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Properties

Label 6240.2.a.h.1.1
Level $6240$
Weight $2$
Character 6240.1
Self dual yes
Analytic conductor $49.827$
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] = [6240,2,Mod(1,6240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6240.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6240.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: \(49.8266508613\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6240.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +5.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +5.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.00000 q^{15} +3.00000 q^{17} -6.00000 q^{19} -5.00000 q^{21} +3.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.00000 q^{29} -1.00000 q^{33} -5.00000 q^{35} -5.00000 q^{37} -1.00000 q^{39} +11.0000 q^{41} +6.00000 q^{43} -1.00000 q^{45} +18.0000 q^{49} -3.00000 q^{51} -9.00000 q^{53} -1.00000 q^{55} +6.00000 q^{57} +12.0000 q^{59} +5.00000 q^{61} +5.00000 q^{63} -1.00000 q^{65} +8.00000 q^{67} -3.00000 q^{69} +13.0000 q^{71} -10.0000 q^{73} -1.00000 q^{75} +5.00000 q^{77} -17.0000 q^{79} +1.00000 q^{81} -4.00000 q^{83} -3.00000 q^{85} -4.00000 q^{87} -11.0000 q^{89} +5.00000 q^{91} +6.00000 q^{95} -7.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 5.00000 1.88982 0.944911 − 0.327327i \(-0.106148\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 − 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −5.00000 −1.09109
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 − 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) −5.00000 −0.845154
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 − 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 11.0000 1.71791 0.858956 − 0.512050i \(-0.171114\pi\)
0.858956 + 0.512050i \(0.171114\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 18.0000 2.57143
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) −9.00000 −1.23625 −0.618123 − 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 − 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 0 0
\(63\) 5.00000 0.629941
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 13.0000 1.54282 0.771408 − 0.636341i \(-0.219553\pi\)
0.771408 + 0.636341i \(0.219553\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 5.00000 0.569803
\(78\) 0 0
\(79\) −17.0000 −1.91265 −0.956325 − 0.292306i \(-0.905577\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) −11.0000 −1.16600 −0.582999 − 0.812473i \(-0.698121\pi\)
−0.582999 + 0.812473i \(0.698121\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −7.00000 −0.710742 −0.355371 − 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 5.00000 0.487950
\(106\) 0 0
\(107\) −13.0000 −1.25676 −0.628379 − 0.777908i \(-0.716281\pi\)
−0.628379 + 0.777908i \(0.716281\pi\)
\(108\) 0 0
\(109\) −8.00000 −0.766261 −0.383131 − 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 0 0
\(111\) 5.00000 0.474579
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 15.0000 1.37505
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) −11.0000 −0.991837
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 − 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 − 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) −30.0000 −2.60133
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 − 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 9.00000 0.763370 0.381685 − 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) −18.0000 −1.48461
\(148\) 0 0
\(149\) 9.00000 0.737309 0.368654 − 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 − 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 − 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 15.0000 1.18217
\(162\) 0 0
\(163\) 1.00000 0.0783260 0.0391630 − 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 0 0
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) 24.0000 1.85718 0.928588 − 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 − 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) 5.00000 0.377964
