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Properties

Label 2415.2.a.f.1.1
Level $2415$
Weight $2$
Character 2415.1
Self dual yes
Analytic conductor $19.284$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2415,2,Mod(1,2415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2415, 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("2415.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2415.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: \(19.2838720881\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 2415.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -2.00000 q^{12} -1.00000 q^{15} +4.00000 q^{16} +2.00000 q^{17} -3.00000 q^{19} +2.00000 q^{20} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{28} -6.00000 q^{29} +1.00000 q^{33} +1.00000 q^{35} -2.00000 q^{36} +6.00000 q^{37} -3.00000 q^{41} -6.00000 q^{43} -2.00000 q^{44} -1.00000 q^{45} -7.00000 q^{47} +4.00000 q^{48} +1.00000 q^{49} +2.00000 q^{51} -7.00000 q^{53} -1.00000 q^{55} -3.00000 q^{57} -3.00000 q^{59} +2.00000 q^{60} +1.00000 q^{61} -1.00000 q^{63} -8.00000 q^{64} -2.00000 q^{67} -4.00000 q^{68} +1.00000 q^{69} +12.0000 q^{71} -14.0000 q^{73} +1.00000 q^{75} +6.00000 q^{76} -1.00000 q^{77} -10.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +4.00000 q^{83} +2.00000 q^{84} -2.00000 q^{85} -6.00000 q^{87} -8.00000 q^{89} -2.00000 q^{92} +3.00000 q^{95} -10.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.00000 0.577350
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 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 \(0.451829\pi\)
\(12\) −2.00000 −0.577350
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 4.00000 1.00000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 − 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 2.00000 0.447214
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 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
\(35\) 1.00000 0.169031
\(36\) −2.00000 −0.333333
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 − 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −2.00000 −0.301511
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −7.00000 −1.02105 −0.510527 − 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 4.00000 0.577350
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −7.00000 −0.961524 −0.480762 − 0.876851i \(-0.659640\pi\)
−0.480762 + 0.876851i \(0.659640\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −3.00000 −0.397360
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 − 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 2.00000 0.258199
\(61\) 1.00000 0.128037 0.0640184 − 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −4.00000 −0.485071
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 6.00000 0.688247
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 2.00000 0.218218
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) 0 0
\(94\) 0 0
\(95\) 3.00000 0.307794
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −2.00000 −0.200000
\(101\) −11.0000 −1.09454 −0.547270 − 0.836956i \(-0.684333\pi\)
−0.547270 + 0.836956i \(0.684333\pi\)
\(102\) 0 0
\(103\) −7.00000 −0.689730 −0.344865 − 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −2.00000 −0.192450
\(109\) −6.00000 −0.574696 −0.287348 − 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) −4.00000 −0.377964
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 12.0000 1.11417
\(117\) 0 0
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) −3.00000 −0.270501
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 7.00000 0.621150 0.310575 − 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −9.00000 −0.786334 −0.393167 − 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) −2.00000 −0.174078
\(133\) 3.00000 0.260133
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −3.00000 −0.256307 −0.128154 − 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) −2.00000 −0.169031
\(141\) −7.00000 −0.589506
\(142\) 0 0
\(143\) 0 0
\(144\) 4.00000 0.333333
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) −12.0000 −0.986394
\(149\) 1.00000 0.0819232 0.0409616 − 0.999161i \(-0.486958\pi\)
0.0409616 + 0.999161i \(0.486958\pi\)
\(150\) 0 0
\(151\) −3.00000 −0.244137 −0.122068 − 0.992522i \(-0.538953\pi\)
−0.122068 + 0.992522i \(0.538953\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.00000 0.558661 0.279330 − 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 0 0
\(159\) −7.00000 −0.555136
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −17.0000 −1.33154 −0.665771 − 0.746156i \(-0.731897\pi\)
