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Properties

Label 175.2.a.b.1.1
Level $175$
Weight $2$
Character 175.1
Self dual yes
Analytic conductor $1.397$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,2,Mod(1,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.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: \(1.39738203537\)
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 \(\chi\) \(=\) 175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} +2.00000 q^{12} -5.00000 q^{13} +4.00000 q^{16} -3.00000 q^{17} +2.00000 q^{19} +1.00000 q^{21} +6.00000 q^{23} +5.00000 q^{27} +2.00000 q^{28} +3.00000 q^{29} -4.00000 q^{31} +3.00000 q^{33} +4.00000 q^{36} -2.00000 q^{37} +5.00000 q^{39} -12.0000 q^{41} +10.0000 q^{43} +6.00000 q^{44} -9.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} +3.00000 q^{51} +10.0000 q^{52} -12.0000 q^{53} -2.00000 q^{57} +8.00000 q^{61} +2.00000 q^{63} -8.00000 q^{64} +4.00000 q^{67} +6.00000 q^{68} -6.00000 q^{69} -2.00000 q^{73} -4.00000 q^{76} +3.00000 q^{77} -1.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -2.00000 q^{84} -3.00000 q^{87} -12.0000 q^{89} +5.00000 q^{91} -12.0000 q^{92} +4.00000 q^{93} +1.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 \(0.5\pi\)
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 2.00000 0.577350
\(13\) −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 \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 2.00000 0.377964
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 4.00000 0.666667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 5.00000 0.800641
\(40\) 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 \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) −4.00000 −0.577350
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 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
\(55\) 0 0
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 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 \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 6.00000 0.727607
\(69\) −6.00000 −0.722315
\(70\) 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
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 0 0
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) −12.0000 −1.25109
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 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 \(0.593662\pi\)
\(108\) −10.0000 −0.962250
\(109\) −7.00000 −0.670478 −0.335239 − 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −4.00000 −0.377964
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 10.0000 0.924500
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 12.0000 1.08200
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) −6.00000 −0.522233
\(133\) −2.00000 −0.173422
\(134\) 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 \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) 0 0
\(143\) 15.0000 1.25436
\(144\) −8.00000 −0.666667
\(145\) 0 0
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 4.00000 0.328798
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −1.00000 −0.0813788 −0.0406894 − 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) −10.0000 −0.800641
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 − 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 24.0000 1.87409
\(165\) 0 0
\(166\) 0 0
\(167\) 3.00000 0.232147 0.116073 − 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) −20.0000 −1.52499
\(173\) 9.00000 0.684257 0.342129 − 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 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 \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 − 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) 18.0000 1.31278
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 9.00000 0.651217 0.325609 − 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(192\) 8.00000 0.577350
\(193\) 4.00000 0.287926 0.143963 − 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) −3.00000 −0.210559
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 0 0
\(207\) −12.0000 −0.834058
\(208\) −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
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 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 \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 4.00000 0.264906
