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Properties

Label 1400.2.a.n.1.1
Level $1400$
Weight $2$
Character 1400.1
Self dual yes
Analytic conductor $11.179$
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] = [1400,2,Mod(1,1400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1400, 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("1400.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.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: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -1.00000 q^{7} +6.00000 q^{9} -5.00000 q^{11} +5.00000 q^{13} +7.00000 q^{17} -2.00000 q^{19} -3.00000 q^{21} +2.00000 q^{23} +9.00000 q^{27} +7.00000 q^{29} +4.00000 q^{31} -15.0000 q^{33} +6.00000 q^{37} +15.0000 q^{39} -12.0000 q^{41} +2.00000 q^{43} -1.00000 q^{47} +1.00000 q^{49} +21.0000 q^{51} -6.00000 q^{57} -4.00000 q^{59} +4.00000 q^{61} -6.00000 q^{63} -8.00000 q^{67} +6.00000 q^{69} -6.00000 q^{73} +5.00000 q^{77} -3.00000 q^{79} +9.00000 q^{81} +4.00000 q^{83} +21.0000 q^{87} -5.00000 q^{91} +12.0000 q^{93} -13.0000 q^{97} -30.0000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.00000 1.69775 0.848875 − 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) 7.00000 1.29987 0.649934 − 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) −15.0000 −2.61116
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 15.0000 2.40192
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 − 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.00000 −0.145865 −0.0729325 − 0.997337i \(-0.523236\pi\)
−0.0729325 + 0.997337i \(0.523236\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 21.0000 2.94059
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) −6.00000 −0.755929
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.00000 0.569803
\(78\) 0 0
\(79\) −3.00000 −0.337526 −0.168763 − 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 21.0000 2.25144
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −5.00000 −0.524142
\(92\) 0 0
\(93\) 12.0000 1.24434
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 − 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 0 0
\(99\) −30.0000 −3.01511
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 − 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) −13.0000 −1.28093 −0.640464 − 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −18.0000 −1.74013 −0.870063 − 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 − 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 18.0000 1.70848
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 30.0000 2.77350
\(118\) 0 0
\(119\) −7.00000 −0.641689
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) −36.0000 −3.24601
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 − 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.00000 0.683486 0.341743 − 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 0 0
\(139\) 18.0000 1.52674 0.763370 − 0.645961i \(-0.223543\pi\)
0.763370 + 0.645961i \(0.223543\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 0 0
\(143\) −25.0000 −2.09061
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) −22.0000 −1.80231 −0.901155 − 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) −19.0000 −1.54620 −0.773099 − 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 42.0000 3.39550
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 14.0000 1.09656 0.548282 − 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 0 0
\(165\) 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
\(171\) −12.0000 −0.917663
\(172\) 0 0
\(173\) 7.00000 0.532200 0.266100 − 0.963945i \(-0.414265\pi\)
0.266100 + 0.963945i \(0.414265\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 − 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 12.0000 0.887066
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −35.0000 −2.55945
\(188\) 0 0
\(189\) −9.00000 −0.654654
\(190\) 0 0
\(191\) −13.0000 −0.940647 −0.470323 − 0.882494i \(-0.655863\pi\)
−0.470323 + 0.882494i \(0.655863\pi\)
\(192\) 0 0
\(193\) 8.00000 0.575853 0.287926 − 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.00000 0.569976 0.284988 − 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −24.0000 −1.69283
\(202\) 0 0
\(203\) −7.00000 −0.491304
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 12.0000 0.834058
\(208\) 0 0
\(209\) 10.0000 0.691714
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 − 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) −18.0000 −1.21633
\(220\) 0 0
\(221\) 35.0000 2.35435
\(222\) 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
\(227\) −9.00000 −0.597351 −0.298675 − 0.954355i \(-0.596545\pi\)
−0.298675 + 0.954355i \(0.596545\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 − 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 15.0000 0.986928
