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Properties

Label 3450.2.a.b.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
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] = [3450,2,Mod(1,3450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3450, 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("3450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.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: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} +2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +4.00000 q^{26} -1.00000 q^{27} +4.00000 q^{29} -1.00000 q^{32} +2.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +8.00000 q^{38} +4.00000 q^{39} -2.00000 q^{41} -2.00000 q^{43} -2.00000 q^{44} -1.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +6.00000 q^{51} -4.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +8.00000 q^{57} -4.00000 q^{58} +8.00000 q^{59} +2.00000 q^{61} +1.00000 q^{64} -2.00000 q^{66} +6.00000 q^{67} -6.00000 q^{68} -1.00000 q^{69} +10.0000 q^{71} -1.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} -8.00000 q^{76} -4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +2.00000 q^{82} -8.00000 q^{83} +2.00000 q^{86} -4.00000 q^{87} +2.00000 q^{88} -12.0000 q^{89} +1.00000 q^{92} -12.0000 q^{94} +1.00000 q^{96} +16.0000 q^{97} +7.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 8.00000 1.29777
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) −4.00000 −0.554700
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) −4.00000 −0.525226
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 6.00000 0.733017 0.366508 − 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) −6.00000 −0.727607
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 − 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −8.00000 −0.878114 −0.439057 − 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) −4.00000 −0.428845
\(88\) 2.00000 0.213201
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 16.0000 1.62455 0.812277 − 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) 7.00000 0.707107
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 − 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) −6.00000 −0.594089
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 − 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) −4.00000 −0.369800
\(118\) −8.00000 −0.736460
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 − 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 − 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) −6.00000 −0.518321
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −2.00000 −0.170872 −0.0854358 − 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 1.00000 0.0851257
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) −10.0000 −0.839181
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 7.00000 0.577350
\(148\) 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
\(151\) −4.00000 −0.325515 −0.162758 − 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 8.00000 0.648886
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) 10.0000 0.798087 0.399043 − 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 8.00000 0.636446
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 16.0000 1.23812 0.619059 − 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) −2.00000 −0.152499
\(173\) 10.0000 0.760286 0.380143 − 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) −8.00000 −0.601317
\(178\) 12.0000 0.899438
\(179\) 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 \(0.378793\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −26.0000 −1.87152 −0.935760 − 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 2.00000 0.142134
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) −12.0000 −0.844317
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) −4.00000 −0.277350
\(209\) 16.0000 1.10674
\(210\) 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
\(213\) −10.0000 −0.685189
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 2.00000 0.134231
\(223\) 2.00000 0.133930 0.0669650 − 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) −28.0000 −1.85843 −0.929213 − 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 8.00000 0.529813
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −14.0000 −0.905585 −0.452792 − 0.891616i \(-0.649572\pi\)
−0.452792 + 0.891616i \(0.649572\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 − 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 32.0000 2.03611
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 − 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 − 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) −2.00000 −0.124515