\(229\) −4.00000 −0.264327 −0.132164 − 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 − 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.00000 0.0649570
\(238\) 0 0
\(239\) −21.0000 −1.35838 −0.679189 − 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) −16.0000 −1.02430
\(245\) 0 0
\(246\) 0 0
\(247\) −10.0000 −0.636285
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 − 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) −4.00000 −0.251976
\(253\) −18.0000 −1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −30.0000 −1.87135 −0.935674 − 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −6.00000 −0.369976 −0.184988 − 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) −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 \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −12.0000 −0.727607
\(273\) −5.00000 −0.302614
\(274\) 0 0
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 − 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 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
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) 4.00000 0.234082
\(293\) 21.0000 1.22683 0.613417 − 0.789760i \(-0.289795\pi\)
0.613417 + 0.789760i \(0.289795\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −15.0000 −0.870388
\(298\) 0 0
\(299\) −30.0000 −1.73494
\(300\) 0 0
\(301\) −10.0000 −0.576390
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) 0 0
\(307\) −11.0000 −0.627803 −0.313902 − 0.949456i \(-0.601636\pi\)
−0.313902 + 0.949456i \(0.601636\pi\)
\(308\) −6.00000 −0.341882
\(309\) 5.00000 0.284440
\(310\) 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
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) −2.00000 −0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) 7.00000 0.387101
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) 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
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) −14.0000 −0.762629 −0.381314 − 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.0000 0.966291 0.483145 − 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 6.00000 0.321634
\(349\) 26.0000 1.39175 0.695874 − 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −25.0000 −1.33440
\(352\) 0 0
\(353\) −15.0000 −0.798369 −0.399185 − 0.916871i \(-0.630707\pi\)
−0.399185 + 0.916871i \(0.630707\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 24.0000 1.27200
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 − 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) −10.0000 −0.524142
\(365\) 0 0
\(366\) 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
\(369\) 24.0000 1.24939
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) −8.00000 −0.414781
\(373\) 4.00000 0.207112 0.103556 − 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 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 \(0.328287\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) −12.0000 −0.613171 −0.306586 − 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −20.0000 −1.01666
\(388\) −2.00000 −0.101535
\(389\) −3.00000 −0.152106 −0.0760530 − 0.997104i \(-0.524232\pi\)
−0.0760530 + 0.997104i \(0.524232\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 6.00000 0.302660
\(394\) 0 0
\(395\) 0 0
\(396\) −12.0000 −0.603023
\(397\) 25.0000 1.25471 0.627357 − 0.778732i \(-0.284137\pi\)
0.627357 + 0.778732i \(0.284137\pi\)
\(398\) 0 0
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) −15.0000 −0.749064 −0.374532 − 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 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 \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 10.0000 0.492665
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −14.0000 −0.685583
\(418\) 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
\(423\) 18.0000 0.875190
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) 12.0000 0.580042
\(429\) −15.0000 −0.724207
\(430\) 0 0
\(431\) 21.0000 1.01153 0.505767 − 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) 20.0000 0.962250
\(433\) −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 \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) 18.0000 0.855206 0.427603 − 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 8.00000 0.377964
\(449\) −9.00000 −0.424736 −0.212368 − 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 36.0000 1.69517
\(452\) 12.0000 0.564433
\(453\) 1.00000 0.0469841
