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Properties

Label 2205.2.a.e.1.1
Level $2205$
Weight $2$
Character 2205.1
Self dual yes
Analytic conductor $17.607$
Analytic rank $1$
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] = [2205,2,Mod(1,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.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: \(17.6070136457\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2205.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} -1.00000 q^{5} +3.00000 q^{11} -5.00000 q^{13} +4.00000 q^{16} +3.00000 q^{17} -2.00000 q^{19} +2.00000 q^{20} +6.00000 q^{23} +1.00000 q^{25} -3.00000 q^{29} +4.00000 q^{31} +2.00000 q^{37} -12.0000 q^{41} -10.0000 q^{43} -6.00000 q^{44} +9.00000 q^{47} +10.0000 q^{52} -12.0000 q^{53} -3.00000 q^{55} -8.00000 q^{61} -8.00000 q^{64} +5.00000 q^{65} -4.00000 q^{67} -6.00000 q^{68} -2.00000 q^{73} +4.00000 q^{76} -1.00000 q^{79} -4.00000 q^{80} +12.0000 q^{83} -3.00000 q^{85} -12.0000 q^{89} -12.0000 q^{92} +2.00000 q^{95} +1.00000 q^{97} +O(q^{100})\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 − 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 10.0000 1.38675
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 5.00000 0.620174
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −4.00000 −0.447214
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −12.0000 −1.25109
\(93\) 0 0
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.00000 −0.200000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −5.00000 −0.492665 −0.246332 − 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 − 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 − 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) −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
\(129\) 0 0
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 − 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 − 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −15.0000 −1.25436
\(144\) 0 0
\(145\) 3.00000 0.249136
\(146\) 0 0
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −1.00000 −0.0813788 −0.0406894 − 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 − 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 24.0000 1.87409
\(165\) 0 0
\(166\) 0 0
\(167\) −3.00000 −0.232147 −0.116073 − 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 20.0000 1.52499
\(173\) −9.00000 −0.684257 −0.342129 − 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 12.0000 0.904534
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 − 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) −18.0000 −1.31278
\(189\) 0 0
\(190\) 0 0
\(191\) −9.00000 −0.651217 −0.325609 − 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 − 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) −20.0000 −1.38675
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 − 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 24.0000 1.64833
\(213\) 0 0
\(214\) 0 0
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 6.00000 0.404520
\(221\) −15.0000 −1.00901
\(222\) 0 0
\(223\) 19.0000 1.27233 0.636167 − 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.00000 −0.199117 −0.0995585 − 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 − 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 − 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.0000 1.35838 0.679189 − 0.733964i \(-0.262332\pi\)
0.679189 + 0.733964i \(0.262332\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 16.0000 1.02430
\(245\) 0 0
\(246\) 0 0
\(247\) 10.0000 0.636285
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 − 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 30.0000 1.87135 0.935674 − 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −10.0000 −0.620174
\(261\) 0 0
\(262\) 0 0
\(263\) −6.00000 −0.369976 −0.184988 − 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 12.0000 0.727607
\(273\) 0 0
\(274\) 0 0
\(275\) 3.00000 0.180907
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.00000 −0.178965 −0.0894825 − 0.995988i \(-0.528521\pi\)
−0.0894825 + 0.995988i \(0.528521\pi\)
\(282\) 0 0
\(283\) 13.0000 0.772770 0.386385 − 0.922338i \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) −21.0000 −1.22683 −0.613417 − 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −30.0000 −1.73494
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −8.00000 −0.458831
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) −11.0000 −0.627803 −0.313902 − 0.949456i \(-0.601636\pi\)
−0.313902 + 0.949456i \(0.601636\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 − 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 19.0000 1.07394 0.536972 − 0.843600i \(-0.319568\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −9.00000 −0.503903