\(259\) 0 0
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 4.00000 0.247121
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 6.00000 0.366508
\(269\) 8.00000 0.487769 0.243884 − 0.969804i \(-0.421578\pi\)
0.243884 + 0.969804i \(0.421578\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 − 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 28.0000 1.68236 0.841178 − 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 − 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 12.0000 0.714590
\(283\) −14.0000 −0.832214 −0.416107 − 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) 2.00000 0.117041
\(293\) 14.0000 0.817889 0.408944 − 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 2.00000 0.116052
\(298\) −14.0000 −0.810998
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) 0 0
\(302\) 4.00000 0.230174
\(303\) −12.0000 −0.689382
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −28.0000 −1.59804 −0.799022 − 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 26.0000 1.47432 0.737162 − 0.675716i \(-0.236165\pi\)
0.737162 + 0.675716i \(0.236165\pi\)
\(312\) −4.00000 −0.226455
\(313\) −28.0000 −1.58265 −0.791327 − 0.611393i \(-0.790609\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 6.00000 0.336463
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 48.0000 2.67079
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) 10.0000 0.553001
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 − 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −8.00000 −0.439057
\(333\) 2.00000 0.109599
\(334\) −16.0000 −0.875481
\(335\) 0 0
\(336\) 0 0
\(337\) 28.0000 1.52526 0.762629 − 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) −3.00000 −0.163178
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 8.00000 0.432590
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −4.00000 −0.214423
\(349\) 26.0000 1.39175 0.695874 − 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 2.00000 0.106600
\(353\) −10.0000 −0.532246 −0.266123 − 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) −12.0000 −0.635999
\(357\) 0 0
\(358\) −16.0000 −0.845626
\(359\) −16.0000 −0.844448 −0.422224 − 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −10.0000 −0.525588
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) −16.0000 −0.835193 −0.417597 − 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 1.00000 0.0521286
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 − 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) −16.0000 −0.824042
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 − 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) −24.0000 −1.22795
\(383\) 32.0000 1.63512 0.817562 − 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 26.0000 1.32337
\(387\) −2.00000 −0.101666
\(388\) 16.0000 0.812277
\(389\) 26.0000 1.31825 0.659126 − 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 7.00000 0.353553
\(393\) 4.00000 0.201773
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) −28.0000 −1.40528 −0.702640 − 0.711546i \(-0.747995\pi\)
−0.702640 + 0.711546i \(0.747995\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) −8.00000 −0.399501 −0.199750 − 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) 6.00000 0.299253
\(403\) 0 0
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) −6.00000 −0.297044
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) −4.00000 −0.195881
\(418\) −16.0000 −0.782586
\(419\) 30.0000 1.46560 0.732798 − 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 − 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) −4.00000 −0.194717
\(423\) 12.0000 0.583460
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 10.0000 0.484502
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 − 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −16.0000 −0.768911 −0.384455 − 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) −8.00000 −0.382692
\(438\) 2.00000 0.0955637
\(439\) 28.0000 1.33637 0.668184 − 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) −24.0000 −1.14156
\(443\) 4.00000 0.190046 0.0950229 − 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) −14.0000 −0.662177
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 − 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) −18.0000 −0.846649
\(453\) 4.00000 0.187936
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) −8.00000 −0.374634
\(457\) 8.00000 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 10.0000 0.467269
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 − 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 26.0000 1.20832 0.604161 − 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 24.0000 1.11059 0.555294 − 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) −4.00000 −0.184900