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) 18.0000 1.34538 0.672692 − 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) −23.0000 −1.70958 −0.854788 − 0.518977i \(-0.826313\pi\)
−0.854788 + 0.518977i \(0.826313\pi\)
\(182\) 0 0
\(183\) −5.00000 −0.369611
\(184\) 0 0
\(185\) 5.00000 0.367607
\(186\) 0 0
\(187\) 3.00000 0.219382
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 − 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) −21.0000 −1.51161 −0.755807 − 0.654795i \(-0.772755\pi\)
−0.755807 + 0.654795i \(0.772755\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 − 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) 20.0000 1.40372
\(204\) 0 0
\(205\) −11.0000 −0.768273
\(206\) 0 0
\(207\) 3.00000 0.208514
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) −13.0000 −0.890745
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 22.0000 1.46019 0.730096 − 0.683345i \(-0.239475\pi\)
0.730096 + 0.683345i \(0.239475\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 − 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) 0 0
\(233\) −3.00000 −0.196537 −0.0982683 − 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 17.0000 1.10427
\(238\) 0 0
\(239\) 5.00000 0.323423 0.161712 − 0.986838i \(-0.448299\pi\)
0.161712 + 0.986838i \(0.448299\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 − 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −18.0000 −1.14998
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 3.00000 0.188608
\(254\) 0 0
\(255\) 3.00000 0.187867
\(256\) 0 0
\(257\) −10.0000 −0.623783 −0.311891 − 0.950118i \(-0.600963\pi\)
−0.311891 + 0.950118i \(0.600963\pi\)
\(258\) 0 0
\(259\) −25.0000 −1.55342
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 0 0
\(267\) 11.0000 0.673189
\(268\) 0 0
\(269\) −4.00000 −0.243884 −0.121942 − 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 − 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 0 0
\(273\) −5.00000 −0.302614
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 16.0000 0.961347 0.480673 − 0.876900i \(-0.340392\pi\)
0.480673 + 0.876900i \(0.340392\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 − 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 24.0000 1.42665 0.713326 − 0.700832i \(-0.247188\pi\)
0.713326 + 0.700832i \(0.247188\pi\)
\(284\) 0 0
\(285\) −6.00000 −0.355409
\(286\) 0 0
\(287\) 55.0000 3.24655
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 7.00000 0.410347
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 − 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 30.0000 1.72917
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5.00000 −0.286299
\(306\) 0 0
\(307\) 25.0000 1.42683 0.713413 − 0.700744i \(-0.247149\pi\)
0.713413 + 0.700744i \(0.247149\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −10.0000 −0.567048 −0.283524 − 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 − 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) −5.00000 −0.281718
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) 13.0000 0.725589
\(322\) 0 0
\(323\) −18.0000 −1.00155
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 8.00000 0.442401
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −14.0000 −0.769510 −0.384755 − 0.923019i \(-0.625714\pi\)
−0.384755 + 0.923019i \(0.625714\pi\)
\(332\) 0 0
\(333\) −5.00000 −0.273998
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 − 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 0 0
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 55.0000 2.96972
\(344\) 0 0
\(345\) 3.00000 0.161515
\(346\) 0 0
\(347\) −17.0000 −0.912608 −0.456304 − 0.889824i \(-0.650827\pi\)
−0.456304 + 0.889824i \(0.650827\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 − 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 4.00000 0.212899 0.106449 − 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) 0 0
\(355\) −13.0000 −0.689968
\(356\) 0 0
\(357\) −15.0000 −0.793884
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) 22.0000 1.14839 0.574195 − 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) 0 0
\(369\) 11.0000 0.572637
\(370\) 0 0
\(371\) −45.0000 −2.33628