\(232\) 0 0
\(233\) 24.0000 1.57229 0.786146 − 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −9.00000 −0.584613
\(238\) 0 0
\(239\) 9.00000 0.582162 0.291081 − 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 − 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −10.0000 −0.636285
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 − 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 0 0
\(253\) −10.0000 −0.628695
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 42.0000 2.59973
\(262\) 0 0
\(263\) 30.0000 1.84988 0.924940 − 0.380114i \(-0.124115\pi\)
0.924940 + 0.380114i \(0.124115\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 26.0000 1.58525 0.792624 − 0.609711i \(-0.208714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 − 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) −15.0000 −0.907841
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) 19.0000 1.13344 0.566722 − 0.823909i \(-0.308211\pi\)
0.566722 + 0.823909i \(0.308211\pi\)
\(282\) 0 0
\(283\) 25.0000 1.48610 0.743048 − 0.669238i \(-0.233379\pi\)
0.743048 + 0.669238i \(0.233379\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) −39.0000 −2.28622
\(292\) 0 0
\(293\) −13.0000 −0.759468 −0.379734 − 0.925096i \(-0.623985\pi\)
−0.379734 + 0.925096i \(0.623985\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −45.0000 −2.61116
\(298\) 0 0
\(299\) 10.0000 0.578315
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 0 0
\(303\) −54.0000 −3.10222
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 17.0000 0.970241 0.485121 − 0.874447i \(-0.338776\pi\)
0.485121 + 0.874447i \(0.338776\pi\)
\(308\) 0 0
\(309\) −39.0000 −2.21863
\(310\) 0 0
\(311\) −34.0000 −1.92796 −0.963982 − 0.265969i \(-0.914308\pi\)
−0.963982 + 0.265969i \(0.914308\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 − 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) −35.0000 −1.95962
\(320\) 0 0
\(321\) −54.0000 −3.01399
\(322\) 0 0
\(323\) −14.0000 −0.778981
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 15.0000 0.829502
\(328\) 0 0
\(329\) 1.00000 0.0551318
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 − 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) 36.0000 1.97279
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −34.0000 −1.85210 −0.926049 − 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −26.0000 −1.39575 −0.697877 − 0.716218i \(-0.745872\pi\)
−0.697877 + 0.716218i \(0.745872\pi\)
\(348\) 0 0
\(349\) 18.0000 0.963518 0.481759 − 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) 45.0000 2.40192
\(352\) 0 0
\(353\) −5.00000 −0.266123 −0.133062 − 0.991108i \(-0.542481\pi\)
−0.133062 + 0.991108i \(0.542481\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −21.0000 −1.11144
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 42.0000 2.20443
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7.00000 0.365397 0.182699 − 0.983169i \(-0.441517\pi\)
0.182699 + 0.983169i \(0.441517\pi\)
\(368\) 0 0
\(369\) −72.0000 −3.74817
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −16.0000 −0.828449 −0.414224 − 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 35.0000 1.80259
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 36.0000 1.84434
\(382\) 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
\(387\) 12.0000 0.609994
\(388\) 0 0
\(389\) −15.0000 −0.760530 −0.380265 − 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) 14.0000 0.708010
\(392\) 0 0
\(393\) −18.0000 −0.907980
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 23.0000 1.15434 0.577168 − 0.816625i \(-0.304158\pi\)
0.577168 + 0.816625i \(0.304158\pi\)
\(398\) 0 0
\(399\) 6.00000 0.300376
\(400\) 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
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −30.0000 −1.48704
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 − 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 24.0000 1.18383
\(412\) 0 0
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 54.0000 2.64439
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 − 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −3.00000 −0.146211 −0.0731055 − 0.997324i \(-0.523291\pi\)
−0.0731055 + 0.997324i \(0.523291\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 0 0
\(429\) −75.0000 −3.62103
\(430\) 0 0
\(431\) −25.0000 −1.20421 −0.602104 − 0.798418i \(-0.705671\pi\)
−0.602104 + 0.798418i \(0.705671\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 − 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 − 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) −6.00000 −0.285069 −0.142534 − 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −66.0000 −3.12169
\(448\) 0 0
