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Properties

Label 2394.2.a.e.1.1
Level $2394$
Weight $2$
Character 2394.1
Self dual yes
Analytic conductor $19.116$
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] = [2394,2,Mod(1,2394)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2394, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2394.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.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: \(19.1161862439\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2394.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{10} -2.00000 q^{11} -6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{19} +2.00000 q^{20} +2.00000 q^{22} +4.00000 q^{23} -1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{28} +2.00000 q^{29} -6.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -2.00000 q^{35} -4.00000 q^{37} -1.00000 q^{38} -2.00000 q^{40} -6.00000 q^{41} -4.00000 q^{43} -2.00000 q^{44} -4.00000 q^{46} +6.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} -6.00000 q^{53} -4.00000 q^{55} +1.00000 q^{56} -2.00000 q^{58} -4.00000 q^{59} -6.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} -14.0000 q^{67} +4.00000 q^{68} +2.00000 q^{70} -8.00000 q^{71} +10.0000 q^{73} +4.00000 q^{74} +1.00000 q^{76} +2.00000 q^{77} +2.00000 q^{80} +6.00000 q^{82} -8.00000 q^{83} +8.00000 q^{85} +4.00000 q^{86} +2.00000 q^{88} -6.00000 q^{89} +6.00000 q^{91} +4.00000 q^{92} -6.00000 q^{94} +2.00000 q^{95} +16.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 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
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 − 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) −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 \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 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
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −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 \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) −14.0000 −1.71037 −0.855186 − 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.00000 0.223607
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) −8.00000 −0.878114 −0.439057 − 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) 16.0000 1.62455 0.812277 − 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 2.00000 0.199007 0.0995037 − 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.985329 −0.492665 − 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 6.00000 0.588348
\(105\) 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
\(109\) 16.0000 1.53252 0.766261 − 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 12.0000 1.05247
\(131\) −4.00000 −0.349482 −0.174741 − 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) −10.0000 −0.854358 −0.427179 − 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 − 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) −2.00000 −0.161165
\(155\) −12.0000 −0.963863
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −2.00000 −0.158114
\(161\) −4.00000 −0.315244
\(162\) 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
\(166\) 8.00000 0.620920
\(167\) −16.0000 −1.23812 −0.619059 − 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −8.00000 −0.613572
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −24.0000 −1.79384 −0.896922 − 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 − 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) −6.00000 −0.444750
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) 12.0000 0.868290 0.434145 − 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 16.0000 1.13995 0.569976 − 0.821661i \(-0.306952\pi\)
0.569976 + 0.821661i \(0.306952\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −2.00000 −0.140720
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 10.0000 0.696733
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 − 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) −16.0000 −1.08366
\(219\) 0 0
\(220\) −4.00000 −0.269680
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 − 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 − 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −14.0000 −0.917170 −0.458585 − 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 4.00000 0.259281
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 − 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) 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
\(259\) 4.00000 0.248548
\(260\) −12.0000 −0.744208
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 1.00000 0.0613139
\(267\) 0 0
\(268\) −14.0000 −0.855186
\(269\) 26.0000 1.58525 0.792624 − 0.609711i \(-0.208714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 − 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 − 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) −4.00000 −0.234888
\(291\) 0 0
\(292\) 10.0000 0.585206
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 4.00000 0.232495
\(297\) 0 0
\(298\) 0 0
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 − 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 2.00000 0.113961
\(309\) 0 0
\(310\) 12.0000 0.681554
