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Properties

Label 7245.2.a.n.1.1
Level $7245$
Weight $2$
Character 7245.1
Self dual yes
Analytic conductor $57.852$
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] = [7245,2,Mod(1,7245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7245, 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("7245.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7245 = 3^{2} \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7245.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: \(57.8516162644\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2415)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{10} -4.00000 q^{11} -6.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{20} -4.00000 q^{22} +1.00000 q^{23} +1.00000 q^{25} -6.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} +5.00000 q^{32} +2.00000 q^{34} -1.00000 q^{35} +2.00000 q^{37} -3.00000 q^{40} -6.00000 q^{41} +4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{46} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{52} -6.00000 q^{53} -4.00000 q^{55} +3.00000 q^{56} -6.00000 q^{58} +12.0000 q^{59} +10.0000 q^{61} +7.00000 q^{64} -6.00000 q^{65} +4.00000 q^{67} -2.00000 q^{68} -1.00000 q^{70} -4.00000 q^{71} -2.00000 q^{73} +2.00000 q^{74} +4.00000 q^{77} -1.00000 q^{80} -6.00000 q^{82} +2.00000 q^{85} +4.00000 q^{86} +12.0000 q^{88} +10.0000 q^{89} +6.00000 q^{91} -1.00000 q^{92} -2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −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 \(0.5\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 6.00000 0.832050
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 12.0000 1.27920
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 14.0000 1.39305 0.696526 − 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 18.0000 1.76505
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 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
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) 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 \(0.443208\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) −6.00000 −0.526235
\(131\) −4.00000 −0.349482 −0.174741 − 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 − 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) 24.0000 2.00698
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 14.0000 1.14692 0.573462 − 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 − 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) 16.0000 1.19590 0.597948 − 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 6.00000 0.444750
\(183\) 0 0
\(184\) −3.00000 −0.221163
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 − 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) 14.0000 0.985037
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 − 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) −5.00000 −0.334077
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 24.0000 1.59294 0.796468 − 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 − 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 1.00000 0.0659380
\(231\) 0 0
\(232\) 18.0000 1.18176
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) −2.00000 −0.129641
\(239\) −4.00000 −0.258738 −0.129369 − 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 − 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −28.0000 −1.76734 −0.883672 − 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 22.0000 1.37232 0.686161 − 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) −4.00000 −0.247121
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −2.00000 −0.121942 −0.0609711 − 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 − 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 − 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) 3.00000 0.179284
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) 24.0000 1.41915
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 14.0000 0.810998
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 − 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 − 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 − 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 7.00000 0.391312
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) 0 0
\(325\) −6.00000 −0.332820
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) 18.0000 0.993884
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 − 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 − 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 23.0000 1.25104
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −20.0000 −1.06600
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) 16.0000 0.845626
\(359\) 16.0000 0.844448 0.422224 − 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 10.0000 0.525588
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 − 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) 2.00000 0.103975
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 − 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) 36.0000 1.85409
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 − 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −28.0000 −1.43073 −0.715367 − 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) −6.00000 −0.305392
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) −10.0000 −0.507020 −0.253510 − 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 − 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 − 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) −30.0000 −1.47087
\(417\) 0 0
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 − 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 0 0
\(439\) 24.0000 1.14546 0.572729 − 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 12.0000 0.572078
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) 36.0000 1.71041 0.855206 − 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 24.0000 1.13643
\(447\) 0 0
\(448\) −7.00000 −0.330719
\(449\) 30.0000 1.41579 0.707894 − 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 − 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 10.0000 0.467269
\(459\) 0 0
\(460\) −1.00000 −0.0466252
\(461\) −26.0000 −1.21094 −0.605470 − 0.795868i \(-0.707015\pi\)
−0.605470 + 0.795868i \(0.707015\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 − 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −24.0000 −1.11059 −0.555294 − 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) −36.0000 −1.65703
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) −4.00000 −0.182956
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 − 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) −30.0000 −1.35804
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) −40.0000 −1.80517 −0.902587 − 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.00000 0.179425
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 − 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −28.0000 −1.24970
\(503\) 12.0000 0.535054 0.267527 − 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) −4.00000 −0.177471
\(509\) −42.0000 −1.86162 −0.930809 − 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) −2.00000 −0.0878750
\(519\) 0 0
\(520\) 18.0000 0.789352
\(521\) 42.0000 1.84005 0.920027 − 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) 44.0000 1.92399 0.961993 − 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) −2.00000 −0.0862261
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 − 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) −24.0000 −1.03089
\(543\) 0 0
\(544\) 10.0000 0.428746
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) 40.0000 1.71028 0.855138 − 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) 2.00000 0.0847427 0.0423714 − 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 16.0000 0.674320 0.337160 − 0.941447i \(-0.390534\pi\)
0.337160 + 0.941447i \(0.390534\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) −22.0000 −0.922288 −0.461144 − 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 − 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −24.0000 −1.00349
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −34.0000 −1.41544 −0.707719 − 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 36.0000 1.48588 0.742940 − 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 12.0000 0.494032