−0.665771 + 0.746156i \(0.731897\pi\)
\(164\) 6.00000 0.468521
\(165\) −1.00000 −0.0778499
\(166\) 0 0
\(167\) 13.0000 1.00597 0.502985 − 0.864295i \(-0.332235\pi\)
0.502985 + 0.864295i \(0.332235\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) 12.0000 0.914991
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 4.00000 0.301511
\(177\) −3.00000 −0.225494
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 − 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 2.00000 0.149071
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 1.00000 0.0739221
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 14.0000 1.02105
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 − 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) −8.00000 −0.577350
\(193\) 19.0000 1.36765 0.683825 − 0.729646i \(-0.260315\pi\)
0.683825 + 0.729646i \(0.260315\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) −8.00000 −0.569976 −0.284988 − 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 0 0
\(199\) −23.0000 −1.63043 −0.815213 − 0.579161i \(-0.803380\pi\)
−0.815213 + 0.579161i \(0.803380\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) −4.00000 −0.280056
\(205\) 3.00000 0.209529
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 − 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) 14.0000 0.961524
\(213\) 12.0000 0.822226
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −14.0000 −0.946032
\(220\) 2.00000 0.134840
\(221\) 0 0
\(222\) 0 0
\(223\) −20.0000 −1.33930 −0.669650 − 0.742677i \(-0.733556\pi\)
−0.669650 + 0.742677i \(0.733556\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 − 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 6.00000 0.397360
\(229\) 9.00000 0.594737 0.297368 − 0.954763i \(-0.403891\pi\)
0.297368 + 0.954763i \(0.403891\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) 2.00000 0.131024 0.0655122 − 0.997852i \(-0.479132\pi\)
0.0655122 + 0.997852i \(0.479132\pi\)
\(234\) 0 0
\(235\) 7.00000 0.456630
\(236\) 6.00000 0.390567
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) 18.0000 1.16432 0.582162 − 0.813073i \(-0.302207\pi\)
0.582162 + 0.813073i \(0.302207\pi\)
\(240\) −4.00000 −0.258199
\(241\) −5.00000 −0.322078 −0.161039 − 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 28.0000 1.76734 0.883672 − 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 2.00000 0.125988
\(253\) 1.00000 0.0628695
\(254\) 0 0
\(255\) −2.00000 −0.125245
\(256\) 16.0000 1.00000
\(257\) −17.0000 −1.06043 −0.530215 − 0.847863i \(-0.677889\pi\)
−0.530215 + 0.847863i \(0.677889\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 27.0000 1.66489 0.832446 − 0.554107i \(-0.186940\pi\)
0.832446 + 0.554107i \(0.186940\pi\)
\(264\) 0 0
\(265\) 7.00000 0.430007
\(266\) 0 0
\(267\) −8.00000 −0.489592
\(268\) 4.00000 0.244339
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 − 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 8.00000 0.485071
\(273\) 0 0
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) −2.00000 −0.120386
\(277\) −7.00000 −0.420589 −0.210295 − 0.977638i \(-0.567442\pi\)
−0.210295 + 0.977638i \(0.567442\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 14.0000 0.835170 0.417585 − 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) 0 0
\(283\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −24.0000 −1.42414
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) 3.00000 0.177084
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 28.0000 1.63858
\(293\) 24.0000 1.40209 0.701047 − 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 3.00000 0.174667
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 0 0
\(300\) −2.00000 −0.115470
\(301\) 6.00000 0.345834
\(302\) 0 0
\(303\) −11.0000 −0.631933
\(304\) −12.0000 −0.688247
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 2.00000 0.113961
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) 17.0000 0.963982 0.481991 − 0.876176i \(-0.339914\pi\)
0.481991 + 0.876176i \(0.339914\pi\)
\(312\) 0 0
\(313\) −5.00000 −0.282617 −0.141308 − 0.989966i \(-0.545131\pi\)
−0.141308 + 0.989966i \(0.545131\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 20.0000 1.12509
\(317\) −10.0000 −0.561656 −0.280828 − 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 8.00000 0.447214
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) −2.00000 −0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) −6.00000 −0.331801
\(328\) 0 0
\(329\) 7.00000 0.385922
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 − 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) −8.00000 −0.439057