\(311\) 2.00000 0.113410 0.0567048 − 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 − 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000 0.336994 0.168497 − 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) 6.00000 0.332820
\(326\) 16.0000 0.886158
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −34.0000 −1.86881 −0.934405 − 0.356214i \(-0.884068\pi\)
−0.934405 + 0.356214i \(0.884068\pi\)
\(332\) −8.00000 −0.439057
\(333\) 0 0
\(334\) 16.0000 0.875481
\(335\) −28.0000 −1.52980
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 − 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) 8.00000 0.433861
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) 18.0000 0.966291 0.483145 − 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 0 0
\(349\) 22.0000 1.17763 0.588817 − 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) 24.0000 1.27739 0.638696 − 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) 32.0000 1.68890 0.844448 − 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 6.00000 0.315353
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) 20.0000 1.04685
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 − 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) −8.00000 −0.414224 −0.207112 − 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −26.0000 −1.33553 −0.667765 − 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) −32.0000 −1.63512 −0.817562 − 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) 16.0000 0.812277
\(389\) 8.00000 0.405616 0.202808 − 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −16.0000 −0.806068
\(395\) 0 0
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 − 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 36.0000 1.79329
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −16.0000 −0.791149 −0.395575 − 0.918434i \(-0.629455\pi\)
−0.395575 + 0.918434i \(0.629455\pi\)
\(410\) 12.0000 0.592638
\(411\) 0 0
\(412\) −10.0000 −0.492665
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) −16.0000 −0.785409
\(416\) 6.00000 0.294174
\(417\) 0 0
\(418\) 2.00000 0.0978232
\(419\) 24.0000 1.17248 0.586238 − 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 10.0000 0.486792
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) 24.0000 1.15604 0.578020 − 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 32.0000 1.53782 0.768911 − 0.639356i \(-0.220799\pi\)
0.768911 + 0.639356i \(0.220799\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) 16.0000 0.766261
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) −6.00000 −0.286364 −0.143182 − 0.989696i \(-0.545733\pi\)
−0.143182 + 0.989696i \(0.545733\pi\)
\(440\) 4.00000 0.190693
\(441\) 0 0
\(442\) 24.0000 1.14156
\(443\) −26.0000 −1.23530 −0.617649 − 0.786454i \(-0.711915\pi\)
−0.617649 + 0.786454i \(0.711915\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) −14.0000 −0.662919
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −34.0000 −1.60456 −0.802280 − 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 2.00000 0.0940721
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 − 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 26.0000 1.21490
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −34.0000 −1.58354 −0.791769 − 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 − 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) 8.00000 0.370196 0.185098 − 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 14.0000 0.646460
\(470\) −12.0000 −0.553519
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) 0 0
\(479\) −2.00000 −0.0913823 −0.0456912 − 0.998956i \(-0.514549\pi\)
−0.0456912 + 0.998956i \(0.514549\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) 4.00000 0.182195
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 32.0000 1.45305
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 − 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) −2.00000 −0.0903508
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 0 0
\(493\) 8.00000 0.360302
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) 42.0000 1.87269 0.936344 − 0.351085i \(-0.114187\pi\)
0.936344 + 0.351085i \(0.114187\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) 8.00000 0.355643
\(507\) 0 0
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 − 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) −20.0000 −0.881305
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) −4.00000 −0.175750
\(519\) 0 0
\(520\) 12.0000 0.526235
\(521\) 42.0000 1.84005 0.920027 − 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 12.0000 0.521247
\(531\) 0 0
\(532\) −1.00000 −0.0433555
\(533\) 36.0000 1.55933
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) 14.0000 0.604708
\(537\) 0 0
\(538\) −26.0000 −1.12094
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 − 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 24.0000 1.03089
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) 32.0000 1.37073
\(546\) 0 0
\(547\) −6.00000 −0.256541 −0.128271 − 0.991739i \(-0.540943\pi\)
−0.128271 + 0.991739i \(0.540943\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) 0 0
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 32.0000 1.35588 0.677942 − 0.735116i \(-0.262872\pi\)
0.677942 + 0.735116i \(0.262872\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) −10.0000 −0.421825
