i \(-0.567796\pi\)
−0.211382 + 0.977403i \(0.567796\pi\)
\(678\) 0 0
\(679\) −5.00000 −0.191882
\(680\) 0 0
\(681\) 9.00000 0.344881
\(682\) 0 0
\(683\) −14.0000 −0.535695 −0.267848 0.963461i \(-0.586312\pi\)
−0.267848 + 0.963461i \(0.586312\pi\)
\(684\) 0 0
\(685\) 22.0000 0.840577
\(686\) 0 0
\(687\) −30.0000 −1.14457
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) 0 0
\(693\) −6.00000 −0.227921
\(694\) 0 0
\(695\) 14.0000 0.531050
\(696\) 0 0
\(697\) −30.0000 −1.13633
\(698\) 0 0
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) −3.00000 −0.113308 −0.0566542 0.998394i \(-0.518043\pi\)
−0.0566542 + 0.998394i \(0.518043\pi\)
\(702\) 0 0
\(703\) 64.0000 2.41381
\(704\) 0 0
\(705\) −9.00000 −0.338960
\(706\) 0 0
\(707\) 4.00000 0.150435
\(708\) 0 0
\(709\) 5.00000 0.187779 0.0938895 0.995583i \(-0.470070\pi\)
0.0938895 + 0.995583i \(0.470070\pi\)
\(710\) 0 0
\(711\) 78.0000 2.92523
\(712\) 0 0
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 1.00000 0.0373979
\(716\) 0 0
\(717\) −81.0000 −3.02500
\(718\) 0 0
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) −13.0000 −0.484145
\(722\) 0 0
\(723\) −84.0000 −3.12399
\(724\) 0 0
\(725\) −3.00000 −0.111417
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 36.0000 1.33151
\(732\) 0 0
\(733\) −19.0000 −0.701781 −0.350891 0.936416i \(-0.614121\pi\)
−0.350891 + 0.936416i \(0.614121\pi\)
\(734\) 0 0
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) 41.0000 1.50821 0.754105 0.656754i \(-0.228071\pi\)
0.754105 + 0.656754i \(0.228071\pi\)
\(740\) 0 0
\(741\) 24.0000 0.881662
\(742\) 0 0
\(743\) −32.0000 −1.17397 −0.586983 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 41.0000 1.49611 0.748056 0.663636i \(-0.230988\pi\)
0.748056 + 0.663636i \(0.230988\pi\)
\(752\) 0 0
\(753\) 30.0000 1.09326
\(754\) 0 0
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) −20.0000 −0.726912 −0.363456 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) 0 0
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 1.00000 0.0362024
\(764\) 0 0
\(765\) −18.0000 −0.650791
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) −42.0000 −1.51259
\(772\) 0 0
\(773\) −27.0000 −0.971123 −0.485561 0.874203i \(-0.661385\pi\)
−0.485561 + 0.874203i \(0.661385\pi\)
\(774\) 0 0
\(775\) 6.00000 0.215526
\(776\) 0 0
\(777\) −24.0000 −0.860995
\(778\) 0 0
\(779\) 80.0000 2.86630
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) −27.0000 −0.964901
\(784\) 0 0
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) −29.0000 −1.03374 −0.516869 0.856064i \(-0.672903\pi\)
−0.516869 + 0.856064i \(0.672903\pi\)
\(788\) 0 0
\(789\) 54.0000 1.92245
\(790\) 0 0
\(791\) 16.0000 0.568895
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) −36.0000 −1.27679
\(796\) 0 0
\(797\) −5.00000 −0.177109 −0.0885545 0.996071i \(-0.528225\pi\)
−0.0885545 + 0.996071i \(0.528225\pi\)
\(798\) 0 0
\(799\) 9.00000 0.318397
\(800\) 0 0
\(801\) −36.0000 −1.27200
\(802\) 0 0
\(803\) −10.0000 −0.352892
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) 49.0000 1.72275 0.861374 0.507971i \(-0.169604\pi\)
0.861374 + 0.507971i \(0.169604\pi\)
\(810\) 0 0
\(811\) −50.0000 −1.75574 −0.877869 0.478901i \(-0.841035\pi\)
−0.877869 + 0.478901i \(0.841035\pi\)
\(812\) 0 0
\(813\) 6.00000 0.210429
\(814\) 0 0
\(815\) −14.0000 −0.490399
\(816\) 0 0
\(817\) −96.0000 −3.35861
\(818\) 0 0
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 43.0000 1.50071 0.750355 0.661035i \(-0.229882\pi\)
0.750355 + 0.661035i \(0.229882\pi\)
\(822\) 0 0
\(823\) 26.0000 0.906303 0.453152 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) 0 0
\(825\) 3.00000 0.104447
\(826\) 0 0
\(827\) 30.0000 1.04320 0.521601 0.853189i \(-0.325335\pi\)
0.521601 + 0.853189i \(0.325335\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 66.0000 2.28951
\(832\) 0 0
\(833\) −3.00000 −0.103944
\(834\) 0 0
\(835\) 21.0000 0.726735
\(836\) 0 0
\(837\) 54.0000 1.86651
\(838\) 0 0
\(839\) −4.00000 −0.138095 −0.0690477 0.997613i \(-0.521996\pi\)
−0.0690477 + 0.997613i \(0.521996\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 45.0000 1.54988
\(844\) 0 0
