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Properties

Label 3465.2.a.l.1.1
Level $3465$
Weight $2$
Character 3465.1
Self dual yes
Analytic conductor $27.668$
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] = [3465,2,Mod(1,3465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3465, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3465.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3465.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: \(27.6681643004\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1155)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3465.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} +6.00000 q^{17} -4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} -8.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} +8.00000 q^{31} +5.00000 q^{32} +6.00000 q^{34} +1.00000 q^{35} +6.00000 q^{37} -4.00000 q^{38} +3.00000 q^{40} +6.00000 q^{41} -4.00000 q^{43} +1.00000 q^{44} -8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +10.0000 q^{53} +1.00000 q^{55} +3.00000 q^{56} -6.00000 q^{58} +12.0000 q^{59} -10.0000 q^{61} +8.00000 q^{62} +7.00000 q^{64} +2.00000 q^{65} -12.0000 q^{67} -6.00000 q^{68} +1.00000 q^{70} +8.00000 q^{71} -6.00000 q^{73} +6.00000 q^{74} +4.00000 q^{76} +1.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} +6.00000 q^{82} +4.00000 q^{83} -6.00000 q^{85} -4.00000 q^{86} +3.00000 q^{88} -10.0000 q^{89} +2.00000 q^{91} +8.00000 q^{92} +8.00000 q^{94} +4.00000 q^{95} -14.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 10.0000 1.37361 0.686803 − 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 14.0000 1.31701 0.658505 − 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −18.0000 −1.54349
\(137\) 22.0000 1.87959 0.939793 − 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 18.0000 1.47462 0.737309 − 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 12.0000 0.973329
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) −5.00000 −0.395285
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 − 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 16.0000 1.23812 0.619059 − 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 − 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) 24.0000 1.76930
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) −16.0000 −1.15772 −0.578860 − 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 − 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) −22.0000 −1.56744 −0.783718 − 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −5.00000 −0.334077
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 18.0000 1.18176
\(233\) 22.0000 1.44127 0.720634 − 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) −6.00000 −0.388922
\(239\) 16.0000 1.03495 0.517477 − 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 − 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) −24.0000 −1.52400
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −20.0000 −1.26239 −0.631194 − 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 30.0000 1.87135 0.935674 − 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) −16.0000 −0.986602 −0.493301 − 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) −14.0000 −0.853595 −0.426798 − 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 22.0000 1.32907
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −18.0000 −1.08152 −0.540758 − 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) −18.0000 −1.07379 −0.536895 − 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) 6.00000 0.351123
\(293\) 18.0000 1.05157 0.525786 − 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) −18.0000 −1.04623
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) 16.0000 0.907277 0.453638 − 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 − 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 34.0000 1.90963 0.954815 − 0.297200i \(-0.0960529\pi\)
0.954815 + 0.297200i \(0.0960529\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) −7.00000 −0.391312
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 20.0000 1.10770
\(327\) 0 0
\(328\) −18.0000 −0.993884
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 − 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) 16.0000 0.875481
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 − 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 6.00000 0.325396
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −20.0000 −1.07366 −0.536828 − 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −5.00000 −0.266501
\(353\) −34.0000 −1.80964 −0.904819 − 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −8.00000 −0.422224 −0.211112 − 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 6.00000 0.315353
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 − 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) −6.00000 −0.311925
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) −24.0000 −1.23771
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) −22.0000 −1.11977
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) 10.0000 0.507020 0.253510 − 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) −48.0000 −2.42746
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) −22.0000 −1.10834
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 − 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −2.00000 −0.0998752 −0.0499376 − 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) −16.0000 −0.797017
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 − 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) −16.0000 −0.788263
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) −10.0000 −0.490290
\(417\) 0 0
\(418\) 4.00000 0.195646