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 2.00000 0.109272
\(336\) −4.00000 −0.218218
\(337\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 4.00000 0.216930
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) 14.0000 0.751559 0.375780 − 0.926709i \(-0.377375\pi\)
0.375780 + 0.926709i \(0.377375\pi\)
\(348\) 12.0000 0.643268
\(349\) 4.00000 0.214115 0.107058 − 0.994253i \(-0.465857\pi\)
0.107058 + 0.994253i \(0.465857\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 − 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 16.0000 0.847998
\(357\) −2.00000 −0.105851
\(358\) 0 0
\(359\) 8.00000 0.422224 0.211112 − 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 0 0
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) −29.0000 −1.51379 −0.756894 − 0.653538i \(-0.773284\pi\)
−0.756894 + 0.653538i \(0.773284\pi\)
\(368\) 4.00000 0.208514
\(369\) −3.00000 −0.156174
\(370\) 0 0
\(371\) 7.00000 0.363422
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 − 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 − 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) −6.00000 −0.307794
\(381\) 7.00000 0.358621
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 − 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 0 0
\(387\) −6.00000 −0.304997
\(388\) 20.0000 1.01535
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) −9.00000 −0.453990
\(394\) 0 0
\(395\) 10.0000 0.503155
\(396\) −2.00000 −0.100504
\(397\) −32.0000 −1.60603 −0.803017 − 0.595956i \(-0.796773\pi\)
−0.803017 + 0.595956i \(0.796773\pi\)
\(398\) 0 0
\(399\) 3.00000 0.150188
\(400\) 4.00000 0.200000
\(401\) 5.00000 0.249688 0.124844 − 0.992176i \(-0.460157\pi\)
0.124844 + 0.992176i \(0.460157\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 22.0000 1.09454
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 − 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) −3.00000 −0.147979
\(412\) 14.0000 0.689730
\(413\) 3.00000 0.147620
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) 20.0000 0.977064 0.488532 − 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) −2.00000 −0.0975900
\(421\) −6.00000 −0.292422 −0.146211 − 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) −7.00000 −0.340352
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −1.00000 −0.0483934
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) −15.0000 −0.722525 −0.361262 − 0.932464i \(-0.617654\pi\)
−0.361262 + 0.932464i \(0.617654\pi\)
\(432\) 4.00000 0.192450
\(433\) −7.00000 −0.336399 −0.168199 − 0.985753i \(-0.553795\pi\)
−0.168199 + 0.985753i \(0.553795\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 12.0000 0.574696
\(437\) −3.00000 −0.143509
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 − 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −14.0000 −0.665160 −0.332580 − 0.943075i \(-0.607919\pi\)
−0.332580 + 0.943075i \(0.607919\pi\)
\(444\) −12.0000 −0.569495
\(445\) 8.00000 0.379236
\(446\) 0 0
\(447\) 1.00000 0.0472984
\(448\) 8.00000 0.377964
\(449\) −6.00000 −0.283158 −0.141579 − 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) −3.00000 −0.141264
\(452\) −12.0000 −0.564433
\(453\) −3.00000 −0.140952
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 2.00000 0.0932505
\(461\) 26.0000 1.21094 0.605470 − 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 0 0
\(463\) −5.00000 −0.232370 −0.116185 − 0.993228i \(-0.537067\pi\)
−0.116185 + 0.993228i \(0.537067\pi\)
\(464\) −24.0000 −1.11417
\(465\) 0 0
\(466\) 0 0
\(467\) −38.0000 −1.75843 −0.879215 − 0.476425i \(-0.841932\pi\)
−0.879215 + 0.476425i \(0.841932\pi\)
\(468\) 0 0
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) 7.00000 0.322543
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) −3.00000 −0.137649
\(476\) 4.00000 0.183340
\(477\) −7.00000 −0.320508
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 20.0000 0.909091
\(485\) 10.0000 0.454077
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 − 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) −17.0000 −0.768767
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 6.00000 0.270501
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 − 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 2.00000 0.0894427
\(501\) 13.0000 0.580797
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 − 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 11.0000 0.489494
\(506\) 0 0
\(507\) −13.0000 −0.577350
\(508\) −14.0000 −0.621150
\(509\) 5.00000 0.221621 0.110811 − 0.993842i \(-0.464655\pi\)
