↑

Properties

Label 7350.2.a.br.1.1
Level $7350$
Weight $2$
Character 7350.1
Self dual yes
Analytic conductor $58.690$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7350,2,Mod(1,7350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7350, 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("7350.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7350.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: \(58.6900454856\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 294)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7350.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} -4.00000 q^{22} -1.00000 q^{24} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -4.00000 q^{39} -4.00000 q^{43} -4.00000 q^{44} -8.00000 q^{47} -1.00000 q^{48} +4.00000 q^{52} +10.0000 q^{53} -1.00000 q^{54} +4.00000 q^{57} +2.00000 q^{58} -4.00000 q^{59} +4.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{66} -4.00000 q^{67} +8.00000 q^{71} +1.00000 q^{72} -16.0000 q^{73} +6.00000 q^{74} -4.00000 q^{76} -4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -4.00000 q^{86} -2.00000 q^{87} -4.00000 q^{88} -8.00000 q^{89} +8.00000 q^{93} -8.00000 q^{94} -1.00000 q^{96} +8.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) 10.0000 1.37361 0.686803 − 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 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
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) −16.0000 −1.87266 −0.936329 − 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 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
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −2.00000 −0.214423
\(88\) −4.00000 −0.426401
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −4.00000 −0.398015 −0.199007 − 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −14.0000 −1.34096 −0.670478 − 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 − 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 4.00000 0.369800
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 4.00000 0.362143
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 − 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 − 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 8.00000 0.671345
\(143\) −16.0000 −1.33799
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −16.0000 −1.32417
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 4.00000 0.319235 0.159617 − 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) −8.00000 −0.636446
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −12.0000 −0.939913 −0.469956 − 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 8.00000 0.619059 0.309529 − 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) −4.00000 −0.304997
\(173\) −4.00000 −0.304114 −0.152057 − 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 4.00000 0.300658
\(178\) −8.00000 −0.599625
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 − 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 0 0
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −2.00000 −0.143963 −0.0719816 − 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −4.00000 −0.284268
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) −4.00000 −0.281439
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 − 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) 10.0000 0.686803
\(213\) −8.00000 −0.548151
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −14.0000 −0.948200
\(219\) 16.0000 1.08118
\(220\) 0 0
\(221\) 0 0
\(222\) −6.00000 −0.402694
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) 20.0000 1.32745 0.663723 − 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 4.00000 0.264906
\(229\) −4.00000 −0.264327 −0.132164 − 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) 10.0000 0.655122 0.327561 − 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 − 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 − 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) −8.00000 −0.508001
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 − 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.00000 0.499026 0.249513 − 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 12.0000 0.741362
\(263\) −8.00000 −0.493301 −0.246651 − 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) −4.00000 −0.244339
\(269\) −28.0000 −1.70719 −0.853595 − 0.520937i \(-0.825583\pi\)
−0.853595 + 0.520937i \(0.825583\pi\)
\(270\) 0 0
\(271\) 32.0000 1.94386 0.971931 − 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 0 0
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 − 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −12.0000 −0.719712
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 8.00000 0.476393
\(283\) −28.0000 −1.66443 −0.832214 − 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −16.0000 −0.946100
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −8.00000 −0.468968
\(292\) −16.0000 −0.936329
\(293\) −12.0000 −0.701047 −0.350524 − 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 4.00000 0.232104
\(298\) −10.0000 −0.579284
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) 4.00000 0.229794
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −32.0000 −1.81455 −0.907277 − 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) −4.00000 −0.226455
\(313\) 24.0000 1.35656 0.678280 − 0.734803i \(-0.262726\pi\)