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) 0 0
\(849\) 87.0000 2.98583
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 0 0
\(855\) 48.0000 1.64157
\(856\) 0 0
\(857\) −2.00000 −0.0683187 −0.0341593 0.999416i \(-0.510875\pi\)
−0.0341593 + 0.999416i \(0.510875\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) −30.0000 −1.02240
\(862\) 0 0
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) 11.0000 0.374011
\(866\) 0 0
\(867\) −24.0000 −0.815083
\(868\) 0 0
\(869\) 13.0000 0.440995
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) 30.0000 1.01535
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 20.0000 0.675352 0.337676 0.941262i \(-0.390359\pi\)
0.337676 + 0.941262i \(0.390359\pi\)
\(878\) 0 0
\(879\) −99.0000 −3.33919
\(880\) 0 0
\(881\) 24.0000 0.808581 0.404290 0.914631i \(-0.367519\pi\)
0.404290 + 0.914631i \(0.367519\pi\)
\(882\) 0 0
\(883\) 38.0000 1.27880 0.639401 0.768874i \(-0.279182\pi\)
0.639401 + 0.768874i \(0.279182\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 9.00000 0.301511
\(892\) 0 0
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −12.0000 −0.400668
\(898\) 0 0
\(899\) −18.0000 −0.600334
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 36.0000 1.19800
\(904\) 0 0
\(905\) −12.0000 −0.398893
\(906\) 0 0
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) 0 0
\(909\) −24.0000 −0.796030
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −6.00000 −0.198354
\(916\) 0 0
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) 9.00000 0.296883 0.148441 0.988921i \(-0.452574\pi\)
0.148441 + 0.988921i \(0.452574\pi\)
\(920\) 0 0
\(921\) −21.0000 −0.691974
\(922\) 0 0
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 0 0
\(927\) 78.0000 2.56186
\(928\) 0 0
\(929\) 16.0000 0.524943 0.262471 0.964940i \(-0.415462\pi\)
0.262471 + 0.964940i \(0.415462\pi\)
\(930\) 0 0
\(931\) 8.00000 0.262189
\(932\) 0 0
\(933\) −24.0000 −0.785725
\(934\) 0 0
\(935\) −3.00000 −0.0981105
\(936\) 0 0
\(937\) 9.00000 0.294017 0.147009 0.989135i \(-0.453036\pi\)
0.147009 + 0.989135i \(0.453036\pi\)
\(938\) 0 0
\(939\) −3.00000 −0.0979013
\(940\) 0 0
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) 0 0
\(943\) −40.0000 −1.30258
\(944\) 0 0
\(945\) −9.00000 −0.292770
\(946\) 0 0
\(947\) −38.0000 −1.23483 −0.617417 0.786636i \(-0.711821\pi\)
−0.617417 + 0.786636i \(0.711821\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 0 0
\(955\) −1.00000 −0.0323592
\(956\) 0 0
\(957\) −9.00000 −0.290929
\(958\) 0 0
\(959\) −22.0000 −0.710417
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) −36.0000 −1.16008
\(964\) 0 0
\(965\) −18.0000 −0.579441
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 0 0
\(969\) −72.0000 −2.31297
\(970\) 0 0
\(971\) −34.0000 −1.09111 −0.545556 0.838074i \(-0.683681\pi\)
−0.545556 + 0.838074i \(0.683681\pi\)
\(972\) 0 0
\(973\) −14.0000 −0.448819
\(974\) 0 0
\(975\) 3.00000 0.0960769
\(976\) 0 0
\(977\) 32.0000 1.02377 0.511885 0.859054i \(-0.328947\pi\)
0.511885 + 0.859054i \(0.328947\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 0 0
\(983\) −9.00000 −0.287055 −0.143528 0.989646i \(-0.545845\pi\)
−0.143528 + 0.989646i \(0.545845\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 9.00000 0.286473
\(988\) 0 0
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) 24.0000 0.762385 0.381193 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(992\) 0 0
\(993\) 12.0000 0.380808
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) 47.0000 1.48850 0.744252 0.667898i \(-0.232806\pi\)
0.744252 + 0.667898i \(0.232806\pi\)
\(998\) 0 0
\(999\) 72.0000 2.27798
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 2240.2.a.ba.1.1 1
4.3 odd 2 2240.2.a.d.1.1 1
8.3 odd 2 1120.2.a.o.1.1 yes 1
8.5 even 2 1120.2.a.a.1.1 1
40.19 odd 2 5600.2.a.a.1.1 1
40.29 even 2 5600.2.a.v.1.1 1
56.13 odd 2 7840.2.a.z.1.1 1
56.27 even 2 7840.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1120.2.a.a.1.1 1 8.5 even 2
1120.2.a.o.1.1 yes 1 8.3 odd 2
2240.2.a.d.1.1 1 4.3 odd 2
2240.2.a.ba.1.1 1 1.1 even 1 trivial
5600.2.a.a.1.1 1 40.19 odd 2
5600.2.a.v.1.1 1 40.29 even 2
7840.2.a.b.1.1 1 56.27 even 2
7840.2.a.z.1.1 1 56.13 odd 2