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 − 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 − 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 18.0000 0.919757 0.459879 − 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) 0 0
\(385\) −5.00000 −0.254824
\(386\) 0 0
\(387\) 6.00000 0.304997
\(388\) 0 0
\(389\) 8.00000 0.405616 0.202808 − 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 0 0
\(391\) 9.00000 0.455150
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) 17.0000 0.855363
\(396\) 0 0
\(397\) 35.0000 1.75660 0.878300 − 0.478110i \(-0.158678\pi\)
0.878300 + 0.478110i \(0.158678\pi\)
\(398\) 0 0
\(399\) 30.0000 1.50188
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −5.00000 −0.247841
\(408\) 0 0
\(409\) −32.0000 −1.58230 −0.791149 − 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) 60.0000 2.95241
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) −9.00000 −0.440732
\(418\) 0 0
\(419\) 18.0000 0.879358 0.439679 − 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 0 0
\(421\) 16.0000 0.779792 0.389896 − 0.920859i \(-0.372511\pi\)
0.389896 + 0.920859i \(0.372511\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.00000 0.145521
\(426\) 0 0
\(427\) 25.0000 1.20983
\(428\) 0 0
\(429\) −1.00000 −0.0482805
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 − 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) −22.0000 −1.05725 −0.528626 − 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) 0 0
\(437\) −18.0000 −0.861057
\(438\) 0 0
\(439\) −7.00000 −0.334092 −0.167046 − 0.985949i \(-0.553423\pi\)
−0.167046 + 0.985949i \(0.553423\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) 0 0
\(443\) −17.0000 −0.807694 −0.403847 − 0.914826i \(-0.632327\pi\)
−0.403847 + 0.914826i \(0.632327\pi\)
\(444\) 0 0
\(445\) 11.0000 0.521450
\(446\) 0 0
\(447\) −9.00000 −0.425685
\(448\) 0 0
\(449\) 7.00000 0.330350 0.165175 − 0.986264i \(-0.447181\pi\)
0.165175 + 0.986264i \(0.447181\pi\)
\(450\) 0 0
\(451\) 11.0000 0.517970
\(452\) 0 0
\(453\) 4.00000 0.187936
\(454\) 0 0
\(455\) −5.00000 −0.234404
\(456\) 0 0
\(457\) 5.00000 0.233890 0.116945 − 0.993138i \(-0.462690\pi\)
0.116945 + 0.993138i \(0.462690\pi\)
\(458\) 0 0
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) 11.0000 0.512321 0.256161 − 0.966634i \(-0.417542\pi\)
0.256161 + 0.966634i \(0.417542\pi\)
\(462\) 0 0
\(463\) 37.0000 1.71954 0.859768 − 0.510685i \(-0.170608\pi\)
0.859768 + 0.510685i \(0.170608\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −41.0000 −1.89725 −0.948627 − 0.316397i \(-0.897527\pi\)
−0.948627 + 0.316397i \(0.897527\pi\)
\(468\) 0 0
\(469\) 40.0000 1.84703
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) 0 0
\(479\) −17.0000 −0.776750 −0.388375 − 0.921501i \(-0.626963\pi\)
−0.388375 + 0.921501i \(0.626963\pi\)
\(480\) 0 0
\(481\) −5.00000 −0.227980
\(482\) 0 0
\(483\) −15.0000 −0.682524
\(484\) 0 0
\(485\) 7.00000 0.317854
\(486\) 0 0
\(487\) 15.0000 0.679715 0.339857 − 0.940477i \(-0.389621\pi\)
0.339857 + 0.940477i \(0.389621\pi\)
\(488\) 0 0
\(489\) −1.00000 −0.0452216
\(490\) 0 0
\(491\) −26.0000 −1.17336 −0.586682 − 0.809818i \(-0.699566\pi\)
−0.586682 + 0.809818i \(0.699566\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) 65.0000 2.91565
\(498\) 0 0
\(499\) 14.0000 0.626726 0.313363 − 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 9.00000 0.398918 0.199459 − 0.979906i \(-0.436082\pi\)
0.199459 + 0.979906i \(0.436082\pi\)
\(510\) 0 0
\(511\) −50.0000 −2.21187
\(512\) 0 0
\(513\) 6.00000 0.264906
\(514\) 0 0
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) 2.00000 0.0876216 0.0438108 − 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) 0 0
\(523\) 14.0000 0.612177 0.306089 − 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) 0 0
\(525\) −5.00000 −0.218218
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 11.0000 0.476463
\(534\) 0 0
\(535\) 13.0000 0.562039
\(536\) 0 0
\(537\) −18.0000 −0.776757
\(538\) 0 0
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 − 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 0 0
\(543\) 23.0000 0.987024
\(544\) 0 0