0.678280 + 0.734803i \(0.262726\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −30.0000 −1.68497 −0.842484 − 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) −10.0000 −0.560772
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) 14.0000 0.774202
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −12.0000 −0.658586
\(333\) 6.00000 0.328798
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 − 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 3.00000 0.163178
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 32.0000 1.73290
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −4.00000 −0.215041
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −2.00000 −0.107211
\(349\) 12.0000 0.642345 0.321173 − 0.947021i \(-0.395923\pi\)
0.321173 + 0.947021i \(0.395923\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) −4.00000 −0.213201
\(353\) −24.0000 −1.27739 −0.638696 − 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −8.00000 −0.423999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 16.0000 0.844448 0.422224 − 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −20.0000 −1.05118
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 0 0
\(366\) −4.00000 −0.209083
\(367\) −16.0000 −0.835193 −0.417597 − 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) −22.0000 −1.13912 −0.569558 − 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) 36.0000 1.84920 0.924598 − 0.380945i \(-0.124401\pi\)
0.924598 + 0.380945i \(0.124401\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −4.00000 −0.203331
\(388\) 8.00000 0.406138
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 4.00000 0.200754 0.100377 − 0.994949i \(-0.467995\pi\)
0.100377 + 0.994949i \(0.467995\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 4.00000 0.199502
\(403\) −32.0000 −1.59403
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) 24.0000 1.18672 0.593362 − 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 12.0000 0.587643
\(418\) 16.0000 0.782586
\(419\) −28.0000 −1.36789 −0.683945 − 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 − 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) −28.0000 −1.36302
\(423\) −8.00000 −0.388973
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) 16.0000 0.772487
\(430\) 0 0
\(431\) 40.0000 1.92673 0.963366 − 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −8.00000 −0.384455 −0.192228 − 0.981350i \(-0.561571\pi\)
−0.192228 + 0.981350i \(0.561571\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) 0 0
\(438\) 16.0000 0.764510
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 − 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 14.0000 0.658505
\(453\) 8.00000 0.375873
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 22.0000 1.02912 0.514558 − 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) −4.00000 −0.186908
\(459\) 0 0
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 − 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) −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
\(466\) 10.0000 0.463241
\(467\) 20.0000 0.925490 0.462745 − 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 4.00000 0.184900
\(469\) 0 0
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) −4.00000 −0.184115
\(473\) 16.0000 0.735681
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) −24.0000 −1.09773
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) −8.00000 −0.364390
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 8.00000 0.362515 0.181257 − 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 4.00000 0.181071
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) −12.0000 −0.537194 −0.268597 − 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) 20.0000 0.892644
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.00000 −0.133235
\(508\) 16.0000 0.709885
\(509\) 4.00000 0.177297 0.0886484 − 0.996063i \(-0.471745\pi\)
0.0886484 + 0.996063i \(0.471745\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) 8.00000 0.352865
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 32.0000 1.40736
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) −32.0000 −1.40195 −0.700973 − 0.713188i \(-0.747251\pi\)
−0.700973 + 0.713188i \(0.747251\pi\)
\(522\) 2.00000 0.0875376
\(523\) 4.00000 0.174908 0.0874539 − 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 0 0
\(534\) 8.00000 0.346194
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) −12.0000 −0.517838
\(538\) −28.0000 −1.20717
\(539\) 0 0
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 − 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 32.0000 1.37452
\(543\) 20.0000 0.858282
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 − 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 10.0000 0.427179
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 0 0
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −8.00000 −0.338667
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −4.00000 −0.168580 −0.0842900 − 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) 6.00000 0.251533 0.125767 − 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 − 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) −16.0000 −0.668994
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −16.0000 −0.666089 −0.333044 − 0.942911i \(-0.608076\pi\)
−0.333044 + 0.942911i \(0.608076\pi\)
\(578\) −17.0000 −0.707107
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) 0 0
