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Properties

Label 2450.2.a.k.1.1
Level $2450$
Weight $2$
Character 2450.1
Self dual yes
Analytic conductor $19.563$
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] = [2450,2,Mod(1,2450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.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: \(19.5633484952\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{8} -3.00000 q^{9} +3.00000 q^{11} +5.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} +3.00000 q^{18} -5.00000 q^{19} -3.00000 q^{22} +7.00000 q^{23} -5.00000 q^{26} -4.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} -3.00000 q^{36} -1.00000 q^{37} +5.00000 q^{38} +3.00000 q^{41} -2.00000 q^{43} +3.00000 q^{44} -7.00000 q^{46} +7.00000 q^{47} +5.00000 q^{52} -9.00000 q^{53} +4.00000 q^{58} -4.00000 q^{59} +6.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} -2.00000 q^{67} +2.00000 q^{68} -6.00000 q^{71} +3.00000 q^{72} +16.0000 q^{73} +1.00000 q^{74} -5.00000 q^{76} +14.0000 q^{79} +9.00000 q^{81} -3.00000 q^{82} +6.00000 q^{83} +2.00000 q^{86} -3.00000 q^{88} +2.00000 q^{89} +7.00000 q^{92} -7.00000 q^{94} +12.0000 q^{97} -9.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 3.00000 0.707107
\(19\) −5.00000 −1.14708 −0.573539 − 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) 7.00000 1.45960 0.729800 − 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −5.00000 −0.980581
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 5.00000 0.811107
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −7.00000 −1.03209
\(47\) 7.00000 1.02105 0.510527 − 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 5.00000 0.693375
\(53\) −9.00000 −1.23625 −0.618123 − 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 4.00000 0.525226
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 3.00000 0.353553
\(73\) 16.0000 1.87266 0.936329 − 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −5.00000 −0.573539
\(77\) 0 0
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 − 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −3.00000 −0.331295
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 7.00000 0.729800
\(93\) 0 0
\(94\) −7.00000 −0.721995
\(95\) 0 0
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 0 0
\(99\) −9.00000 −0.904534
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −5.00000 −0.490290
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) 16.0000 1.54678 0.773389 − 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 − 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) −15.0000 −1.38675
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −6.00000 −0.543214
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 − 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −1.00000 −0.0873704 −0.0436852 − 0.999045i \(-0.513910\pi\)
−0.0436852 + 0.999045i \(0.513910\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 8.00000 0.683486 0.341743 − 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 − 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) 15.0000 1.25436
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) −16.0000 −1.32417
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) −18.0000 −1.47462 −0.737309 − 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −6.00000 −0.488273 −0.244137 − 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) 5.00000 0.405554
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 9.00000 0.718278 0.359139 − 0.933284i \(-0.383070\pi\)
0.359139 + 0.933284i \(0.383070\pi\)
\(158\) −14.0000 −1.11378
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −9.00000 −0.707107
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) −15.0000 −1.16073 −0.580367 − 0.814355i \(-0.697091\pi\)
−0.580367 + 0.814355i \(0.697091\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 15.0000 1.14708