\(545\) 8.00000 0.342682
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 − 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 0 0
\(549\) 5.00000 0.213395
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) −85.0000 −3.61457
\(554\) 0 0
\(555\) −5.00000 −0.212238
\(556\) 0 0
\(557\) 24.0000 1.01691 0.508456 − 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) 0 0
\(563\) 3.00000 0.126435 0.0632175 − 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) 0 0
\(567\) 5.00000 0.209980
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 23.0000 0.962520 0.481260 − 0.876578i \(-0.340179\pi\)
0.481260 + 0.876578i \(0.340179\pi\)
\(572\) 0 0
\(573\) −18.0000 −0.751961
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) 0 0
\(577\) −31.0000 −1.29055 −0.645273 − 0.763952i \(-0.723257\pi\)
−0.645273 + 0.763952i \(0.723257\pi\)
\(578\) 0 0
\(579\) 21.0000 0.872730
\(580\) 0 0
\(581\) −20.0000 −0.829740
\(582\) 0 0
\(583\) −9.00000 −0.372742
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) 30.0000 1.23823 0.619116 − 0.785299i \(-0.287491\pi\)
0.619116 + 0.785299i \(0.287491\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) 20.0000 0.821302 0.410651 − 0.911793i \(-0.365302\pi\)
0.410651 + 0.911793i \(0.365302\pi\)
\(594\) 0 0
\(595\) −15.0000 −0.614940
\(596\) 0 0
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) 8.00000 0.326871 0.163436 − 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 − 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) 0 0
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) −2.00000 −0.0811775 −0.0405887 − 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) 0 0
\(609\) −20.0000 −0.810441
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 17.0000 0.686624 0.343312 − 0.939222i \(-0.388451\pi\)
0.343312 + 0.939222i \(0.388451\pi\)
\(614\) 0 0
\(615\) 11.0000 0.443563
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 − 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) −36.0000 −1.44696 −0.723481 − 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 0 0
\(621\) −3.00000 −0.120386
\(622\) 0 0
\(623\) −55.0000 −2.20353
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.00000 0.239617
\(628\) 0 0
\(629\) −15.0000 −0.598089
\(630\) 0 0
\(631\) 48.0000 1.91085 0.955425 − 0.295234i \(-0.0953977\pi\)
0.955425 + 0.295234i \(0.0953977\pi\)
\(632\) 0 0
\(633\) −8.00000 −0.317971
\(634\) 0 0
\(635\) −4.00000 −0.158735
\(636\) 0 0
\(637\) 18.0000 0.713186
\(638\) 0 0
\(639\) 13.0000 0.514272
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 − 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 0 0
\(643\) −11.0000 −0.433798 −0.216899 − 0.976194i \(-0.569594\pi\)
−0.216899 + 0.976194i \(0.569594\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) −19.0000 −0.746967 −0.373484 − 0.927637i \(-0.621837\pi\)
−0.373484 + 0.927637i \(0.621837\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.0000 0.704394 0.352197 − 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) 0 0
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) 20.0000 0.779089 0.389545 − 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 − 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 0 0
\(663\) −3.00000 −0.116510
\(664\) 0 0
\(665\) 30.0000 1.16335
\(666\) 0 0
\(667\) 12.0000 0.464642
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 5.00000 0.193023
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 − 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 27.0000 1.03769 0.518847 − 0.854867i \(-0.326361\pi\)
0.518847 + 0.854867i \(0.326361\pi\)
\(678\) 0 0
\(679\) −35.0000 −1.34318
\(680\) 0 0
\(681\) −22.0000 −0.843042
\(682\) 0 0
\(683\) −10.0000 −0.382639 −0.191320 − 0.981528i \(-0.561277\pi\)
−0.191320 + 0.981528i \(0.561277\pi\)
\(684\) 0 0
\(685\) −2.00000 −0.0764161
\(686\) 0 0
\(687\) 16.0000 0.610438
\(688\) 0 0
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) −18.0000 −0.684752 −0.342376 − 0.939563i \(-0.611232\pi\)
−0.342376 + 0.939563i \(0.611232\pi\)
\(692\) 0 0
\(693\) 5.00000 0.189934
\(694\) 0 0
\(695\) −9.00000 −0.341389
\(696\) 0 0
\(697\) 33.0000 1.24996