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 − 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 0 0
\(459\) −15.0000 −0.700140
\(460\) 0 0
\(461\) −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 \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 12.0000 0.557086
\(465\) 0 0
\(466\) 0 0
\(467\) −15.0000 −0.694117 −0.347059 − 0.937843i \(-0.612820\pi\)
−0.347059 + 0.937843i \(0.612820\pi\)
\(468\) −20.0000 −0.924500
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) −30.0000 −1.37940
\(474\) 0 0
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) 24.0000 1.09888
\(478\) 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
\(483\) 6.00000 0.273009
\(484\) 4.00000 0.181818
\(485\) 0 0
\(486\) 0 0
\(487\) −38.0000 −1.72194 −0.860972 − 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) 0 0
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) 15.0000 0.676941 0.338470 − 0.940977i \(-0.390091\pi\)
0.338470 + 0.940977i \(0.390091\pi\)
\(492\) −24.0000 −1.08200
\(493\) −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 \(0.744098\pi\)
\(500\) 0 0
\(501\) −3.00000 −0.134030
\(502\) 0 0
\(503\) −27.0000 −1.20387 −0.601935 − 0.798545i \(-0.705603\pi\)
−0.601935 + 0.798545i \(0.705603\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) −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
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) 10.0000 0.441511
\(514\) 0 0
\(515\) 0 0
\(516\) 20.0000 0.880451
\(517\) 27.0000 1.18746
\(518\) 0 0
\(519\) −9.00000 −0.395056
\(520\) 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
\(528\) 12.0000 0.522233
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) 60.0000 2.59889
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 − 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 0 0
\(543\) −20.0000 −0.858282
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 − 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −24.0000 −1.02523
\(549\) −16.0000 −0.682863
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 1.00000 0.0425243
\(554\) 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
\(561\) −9.00000 −0.379980
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 − 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) −18.0000 −0.757937
\(565\) 0 0
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 − 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 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
\(573\) −9.00000 −0.375980
\(574\) 0 0
\(575\) 0 0
\(576\) 16.0000 0.666667
\(577\) 7.00000 0.291414 0.145707 − 0.989328i \(-0.453454\pi\)
0.145707 + 0.989328i \(0.453454\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 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 \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 2.00000 0.0824786
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) 39.0000 1.60154 0.800769 − 0.598973i \(-0.204424\pi\)
0.800769 + 0.598973i \(0.204424\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 45.0000 1.83865 0.919325 − 0.393499i \(-0.128735\pi\)
0.919325 + 0.393499i \(0.128735\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 2.00000 0.0813788
\(605\) 0 0
\(606\) 0 0
\(607\) 13.0000 0.527654 0.263827 − 0.964570i \(-0.415015\pi\)
0.263827 + 0.964570i \(0.415015\pi\)
\(608\) 0 0
\(609\) 3.00000 0.121566
\(610\) 0 0
\(611\) 45.0000 1.82051
\(612\) −12.0000 −0.485071
\(613\) −2.00000 −0.0807792 −0.0403896 − 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 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 \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 30.0000 1.20386
\(622\) 0 0
\(623\) 12.0000 0.480770
\(624\) 20.0000 0.800641
\(625\) 0 0
\(626\) 0 0
\(627\) 6.00000 0.239617
\(628\) 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
\(633\) 13.0000 0.516704
\(634\) 0 0
\(635\) 0 0
\(636\) −24.0000 −0.951662
\(637\) −5.00000 −0.198107
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −41.0000 −1.61688 −0.808441 − 0.588577i \(-0.799688\pi\)
−0.808441 + 0.588577i \(0.799688\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 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
\(655\) 0 0
\(656\) −48.0000 −1.87409
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) −15.0000 −0.584317 −0.292159 − 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) 0 0
\(661\) 32.0000 1.24466 0.622328 − 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) 0 0