\(172\) −2.00000 −0.152499
\(173\) −9.00000 −0.684257 −0.342129 − 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) −2.00000 −0.149906
\(179\) 13.0000 0.971666 0.485833 − 0.874052i \(-0.338516\pi\)
0.485833 + 0.874052i \(0.338516\pi\)
\(180\) 0 0
\(181\) 26.0000 1.93256 0.966282 − 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −7.00000 −0.516047
\(185\) 0 0
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) 7.00000 0.510527
\(189\) 0 0
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 − 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 − 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 0 0
\(197\) 5.00000 0.356235 0.178118 − 0.984009i \(-0.442999\pi\)
0.178118 + 0.984009i \(0.442999\pi\)
\(198\) 9.00000 0.639602
\(199\) 18.0000 1.27599 0.637993 − 0.770042i \(-0.279765\pi\)
0.637993 + 0.770042i \(0.279765\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) −21.0000 −1.45960
\(208\) 5.00000 0.346688
\(209\) −15.0000 −1.03757
\(210\) 0 0
\(211\) −9.00000 −0.619586 −0.309793 − 0.950804i \(-0.600260\pi\)
−0.309793 + 0.950804i \(0.600260\pi\)
\(212\) −9.00000 −0.618123
\(213\) 0 0
\(214\) −16.0000 −1.09374
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) 0 0
\(221\) 10.0000 0.672673
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) 6.00000 0.398234 0.199117 − 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 − 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4.00000 0.262613
\(233\) 8.00000 0.524097 0.262049 − 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 15.0000 0.980581
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 − 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) −9.00000 −0.579741 −0.289870 − 0.957066i \(-0.593612\pi\)
−0.289870 + 0.957066i \(0.593612\pi\)
\(242\) 2.00000 0.128565
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) 0 0
\(247\) −25.0000 −1.59071
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) 0 0
\(251\) 5.00000 0.315597 0.157799 − 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(252\) 0 0
\(253\) 21.0000 1.32026
\(254\) 7.00000 0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 1.00000 0.0617802
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −2.00000 −0.122169
\(269\) −10.0000 −0.609711 −0.304855 − 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 − 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −8.00000 −0.483298
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 − 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −16.0000 −0.959616
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 9.00000 0.536895 0.268447 − 0.963294i \(-0.413489\pi\)
0.268447 + 0.963294i \(0.413489\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 − 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) −15.0000 −0.886969
\(287\) 0 0
\(288\) 3.00000 0.176777
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 16.0000 0.936329
\(293\) 9.00000 0.525786 0.262893 − 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) 35.0000 2.02410
\(300\) 0 0
\(301\) 0 0
\(302\) 6.00000 0.345261
\(303\) 0 0
\(304\) −5.00000 −0.286770
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 22.0000 1.25561 0.627803 − 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 − 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 − 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) −9.00000 −0.507899
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) 2.00000 0.112331 0.0561656 − 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −10.0000 −0.556415
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) −3.00000 −0.165647
\(329\) 0 0
\(330\) 0 0
\(331\) 5.00000 0.274825 0.137412 − 0.990514i \(-0.456121\pi\)
0.137412 + 0.990514i \(0.456121\pi\)
\(332\) 6.00000 0.329293
\(333\) 3.00000 0.164399
\(334\) 15.0000 0.820763
\(335\) 0 0