\(698\) 0 0
\(699\) 3.00000 0.113470
\(700\) 0 0
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 30.0000 1.13147
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 52.0000 1.95290 0.976450 − 0.215742i \(-0.0692169\pi\)
0.976450 + 0.215742i \(0.0692169\pi\)
\(710\) 0 0
\(711\) −17.0000 −0.637550
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −1.00000 −0.0373979
\(716\) 0 0
\(717\) −5.00000 −0.186728
\(718\) 0 0
\(719\) 48.0000 1.79010 0.895049 − 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) 20.0000 0.744839
\(722\) 0 0
\(723\) −8.00000 −0.297523
\(724\) 0 0
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) 46.0000 1.70605 0.853023 − 0.521874i \(-0.174767\pi\)
0.853023 + 0.521874i \(0.174767\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 18.0000 0.665754
\(732\) 0 0
\(733\) 27.0000 0.997268 0.498634 − 0.866813i \(-0.333835\pi\)
0.498634 + 0.866813i \(0.333835\pi\)
\(734\) 0 0
\(735\) 18.0000 0.663940
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) −10.0000 −0.367856 −0.183928 − 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) 6.00000 0.220416
\(742\) 0 0
\(743\) −38.0000 −1.39408 −0.697042 − 0.717030i \(-0.745501\pi\)
−0.697042 + 0.717030i \(0.745501\pi\)
\(744\) 0 0
\(745\) −9.00000 −0.329734
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) −65.0000 −2.37505
\(750\) 0 0
\(751\) −7.00000 −0.255434 −0.127717 − 0.991811i \(-0.540765\pi\)
−0.127717 + 0.991811i \(0.540765\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 4.00000 0.145575
\(756\) 0 0
\(757\) −8.00000 −0.290765 −0.145382 − 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) 0 0
\(759\) −3.00000 −0.108893
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 − 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) −40.0000 −1.44810
\(764\) 0 0
\(765\) −3.00000 −0.108465
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) −54.0000 −1.94729 −0.973645 − 0.228069i \(-0.926759\pi\)
−0.973645 + 0.228069i \(0.926759\pi\)
\(770\) 0 0
\(771\) 10.0000 0.360141
\(772\) 0 0
\(773\) −36.0000 −1.29483 −0.647415 − 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 25.0000 0.896870
\(778\) 0 0
\(779\) −66.0000 −2.36470
\(780\) 0 0
\(781\) 13.0000 0.465177
\(782\) 0 0
\(783\) −4.00000 −0.142948
\(784\) 0 0
\(785\) −18.0000 −0.642448
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 − 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 0 0
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) 10.0000 0.355559
\(792\) 0 0
\(793\) 5.00000 0.177555
\(794\) 0 0
\(795\) −9.00000 −0.319197
\(796\) 0 0
\(797\) −27.0000 −0.956389 −0.478195 − 0.878254i \(-0.658709\pi\)
−0.478195 + 0.878254i \(0.658709\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −11.0000 −0.388666
\(802\) 0 0
\(803\) −10.0000 −0.352892
\(804\) 0 0
\(805\) −15.0000 −0.528681
\(806\) 0 0
\(807\) 4.00000 0.140807
\(808\) 0 0
\(809\) 50.0000 1.75791 0.878953 − 0.476908i \(-0.158243\pi\)
0.878953 + 0.476908i \(0.158243\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 − 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 0 0
\(813\) −2.00000 −0.0701431
\(814\) 0 0
\(815\) −1.00000 −0.0350285
\(816\) 0 0
\(817\) −36.0000 −1.25948
\(818\) 0 0
\(819\) 5.00000 0.174714
\(820\) 0 0
\(821\) −1.00000 −0.0349002 −0.0174501 − 0.999848i \(-0.505555\pi\)
−0.0174501 + 0.999848i \(0.505555\pi\)
\(822\) 0 0
\(823\) 12.0000 0.418294 0.209147 − 0.977884i \(-0.432931\pi\)
0.209147 + 0.977884i \(0.432931\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) −24.0000 −0.834562 −0.417281 − 0.908778i \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 − 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) −16.0000 −0.555034
\(832\) 0 0
\(833\) 54.0000 1.87099
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −23.0000 −0.794048 −0.397024 − 0.917808i \(-0.629957\pi\)
−0.397024 + 0.917808i \(0.629957\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) −30.0000 −1.03325
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −50.0000 −1.71802
\(848\) 0 0
\(849\) −24.0000 −0.823678
\(850\) 0 0
\(851\) −15.0000 −0.514193
\(852\) 0 0