0.110811 + 0.993842i \(0.464655\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 0 0
\(513\) −3.00000 −0.132453
\(514\) 0 0
\(515\) 7.00000 0.308457
\(516\) 12.0000 0.528271
\(517\) −7.00000 −0.307860
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 − 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) 15.0000 0.655904 0.327952 − 0.944694i \(-0.393642\pi\)
0.327952 + 0.944694i \(0.393642\pi\)
\(524\) 18.0000 0.786334
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −3.00000 −0.130189
\(532\) −6.00000 −0.260133
\(533\) 0 0
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 0 0
\(537\) −20.0000 −0.863064
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 2.00000 0.0860663
\(541\) 35.0000 1.50477 0.752384 − 0.658725i \(-0.228904\pi\)
0.752384 + 0.658725i \(0.228904\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 − 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 6.00000 0.256307
\(549\) 1.00000 0.0426790
\(550\) 0 0
\(551\) 18.0000 0.766826
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) 0 0
\(555\) −6.00000 −0.254686
\(556\) −24.0000 −1.01783
\(557\) 6.00000 0.254228 0.127114 − 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 4.00000 0.169031
\(561\) 2.00000 0.0844401
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 14.0000 0.589506
\(565\) −6.00000 −0.252422
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 29.0000 1.21574 0.607872 − 0.794035i \(-0.292024\pi\)
0.607872 + 0.794035i \(0.292024\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 − 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 3.00000 0.125327
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) −8.00000 −0.333333
\(577\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) 19.0000 0.789613
\(580\) −12.0000 −0.498273
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) −7.00000 −0.289910
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.00000 0.371470 0.185735 − 0.982600i \(-0.440533\pi\)
0.185735 + 0.982600i \(0.440533\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 0 0
\(590\) 0 0
\(591\) −8.00000 −0.329076
\(592\) 24.0000 0.986394
\(593\) −22.0000 −0.903432 −0.451716 − 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) −2.00000 −0.0819232
\(597\) −23.0000 −0.941327
\(598\) 0 0
\(599\) −4.00000 −0.163436 −0.0817178 − 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 − 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 6.00000 0.244137
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) 30.0000 1.21766 0.608831 − 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 0 0
\(612\) −4.00000 −0.161690
\(613\) 4.00000 0.161558 0.0807792 − 0.996732i \(-0.474259\pi\)
0.0807792 + 0.996732i \(0.474259\pi\)
\(614\) 0 0
\(615\) 3.00000 0.120972
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 − 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 − 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 8.00000 0.320513
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −3.00000 −0.119808
\(628\) −14.0000 −0.558661
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 28.0000 1.11466 0.557331 − 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) 0 0
\(633\) −3.00000 −0.119239
\(634\) 0 0
\(635\) −7.00000 −0.277787
\(636\) 14.0000 0.555136
\(637\) 0 0
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −15.0000 −0.592464 −0.296232 − 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 0 0
\(643\) −47.0000 −1.85350 −0.926750 − 0.375680i \(-0.877409\pi\)
−0.926750 + 0.375680i \(0.877409\pi\)
\(644\) 2.00000 0.0788110
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 − 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) −3.00000 −0.117760
\(650\) 0 0
\(651\) 0 0
\(652\) 34.0000 1.33154
\(653\) 16.0000 0.626128 0.313064 − 0.949732i \(-0.398644\pi\)
0.313064 + 0.949732i \(0.398644\pi\)
\(654\) 0 0
\(655\) 9.00000 0.351659
\(656\) −12.0000 −0.468521
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 − 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 2.00000 0.0778499
\(661\) 1.00000 0.0388955 0.0194477 − 0.999811i \(-0.493809\pi\)
0.0194477 + 0.999811i \(0.493809\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.00000 −0.116335
\(666\) 0 0
\(667\) −6.00000 −0.232321
\(668\) −26.0000 −1.00597
\(669\) −20.0000 −0.773245
\(670\) 0 0
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) −29.0000 −1.11787 −0.558934 − 0.829212i \(-0.688789\pi\)
−0.558934 + 0.829212i \(0.688789\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 26.0000 1.00000
\(677\) −20.0000 −0.768662 −0.384331 − 0.923195i \(-0.625568\pi\)