\(419\) 36.0000 1.75872 0.879358 − 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 − 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) −30.0000 −1.45693
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 16.0000 0.770693 0.385346 − 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 − 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 32.0000 1.53077
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 − 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) −3.00000 −0.143019
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) −7.00000 −0.330719
\(449\) −2.00000 −0.0943858 −0.0471929 − 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 − 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) −8.00000 −0.373002
\(461\) −14.0000 −0.652045 −0.326023 − 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 22.0000 1.01913
\(467\) −28.0000 −1.29569 −0.647843 − 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) −36.0000 −1.65703
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 6.00000 0.275010
\(477\) 0 0
\(478\) 16.0000 0.731823
\(479\) 16.0000 0.731059 0.365529 − 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) 24.0000 1.08754 0.543772 − 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) 30.0000 1.35804
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 20.0000 0.902587 0.451294 − 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −12.0000 −0.537194 −0.268597 − 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −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
\(505\) 6.00000 0.266996
\(506\) 8.00000 0.355643
\(507\) 0 0
\(508\) 0 0
\(509\) −30.0000 −1.32973 −0.664863 − 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) 30.0000 1.32324
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) −6.00000 −0.263625
\(519\) 0 0
\(520\) −6.00000 −0.263117
\(521\) 6.00000 0.262865 0.131432 − 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) −36.0000 −1.57417 −0.787085 − 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 48.0000 2.09091
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −10.0000 −0.434372
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 36.0000 1.55496
\(537\) 0 0
\(538\) −14.0000 −0.603583
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 − 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) 30.0000 1.28624
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 − 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −22.0000 −0.939793
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) −18.0000 −0.764747
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) 4.00000 0.168580 0.0842900 − 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) −24.0000 −1.00702
\(569\) −18.0000 −0.754599 −0.377300 − 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 − 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 − 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) −10.0000 −0.414158
\(584\) 18.0000 0.744845
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) −12.0000 −0.494032
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) −18.0000 −0.737309
\(597\) 0 0
\(598\) 16.0000 0.654289
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 − 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 4.00000 0.163028
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) −20.0000 −0.811107
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 − 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) −42.0000 −1.69086 −0.845428 − 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 16.0000 0.641542
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 36.0000 1.43541
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −24.0000 −0.954669
\(633\) 0 0
\(634\) 34.0000 1.35031
\(635\) 0 0
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) 3.00000 0.118585
\(641\) −2.00000 −0.0789953 −0.0394976 − 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 − 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −20.0000 −0.783260
\(653\) 18.0000 0.704394 0.352197 − 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) −8.00000 −0.311872
\(659\) −36.0000 −1.40236 −0.701180 − 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −42.0000 −1.63361 −0.816805 − 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) −4.00000 −0.155113
\(666\) 0 0
\(667\) 48.0000 1.85857
\(668\) −16.0000 −0.619059
\(669\) 0 0
\(670\) 12.0000 0.463600
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) −38.0000 −1.46479 −0.732396 − 0.680879i \(-0.761598\pi\)
−0.732396 + 0.680879i \(0.761598\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) 18.0000 0.690268
\(681\) 0 0
\(682\) −8.00000 −0.306336
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) −22.0000 −0.840577
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) −10.0000 −0.378506
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 42.0000 1.58632 0.793159 − 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) −7.00000 −0.263822
\(705\) 0 0
\(706\) −34.0000 −1.27961
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 − 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) 30.0000 1.12430
\(713\) −64.0000 −2.39682
\(714\) 0 0
\(715\) −2.00000 −0.0747958
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) −8.00000 −0.298557
\(719\) 8.00000 0.298350 0.149175 − 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) −6.00000 −0.222988
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 − 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) −6.00000 −0.222375
\(729\) 0 0
\(730\) 6.00000 0.222070
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 46.0000 1.69905 0.849524 − 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −40.0000 −1.47442