\(469\) 0 0
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) −8.00000 −0.368230
\(473\) 4.00000 0.183920
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 14.0000 0.640345
\(479\) −16.0000 −0.731059 −0.365529 − 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 6.00000 0.273293
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 26.0000 1.17817 0.589086 − 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 − 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 2.00000 0.0901670
\(493\) −24.0000 −1.08091
\(494\) −32.0000 −1.43975
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −8.00000 −0.358489
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) −2.00000 −0.0892644
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) −3.00000 −0.133235
\(508\) −2.00000 −0.0887357
\(509\) 20.0000 0.886484 0.443242 − 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 8.00000 0.353209
\(514\) 22.0000 0.970378
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) −24.0000 −1.05552
\(518\) 0 0
\(519\) −10.0000 −0.438951
\(520\) 0 0
\(521\) −20.0000 −0.876216 −0.438108 − 0.898922i \(-0.644351\pi\)
−0.438108 + 0.898922i \(0.644351\pi\)
\(522\) −4.00000 −0.175075
\(523\) −30.0000 −1.31181 −0.655904 − 0.754844i \(-0.727712\pi\)
−0.655904 + 0.754844i \(0.727712\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) −6.00000 −0.259161
\(537\) −16.0000 −0.690451
\(538\) −8.00000 −0.344904
\(539\) 14.0000 0.603023
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 − 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) −24.0000 −1.03089
\(543\) −10.0000 −0.429141
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 − 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −32.0000 −1.36325
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −28.0000 −1.18961
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) −38.0000 −1.61011 −0.805056 − 0.593199i \(-0.797865\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) −12.0000 −0.506189
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) −10.0000 −0.419591
\(569\) −24.0000 −1.00613 −0.503066 − 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 − 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 8.00000 0.334497
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −38.0000 −1.58196 −0.790980 − 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) −19.0000 −0.790296
\(579\) 26.0000 1.08052
\(580\) 0 0
\(581\) 0 0
\(582\) 16.0000 0.663221
\(583\) −12.0000 −0.496989
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) −4.00000 −0.165098 −0.0825488 − 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) 7.00000 0.288675
\(589\) 0 0
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 2.00000 0.0821995
\(593\) −6.00000 −0.246390 −0.123195 − 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) 16.0000 0.654836
\(598\) 4.00000 0.163572
\(599\) −34.0000 −1.38920 −0.694601 − 0.719395i \(-0.744419\pi\)
−0.694601 + 0.719395i \(0.744419\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 − 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 6.00000 0.244339
\(604\) −4.00000 −0.162758
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) 10.0000 0.405887 0.202944 − 0.979190i \(-0.434949\pi\)
0.202944 + 0.979190i \(0.434949\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) −6.00000 −0.242536
\(613\) 30.0000 1.21169 0.605844 − 0.795583i \(-0.292835\pi\)
0.605844 + 0.795583i \(0.292835\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) 32.0000 1.28619 0.643094 − 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −26.0000 −1.04251
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) 28.0000 1.11911
\(627\) −16.0000 −0.638978
\(628\) 10.0000 0.399043
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 − 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 8.00000 0.318223
\(633\) −4.00000 −0.158986
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 28.0000 1.10940
\(638\) 8.00000 0.316723
\(639\) 10.0000 0.395594
\(640\) 0 0
\(641\) −4.00000 −0.157991 −0.0789953 − 0.996875i \(-0.525171\pi\)
−0.0789953 + 0.996875i \(0.525171\pi\)
\(642\) 12.0000 0.473602
\(643\) 10.0000 0.394362 0.197181 − 0.980367i \(-0.436821\pi\)
0.197181 + 0.980367i \(0.436821\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −48.0000 −1.88853
\(647\) 28.0000 1.10079 0.550397 − 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) −2.00000 −0.0782660 −0.0391330 − 0.999234i \(-0.512460\pi\)
−0.0391330 + 0.999234i \(0.512460\pi\)
\(654\) −10.0000 −0.391031
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −26.0000 −1.01282 −0.506408 − 0.862294i \(-0.669027\pi\)
−0.506408 + 0.862294i \(0.669027\pi\)