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) −14.0000 −0.573462
\(597\) 0 0
\(598\) −6.00000 −0.245358
\(599\) 4.00000 0.163436 0.0817178 − 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 − 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) 0 0
\(613\) 42.0000 1.69636 0.848182 − 0.529705i \(-0.177697\pi\)
0.848182 + 0.529705i \(0.177697\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) 30.0000 1.20775 0.603877 − 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) −16.0000 −0.643094 −0.321547 − 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 8.00000 0.320771
\(623\) −10.0000 −0.400642
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 − 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 2.00000 0.0794301
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 24.0000 0.950169
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) 10.0000 0.394976 0.197488 − 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 − 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 0 0
\(647\) −8.00000 −0.314512 −0.157256 − 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) 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
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 − 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 − 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) −12.0000 −0.466393
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −6.00000 −0.232321
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 − 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) −23.0000 −0.884615
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 − 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 − 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −20.0000 −0.758643
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) −2.00000 −0.0757011
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −28.0000 −1.05529
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) −18.0000 −0.676004 −0.338002 − 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) −4.00000 −0.150117
\(711\) 0 0
\(712\) −30.0000 −1.12430
\(713\) 0 0
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) −16.0000 −0.597948
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) 40.0000 1.49175 0.745874 − 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) −19.0000 −0.707107
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) −18.0000 −0.667124
\(729\) 0 0
\(730\) −2.00000 −0.0740233
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 2.00000 0.0738717 0.0369358 − 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 5.00000 0.184302
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) −22.0000 −0.805477
\(747\) 0 0
\(748\) 8.00000 0.292509
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 − 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 36.0000 1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) 50.0000 1.81728 0.908640 − 0.417579i \(-0.137121\pi\)
0.908640 + 0.417579i \(0.137121\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) 0 0
\(766\) −28.0000 −1.01168
\(767\) −72.0000 −2.59977
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 − 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 4.00000 0.144150
\(771\) 0 0
\(772\) 6.00000 0.215945
\(773\) −18.0000 −0.647415 −0.323708 − 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) −10.0000 −0.358517
\(779\) 0 0
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) 2.00000 0.0715199
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) −44.0000 −1.56843 −0.784215 − 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) −60.0000 −2.13066
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) −18.0000 −0.637593 −0.318796 − 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) 8.00000 0.282314
\(804\) 0 0
\(805\) −1.00000 −0.0352454
\(806\) 0 0
\(807\) 0 0
\(808\) −42.0000 −1.47755
\(809\) 30.0000 1.05474 0.527372 − 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 − 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) 0 0
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 34.0000 1.18661 0.593304 − 0.804978i \(-0.297823\pi\)
0.593304 + 0.804978i \(0.297823\pi\)
\(822\) 0 0
\(823\) 36.0000 1.25488 0.627441 − 0.778664i \(-0.284103\pi\)
0.627441 + 0.778664i \(0.284103\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 36.0000 1.25184 0.625921 − 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) −50.0000 −1.73657 −0.868286 − 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −42.0000 −1.45609
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 0 0
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) 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
\(842\) −10.0000 −0.344623
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) 23.0000 0.791224
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) −14.0000 −0.479351 −0.239675 − 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) −10.0000 −0.341593 −0.170797 − 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 − 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) −2.00000 −0.0679628
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −10.0000 −0.337676 −0.168838 − 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 24.0000 0.809961
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) −14.0000 −0.471672 −0.235836 − 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) 24.0000 0.807664 0.403832 − 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) −56.0000 −1.88030 −0.940148 − 0.340766i \(-0.889313\pi\)
−0.940148 + 0.340766i \(0.889313\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 10.0000 0.335201
\(891\) 0 0
\(892\) −24.0000 −0.803579
\(893\) 0 0
\(894\) 0 0
\(895\) 16.0000 0.534821
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) 0 0
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 24.0000 0.799113
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) 52.0000 1.72663 0.863316 − 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) −24.0000 −0.796468
\(909\) 0 0
\(910\) 6.00000 0.198898
\(911\) −56.0000 −1.85536 −0.927681 − 0.373373i \(-0.878201\pi\)
−0.927681 + 0.373373i \(0.878201\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 − 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 0 0
\(922\) −26.0000 −0.856264
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) −4.00000 −0.131448
\(927\) 0 0
\(928\) −30.0000 −0.984798
\(929\) 26.0000 0.853032 0.426516 − 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) −24.0000 −0.785304
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) 6.00000 0.196011 0.0980057 − 0.995186i \(-0.468754\pi\)
0.0980057 + 0.995186i \(0.468754\pi\)
\(938\) −4.00000 −0.130605
\(939\) 0 0
\(940\) 0 0
\(941\) −50.0000 −1.62995 −0.814977 − 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 6.00000 0.194461
\(953\) 46.0000 1.49009 0.745043 − 0.667016i \(-0.232429\pi\)
0.745043 + 0.667016i \(0.232429\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 4.00000 0.129369
\(957\) 0 0
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) −14.0000 −0.450910
\(965\) −6.00000 −0.193147
\(966\) 0 0
\(967\) 20.0000 0.643157 0.321578 − 0.946883i \(-0.395787\pi\)
0.321578 + 0.946883i \(0.395787\pi\)
\(968\) −15.0000 −0.482118
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) −12.0000 −0.385098 −0.192549 − 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 20.0000 0.641171
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −42.0000 −1.34370 −0.671850 − 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) −40.0000 −1.27841
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) −40.0000 −1.27645
\(983\) 4.00000 0.127580 0.0637901 − 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −56.0000 −1.77890 −0.889449 − 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 4.00000 0.126872
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) 18.0000 0.570066 0.285033 − 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) 28.0000 0.886325
\(999\) 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 7245.2.a.n.1.1 1
3.2 odd 2 2415.2.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2415.2.a.a.1.1 ✓ 1 3.2 odd 2
7245.2.a.n.1.1 1 1.1 even 1 trivial