\(449\) −9.00000 −0.424736 −0.212368 − 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 60.0000 2.82529
\(452\) 0 0
\(453\) −57.0000 −2.67809
\(454\) 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
\(459\) 63.0000 2.94059
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 36.0000 1.67306 0.836531 − 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −11.0000 −0.509019 −0.254510 − 0.967070i \(-0.581914\pi\)
−0.254510 + 0.967070i \(0.581914\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −30.0000 −1.38233
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 34.0000 1.55350 0.776750 − 0.629809i \(-0.216867\pi\)
0.776750 + 0.629809i \(0.216867\pi\)
\(480\) 0 0
\(481\) 30.0000 1.36788
\(482\) 0 0
\(483\) −6.00000 −0.273009
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −22.0000 −0.996915 −0.498458 − 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) 0 0
\(489\) 42.0000 1.89931
\(490\) 0 0
\(491\) 9.00000 0.406164 0.203082 − 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 0 0
\(493\) 49.0000 2.20685
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.00000 −0.0447661 −0.0223831 − 0.999749i \(-0.507125\pi\)
−0.0223831 + 0.999749i \(0.507125\pi\)
\(500\) 0 0
\(501\) 9.00000 0.402090
\(502\) 0 0
\(503\) −3.00000 −0.133763 −0.0668817 − 0.997761i \(-0.521305\pi\)
−0.0668817 + 0.997761i \(0.521305\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 36.0000 1.59882
\(508\) 0 0
\(509\) −14.0000 −0.620539 −0.310270 − 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) −18.0000 −0.794719
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5.00000 0.219900
\(518\) 0 0
\(519\) 21.0000 0.921798
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 − 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) 44.0000 1.92399 0.961993 − 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 28.0000 1.21970
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) −24.0000 −1.04151
\(532\) 0 0
\(533\) −60.0000 −2.59889
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) 7.00000 0.300954 0.150477 − 0.988614i \(-0.451919\pi\)
0.150477 + 0.988614i \(0.451919\pi\)
\(542\) 0 0
\(543\) 24.0000 1.02994
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 − 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) 24.0000 1.02430
\(550\) 0 0
\(551\) −14.0000 −0.596420
\(552\) 0 0
\(553\) 3.00000 0.127573
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.0000 0.677942 0.338971 − 0.940797i \(-0.389921\pi\)
0.338971 + 0.940797i \(0.389921\pi\)
\(558\) 0 0
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) −105.000 −4.43310
\(562\) 0 0
\(563\) −4.00000 −0.168580 −0.0842900 − 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −9.00000 −0.377964
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 − 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 − 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) −39.0000 −1.62925
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 37.0000 1.54033 0.770165 − 0.637845i \(-0.220174\pi\)
0.770165 + 0.637845i \(0.220174\pi\)
\(578\) 0 0
\(579\) 24.0000 0.997406
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) 0 0
\(593\) −27.0000 −1.10876 −0.554379 − 0.832265i \(-0.687044\pi\)
−0.554379 + 0.832265i \(0.687044\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −12.0000 −0.491127
\(598\) 0 0
\(599\) 15.0000 0.612883 0.306442 − 0.951889i \(-0.400862\pi\)
0.306442 + 0.951889i \(0.400862\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 − 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) −48.0000 −1.95471
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −3.00000 −0.121766 −0.0608831 − 0.998145i \(-0.519392\pi\)
−0.0608831 + 0.998145i \(0.519392\pi\)
\(608\) 0 0
\(609\) −21.0000 −0.850963
\(610\) 0 0
\(611\) −5.00000 −0.202278
\(612\) 0 0
\(613\) 30.0000 1.21169 0.605844 − 0.795583i \(-0.292835\pi\)
0.605844 + 0.795583i \(0.292835\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 − 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 18.0000 0.722315
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 30.0000 1.19808
\(628\) 0 0
\(629\) 42.0000 1.67465
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 − 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) 15.0000 0.596196
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 5.00000 0.198107
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) −21.0000 −0.828159 −0.414080 − 0.910241i \(-0.635896\pi\)
−0.414080 + 0.910241i \(0.635896\pi\)
\(644\) 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
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) −12.0000 −0.470317
\(652\) 0 0
\(653\) −38.0000 −1.48705 −0.743527 − 0.668705i \(-0.766849\pi\)
−0.743527 + 0.668705i \(0.766849\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −36.0000 −1.40449