\(320\) 8.00000 0.447214
\(321\) 0 0
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) −5.00000 −0.277350
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 − 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −24.0000 −1.31717
\(333\) 0 0
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 6.00000 0.325396
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.0000 0.966291 0.483145 − 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 − 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 15.0000 0.798369 0.399185 − 0.916871i \(-0.369293\pi\)
0.399185 + 0.916871i \(0.369293\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 24.0000 1.27200
\(357\) 0 0
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) −17.0000 −0.887393 −0.443696 − 0.896177i \(-0.646333\pi\)
−0.443696 + 0.896177i \(0.646333\pi\)
\(368\) 24.0000 1.25109
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 15.0000 0.772539
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 − 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 3.00000 0.152106 0.0760530 − 0.997104i \(-0.475768\pi\)
0.0760530 + 0.997104i \(0.475768\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.00000 0.0503155
\(396\) 0 0
\(397\) 25.0000 1.25471 0.627357 − 0.778732i \(-0.284137\pi\)
0.627357 + 0.778732i \(0.284137\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 15.0000 0.749064 0.374532 − 0.927214i \(-0.377803\pi\)
0.374532 + 0.927214i \(0.377803\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 − 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 10.0000 0.492665
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 17.0000 0.828529 0.414265 − 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.00000 0.145521
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) −21.0000 −1.01153 −0.505767 − 0.862670i \(-0.668791\pi\)
−0.505767 + 0.862670i \(0.668791\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 − 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.0000 0.855206 0.427603 − 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.00000 0.424736 0.212368 − 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) 12.0000 0.564433
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 12.0000 0.559503
\(461\) −24.0000 −1.11779 −0.558896 − 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 − 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) −12.0000 −0.557086
\(465\) 0 0
\(466\) 0 0
\(467\) 15.0000 0.694117 0.347059 − 0.937843i \(-0.387180\pi\)
0.347059 + 0.937843i \(0.387180\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −30.0000 −1.37940
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −30.0000 −1.37073 −0.685367 − 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) −10.0000 −0.455961
\(482\) 0 0
\(483\) 0 0
\(484\) 4.00000 0.181818
\(485\) −1.00000 −0.0454077
\(486\) 0 0
\(487\) 38.0000 1.72194 0.860972 − 0.508652i \(-0.169856\pi\)
0.860972 + 0.508652i \(0.169856\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 − 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) 0 0
\(496\) 16.0000 0.718421
\(497\) 0 0
\(498\) 0 0
\(499\) −31.0000 −1.38775 −0.693875 − 0.720095i \(-0.744098\pi\)
−0.693875 + 0.720095i \(0.744098\pi\)
\(500\) 2.00000 0.0894427
\(501\) 0 0
\(502\) 0 0
\(503\) 27.0000 1.20387 0.601935 − 0.798545i \(-0.294397\pi\)
0.601935 + 0.798545i \(0.294397\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 32.0000 1.41977
\(509\) −6.00000 −0.265945 −0.132973 − 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.00000 0.220326
\(516\) 0 0
\(517\) 27.0000 1.18746
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 − 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 60.0000 2.59889
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 − 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.00000 0.299847
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 − 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −24.0000 −1.02523
\(549\) 0 0
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 28.0000 1.18746
\(557\) 24.0000 1.01691 0.508456 − 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) 50.0000 2.11477
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 − 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 − 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 30.0000 1.25436
\(573\) 0 0
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) 7.00000 0.291414 0.145707 − 0.989328i \(-0.453454\pi\)
0.145707 + 0.989328i \(0.453454\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) 0 0
\(582\) 0 0
\(583\) −36.0000 −1.49097