−0.384331 + 0.923195i \(0.625568\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 − 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 6.00000 0.229416
\(685\) 3.00000 0.114624
\(686\) 0 0
\(687\) 9.00000 0.343371
\(688\) −24.0000 −0.914991
\(689\) 0 0
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 − 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) −12.0000 −0.456172
\(693\) −1.00000 −0.0379869
\(694\) 0 0
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) 0 0
\(699\) 2.00000 0.0756469
\(700\) 2.00000 0.0755929
\(701\) 39.0000 1.47301 0.736505 − 0.676432i \(-0.236475\pi\)
0.736505 + 0.676432i \(0.236475\pi\)
\(702\) 0 0
\(703\) −18.0000 −0.678883
\(704\) −8.00000 −0.301511
\(705\) 7.00000 0.263635
\(706\) 0 0
\(707\) 11.0000 0.413698
\(708\) 6.00000 0.225494
\(709\) −28.0000 −1.05156 −0.525781 − 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 40.0000 1.49487
\(717\) 18.0000 0.672222
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 − 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −4.00000 −0.149071
\(721\) 7.00000 0.260694
\(722\) 0 0
\(723\) −5.00000 −0.185952
\(724\) −20.0000 −0.743294
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −47.0000 −1.74313 −0.871567 − 0.490277i \(-0.836896\pi\)
−0.871567 + 0.490277i \(0.836896\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) −2.00000 −0.0739221
\(733\) 26.0000 0.960332 0.480166 − 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) −2.00000 −0.0736709
\(738\) 0 0
\(739\) 32.0000 1.17714 0.588570 − 0.808447i \(-0.299691\pi\)
0.588570 + 0.808447i \(0.299691\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) 0 0
\(743\) 7.00000 0.256805 0.128403 − 0.991722i \(-0.459015\pi\)
0.128403 + 0.991722i \(0.459015\pi\)
\(744\) 0 0
\(745\) −1.00000 −0.0366372
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) −4.00000 −0.146254
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 − 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −28.0000 −1.02105
\(753\) 28.0000 1.02038
\(754\) 0 0
\(755\) 3.00000 0.109181
\(756\) 2.00000 0.0727393
\(757\) 20.0000 0.726912 0.363456 − 0.931611i \(-0.381597\pi\)
0.363456 + 0.931611i \(0.381597\pi\)
\(758\) 0 0
\(759\) 1.00000 0.0362977
\(760\) 0 0
\(761\) −21.0000 −0.761249 −0.380625 − 0.924730i \(-0.624291\pi\)
−0.380625 + 0.924730i \(0.624291\pi\)
\(762\) 0 0
\(763\) 6.00000 0.217215
\(764\) −6.00000 −0.217072
\(765\) −2.00000 −0.0723102
\(766\) 0 0
\(767\) 0 0
\(768\) 16.0000 0.577350
\(769\) 26.0000 0.937584 0.468792 − 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) −17.0000 −0.612240
\(772\) −38.0000 −1.36765
\(773\) 4.00000 0.143870 0.0719350 − 0.997409i \(-0.477083\pi\)
0.0719350 + 0.997409i \(0.477083\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −6.00000 −0.215249
\(778\) 0 0
\(779\) 9.00000 0.322458
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 4.00000 0.142857
\(785\) −7.00000 −0.249841
\(786\) 0 0
\(787\) −7.00000 −0.249523 −0.124762 − 0.992187i \(-0.539817\pi\)
−0.124762 + 0.992187i \(0.539817\pi\)
\(788\) 16.0000 0.569976
\(789\) 27.0000 0.961225
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 7.00000 0.248264
\(796\) 46.0000 1.63043
\(797\) −28.0000 −0.991811 −0.495905 − 0.868377i \(-0.665164\pi\)
−0.495905 + 0.868377i \(0.665164\pi\)
\(798\) 0 0
\(799\) −14.0000 −0.495284
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) 0 0
\(803\) −14.0000 −0.494049
\(804\) 4.00000 0.141069
\(805\) 1.00000 0.0352454
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 0 0
\(809\) −4.00000 −0.140633 −0.0703163 − 0.997525i \(-0.522401\pi\)
−0.0703163 + 0.997525i \(0.522401\pi\)
\(810\) 0 0
\(811\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) −12.0000 −0.421117
\(813\) 2.00000 0.0701431
\(814\) 0 0
\(815\) 17.0000 0.595484
\(816\) 8.00000 0.280056
\(817\) 18.0000 0.629740
\(818\) 0 0
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 30.0000 1.04701 0.523504 − 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) −11.0000 −0.383436 −0.191718 − 0.981450i \(-0.561406\pi\)
−0.191718 + 0.981450i \(0.561406\pi\)
\(824\) 0 0
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) 27.0000 0.938882 0.469441 − 0.882964i \(-0.344455\pi\)
0.469441 + 0.882964i \(0.344455\pi\)
\(828\) −2.00000 −0.0695048
\(829\) 38.0000 1.31979 0.659897 − 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) −7.00000 −0.242827
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −13.0000 −0.449884
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 − 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 14.0000 0.482186