\(663\) −15.0000 −0.582552
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 18.0000 0.696963
\(668\) −6.00000 −0.232147
\(669\) −19.0000 −0.734582
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 − 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) −45.0000 −1.72949 −0.864745 − 0.502211i \(-0.832520\pi\)
−0.864745 + 0.502211i \(0.832520\pi\)
\(678\) 0 0
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) −3.00000 −0.114960
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 − 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 8.00000 0.305888
\(685\) 0 0
\(686\) 0 0
\(687\) 4.00000 0.152610
\(688\) 40.0000 1.52499
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −18.0000 −0.684257
\(693\) −6.00000 −0.227921
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 0 0
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) −9.00000 −0.339925 −0.169963 − 0.985451i \(-0.554365\pi\)
−0.169963 + 0.985451i \(0.554365\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 24.0000 0.904534
\(705\) 0 0
\(706\) 0 0
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 35.0000 1.31445 0.657226 − 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) 2.00000 0.0750059
\(712\) 0 0
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 21.0000 0.784259
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 5.00000 0.186210
\(722\) 0 0
\(723\) 10.0000 0.371904
\(724\) −40.0000 −1.48659
\(725\) 0 0
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −30.0000 −1.10959
\(732\) 16.0000 0.591377
\(733\) 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 \(0.790382\pi\)
\(740\) 0 0
\(741\) 10.0000 0.367359
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 − 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 24.0000 0.878114
\(748\) −18.0000 −0.658145
\(749\) 6.00000 0.219235
\(750\) 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
\(753\) −18.0000 −0.655956
\(754\) 0 0
\(755\) 0 0
\(756\) 10.0000 0.363696
\(757\) 16.0000 0.581530 0.290765 − 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) 0 0
\(759\) 18.0000 0.653359
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 − 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 7.00000 0.253417
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −16.0000 −0.577350
\(769\) 14.0000 0.504853 0.252426 − 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) −8.00000 −0.287926
\(773\) −21.0000 −0.755318 −0.377659 − 0.925945i \(-0.623271\pi\)
−0.377659 + 0.925945i \(0.623271\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −2.00000 −0.0717496
\(778\) 0 0
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 15.0000 0.536056
\(784\) 4.00000 0.142857
\(785\) 0 0
\(786\) 0 0
\(787\) −5.00000 −0.178231 −0.0891154 − 0.996021i \(-0.528404\pi\)
−0.0891154 + 0.996021i \(0.528404\pi\)
\(788\) 0 0
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 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 \(-0.585591\pi\)
−0.265664 + 0.964066i \(0.585591\pi\)
\(798\) 0 0
\(799\) 27.0000 0.955191
\(800\) 0 0
\(801\) 24.0000 0.847998
\(802\) 0 0
\(803\) 6.00000 0.211735
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) −15.0000 −0.527372 −0.263686 − 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 − 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 6.00000 0.210559
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) 20.0000 0.699711
\(818\) 0 0
\(819\) −10.0000 −0.349428
\(820\) 0 0
\(821\) −27.0000 −0.942306 −0.471153 − 0.882051i \(-0.656162\pi\)
−0.471153 + 0.882051i \(0.656162\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 − 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −54.0000 −1.87776 −0.938882 − 0.344239i \(-0.888137\pi\)
−0.938882 + 0.344239i \(0.888137\pi\)
\(828\) 24.0000 0.834058
\(829\) −52.0000 −1.80603 −0.903017 − 0.429604i \(-0.858653\pi\)
−0.903017 + 0.429604i \(0.858653\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 40.0000 1.38675
\(833\) −3.00000 −0.103944
\(834\) 0 0
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) −20.0000 −0.691301
\(838\) 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
\(843\) −3.00000 −0.103325
\(844\) 26.0000 0.894957
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) −48.0000 −1.64833
\(849\) −13.0000 −0.446159
\(850\) 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 \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) −8.00000 −0.271538