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 − 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) −12.0000 −0.652714
\(339\) 0 0
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) −15.0000 −0.811107
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) 9.00000 0.483843
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −12.0000 −0.642345 −0.321173 − 0.947021i \(-0.604077\pi\)
−0.321173 + 0.947021i \(0.604077\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) −24.0000 −1.27739 −0.638696 − 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) −13.0000 −0.687071
\(359\) 16.0000 0.844448 0.422224 − 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) −26.0000 −1.36653
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −13.0000 −0.678594 −0.339297 − 0.940679i \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(368\) 7.00000 0.364900
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 − 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) −7.00000 −0.360997
\(377\) −20.0000 −1.03005
\(378\) 0 0
\(379\) −29.0000 −1.48963 −0.744815 − 0.667271i \(-0.767462\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 20.0000 1.02329
\(383\) −21.0000 −1.07305 −0.536525 − 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 6.00000 0.304997
\(388\) 12.0000 0.609208
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 14.0000 0.708010
\(392\) 0 0
\(393\) 0 0
\(394\) −5.00000 −0.251896
\(395\) 0 0
\(396\) −9.00000 −0.452267
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −18.0000 −0.902258
\(399\) 0 0
\(400\) 0 0
\(401\) −15.0000 −0.749064 −0.374532 − 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) −10.0000 −0.498135
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.00000 −0.148704
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 − 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 21.0000 1.03209
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 15.0000 0.733674
\(419\) 35.0000 1.70986 0.854931 − 0.518742i \(-0.173599\pi\)
0.854931 + 0.518742i \(0.173599\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 − 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 9.00000 0.438113
\(423\) −21.0000 −1.02105
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) 0 0
\(431\) 2.00000 0.0963366 0.0481683 − 0.998839i \(-0.484662\pi\)
0.0481683 + 0.998839i \(0.484662\pi\)
\(432\) 0 0
\(433\) −28.0000 −1.34559 −0.672797 − 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −35.0000 −1.67428
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 − 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −10.0000 −0.475651
\(443\) −30.0000 −1.42534 −0.712672 − 0.701498i \(-0.752515\pi\)
−0.712672 + 0.701498i \(0.752515\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) 0 0
\(449\) 5.00000 0.235965 0.117982 − 0.993016i \(-0.462357\pi\)
0.117982 + 0.993016i \(0.462357\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −16.0000 −0.747631
\(459\) 0 0
\(460\) 0 0
\(461\) −32.0000 −1.49039 −0.745194 − 0.666847i \(-0.767643\pi\)
−0.745194 + 0.666847i \(0.767643\pi\)
\(462\) 0 0
\(463\) 17.0000 0.790057 0.395029 − 0.918669i \(-0.370735\pi\)
0.395029 + 0.918669i \(0.370735\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −8.00000 −0.370593
\(467\) −34.0000 −1.57333 −0.786666 − 0.617379i \(-0.788195\pi\)
−0.786666 + 0.617379i \(0.788195\pi\)
\(468\) −15.0000 −0.693375
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 27.0000 1.23625
\(478\) −20.0000 −0.914779
\(479\) 36.0000 1.64488 0.822441 − 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) −5.00000 −0.227980
\(482\) 9.00000 0.409939
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 − 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −6.00000 −0.271607
\(489\) 0 0
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 − 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) −8.00000 −0.360302