\(844\) 6.00000 0.206529
\(845\) 13.0000 0.447214
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) −28.0000 −0.961524
\(849\) 0 0
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) −24.0000 −0.822226
\(853\) −56.0000 −1.91740 −0.958702 − 0.284413i \(-0.908201\pi\)
−0.958702 + 0.284413i \(0.908201\pi\)
\(854\) 0 0
\(855\) 3.00000 0.102598
\(856\) 0 0
\(857\) −33.0000 −1.12726 −0.563629 − 0.826028i \(-0.690595\pi\)
−0.563629 + 0.826028i \(0.690595\pi\)
\(858\) 0 0
\(859\) −2.00000 −0.0682391 −0.0341196 − 0.999418i \(-0.510863\pi\)
−0.0341196 + 0.999418i \(0.510863\pi\)
\(860\) −12.0000 −0.409197
\(861\) 3.00000 0.102240
\(862\) 0 0
\(863\) 4.00000 0.136162 0.0680808 − 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −10.0000 −0.339227
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 28.0000 0.946032
\(877\) 33.0000 1.11433 0.557165 − 0.830402i \(-0.311889\pi\)
0.557165 + 0.830402i \(0.311889\pi\)
\(878\) 0 0
\(879\) 24.0000 0.809500
\(880\) −4.00000 −0.134840
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 − 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 3.00000 0.100844
\(886\) 0 0
\(887\) −36.0000 −1.20876 −0.604381 − 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) −7.00000 −0.234772
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 40.0000 1.33930
\(893\) 21.0000 0.702738
\(894\) 0 0
\(895\) 20.0000 0.668526
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −2.00000 −0.0666667
\(901\) −14.0000 −0.466408
\(902\) 0 0
\(903\) 6.00000 0.199667
\(904\) 0 0
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) −32.0000 −1.06254 −0.531271 − 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) 8.00000 0.265489
\(909\) −11.0000 −0.364847
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 − 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) −12.0000 −0.397360
\(913\) 4.00000 0.132381
\(914\) 0 0
\(915\) −1.00000 −0.0330590
\(916\) −18.0000 −0.594737
\(917\) 9.00000 0.297206
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 − 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 2.00000 0.0657952
\(925\) 6.00000 0.197279
\(926\) 0 0
\(927\) −7.00000 −0.229910
\(928\) 0 0
\(929\) 42.0000 1.37798 0.688988 − 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) −4.00000 −0.131024
\(933\) 17.0000 0.556555
\(934\) 0 0
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) 25.0000 0.816714 0.408357 − 0.912822i \(-0.366102\pi\)
0.408357 + 0.912822i \(0.366102\pi\)
\(938\) 0 0
\(939\) −5.00000 −0.163169
\(940\) −14.0000 −0.456630
\(941\) −60.0000 −1.95594 −0.977972 − 0.208736i \(-0.933065\pi\)
−0.977972 + 0.208736i \(0.933065\pi\)
\(942\) 0 0
\(943\) −3.00000 −0.0976934
\(944\) −12.0000 −0.390567
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) 2.00000 0.0649913 0.0324956 − 0.999472i \(-0.489654\pi\)
0.0324956 + 0.999472i \(0.489654\pi\)
\(948\) 20.0000 0.649570
\(949\) 0 0
\(950\) 0 0
\(951\) −10.0000 −0.324272
\(952\) 0 0
\(953\) 13.0000 0.421111 0.210556 − 0.977582i \(-0.432473\pi\)
0.210556 + 0.977582i \(0.432473\pi\)
\(954\) 0 0
\(955\) −3.00000 −0.0970777
\(956\) −36.0000 −1.16432
\(957\) −6.00000 −0.193952
\(958\) 0 0
\(959\) 3.00000 0.0968751
\(960\) 8.00000 0.258199
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) 10.0000 0.322078
\(965\) −19.0000 −0.611632
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 − 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 0 0
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 − 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) −2.00000 −0.0641500
\(973\) −12.0000 −0.384702
\(974\) 0 0
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) −35.0000 −1.11975 −0.559875 − 0.828577i \(-0.689151\pi\)
−0.559875 + 0.828577i \(0.689151\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 2.00000 0.0638877
\(981\) −6.00000 −0.191565
\(982\) 0 0
\(983\) 30.0000 0.956851 0.478426 − 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 0 0
\(985\) 8.00000 0.254901
\(986\) 0 0
\(987\) 7.00000 0.222812
\(988\) 0 0
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) 57.0000 1.81066 0.905332 − 0.424704i \(-0.139622\pi\)
0.905332 + 0.424704i \(0.139622\pi\)
\(992\) 0 0
\(993\) 17.0000 0.539479
\(994\) 0 0
\(995\) 23.0000 0.729149
\(996\) −8.00000 −0.253490
\(997\) −22.0000 −0.696747 −0.348373 − 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) 0 0
\(999\) 6.00000 0.189832
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 2415.2.a.f.1.1 ✓ 1
3.2 odd 2 7245.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2415.2.a.f.1.1 ✓ 1 1.1 even 1 trivial
7245.2.a.i.1.1 1 3.2 odd 2