\(658\) 0 0
\(659\) 15.0000 0.584317 0.292159 − 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 0 0
\(661\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) 105.000 4.07786
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 14.0000 0.542082
\(668\) 0 0
\(669\) 57.0000 2.20375
\(670\) 0 0
\(671\) −20.0000 −0.772091
\(672\) 0 0
\(673\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 29.0000 1.11456 0.557280 − 0.830324i \(-0.311845\pi\)
0.557280 + 0.830324i \(0.311845\pi\)
\(678\) 0 0
\(679\) 13.0000 0.498894
\(680\) 0 0
\(681\) −27.0000 −1.03464
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 12.0000 0.457829
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 − 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 0 0
\(693\) 30.0000 1.13961
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −84.0000 −3.18173
\(698\) 0 0
\(699\) 72.0000 2.72329
\(700\) 0 0
\(701\) 35.0000 1.32193 0.660966 − 0.750416i \(-0.270147\pi\)
0.660966 + 0.750416i \(0.270147\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.0000 0.676960
\(708\) 0 0
\(709\) −25.0000 −0.938895 −0.469447 − 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 0 0
\(711\) −18.0000 −0.675053
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 27.0000 1.00833
\(718\) 0 0
\(719\) −14.0000 −0.522112 −0.261056 − 0.965324i \(-0.584071\pi\)
−0.261056 + 0.965324i \(0.584071\pi\)
\(720\) 0 0
\(721\) 13.0000 0.484145
\(722\) 0 0
\(723\) −66.0000 −2.45457
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 40.0000 1.48352 0.741759 − 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 14.0000 0.517809
\(732\) 0 0
\(733\) 33.0000 1.21888 0.609441 − 0.792831i \(-0.291394\pi\)
0.609441 + 0.792831i \(0.291394\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 40.0000 1.47342
\(738\) 0 0
\(739\) 51.0000 1.87607 0.938033 − 0.346547i \(-0.112646\pi\)
0.938033 + 0.346547i \(0.112646\pi\)
\(740\) 0 0
\(741\) −30.0000 −1.10208
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 24.0000 0.878114
\(748\) 0 0
\(749\) 18.0000 0.657706
\(750\) 0 0
\(751\) −3.00000 −0.109472 −0.0547358 − 0.998501i \(-0.517432\pi\)
−0.0547358 + 0.998501i \(0.517432\pi\)
\(752\) 0 0
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 40.0000 1.45382 0.726912 − 0.686730i \(-0.240955\pi\)
0.726912 + 0.686730i \(0.240955\pi\)
\(758\) 0 0
\(759\) −30.0000 −1.08893
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 − 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) −5.00000 −0.181012
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −20.0000 −0.722158
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 − 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) −54.0000 −1.94476
\(772\) 0 0
\(773\) 21.0000 0.755318 0.377659 − 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −18.0000 −0.645746
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 63.0000 2.25144
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −17.0000 −0.605985 −0.302992 − 0.952993i \(-0.597986\pi\)
−0.302992 + 0.952993i \(0.597986\pi\)
\(788\) 0 0
\(789\) 90.0000 3.20408
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 20.0000 0.710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 39.0000 1.38145 0.690725 − 0.723117i \(-0.257291\pi\)
0.690725 + 0.723117i \(0.257291\pi\)
\(798\) 0 0
\(799\) −7.00000 −0.247642
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 30.0000 1.05868
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 78.0000 2.74573
\(808\) 0 0
\(809\) −39.0000 −1.37117 −0.685583 − 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) −10.0000 −0.351147 −0.175574 − 0.984466i \(-0.556178\pi\)
−0.175574 + 0.984466i \(0.556178\pi\)
\(812\) 0 0
\(813\) 36.0000 1.26258
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) 0 0
\(819\) −30.0000 −1.04828
\(820\) 0 0
\(821\) 1.00000 0.0349002 0.0174501 − 0.999848i \(-0.494445\pi\)
0.0174501 + 0.999848i \(0.494445\pi\)
\(822\) 0 0
\(823\) −28.0000 −0.976019 −0.488009 − 0.872838i \(-0.662277\pi\)
−0.488009 + 0.872838i \(0.662277\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −6.00000 −0.208640 −0.104320 − 0.994544i \(-0.533267\pi\)
−0.104320 + 0.994544i \(0.533267\pi\)
\(828\) 0 0
\(829\) −20.0000 −0.694629 −0.347314 − 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 0 0
\(831\) −6.00000 −0.208138
\(832\) 0 0
\(833\) 7.00000 0.242536
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 36.0000 1.24434
\(838\) 0 0
\(839\) −34.0000 −1.17381 −0.586905 − 0.809656i \(-0.699654\pi\)
−0.586905 + 0.809656i \(0.699654\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 0 0
\(843\) 57.0000 1.96318
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) 0 0
\(849\) 75.0000 2.57399