\(563\) 4.00000 0.168580 0.0842900 − 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 4.00000 0.168281
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −30.0000 −1.25767 −0.628833 − 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 24.0000 1.00437 0.502184 − 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 − 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) 12.0000 0.496989
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 28.0000 1.15568 0.577842 − 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) −6.00000 −0.247226
\(590\) 8.00000 0.329355
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) 36.0000 1.47834 0.739171 − 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) 0 0
\(595\) −8.00000 −0.327968
\(596\) 0 0
\(597\) 0 0
\(598\) 24.0000 0.981433
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 44.0000 1.79480 0.897399 − 0.441221i \(-0.145454\pi\)
0.897399 + 0.441221i \(0.145454\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) −14.0000 −0.569181
\(606\) 0 0
\(607\) 42.0000 1.70473 0.852364 − 0.522949i \(-0.175168\pi\)
0.852364 + 0.522949i \(0.175168\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 12.0000 0.485866
\(611\) −36.0000 −1.45640
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 − 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) 34.0000 1.36879 0.684394 − 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) −12.0000 −0.481932
\(621\) 0 0
\(622\) −2.00000 −0.0801927
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) −2.00000 −0.0789953 −0.0394976 − 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) 20.0000 0.788723 0.394362 − 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) 6.00000 0.235884 0.117942 − 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −36.0000 −1.40879 −0.704394 − 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 6.00000 0.233904
\(659\) 16.0000 0.623272 0.311636 − 0.950202i \(-0.399123\pi\)
0.311636 + 0.950202i \(0.399123\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 − 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 34.0000 1.32145
\(663\) 0 0
\(664\) 8.00000 0.310460
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) 8.00000 0.309761
\(668\) −16.0000 −0.619059
\(669\) 0 0
\(670\) 28.0000 1.08173
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) 50.0000 1.92736 0.963679 − 0.267063i \(-0.0860531\pi\)
0.963679 + 0.267063i \(0.0860531\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −42.0000 −1.61419 −0.807096 − 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) −16.0000 −0.614024
\(680\) −8.00000 −0.306786
\(681\) 0 0
\(682\) −12.0000 −0.459504
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −20.0000 −0.764161
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) −18.0000 −0.683271
\(695\) −40.0000 −1.51729
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) −22.0000 −0.832712
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −16.0000 −0.604312 −0.302156 − 0.953259i \(-0.597706\pi\)
−0.302156 + 0.953259i \(0.597706\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) −2.00000 −0.0752177
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 − 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 16.0000 0.600469
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) −32.0000 −1.19423
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 10.0000 0.372419
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) −6.00000 −0.222988
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 − 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) −6.00000 −0.222375
\(729\) 0 0
\(730\) −20.0000 −0.740233
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) 30.0000 1.10808 0.554038 − 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 28.0000 1.03139
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 − 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 8.00000 0.292901
\(747\) 0 0
\(748\) −8.00000 −0.292509
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 12.0000 0.437886 0.218943 − 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 − 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 26.0000 0.944363
\(759\) 0 0
\(760\) −2.00000 −0.0725476
\(761\) −8.00000 −0.290000 −0.145000 − 0.989432i \(-0.546318\pi\)
−0.145000 + 0.989432i \(0.546318\pi\)
\(762\) 0 0
\(763\) −16.0000 −0.579239
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 32.0000 1.15621
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) 10.0000 0.360609 0.180305 − 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) −4.00000 −0.144150
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 26.0000 0.935155 0.467578 − 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) 0 0
\(775\) 6.00000 0.215526
\(776\) −16.0000 −0.574367
\(777\) 0 0
\(778\) −8.00000 −0.286814
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) −16.0000 −0.572159
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) 48.0000 1.71102 0.855508 − 0.517790i \(-0.173245\pi\)
0.855508 + 0.517790i \(0.173245\pi\)
\(788\) 16.0000 0.569976
\(789\) 0 0
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) 36.0000 1.27840
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) −20.0000 −0.705785