\(869\) 3.00000 0.101768
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) −50.0000 −1.68838 −0.844190 − 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) 0 0
\(879\) −21.0000 −0.708312
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 − 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −30.0000 −1.00901
\(885\) 0 0
\(886\) 0 0
\(887\) 24.0000 0.805841 0.402921 − 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) −38.0000 −1.27233
\(893\) −18.0000 −0.602347
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 30.0000 1.00167
\(898\) 0 0
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 10.0000 0.332779
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −26.0000 −0.863316 −0.431658 − 0.902037i \(-0.642071\pi\)
−0.431658 + 0.902037i \(0.642071\pi\)
\(908\) −6.00000 −0.199117
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) −8.00000 −0.264906
\(913\) 36.0000 1.19143
\(914\) 0 0
\(915\) 0 0
\(916\) 8.00000 0.264327
\(917\) 6.00000 0.198137
\(918\) 0 0
\(919\) 11.0000 0.362857 0.181428 − 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) 0 0
\(921\) 11.0000 0.362462
\(922\) 0 0
\(923\) 0 0
\(924\) 6.00000 0.197386
\(925\) 0 0
\(926\) 0 0
\(927\) 10.0000 0.328443
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 − 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 48.0000 1.57229
\(933\) −18.0000 −0.589294
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −47.0000 −1.53542 −0.767712 − 0.640796i \(-0.778605\pi\)
−0.767712 + 0.640796i \(0.778605\pi\)
\(938\) 0 0
\(939\) −19.0000 −0.620042
\(940\) 0 0
\(941\) 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 \(0.5\pi\)
\(948\) −2.00000 −0.0649570
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 42.0000 1.35838
\(957\) 9.00000 0.290929
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 20.0000 0.644157
\(965\) 0 0
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 − 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) −48.0000 −1.54039 −0.770197 − 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 32.0000 1.02640
\(973\) −14.0000 −0.448819
\(974\) 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
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) 21.0000 0.669796 0.334898 − 0.942254i \(-0.391298\pi\)
0.334898 + 0.942254i \(0.391298\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −9.00000 −0.286473
\(988\) 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
\(993\) 28.0000 0.888553
\(994\) 0 0
\(995\) 0 0
\(996\) −24.0000 −0.760469
\(997\) 37.0000 1.17180 0.585901 − 0.810383i \(-0.300741\pi\)
0.585901 + 0.810383i \(0.300741\pi\)
\(998\) 0 0
\(999\) −10.0000 −0.316386
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 175.2.a.b.1.1 1
3.2 odd 2 1575.2.a.f.1.1 1
4.3 odd 2 2800.2.a.z.1.1 1
5.2 odd 4 175.2.b.a.99.2 2
5.3 odd 4 175.2.b.a.99.1 2
5.4 even 2 35.2.a.a.1.1 ✓ 1
7.6 odd 2 1225.2.a.e.1.1 1
15.2 even 4 1575.2.d.c.1324.1 2
15.8 even 4 1575.2.d.c.1324.2 2
15.14 odd 2 315.2.a.b.1.1 1
20.3 even 4 2800.2.g.l.449.2 2
20.7 even 4 2800.2.g.l.449.1 2
20.19 odd 2 560.2.a.b.1.1 1
35.4 even 6 245.2.e.a.226.1 2
35.9 even 6 245.2.e.a.116.1 2
35.13 even 4 1225.2.b.d.99.2 2
35.19 odd 6 245.2.e.b.116.1 2
35.24 odd 6 245.2.e.b.226.1 2
35.27 even 4 1225.2.b.d.99.1 2
35.34 odd 2 245.2.a.c.1.1 1
40.19 odd 2 2240.2.a.u.1.1 1
40.29 even 2 2240.2.a.k.1.1 1
55.54 odd 2 4235.2.a.c.1.1 1
60.59 even 2 5040.2.a.v.1.1 1
65.64 even 2 5915.2.a.f.1.1 1
105.104 even 2 2205.2.a.e.1.1 1
140.139 even 2 3920.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.a.1.1 ✓ 1 5.4 even 2
175.2.a.b.1.1 1 1.1 even 1 trivial
175.2.b.a.99.1 2 5.3 odd 4
175.2.b.a.99.2 2 5.2 odd 4
245.2.a.c.1.1 1 35.34 odd 2
245.2.e.a.116.1 2 35.9 even 6
245.2.e.a.226.1 2 35.4 even 6
245.2.e.b.116.1 2 35.19 odd 6
245.2.e.b.226.1 2 35.24 odd 6
315.2.a.b.1.1 1 15.14 odd 2
560.2.a.b.1.1 1 20.19 odd 2
1225.2.a.e.1.1 1 7.6 odd 2
1225.2.b.d.99.1 2 35.27 even 4
1225.2.b.d.99.2 2 35.13 even 4
1575.2.a.f.1.1 1 3.2 odd 2
1575.2.d.c.1324.1 2 15.2 even 4
1575.2.d.c.1324.2 2 15.8 even 4
2205.2.a.e.1.1 1 105.104 even 2
2240.2.a.k.1.1 1 40.29 even 2
2240.2.a.u.1.1 1 40.19 odd 2
2800.2.a.z.1.1 1 4.3 odd 2
2800.2.g.l.449.1 2 20.7 even 4
2800.2.g.l.449.2 2 20.3 even 4
3920.2.a.ba.1.1 1 140.139 even 2
4235.2.a.c.1.1 1 55.54 odd 2
5040.2.a.v.1.1 1 60.59 even 2
5915.2.a.f.1.1 1 65.64 even 2