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 − 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) −10.0000 −0.367112
\(743\) 32.0000 1.17397 0.586983 − 0.809599i \(-0.300316\pi\)
0.586983 + 0.809599i \(0.300316\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 6.00000 0.219382
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 − 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 − 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) −12.0000 −0.435286
\(761\) 6.00000 0.217500 0.108750 − 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 − 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 0 0
\(772\) 22.0000 0.791797
\(773\) 10.0000 0.359675 0.179838 − 0.983696i \(-0.442443\pi\)
0.179838 + 0.983696i \(0.442443\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 42.0000 1.50771
\(777\) 0 0
\(778\) 10.0000 0.358517
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) −48.0000 −1.71648
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 − 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 22.0000 0.783718
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 20.0000 0.710221
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) −30.0000 −1.06265 −0.531327 − 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 5.00000 0.176777
\(801\) 0 0
\(802\) −2.00000 −0.0706225
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) −16.0000 −0.563576
\(807\) 0 0
\(808\) 18.0000 0.633238
\(809\) 30.0000 1.05474 0.527372 − 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) −6.00000 −0.210300
\(815\) −20.0000 −0.700569
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) −8.00000 −0.278862 −0.139431 − 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) −48.0000 −1.67216
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 − 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) −14.0000 −0.485363
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) −16.0000 −0.552381 −0.276191 − 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 6.00000 0.206774
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −10.0000 −0.343401
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) −48.0000 −1.64542
\(852\) 0 0
\(853\) 38.0000 1.30110 0.650548 − 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) −18.0000 −0.614868 −0.307434 − 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 − 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 16.0000 0.544962
\(863\) 16.0000 0.544646 0.272323 − 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) 8.00000 0.271538
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) 32.0000 1.08242
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 32.0000 1.07995
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) −2.00000 −0.0673817 −0.0336909 − 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 − 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 32.0000 1.07445 0.537227 − 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 10.0000 0.335201
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) −32.0000 −1.07084
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) −2.00000 −0.0667409
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) 60.0000 1.99889
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) −42.0000 −1.39690
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) 44.0000 1.46100 0.730498 − 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) −2.00000 −0.0662994
\(911\) −48.0000 −1.59031 −0.795155 − 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) −4.00000 −0.132381
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 − 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) −24.0000 −0.791257
\(921\) 0 0
\(922\) −14.0000 −0.461065
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 16.0000 0.525793
\(927\) 0 0
\(928\) −30.0000 −0.984798
\(929\) 14.0000 0.459325 0.229663 − 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) −22.0000 −0.720634
\(933\) 0 0
\(934\) −28.0000 −0.916188
\(935\) 6.00000 0.196221
\(936\) 0 0
\(937\) −38.0000 −1.24141 −0.620703 − 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 12.0000 0.391814
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) −30.0000 −0.977972 −0.488986 − 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) −48.0000 −1.56310
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 28.0000 0.909878 0.454939 − 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) 18.0000 0.583383
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 16.0000 0.517748
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) 16.0000 0.516937
\(959\) −22.0000 −0.710417
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) 22.0000 0.708572
\(965\) 22.0000 0.708205
\(966\) 0 0
\(967\) 56.0000 1.80084 0.900419 − 0.435023i \(-0.143260\pi\)
0.900419 + 0.435023i \(0.143260\pi\)
\(968\) −3.00000 −0.0964237
\(969\) 0 0
\(970\) 14.0000 0.449513
\(971\) 28.0000 0.898563 0.449281 − 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) 24.0000 0.769010
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −18.0000 −0.575871 −0.287936 − 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 10.0000 0.319601
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 20.0000 0.638226
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 22.0000 0.700978
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 40.0000 1.27000
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −12.0000 −0.379853
\(999\) 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 3465.2.a.l.1.1 1
3.2 odd 2 1155.2.a.e.1.1 ✓ 1
15.14 odd 2 5775.2.a.v.1.1 1
21.20 even 2 8085.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.e.1.1 ✓ 1 3.2 odd 2
3465.2.a.l.1.1 1 1.1 even 1 trivial
5775.2.a.v.1.1 1 15.14 odd 2
8085.2.a.g.1.1 1 21.20 even 2