\(660\) 0 0
\(661\) −50.0000 −1.94477 −0.972387 − 0.233373i \(-0.925024\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) −28.0000 −1.08825
\(663\) −24.0000 −0.932083
\(664\) 8.00000 0.310460
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 4.00000 0.154881
\(668\) 16.0000 0.619059
\(669\) −2.00000 −0.0773245
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 − 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) −28.0000 −1.07852
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 10.0000 0.384331 0.192166 − 0.981363i \(-0.438449\pi\)
0.192166 + 0.981363i \(0.438449\pi\)
\(678\) −18.0000 −0.691286
\(679\) 0 0
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) −2.00000 −0.0762493
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 10.0000 0.380143
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) 12.0000 0.454532
\(698\) −26.0000 −0.984115
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −14.0000 −0.528773 −0.264386 − 0.964417i \(-0.585169\pi\)
−0.264386 + 0.964417i \(0.585169\pi\)
\(702\) −4.00000 −0.150970
\(703\) −16.0000 −0.603451
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) 0 0
\(708\) −8.00000 −0.300658
\(709\) 38.0000 1.42712 0.713560 − 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 12.0000 0.449719
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 16.0000 0.597948
\(717\) 14.0000 0.522840
\(718\) 16.0000 0.597115
\(719\) −2.00000 −0.0745874 −0.0372937 − 0.999304i \(-0.511874\pi\)
−0.0372937 + 0.999304i \(0.511874\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −45.0000 −1.67473
\(723\) 6.00000 0.223142
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) −28.0000 −1.03846 −0.519231 − 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) −2.00000 −0.0739221
\(733\) 42.0000 1.55131 0.775653 − 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −12.0000 −0.442026
\(738\) 2.00000 0.0736210
\(739\) −36.0000 −1.32428 −0.662141 − 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 0 0
\(741\) −32.0000 −1.17555
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 − 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 6.00000 0.219676
\(747\) −8.00000 −0.292705
\(748\) 12.0000 0.438763
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 − 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 12.0000 0.437595
\(753\) −2.00000 −0.0728841
\(754\) 16.0000 0.582686
\(755\) 0 0
\(756\) 0 0
\(757\) 6.00000 0.218074 0.109037 − 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) −12.0000 −0.435860
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 − 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) −2.00000 −0.0724524
\(763\) 0 0
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) −32.0000 −1.15621
\(767\) −32.0000 −1.15545
\(768\) −1.00000 −0.0360844
\(769\) −2.00000 −0.0721218 −0.0360609 − 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 22.0000 0.792311
\(772\) −26.0000 −0.935760
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 2.00000 0.0718885
\(775\) 0 0
\(776\) −16.0000 −0.574367
\(777\) 0 0
\(778\) −26.0000 −0.932145
\(779\) 16.0000 0.573259
\(780\) 0 0
\(781\) −20.0000 −0.715656
\(782\) 6.00000 0.214560
\(783\) −4.00000 −0.142948
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) −22.0000 −0.784215 −0.392108 − 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 2.00000 0.0710669
\(793\) −8.00000 −0.284088
\(794\) 28.0000 0.993683
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) −14.0000 −0.495905 −0.247953 − 0.968772i \(-0.579758\pi\)
−0.247953 + 0.968772i \(0.579758\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 0 0
\(801\) −12.0000 −0.423999
\(802\) 8.00000 0.282490
\(803\) −4.00000 −0.141157
\(804\) −6.00000 −0.211604
\(805\) 0 0
\(806\) 0 0
\(807\) −8.00000 −0.281613
\(808\) −12.0000 −0.422159
\(809\) −46.0000 −1.61727 −0.808637 − 0.588308i \(-0.799794\pi\)
−0.808637 + 0.588308i \(0.799794\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 − 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) −24.0000 −0.841717
\(814\) 4.00000 0.140200
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 16.0000 0.559769
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) −2.00000 −0.0697580
\(823\) 22.0000 0.766872 0.383436 − 0.923567i \(-0.374741\pi\)
0.383436 + 0.923567i \(0.374741\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 1.00000 0.0347524
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) −28.0000 −0.971309
\(832\) −4.00000 −0.138675
\(833\) 42.0000 1.45521
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) −30.0000 −1.03633
\(839\) −12.0000 −0.414286 −0.207143 − 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 14.0000 0.482472
\(843\) −12.0000 −0.413302