\(494\) 25.0000 1.12480
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 − 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −5.00000 −0.223161
\(503\) −40.0000 −1.78351 −0.891756 − 0.452517i \(-0.850526\pi\)
−0.891756 + 0.452517i \(0.850526\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −21.0000 −0.933564
\(507\) 0 0
\(508\) −7.00000 −0.310575
\(509\) −34.0000 −1.50702 −0.753512 − 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 0 0
\(517\) 21.0000 0.923579
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −27.0000 −1.18289 −0.591446 − 0.806345i \(-0.701443\pi\)
−0.591446 + 0.806345i \(0.701443\pi\)
\(522\) −12.0000 −0.525226
\(523\) −16.0000 −0.699631 −0.349816 − 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −1.00000 −0.0436852
\(525\) 0 0
\(526\) 0 0
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 15.0000 0.649722
\(534\) 0 0
\(535\) 0 0
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) 10.0000 0.431131
\(539\) 0 0
\(540\) 0 0
\(541\) −16.0000 −0.687894 −0.343947 − 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) −24.0000 −1.03089
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) 26.0000 1.11168 0.555840 − 0.831289i \(-0.312397\pi\)
0.555840 + 0.831289i \(0.312397\pi\)
\(548\) 8.00000 0.341743
\(549\) −18.0000 −0.768221
\(550\) 0 0
\(551\) 20.0000 0.852029
\(552\) 0 0
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) −23.0000 −0.974541 −0.487271 − 0.873251i \(-0.662007\pi\)
−0.487271 + 0.873251i \(0.662007\pi\)
\(558\) −6.00000 −0.254000
\(559\) −10.0000 −0.422955
\(560\) 0 0
\(561\) 0 0
\(562\) −9.00000 −0.379642
\(563\) −2.00000 −0.0842900 −0.0421450 − 0.999112i \(-0.513419\pi\)
−0.0421450 + 0.999112i \(0.513419\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) −15.0000 −0.628833 −0.314416 − 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 − 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 15.0000 0.627182
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −3.00000 −0.125000
\(577\) 4.00000 0.166522 0.0832611 − 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −27.0000 −1.11823
\(584\) −16.0000 −0.662085
\(585\) 0 0
\(586\) −9.00000 −0.371787
\(587\) −34.0000 −1.40333 −0.701665 − 0.712507i \(-0.747560\pi\)
−0.701665 + 0.712507i \(0.747560\pi\)
\(588\) 0 0
\(589\) 10.0000 0.412043
\(590\) 0 0
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) 0 0
\(598\) −35.0000 −1.43126
\(599\) −12.0000 −0.490307 −0.245153 − 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 − 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 6.00000 0.244339
\(604\) −6.00000 −0.244137
\(605\) 0 0
\(606\) 0 0
\(607\) 13.0000 0.527654 0.263827 − 0.964570i \(-0.415015\pi\)
0.263827 + 0.964570i \(0.415015\pi\)
\(608\) 5.00000 0.202777
\(609\) 0 0
\(610\) 0 0
\(611\) 35.0000 1.41595
\(612\) −6.00000 −0.242536
\(613\) 15.0000 0.605844 0.302922 − 0.953015i \(-0.402038\pi\)
0.302922 + 0.953015i \(0.402038\pi\)
\(614\) −22.0000 −0.887848
\(615\) 0 0
\(616\) 0 0
\(617\) −14.0000 −0.563619 −0.281809 − 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 0 0
\(619\) 19.0000 0.763674 0.381837 − 0.924230i \(-0.375291\pi\)
0.381837 + 0.924230i \(0.375291\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −6.00000 −0.240578
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) 9.00000 0.359139
\(629\) −2.00000 −0.0797452
\(630\) 0 0
\(631\) −18.0000 −0.716569 −0.358284 − 0.933613i \(-0.616638\pi\)
−0.358284 + 0.933613i \(0.616638\pi\)
\(632\) −14.0000 −0.556890
\(633\) 0 0
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 12.0000 0.475085
\(639\) 18.0000 0.712069
\(640\) 0 0
\(641\) −5.00000 −0.197488 −0.0987441 − 0.995113i \(-0.531483\pi\)
−0.0987441 + 0.995113i \(0.531483\pi\)
\(642\) 0 0
\(643\) 14.0000 0.552106 0.276053 − 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 10.0000 0.393445