\(853\) −27.0000 −0.924462 −0.462231 − 0.886759i \(-0.652951\pi\)
−0.462231 + 0.886759i \(0.652951\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 0 0
\(857\) −47.0000 −1.60549 −0.802745 − 0.596323i \(-0.796628\pi\)
−0.802745 + 0.596323i \(0.796628\pi\)
\(858\) 0 0
\(859\) 29.0000 0.989467 0.494734 − 0.869045i \(-0.335266\pi\)
0.494734 + 0.869045i \(0.335266\pi\)
\(860\) 0 0
\(861\) −55.0000 −1.87439
\(862\) 0 0
\(863\) 6.00000 0.204242 0.102121 − 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −17.0000 −0.576686
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) −7.00000 −0.236914
\(874\) 0 0
\(875\) −5.00000 −0.169031
\(876\) 0 0
\(877\) −38.0000 −1.28317 −0.641584 − 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 0 0
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) 40.0000 1.34763 0.673817 − 0.738898i \(-0.264654\pi\)
0.673817 + 0.738898i \(0.264654\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) 0 0
\(887\) 25.0000 0.839418 0.419709 − 0.907659i \(-0.362132\pi\)
0.419709 + 0.907659i \(0.362132\pi\)
\(888\) 0 0
\(889\) 20.0000 0.670778
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −18.0000 −0.601674
\(896\) 0 0
\(897\) −3.00000 −0.100167
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −27.0000 −0.899500
\(902\) 0 0
\(903\) −30.0000 −0.998337
\(904\) 0 0
\(905\) 23.0000 0.764546
\(906\) 0 0
\(907\) −18.0000 −0.597680 −0.298840 − 0.954303i \(-0.596600\pi\)
−0.298840 + 0.954303i \(0.596600\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 22.0000 0.728893 0.364446 − 0.931224i \(-0.381258\pi\)
0.364446 + 0.931224i \(0.381258\pi\)
\(912\) 0 0
\(913\) −4.00000 −0.132381
\(914\) 0 0
\(915\) 5.00000 0.165295
\(916\) 0 0
\(917\) −20.0000 −0.660458
\(918\) 0 0
\(919\) −21.0000 −0.692726 −0.346363 − 0.938101i \(-0.612583\pi\)
−0.346363 + 0.938101i \(0.612583\pi\)
\(920\) 0 0
\(921\) −25.0000 −0.823778
\(922\) 0 0
\(923\) 13.0000 0.427900
\(924\) 0 0
\(925\) −5.00000 −0.164399
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) −23.0000 −0.754606 −0.377303 − 0.926090i \(-0.623148\pi\)
−0.377303 + 0.926090i \(0.623148\pi\)
\(930\) 0 0
\(931\) −108.000 −3.53956
\(932\) 0 0
\(933\) 10.0000 0.327385
\(934\) 0 0
\(935\) −3.00000 −0.0981105
\(936\) 0 0
\(937\) −44.0000 −1.43742 −0.718709 − 0.695311i \(-0.755266\pi\)
−0.718709 + 0.695311i \(0.755266\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −11.0000 −0.358590 −0.179295 − 0.983795i \(-0.557382\pi\)
−0.179295 + 0.983795i \(0.557382\pi\)
\(942\) 0 0
\(943\) 33.0000 1.07463
\(944\) 0 0
\(945\) 5.00000 0.162650
\(946\) 0 0
\(947\) −32.0000 −1.03986 −0.519930 − 0.854209i \(-0.674042\pi\)
−0.519930 + 0.854209i \(0.674042\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) −25.0000 −0.809829 −0.404915 − 0.914354i \(-0.632699\pi\)
−0.404915 + 0.914354i \(0.632699\pi\)
\(954\) 0 0
\(955\) −18.0000 −0.582466
\(956\) 0 0
\(957\) −4.00000 −0.129302
\(958\) 0 0
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −13.0000 −0.418919
\(964\) 0 0
\(965\) 21.0000 0.676014
\(966\) 0 0
\(967\) 56.0000 1.80084 0.900419 − 0.435023i \(-0.143260\pi\)
0.900419 + 0.435023i \(0.143260\pi\)
\(968\) 0 0
\(969\) 18.0000 0.578243
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 − 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 0 0
\(973\) 45.0000 1.44263
\(974\) 0 0
\(975\) −1.00000 −0.0320256
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) −11.0000 −0.351562
\(980\) 0 0
\(981\) −8.00000 −0.255420
\(982\) 0 0
\(983\) −16.0000 −0.510321 −0.255160 − 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 18.0000 0.572367
\(990\) 0 0
\(991\) −35.0000 −1.11181 −0.555906 − 0.831245i \(-0.687628\pi\)
−0.555906 + 0.831245i \(0.687628\pi\)
\(992\) 0 0
\(993\) 14.0000 0.444277
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) −56.0000 −1.77354 −0.886769 − 0.462213i \(-0.847056\pi\)
−0.886769 + 0.462213i \(0.847056\pi\)
\(998\) 0 0
\(999\) 5.00000 0.158193
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 6240.2.a.h.1.1 ✓ 1
4.3 odd 2 6240.2.a.q.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6240.2.a.h.1.1 ✓ 1 1.1 even 1 trivial
6240.2.a.q.1.1 yes 1 4.3 odd 2