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) −10.0000 −0.342594
\(853\) −36.0000 −1.23262 −0.616308 − 0.787505i \(-0.711372\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −18.0000 −0.614868 −0.307434 − 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 8.00000 0.273115
\(859\) −28.0000 −0.955348 −0.477674 − 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8.00000 0.272481
\(863\) 20.0000 0.680808 0.340404 − 0.940279i \(-0.389436\pi\)
0.340404 + 0.940279i \(0.389436\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 10.0000 0.338643
\(873\) 16.0000 0.541518
\(874\) 8.00000 0.270604
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) 20.0000 0.675352 0.337676 − 0.941262i \(-0.390359\pi\)
0.337676 + 0.941262i \(0.390359\pi\)
\(878\) −28.0000 −0.944954
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) 4.00000 0.134763 0.0673817 − 0.997727i \(-0.478535\pi\)
0.0673817 + 0.997727i \(0.478535\pi\)
\(882\) 7.00000 0.235702
\(883\) −32.0000 −1.07689 −0.538443 − 0.842662i \(-0.680987\pi\)
−0.538443 + 0.842662i \(0.680987\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) −36.0000 −1.20876 −0.604381 − 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 2.00000 0.0669650
\(893\) −96.0000 −3.21252
\(894\) 14.0000 0.468230
\(895\) 0 0
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) −30.0000 −1.00111
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) −4.00000 −0.133185
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) −4.00000 −0.132891
\(907\) 22.0000 0.730498 0.365249 − 0.930910i \(-0.380984\pi\)
0.365249 + 0.930910i \(0.380984\pi\)
\(908\) −28.0000 −0.929213
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −4.00000 −0.132526 −0.0662630 − 0.997802i \(-0.521108\pi\)
−0.0662630 + 0.997802i \(0.521108\pi\)
\(912\) 8.00000 0.264906
\(913\) 16.0000 0.529523
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) −6.00000 −0.198030
\(919\) 40.0000 1.31948 0.659739 − 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) 12.0000 0.395199
\(923\) −40.0000 −1.31662
\(924\) 0 0
\(925\) 0 0
\(926\) −26.0000 −0.854413
\(927\) 0 0
\(928\) −4.00000 −0.131306
\(929\) 30.0000 0.984268 0.492134 − 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 56.0000 1.83533
\(932\) −6.00000 −0.196537
\(933\) −26.0000 −0.851202
\(934\) −24.0000 −0.785304
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) −32.0000 −1.04539 −0.522697 − 0.852518i \(-0.675074\pi\)
−0.522697 + 0.852518i \(0.675074\pi\)
\(938\) 0 0
\(939\) 28.0000 0.913745
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 − 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 10.0000 0.325818
\(943\) −2.00000 −0.0651290
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −4.00000 −0.129983 −0.0649913 − 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 8.00000 0.259828
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −26.0000 −0.842223 −0.421111 − 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −14.0000 −0.452792
\(957\) 8.00000 0.258603
\(958\) 16.0000 0.516937
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 8.00000 0.257930
\(963\) 12.0000 0.386695
\(964\) −6.00000 −0.193247
\(965\) 0 0
\(966\) 0 0
\(967\) 42.0000 1.35063 0.675314 − 0.737530i \(-0.264008\pi\)
0.675314 + 0.737530i \(0.264008\pi\)
\(968\) 7.00000 0.224989
\(969\) −48.0000 −1.54198
\(970\) 0 0
\(971\) −14.0000 −0.449281 −0.224641 − 0.974442i \(-0.572121\pi\)
−0.224641 + 0.974442i \(0.572121\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −26.0000 −0.833094
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 42.0000 1.34370 0.671850 − 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 16.0000 0.511624
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) −20.0000 −0.638226
\(983\) −48.0000 −1.53096 −0.765481 − 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) 24.0000 0.764316
\(987\) 0 0
\(988\) 32.0000 1.01806
\(989\) −2.00000 −0.0635963
\(990\) 0 0
\(991\) 12.0000 0.381193 0.190596 − 0.981669i \(-0.438958\pi\)
0.190596 + 0.981669i \(0.438958\pi\)
\(992\) 0 0
\(993\) −28.0000 −0.888553
\(994\) 0 0
\(995\) 0 0
\(996\) 8.00000 0.253490
\(997\) 4.00000 0.126681 0.0633406 − 0.997992i \(-0.479825\pi\)
0.0633406 + 0.997992i \(0.479825\pi\)
\(998\) 4.00000 0.126618
\(999\) −2.00000 −0.0632772
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 3450.2.a.b.1.1 1
5.2 odd 4 3450.2.d.m.2899.1 2
5.3 odd 4 3450.2.d.m.2899.2 2
5.4 even 2 690.2.a.j.1.1 ✓ 1
15.14 odd 2 2070.2.a.c.1.1 1
20.19 odd 2 5520.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.j.1.1 ✓ 1 5.4 even 2
2070.2.a.c.1.1 1 15.14 odd 2
3450.2.a.b.1.1 1 1.1 even 1 trivial
3450.2.d.m.2899.1 2 5.2 odd 4
3450.2.d.m.2899.2 2 5.3 odd 4
5520.2.a.l.1.1 1 20.19 odd 2