\(647\) 27.0000 1.06148 0.530740 − 0.847535i \(-0.321914\pi\)
0.530740 + 0.847535i \(0.321914\pi\)
\(648\) −9.00000 −0.353553
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) −3.00000 −0.117399 −0.0586995 − 0.998276i \(-0.518695\pi\)
−0.0586995 + 0.998276i \(0.518695\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.00000 0.117130
\(657\) −48.0000 −1.87266
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 − 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 16.0000 0.622328 0.311164 − 0.950356i \(-0.399281\pi\)
0.311164 + 0.950356i \(0.399281\pi\)
\(662\) −5.00000 −0.194331
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) −3.00000 −0.116248
\(667\) −28.0000 −1.08416
\(668\) −15.0000 −0.580367
\(669\) 0 0
\(670\) 0 0
\(671\) 18.0000 0.694882
\(672\) 0 0
\(673\) −32.0000 −1.23351 −0.616755 − 0.787155i \(-0.711553\pi\)
−0.616755 + 0.787155i \(0.711553\pi\)
\(674\) 10.0000 0.385186
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) 17.0000 0.653363 0.326682 − 0.945134i \(-0.394070\pi\)
0.326682 + 0.945134i \(0.394070\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 6.00000 0.229752
\(683\) 44.0000 1.68361 0.841807 − 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 15.0000 0.573539
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −2.00000 −0.0762493
\(689\) −45.0000 −1.71436
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 − 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) −9.00000 −0.342129
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 6.00000 0.227266
\(698\) 12.0000 0.454207
\(699\) 0 0
\(700\) 0 0
\(701\) 26.0000 0.982006 0.491003 − 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 0 0
\(703\) 5.00000 0.188579
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 0 0
\(708\) 0 0
\(709\) 12.0000 0.450669 0.225335 − 0.974281i \(-0.427652\pi\)
0.225335 + 0.974281i \(0.427652\pi\)
\(710\) 0 0
\(711\) −42.0000 −1.57512
\(712\) −2.00000 −0.0749532
\(713\) −14.0000 −0.524304
\(714\) 0 0
\(715\) 0 0
\(716\) 13.0000 0.485833
\(717\) 0 0
\(718\) −16.0000 −0.597115
\(719\) 26.0000 0.969636 0.484818 − 0.874615i \(-0.338886\pi\)
0.484818 + 0.874615i \(0.338886\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −6.00000 −0.223297
\(723\) 0 0
\(724\) 26.0000 0.966282
\(725\) 0 0
\(726\) 0 0
\(727\) 29.0000 1.07555 0.537775 − 0.843088i \(-0.319265\pi\)
0.537775 + 0.843088i \(0.319265\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) 0 0
\(733\) −41.0000 −1.51437 −0.757185 − 0.653201i \(-0.773426\pi\)
−0.757185 + 0.653201i \(0.773426\pi\)
\(734\) 13.0000 0.479839
\(735\) 0 0
\(736\) −7.00000 −0.258023
\(737\) −6.00000 −0.221013
\(738\) 9.00000 0.331295
\(739\) −29.0000 −1.06678 −0.533391 − 0.845869i \(-0.679083\pi\)
−0.533391 + 0.845869i \(0.679083\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21.0000 0.770415 0.385208 − 0.922830i \(-0.374130\pi\)
0.385208 + 0.922830i \(0.374130\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 26.0000 0.951928
\(747\) −18.0000 −0.658586
\(748\) 6.00000 0.219382
\(749\) 0 0
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 − 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) 7.00000 0.255264
\(753\) 0 0
\(754\) 20.0000 0.728357
\(755\) 0 0
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 − 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 29.0000 1.05333
\(759\) 0 0
\(760\) 0 0
\(761\) 1.00000 0.0362500 0.0181250 − 0.999836i \(-0.494230\pi\)
0.0181250 + 0.999836i \(0.494230\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −20.0000 −0.723575
\(765\) 0 0
\(766\) 21.0000 0.758761
\(767\) −20.0000 −0.722158
\(768\) 0 0
\(769\) −29.0000 −1.04577 −0.522883 − 0.852404i \(-0.675144\pi\)
−0.522883 + 0.852404i \(0.675144\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10.0000 0.359908
\(773\) −45.0000 −1.61854 −0.809269 − 0.587439i \(-0.800136\pi\)