\(582\) −8.00000 −0.331611
\(583\) −40.0000 −1.65663
\(584\) −16.0000 −0.662085
\(585\) 0 0
\(586\) −12.0000 −0.495715
\(587\) 28.0000 1.15568 0.577842 − 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 6.00000 0.246598
\(593\) −24.0000 −0.985562 −0.492781 − 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 32.0000 1.30531 0.652654 − 0.757656i \(-0.273656\pi\)
0.652654 + 0.757656i \(0.273656\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 4.00000 0.162489
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) −32.0000 −1.29458
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 − 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 − 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 8.00000 0.321807
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −32.0000 −1.28308
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 24.0000 0.959233
\(627\) −16.0000 −0.638978
\(628\) 4.00000 0.159617
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −8.00000 −0.318223
\(633\) 28.0000 1.11290
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) 0 0
\(638\) −8.00000 −0.316723
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 − 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 4.00000 0.157867
\(643\) −36.0000 −1.41970 −0.709851 − 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) 46.0000 1.80012 0.900060 − 0.435767i \(-0.143523\pi\)
0.900060 + 0.435767i \(0.143523\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) 0 0
\(657\) −16.0000 −0.624219
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 − 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) −28.0000 −1.08907 −0.544537 − 0.838737i \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 0 0
\(668\) 8.00000 0.309529
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −16.0000 −0.617673
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 − 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 12.0000 0.461197 0.230599 − 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) −14.0000 −0.537667
\(679\) 0 0
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 32.0000 1.22534
\(683\) 44.0000 1.68361 0.841807 − 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) 4.00000 0.152610
\(688\) −4.00000 −0.152499
\(689\) 40.0000 1.52388
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −4.00000 −0.152057
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) 0 0
\(698\) 12.0000 0.454207
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 − 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) −4.00000 −0.150970
\(703\) −24.0000 −0.905177
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) −38.0000 −1.42712 −0.713560 − 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −8.00000 −0.299813
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 24.0000 0.896296
\(718\) 16.0000 0.597115
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 8.00000 0.297523
\(724\) −20.0000 −0.743294
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −4.00000 −0.147844
\(733\) −4.00000 −0.147743 −0.0738717 − 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 − 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) 0 0
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) −8.00000 −0.291730
\(753\) −20.0000 −0.728841
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 36.0000 1.30758
\(759\) 0 0
\(760\) 0 0
\(761\) −16.0000 −0.580000 −0.290000 − 0.957027i \(-0.593655\pi\)
−0.290000 + 0.957027i \(0.593655\pi\)
\(762\) −16.0000 −0.579619
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) −16.0000 −0.577727
\(768\) −1.00000 −0.0360844
\(769\) −40.0000 −1.44244 −0.721218 − 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) −8.00000 −0.288113
\(772\) −2.00000 −0.0719816
\(773\) 36.0000 1.29483 0.647415 − 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 8.00000 0.287183
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 0 0
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) 20.0000 0.712923 0.356462 − 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) −6.00000 −0.213741
\(789\) 8.00000 0.284808
\(790\) 0 0
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) 16.0000 0.568177
\(794\) 4.00000 0.141955
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) 12.0000 0.425062 0.212531 − 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) −18.0000 −0.635602
\(803\) 64.0000 2.25851
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −32.0000 −1.12715
\(807\) 28.0000 0.985647
\(808\) −4.00000 −0.140720
\(809\) 42.0000 1.47664 0.738321 − 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 − 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 0 0
\(813\) −32.0000 −1.12229
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 24.0000 0.839140
\(819\) 0 0
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 − 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) −10.0000 −0.348790
\(823\) −16.0000 −0.557725 −0.278862 − 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 − 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) 52.0000 1.80603 0.903017 − 0.429604i \(-0.141347\pi\)
0.903017 + 0.429604i \(0.141347\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 4.00000 0.138675
\(833\) 0 0
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 8.00000 0.276520
\(838\) −28.0000 −0.967244
\(839\) −24.0000 −0.828572 −0.414286 − 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 6.00000 0.206774