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 − 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6.00000 −0.204956 −0.102478 − 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 36.0000 1.22688
\(862\) 0 0
\(863\) −48.0000 −1.63394 −0.816970 − 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 96.0000 3.26033
\(868\) 0 0
\(869\) 15.0000 0.508840
\(870\) 0 0
\(871\) −40.0000 −1.35535
\(872\) 0 0
\(873\) −78.0000 −2.63990
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 − 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) −39.0000 −1.31544
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 40.0000 1.34611 0.673054 − 0.739594i \(-0.264982\pi\)
0.673054 + 0.739594i \(0.264982\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16.0000 −0.537227 −0.268614 − 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) −45.0000 −1.50756
\(892\) 0 0
\(893\) 2.00000 0.0669274
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 30.0000 1.00167
\(898\) 0 0
\(899\) 28.0000 0.933852
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −6.00000 −0.199667
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 18.0000 0.597680 0.298840 − 0.954303i \(-0.403400\pi\)
0.298840 + 0.954303i \(0.403400\pi\)
\(908\) 0 0
\(909\) −108.000 −3.58213
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 − 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) −20.0000 −0.661903
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.00000 0.198137
\(918\) 0 0
\(919\) −55.0000 −1.81428 −0.907141 − 0.420826i \(-0.861740\pi\)
−0.907141 + 0.420826i \(0.861740\pi\)
\(920\) 0 0
\(921\) 51.0000 1.68051
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −78.0000 −2.56186
\(928\) 0 0
\(929\) −32.0000 −1.04989 −0.524943 − 0.851137i \(-0.675913\pi\)
−0.524943 + 0.851137i \(0.675913\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 0 0
\(933\) −102.000 −3.33933
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −13.0000 −0.424691 −0.212346 − 0.977195i \(-0.568110\pi\)
−0.212346 + 0.977195i \(0.568110\pi\)
\(938\) 0 0
\(939\) 3.00000 0.0979013
\(940\) 0 0
\(941\) 28.0000 0.912774 0.456387 − 0.889781i \(-0.349143\pi\)
0.456387 + 0.889781i \(0.349143\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −40.0000 −1.29983 −0.649913 − 0.760009i \(-0.725195\pi\)
−0.649913 + 0.760009i \(0.725195\pi\)
\(948\) 0 0
\(949\) −30.0000 −0.973841
\(950\) 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
\(956\) 0 0
\(957\) −105.000 −3.39417
\(958\) 0 0
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −108.000 −3.48025
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −6.00000 −0.192947 −0.0964735 − 0.995336i \(-0.530756\pi\)
−0.0964735 + 0.995336i \(0.530756\pi\)
\(968\) 0 0
\(969\) −42.0000 −1.34923
\(970\) 0 0
\(971\) −16.0000 −0.513464 −0.256732 − 0.966483i \(-0.582646\pi\)
−0.256732 + 0.966483i \(0.582646\pi\)
\(972\) 0 0
\(973\) −18.0000 −0.577054
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −26.0000 −0.831814 −0.415907 − 0.909407i \(-0.636536\pi\)
−0.415907 + 0.909407i \(0.636536\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 30.0000 0.957826
\(982\) 0 0
\(983\) −3.00000 −0.0956851 −0.0478426 − 0.998855i \(-0.515235\pi\)
−0.0478426 + 0.998855i \(0.515235\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 3.00000 0.0954911
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 − 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) −36.0000 −1.14243
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.00000 0.0950110 0.0475055 − 0.998871i \(-0.484873\pi\)
0.0475055 + 0.998871i \(0.484873\pi\)
\(998\) 0 0
\(999\) 54.0000 1.70848
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 1400.2.a.n.1.1 1
4.3 odd 2 2800.2.a.c.1.1 1
5.2 odd 4 1400.2.g.a.449.1 2
5.3 odd 4 1400.2.g.a.449.2 2
5.4 even 2 280.2.a.a.1.1 ✓ 1
7.6 odd 2 9800.2.a.a.1.1 1
15.14 odd 2 2520.2.a.i.1.1 1
20.3 even 4 2800.2.g.b.449.1 2
20.7 even 4 2800.2.g.b.449.2 2
20.19 odd 2 560.2.a.f.1.1 1
35.4 even 6 1960.2.q.o.961.1 2
35.9 even 6 1960.2.q.o.361.1 2
35.19 odd 6 1960.2.q.a.361.1 2
35.24 odd 6 1960.2.q.a.961.1 2
35.34 odd 2 1960.2.a.o.1.1 1
40.19 odd 2 2240.2.a.a.1.1 1
40.29 even 2 2240.2.a.z.1.1 1
60.59 even 2 5040.2.a.a.1.1 1
140.139 even 2 3920.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.a.1.1 ✓ 1 5.4 even 2
560.2.a.f.1.1 1 20.19 odd 2
1400.2.a.n.1.1 1 1.1 even 1 trivial
1400.2.g.a.449.1 2 5.2 odd 4
1400.2.g.a.449.2 2 5.3 odd 4
1960.2.a.o.1.1 1 35.34 odd 2
1960.2.q.a.361.1 2 35.19 odd 6
1960.2.q.a.961.1 2 35.24 odd 6
1960.2.q.o.361.1 2 35.9 even 6
1960.2.q.o.961.1 2 35.4 even 6
2240.2.a.a.1.1 1 40.19 odd 2
2240.2.a.z.1.1 1 40.29 even 2
2520.2.a.i.1.1 1 15.14 odd 2
2800.2.a.c.1.1 1 4.3 odd 2
2800.2.g.b.449.1 2 20.3 even 4
2800.2.g.b.449.2 2 20.7 even 4
3920.2.a.c.1.1 1 140.139 even 2
5040.2.a.a.1.1 1 60.59 even 2
9800.2.a.a.1.1 1 7.6 odd 2