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) −36.0000 −1.26805
\(807\) 0 0
\(808\) −2.00000 −0.0703598
\(809\) −22.0000 −0.773479 −0.386739 − 0.922189i \(-0.626399\pi\)
−0.386739 + 0.922189i \(0.626399\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 − 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) −32.0000 −1.12091
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) 16.0000 0.559427
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) −20.0000 −0.698005 −0.349002 − 0.937122i \(-0.613479\pi\)
−0.349002 + 0.937122i \(0.613479\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 − 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 10.0000 0.348367
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) −48.0000 −1.66912 −0.834562 − 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) 0 0
\(829\) 46.0000 1.59765 0.798823 − 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 16.0000 0.555368
\(831\) 0 0
\(832\) −6.00000 −0.208013
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) −32.0000 −1.10741
\(836\) −2.00000 −0.0691714
\(837\) 0 0
\(838\) −24.0000 −0.829066
\(839\) −32.0000 −1.10476 −0.552381 − 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) −10.0000 −0.344214
\(845\) 46.0000 1.58245
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 4.00000 0.137199
\(851\) −16.0000 −0.548473
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −54.0000 −1.84460 −0.922302 − 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 8.00000 0.272323 0.136162 − 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) 0 0
\(865\) 28.0000 0.952029
\(866\) −32.0000 −1.08740
\(867\) 0 0
\(868\) 6.00000 0.203653
\(869\) 0 0
\(870\) 0 0
\(871\) 84.0000 2.84623
\(872\) −16.0000 −0.541828
\(873\) 0 0
\(874\) −4.00000 −0.135302
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) −20.0000 −0.675352 −0.337676 − 0.941262i \(-0.609641\pi\)
−0.337676 + 0.941262i \(0.609641\pi\)
\(878\) 6.00000 0.202490
\(879\) 0 0
\(880\) −4.00000 −0.134840
\(881\) −24.0000 −0.808581 −0.404290 − 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 − 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) 26.0000 0.873487
\(887\) 36.0000 1.20876 0.604381 − 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 12.0000 0.402241
\(891\) 0 0
\(892\) 14.0000 0.468755
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) −48.0000 −1.60446
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 34.0000 1.13459
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) −12.0000 −0.399556
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) −12.0000 −0.398893
\(906\) 0 0
\(907\) −42.0000 −1.39459 −0.697294 − 0.716786i \(-0.745613\pi\)
−0.697294 + 0.716786i \(0.745613\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) −12.0000 −0.397796
\(911\) −48.0000 −1.59031 −0.795155 − 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 26.0000 0.860004
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 − 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) −8.00000 −0.263752
\(921\) 0 0
\(922\) 34.0000 1.11973
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) 8.00000 0.262896
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) 36.0000 1.18112 0.590561 − 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) −14.0000 −0.458585
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) −16.0000 −0.523256
\(936\) 0 0
\(937\) −58.0000 −1.89478 −0.947389 − 0.320085i \(-0.896288\pi\)
−0.947389 + 0.320085i \(0.896288\pi\)
\(938\) −14.0000 −0.457116
\(939\) 0 0
\(940\) 12.0000 0.391397
\(941\) 50.0000 1.62995 0.814977 − 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 38.0000 1.23483 0.617417 − 0.786636i \(-0.288179\pi\)
0.617417 + 0.786636i \(0.288179\pi\)
\(948\) 0 0
\(949\) −60.0000 −1.94768
\(950\) 1.00000 0.0324443
\(951\) 0 0
\(952\) 4.00000 0.129641
\(953\) 10.0000 0.323932 0.161966 − 0.986796i \(-0.448217\pi\)
0.161966 + 0.986796i \(0.448217\pi\)
\(954\) 0 0
\(955\) 24.0000 0.776622
\(956\) 0 0
\(957\) 0 0
\(958\) 2.00000 0.0646171
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −24.0000 −0.773791
\(963\) 0 0
\(964\) −4.00000 −0.128831
\(965\) 28.0000 0.901352
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 − 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) −32.0000 −1.02746
\(971\) 36.0000 1.15529 0.577647 − 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 20.0000 0.641171
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) 2.00000 0.0638877
\(981\) 0 0
\(982\) −30.0000 −0.957338
\(983\) −36.0000 −1.14822 −0.574111 − 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) 32.0000 1.01960
\(986\) −8.00000 −0.254772
\(987\) 0 0
\(988\) −6.00000 −0.190885
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 − 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) 40.0000 1.26809
\(996\) 0 0
\(997\) 38.0000 1.20347 0.601736 − 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) 20.0000 0.633089
\(999\) 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 2394.2.a.e.1.1 1
3.2 odd 2 798.2.a.g.1.1 ✓ 1
12.11 even 2 6384.2.a.t.1.1 1
21.20 even 2 5586.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.a.g.1.1 ✓ 1 3.2 odd 2
2394.2.a.e.1.1 1 1.1 even 1 trivial
5586.2.a.bb.1.1 1 21.20 even 2
6384.2.a.t.1.1 1 12.11 even 2