−0.809269 + 0.587439i \(0.800136\pi\)
\(774\) −6.00000 −0.215666
\(775\) 0 0
\(776\) −12.0000 −0.430775
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −15.0000 −0.537431
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) −14.0000 −0.500639
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −18.0000 −0.641631 −0.320815 − 0.947142i \(-0.603957\pi\)
−0.320815 + 0.947142i \(0.603957\pi\)
\(788\) 5.00000 0.178118
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 9.00000 0.319801
\(793\) 30.0000 1.06533
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 18.0000 0.637993
\(797\) 2.00000 0.0708436 0.0354218 − 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 14.0000 0.495284
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 15.0000 0.529668
\(803\) 48.0000 1.69388
\(804\) 0 0
\(805\) 0 0
\(806\) 10.0000 0.352235
\(807\) 0 0
\(808\) 0 0
\(809\) −5.00000 −0.175791 −0.0878953 − 0.996130i \(-0.528014\pi\)
−0.0878953 + 0.996130i \(0.528014\pi\)
\(810\) 0 0
\(811\) −33.0000 −1.15879 −0.579393 − 0.815048i \(-0.696710\pi\)
−0.579393 + 0.815048i \(0.696710\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 3.00000 0.105150
\(815\) 0 0
\(816\) 0 0
\(817\) 10.0000 0.349856
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 − 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 24.0000 0.836587 0.418294 − 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) 22.0000 0.765015 0.382507 − 0.923952i \(-0.375061\pi\)
0.382507 + 0.923952i \(0.375061\pi\)
\(828\) −21.0000 −0.729800
\(829\) −26.0000 −0.903017 −0.451509 − 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 5.00000 0.173344
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −15.0000 −0.518786
\(837\) 0 0
\(838\) −35.0000 −1.20905
\(839\) −34.0000 −1.17381 −0.586905 − 0.809656i \(-0.699654\pi\)
−0.586905 + 0.809656i \(0.699654\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 20.0000 0.689246
\(843\) 0 0
\(844\) −9.00000 −0.309793
\(845\) 0 0
\(846\) 21.0000 0.721995
\(847\) 0 0
\(848\) −9.00000 −0.309061
\(849\) 0 0
\(850\) 0 0
\(851\) −7.00000 −0.239957
\(852\) 0 0
\(853\) 43.0000 1.47229 0.736146 − 0.676823i \(-0.236644\pi\)
0.736146 + 0.676823i \(0.236644\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −16.0000 −0.546869
\(857\) −8.00000 −0.273275 −0.136637 − 0.990621i \(-0.543630\pi\)
−0.136637 + 0.990621i \(0.543630\pi\)
\(858\) 0 0
\(859\) −12.0000 −0.409435 −0.204717 − 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −2.00000 −0.0681203
\(863\) 11.0000 0.374444 0.187222 − 0.982318i \(-0.440052\pi\)
0.187222 + 0.982318i \(0.440052\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 28.0000 0.951479
\(867\) 0 0
\(868\) 0 0
\(869\) 42.0000 1.42475
\(870\) 0 0
\(871\) −10.0000 −0.338837
\(872\) −2.00000 −0.0677285
\(873\) −36.0000 −1.21842
\(874\) 35.0000 1.18389
\(875\) 0 0
\(876\) 0 0
\(877\) −31.0000 −1.04680 −0.523398 − 0.852088i \(-0.675336\pi\)
−0.523398 + 0.852088i \(0.675336\pi\)
\(878\) 28.0000 0.944954
\(879\) 0 0
\(880\) 0 0
\(881\) −15.0000 −0.505363 −0.252681 − 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 − 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 10.0000 0.336336
\(885\) 0 0
\(886\) 30.0000 1.00787
\(887\) −36.0000 −1.20876 −0.604381 − 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 27.0000 0.904534
\(892\) 8.00000 0.267860
\(893\) −35.0000 −1.17123
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −5.00000 −0.166852
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) −9.00000 −0.299667
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 0 0
\(907\) 40.0000 1.32818 0.664089 − 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(908\) 6.00000 0.199117
\(909\) 0 0
\(910\) 0 0
\(911\) 2.00000 0.0662630 0.0331315 − 0.999451i \(-0.489452\pi\)
0.0331315 + 0.999451i \(0.489452\pi\)
\(912\) 0 0
\(913\) 18.0000 0.595713