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −24.0000 −0.990586 −0.495293 − 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) 8.00000 0.328798
\(593\) −39.0000 −1.60154 −0.800769 − 0.598973i \(-0.795576\pi\)
−0.800769 + 0.598973i \(0.795576\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) 0 0
\(598\) 0 0
\(599\) −45.0000 −1.83865 −0.919325 − 0.393499i \(-0.871265\pi\)
−0.919325 + 0.393499i \(0.871265\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 − 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 2.00000 0.0813788
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 13.0000 0.527654 0.263827 − 0.964570i \(-0.415015\pi\)
0.263827 + 0.964570i \(0.415015\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −45.0000 −1.82051
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 − 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 − 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 28.0000 1.11732
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 − 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 − 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) −41.0000 −1.61688 −0.808441 − 0.588577i \(-0.799688\pi\)
−0.808441 + 0.588577i \(0.799688\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 6.00000 0.234798 0.117399 − 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) −48.0000 −1.87409
\(657\) 0 0
\(658\) 0 0
\(659\) 15.0000 0.584317 0.292159 − 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 − 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −18.0000 −0.696963
\(668\) 6.00000 0.232147
\(669\) 0 0
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) −28.0000 −1.07932 −0.539660 − 0.841883i \(-0.681447\pi\)
−0.539660 + 0.841883i \(0.681447\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) 45.0000 1.72949 0.864745 − 0.502211i \(-0.167480\pi\)
0.864745 + 0.502211i \(0.167480\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 − 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) −40.0000 −1.52499
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 0 0
\(695\) 14.0000 0.531050
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.00000 0.339925 0.169963 − 0.985451i \(-0.445635\pi\)
0.169963 + 0.985451i \(0.445635\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) −24.0000 −0.904534
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 35.0000 1.31445 0.657226 − 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 15.0000 0.560968
\(716\) 24.0000 0.896922
\(717\) 0 0
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 40.0000 1.48659
\(725\) −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
\(729\) 0 0
\(730\) 0 0
\(731\) −30.0000 −1.10959
\(732\) 0 0
\(733\) 31.0000 1.14501 0.572506 − 0.819901i \(-0.305971\pi\)
0.572506 + 0.819901i \(0.305971\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12.0000 −0.442026
\(738\) 0 0
\(739\) −43.0000 −1.58178 −0.790890 − 0.611958i \(-0.790382\pi\)
−0.790890 + 0.611958i \(0.790382\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 − 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) 0 0
\(748\) −18.0000 −0.658145
\(749\) 0 0
\(750\) 0 0
\(751\) 23.0000 0.839282 0.419641 − 0.907690i \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(752\) 36.0000 1.31278
\(753\) 0 0
\(754\) 0 0
\(755\) 1.00000 0.0363937
\(756\) 0 0
\(757\) −16.0000 −0.581530 −0.290765 − 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 − 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 − 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.00000 0.287926
\(773\) 21.0000 0.755318 0.377659 − 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) −5.00000 −0.178231 −0.0891154 − 0.996021i \(-0.528404\pi\)
−0.0891154 + 0.996021i \(0.528404\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) 0 0
\(795\) 0 0
\(796\) −32.0000 −1.13421
\(797\) 15.0000 0.531327 0.265664 − 0.964066i \(-0.414409\pi\)
0.265664 + 0.964066i \(0.414409\pi\)
\(798\) 0 0
\(799\) 27.0000 0.955191
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.00000 −0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 15.0000 0.527372 0.263686 − 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 − 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.00000 −0.0700569
\(816\) 0 0
\(817\) 20.0000 0.699711
\(818\) 0 0
\(819\) 0 0
\(820\) −24.0000 −0.838116
\(821\) 27.0000 0.942306 0.471153 − 0.882051i \(-0.343838\pi\)
0.471153 + 0.882051i \(0.343838\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 − 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −54.0000 −1.87776 −0.938882 − 0.344239i \(-0.888137\pi\)
−0.938882 + 0.344239i \(0.888137\pi\)
\(828\) 0 0
\(829\) 52.0000 1.80603 0.903017 − 0.429604i \(-0.141347\pi\)
0.903017 + 0.429604i \(0.141347\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 40.0000 1.38675