\(843\) −6.00000 −0.206651
\(844\) −28.0000 −0.963800
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) 10.0000 0.343401
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) 0 0
\(852\) −8.00000 −0.274075
\(853\) 52.0000 1.78045 0.890223 − 0.455525i \(-0.150548\pi\)
0.890223 + 0.455525i \(0.150548\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 16.0000 0.546231
\(859\) 36.0000 1.22830 0.614152 − 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 40.0000 1.36241
\(863\) −48.0000 −1.63394 −0.816970 − 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −8.00000 −0.271851
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) −14.0000 −0.474100
\(873\) 8.00000 0.270759
\(874\) 0 0
\(875\) 0 0
\(876\) 16.0000 0.540590
\(877\) −18.0000 −0.607817 −0.303908 − 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 0 0
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) 8.00000 0.269527 0.134763 − 0.990878i \(-0.456973\pi\)
0.134763 + 0.990878i \(0.456973\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 − 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 24.0000 0.805841 0.402921 − 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) −16.0000 −0.535720
\(893\) 32.0000 1.07084
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) −16.0000 −0.533630
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) 20.0000 0.664089 0.332045 − 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 20.0000 0.663723
\(909\) −4.00000 −0.132672
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 − 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 4.00000 0.132453
\(913\) 48.0000 1.58857
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) 0 0
\(918\) 0 0
\(919\) −48.0000 −1.58337 −0.791687 − 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) −12.0000 −0.395199
\(923\) 32.0000 1.05329
\(924\) 0 0
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) −8.00000 −0.262754
\(928\) 2.00000 0.0656532
\(929\) −48.0000 −1.57483 −0.787414 − 0.616424i \(-0.788581\pi\)
−0.787414 + 0.616424i \(0.788581\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) 32.0000 1.04763
\(934\) 20.0000 0.654420
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) −24.0000 −0.783210
\(940\) 0 0
\(941\) 12.0000 0.391189 0.195594 − 0.980685i \(-0.437336\pi\)
0.195594 + 0.980685i \(0.437336\pi\)
\(942\) −4.00000 −0.130327
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) 28.0000 0.909878 0.454939 − 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 8.00000 0.259828
\(949\) −64.0000 −2.07753
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 8.00000 0.258603
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 24.0000 0.773791
\(963\) −4.00000 −0.128898
\(964\) −8.00000 −0.257663
\(965\) 0 0
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 0 0
\(971\) −4.00000 −0.128366 −0.0641831 − 0.997938i \(-0.520444\pi\)
−0.0641831 + 0.997938i \(0.520444\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) −14.0000 −0.447900 −0.223950 − 0.974601i \(-0.571895\pi\)
−0.223950 + 0.974601i \(0.571895\pi\)
\(978\) 12.0000 0.383718
\(979\) 32.0000 1.02272
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) −12.0000 −0.382935
\(983\) −8.00000 −0.255160 −0.127580 − 0.991828i \(-0.540721\pi\)
−0.127580 + 0.991828i \(0.540721\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) 0 0
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 − 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −8.00000 −0.254000
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) 28.0000 0.886769 0.443384 − 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) −12.0000 −0.379853
\(999\) −6.00000 −0.189832
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7350.2.a.br.1.1 1
5.4 even 2 294.2.a.c.1.1 yes 1
7.6 odd 2 7350.2.a.cj.1.1 1
15.14 odd 2 882.2.a.l.1.1 1
20.19 odd 2 2352.2.a.b.1.1 1
35.4 even 6 294.2.e.d.79.1 2
35.9 even 6 294.2.e.d.67.1 2
35.19 odd 6 294.2.e.e.67.1 2
35.24 odd 6 294.2.e.e.79.1 2
35.34 odd 2 294.2.a.b.1.1 ✓ 1
40.19 odd 2 9408.2.a.de.1.1 1
40.29 even 2 9408.2.a.bo.1.1 1
60.59 even 2 7056.2.a.ca.1.1 1
105.44 odd 6 882.2.g.a.361.1 2
105.59 even 6 882.2.g.f.667.1 2
105.74 odd 6 882.2.g.a.667.1 2
105.89 even 6 882.2.g.f.361.1 2
105.104 even 2 882.2.a.f.1.1 1
140.19 even 6 2352.2.q.a.1537.1 2
140.39 odd 6 2352.2.q.y.961.1 2
140.59 even 6 2352.2.q.a.961.1 2
140.79 odd 6 2352.2.q.y.1537.1 2
140.139 even 2 2352.2.a.y.1.1 1
280.69 odd 2 9408.2.a.br.1.1 1
280.139 even 2 9408.2.a.b.1.1 1
420.419 odd 2 7056.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.2.a.b.1.1 ✓ 1 35.34 odd 2
294.2.a.c.1.1 yes 1 5.4 even 2
294.2.e.d.67.1 2 35.9 even 6
294.2.e.d.79.1 2 35.4 even 6
294.2.e.e.67.1 2 35.19 odd 6
294.2.e.e.79.1 2 35.24 odd 6
882.2.a.f.1.1 1 105.104 even 2
882.2.a.l.1.1 1 15.14 odd 2
882.2.g.a.361.1 2 105.44 odd 6
882.2.g.a.667.1 2 105.74 odd 6
882.2.g.f.361.1 2 105.89 even 6
882.2.g.f.667.1 2 105.59 even 6
2352.2.a.b.1.1 1 20.19 odd 2
2352.2.a.y.1.1 1 140.139 even 2
2352.2.q.a.961.1 2 140.59 even 6
2352.2.q.a.1537.1 2 140.19 even 6
2352.2.q.y.961.1 2 140.39 odd 6
2352.2.q.y.1537.1 2 140.79 odd 6
7056.2.a.a.1.1 1 420.419 odd 2
7056.2.a.ca.1.1 1 60.59 even 2
7350.2.a.br.1.1 1 1.1 even 1 trivial
7350.2.a.cj.1.1 1 7.6 odd 2
9408.2.a.b.1.1 1 280.139 even 2
9408.2.a.bo.1.1 1 40.29 even 2
9408.2.a.br.1.1 1 280.69 odd 2
9408.2.a.de.1.1 1 40.19 odd 2