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 16.0000 0.528655
\(917\) 0 0
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 − 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 32.0000 1.05386
\(923\) −30.0000 −0.987462
\(924\) 0 0
\(925\) 0 0
\(926\) −17.0000 −0.558655
\(927\) −24.0000 −0.788263
\(928\) 4.00000 0.131306
\(929\) 21.0000 0.688988 0.344494 − 0.938789i \(-0.388051\pi\)
0.344494 + 0.938789i \(0.388051\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 8.00000 0.262049
\(933\) 0 0
\(934\) 34.0000 1.11251
\(935\) 0 0
\(936\) 15.0000 0.490290
\(937\) 26.0000 0.849383 0.424691 − 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −28.0000 −0.912774 −0.456387 − 0.889781i \(-0.650857\pi\)
−0.456387 + 0.889781i \(0.650857\pi\)
\(942\) 0 0
\(943\) 21.0000 0.683854
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 80.0000 2.59691
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 − 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) −27.0000 −0.874157
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) 0 0
\(958\) −36.0000 −1.16311
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 5.00000 0.161206
\(963\) −48.0000 −1.54678
\(964\) −9.00000 −0.289870
\(965\) 0 0
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 − 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) 0 0
\(971\) −33.0000 −1.05902 −0.529510 − 0.848304i \(-0.677624\pi\)
−0.529510 + 0.848304i \(0.677624\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) −54.0000 −1.72761 −0.863807 − 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) −24.0000 −0.765871
\(983\) 29.0000 0.924956 0.462478 − 0.886631i \(-0.346960\pi\)
0.462478 + 0.886631i \(0.346960\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 8.00000 0.254772
\(987\) 0 0
\(988\) −25.0000 −0.795356
\(989\) −14.0000 −0.445174
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 − 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 38.0000 1.20347 0.601736 − 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) −4.00000 −0.126618
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2450.2.a.k.1.1 1
5.2 odd 4 490.2.c.a.99.1 2
5.3 odd 4 490.2.c.a.99.2 2
5.4 even 2 2450.2.a.ba.1.1 1
7.2 even 3 350.2.e.j.151.1 2
7.4 even 3 350.2.e.j.51.1 2
7.6 odd 2 2450.2.a.j.1.1 1
35.2 odd 12 70.2.i.b.39.1 yes 4
35.3 even 12 490.2.i.a.79.1 4
35.4 even 6 350.2.e.c.51.1 2
35.9 even 6 350.2.e.c.151.1 2
35.12 even 12 490.2.i.a.459.1 4
35.13 even 4 490.2.c.d.99.2 2
35.17 even 12 490.2.i.a.79.2 4
35.18 odd 12 70.2.i.b.9.1 ✓ 4
35.23 odd 12 70.2.i.b.39.2 yes 4
35.27 even 4 490.2.c.d.99.1 2
35.32 odd 12 70.2.i.b.9.2 yes 4
35.33 even 12 490.2.i.a.459.2 4
35.34 odd 2 2450.2.a.bb.1.1 1
105.2 even 12 630.2.u.a.109.2 4
105.23 even 12 630.2.u.a.109.1 4
105.32 even 12 630.2.u.a.289.1 4
105.53 even 12 630.2.u.a.289.2 4
140.23 even 12 560.2.bw.d.529.1 4
140.67 even 12 560.2.bw.d.289.1 4
140.107 even 12 560.2.bw.d.529.2 4
140.123 even 12 560.2.bw.d.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.i.b.9.1 ✓ 4 35.18 odd 12
70.2.i.b.9.2 yes 4 35.32 odd 12
70.2.i.b.39.1 yes 4 35.2 odd 12
70.2.i.b.39.2 yes 4 35.23 odd 12
350.2.e.c.51.1 2 35.4 even 6
350.2.e.c.151.1 2 35.9 even 6
350.2.e.j.51.1 2 7.4 even 3
350.2.e.j.151.1 2 7.2 even 3
490.2.c.a.99.1 2 5.2 odd 4
490.2.c.a.99.2 2 5.3 odd 4
490.2.c.d.99.1 2 35.27 even 4
490.2.c.d.99.2 2 35.13 even 4
490.2.i.a.79.1 4 35.3 even 12
490.2.i.a.79.2 4 35.17 even 12
490.2.i.a.459.1 4 35.12 even 12
490.2.i.a.459.2 4 35.33 even 12
560.2.bw.d.289.1 4 140.67 even 12
560.2.bw.d.289.2 4 140.123 even 12
560.2.bw.d.529.1 4 140.23 even 12
560.2.bw.d.529.2 4 140.107 even 12
630.2.u.a.109.1 4 105.23 even 12
630.2.u.a.109.2 4 105.2 even 12
630.2.u.a.289.1 4 105.32 even 12
630.2.u.a.289.2 4 105.53 even 12
2450.2.a.j.1.1 1 7.6 odd 2
2450.2.a.k.1.1 1 1.1 even 1 trivial
2450.2.a.ba.1.1 1 5.4 even 2
2450.2.a.bb.1.1 1 35.34 odd 2