\(833\) 0 0
\(834\) 0 0
\(835\) 3.00000 0.103819
\(836\) 12.0000 0.415029
\(837\) 0 0
\(838\) 0 0
\(839\) −42.0000 −1.45000 −0.725001 − 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 0 0
\(844\) 26.0000 0.894957
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 0 0
\(848\) −48.0000 −1.64833
\(849\) 0 0
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 − 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −30.0000 −1.02478 −0.512390 − 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 − 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) −20.0000 −0.681994
\(861\) 0 0
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) 9.00000 0.306009
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −3.00000 −0.101768
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 50.0000 1.68838 0.844190 − 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −12.0000 −0.404520
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 30.0000 1.00901
\(885\) 0 0
\(886\) 0 0
\(887\) −24.0000 −0.805841 −0.402921 − 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −38.0000 −1.27233
\(893\) −18.0000 −0.602347
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) 26.0000 0.863316 0.431658 − 0.902037i \(-0.357929\pi\)
0.431658 + 0.902037i \(0.357929\pi\)
\(908\) 6.00000 0.199117
\(909\) 0 0
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 − 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 0 0
\(915\) 0 0
\(916\) −8.00000 −0.264327
\(917\) 0 0
\(918\) 0 0
\(919\) 11.0000 0.362857 0.181428 − 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 − 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 48.0000 1.57229
\(933\) 0 0
\(934\) 0 0
\(935\) −9.00000 −0.294331
\(936\) 0 0
\(937\) −47.0000 −1.53542 −0.767712 − 0.640796i \(-0.778605\pi\)
−0.767712 + 0.640796i \(0.778605\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 18.0000 0.587095
\(941\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) −72.0000 −2.34464
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 9.00000 0.291233
\(956\) −42.0000 −1.35838
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) −20.0000 −0.644157
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 − 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −48.0000 −1.54039 −0.770197 − 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −32.0000 −1.02430
\(977\) −54.0000 −1.72761 −0.863807 − 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) 0 0
\(979\) −36.0000 −1.15056
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −21.0000 −0.669796 −0.334898 − 0.942254i \(-0.608702\pi\)
−0.334898 + 0.942254i \(0.608702\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −20.0000 −0.636285
\(989\) −60.0000 −1.90789
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 − 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) 37.0000 1.17180 0.585901 − 0.810383i \(-0.300741\pi\)
0.585901 + 0.810383i \(0.300741\pi\)
\(998\) 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 2205.2.a.e.1.1 1
3.2 odd 2 245.2.a.c.1.1 1
7.6 odd 2 315.2.a.b.1.1 1
12.11 even 2 3920.2.a.ba.1.1 1
15.2 even 4 1225.2.b.d.99.2 2
15.8 even 4 1225.2.b.d.99.1 2
15.14 odd 2 1225.2.a.e.1.1 1
21.2 odd 6 245.2.e.b.116.1 2
21.5 even 6 245.2.e.a.116.1 2
21.11 odd 6 245.2.e.b.226.1 2
21.17 even 6 245.2.e.a.226.1 2
21.20 even 2 35.2.a.a.1.1 ✓ 1
28.27 even 2 5040.2.a.v.1.1 1
35.13 even 4 1575.2.d.c.1324.1 2
35.27 even 4 1575.2.d.c.1324.2 2
35.34 odd 2 1575.2.a.f.1.1 1
84.83 odd 2 560.2.a.b.1.1 1
105.62 odd 4 175.2.b.a.99.1 2
105.83 odd 4 175.2.b.a.99.2 2
105.104 even 2 175.2.a.b.1.1 1
168.83 odd 2 2240.2.a.u.1.1 1
168.125 even 2 2240.2.a.k.1.1 1
231.230 odd 2 4235.2.a.c.1.1 1
273.272 even 2 5915.2.a.f.1.1 1
420.83 even 4 2800.2.g.l.449.1 2
420.167 even 4 2800.2.g.l.449.2 2
420.419 odd 2 2800.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.a.1.1 ✓ 1 21.20 even 2
175.2.a.b.1.1 1 105.104 even 2
175.2.b.a.99.1 2 105.62 odd 4
175.2.b.a.99.2 2 105.83 odd 4
245.2.a.c.1.1 1 3.2 odd 2
245.2.e.a.116.1 2 21.5 even 6
245.2.e.a.226.1 2 21.17 even 6
245.2.e.b.116.1 2 21.2 odd 6
245.2.e.b.226.1 2 21.11 odd 6
315.2.a.b.1.1 1 7.6 odd 2
560.2.a.b.1.1 1 84.83 odd 2
1225.2.a.e.1.1 1 15.14 odd 2
1225.2.b.d.99.1 2 15.8 even 4
1225.2.b.d.99.2 2 15.2 even 4
1575.2.a.f.1.1 1 35.34 odd 2
1575.2.d.c.1324.1 2 35.13 even 4
1575.2.d.c.1324.2 2 35.27 even 4
2205.2.a.e.1.1 1 1.1 even 1 trivial
2240.2.a.k.1.1 1 168.125 even 2
2240.2.a.u.1.1 1 168.83 odd 2
2800.2.a.z.1.1 1 420.419 odd 2
2800.2.g.l.449.1 2 420.83 even 4
2800.2.g.l.449.2 2 420.167 even 4
3920.2.a.ba.1.1 1 12.11 even 2
4235.2.a.c.1.1 1 231.230 odd 2
5040.2.a.v.1.1 1 28.27 even 2
5915.2.a.f.1.1 1 273.272 even 2