Properties

Label 2310.2.a.h.1.1
Level $2310$
Weight $2$
Character 2310.1
Self dual yes
Analytic conductor $18.445$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2310,2,Mod(1,2310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2310, 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("2310.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2310.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: \(18.4454428669\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2310.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} +1.00000 q^{22} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +6.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +1.00000 q^{55} -1.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} +12.0000 q^{59} -1.00000 q^{60} +14.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +1.00000 q^{66} -16.0000 q^{67} -6.00000 q^{68} +1.00000 q^{70} -1.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -1.00000 q^{77} -2.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +1.00000 q^{84} +6.00000 q^{85} +4.00000 q^{86} -6.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} +1.00000 q^{90} +2.00000 q^{91} -4.00000 q^{93} +4.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.00000 −0.258199
\(16\) 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
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) 1.00000 0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) 6.00000 1.02899
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 4.00000 0.648886
\(39\) 2.00000 0.320256
\(40\) 1.00000 0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −1.00000 −0.154303
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −6.00000 −0.840168
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.00000 0.134840
\(56\) −1.00000 −0.133631
\(57\) −4.00000 −0.529813
\(58\) 6.00000 0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −1.00000 −0.129099
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 4.00000 0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 1.00000 0.123091
\(67\) −16.0000 −1.95471 −0.977356 0.211604i \(-0.932131\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −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
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) −1.00000 −0.113961
\(78\) −2.00000 −0.226455
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 1.00000 0.109109
\(85\) 6.00000 0.650791
\(86\) 4.00000 0.431331
\(87\) −6.00000 −0.643268
\(88\) 1.00000 0.106600
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 6.00000 0.594089
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) −1.00000 −0.0975900
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 2.00000 0.189832
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) −12.0000 −1.10469
\(119\) −6.00000 −0.550019
\(120\) 1.00000 0.0912871
\(121\) 1.00000 0.0909091
\(122\) −14.0000 −1.26750
\(123\) −6.00000 −0.541002
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 2.00000 0.175412
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −4.00000 −0.346844
\(134\) 16.0000 1.38219
\(135\) −1.00000 −0.0860663
\(136\) 6.00000 0.514496
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) −2.00000 −0.165521
\(147\) 1.00000 0.0824786
\(148\) 2.00000 0.164399
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 4.00000 0.324443
\(153\) −6.00000 −0.485071
\(154\) 1.00000 0.0805823
\(155\) 4.00000 0.321288
\(156\) 2.00000 0.160128
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 4.00000 0.318223
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) −6.00000 −0.468521
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) −4.00000 −0.305888
\(172\) −4.00000 −0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 6.00000 0.454859
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 12.0000 0.901975
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −2.00000 −0.148250
\(183\) 14.0000 1.03491
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 4.00000 0.293294
\(187\) 6.00000 0.438763
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) −4.00000 −0.290191
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 10.0000 0.717958
\(195\) −2.00000 −0.143223
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 1.00000 0.0710669
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −16.0000 −1.12855
\(202\) 18.0000 1.26648
\(203\) −6.00000 −0.421117
\(204\) −6.00000 −0.420084
\(205\) 6.00000 0.419058
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 4.00000 0.276686
\(210\) 1.00000 0.0690066
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 4.00000 0.272798
\(216\) −1.00000 −0.0680414
\(217\) −4.00000 −0.271538
\(218\) −2.00000 −0.135457
\(219\) 2.00000 0.135147
\(220\) 1.00000 0.0674200
\(221\) −12.0000 −0.807207
\(222\) −2.00000 −0.134231
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) −4.00000 −0.264906
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 6.00000 0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) −4.00000 −0.259828
\(238\) 6.00000 0.388922
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) 14.0000 0.896258
\(245\) −1.00000 −0.0638877
\(246\) 6.00000 0.382546
\(247\) −8.00000 −0.509028
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 4.00000 0.249029
\(259\) 2.00000 0.124274
\(260\) −2.00000 −0.124035
\(261\) −6.00000 −0.371391
\(262\) −12.0000 −0.741362
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 1.00000 0.0615457
\(265\) 6.00000 0.368577
\(266\) 4.00000 0.245256
\(267\) −6.00000 −0.367194
\(268\) −16.0000 −0.977356
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 1.00000 0.0608581
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −6.00000 −0.363803
\(273\) 2.00000 0.121046
\(274\) −18.0000 −1.08742
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 4.00000 0.239904
\(279\) −4.00000 −0.239474
\(280\) 1.00000 0.0597614
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) 2.00000 0.118262
\(287\) −6.00000 −0.354169
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) −10.0000 −0.586210
\(292\) 2.00000 0.117041
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −12.0000 −0.698667
\(296\) −2.00000 −0.116248
\(297\) −1.00000 −0.0580259
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −4.00000 −0.230556
\(302\) −20.0000 −1.15087
\(303\) −18.0000 −1.03407
\(304\) −4.00000 −0.229416
\(305\) −14.0000 −0.801638
\(306\) 6.00000 0.342997
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 8.00000 0.455104
\(310\) −4.00000 −0.227185
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) −2.00000 −0.113228
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −2.00000 −0.112867
\(315\) −1.00000 −0.0563436
\(316\) −4.00000 −0.225018
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 6.00000 0.336463
\(319\) 6.00000 0.335936
\(320\) −1.00000 −0.0559017
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −8.00000 −0.443079
\(327\) 2.00000 0.110600
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) −1.00000 −0.0550482
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 24.0000 1.31322
\(335\) 16.0000 0.874173
\(336\) 1.00000 0.0545545
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 9.00000 0.489535
\(339\) −6.00000 −0.325875
\(340\) 6.00000 0.325396
\(341\) 4.00000 0.216612
\(342\) 4.00000 0.216295
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −6.00000 −0.321634
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 2.00000 0.106752
\(352\) 1.00000 0.0533002
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −6.00000 −0.317554
\(358\) 12.0000 0.634220
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 22.0000 1.15629
\(363\) 1.00000 0.0524864
\(364\) 2.00000 0.104828
\(365\) −2.00000 −0.104685
\(366\) −14.0000 −0.731792
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 2.00000 0.103975
\(371\) −6.00000 −0.311504
\(372\) −4.00000 −0.207390
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −6.00000 −0.310253
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) −1.00000 −0.0514344
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 4.00000 0.205196
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 1.00000 0.0509647
\(386\) 10.0000 0.508987
\(387\) −4.00000 −0.203331
\(388\) −10.0000 −0.507673
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 2.00000 0.101274
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 12.0000 0.605320
\(394\) 6.00000 0.302276
\(395\) 4.00000 0.201262
\(396\) −1.00000 −0.0502519
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −20.0000 −1.00251
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 16.0000 0.798007
\(403\) −8.00000 −0.398508
\(404\) −18.0000 −0.895533
\(405\) −1.00000 −0.0496904
\(406\) 6.00000 0.297775
\(407\) −2.00000 −0.0991363
\(408\) 6.00000 0.297044
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) −6.00000 −0.296319
\(411\) 18.0000 0.887875
\(412\) 8.00000 0.394132
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −4.00000 −0.195881
\(418\) −4.00000 −0.195646
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 14.0000 0.677507
\(428\) −12.0000 −0.580042
\(429\) −2.00000 −0.0965609
\(430\) −4.00000 −0.192897
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 4.00000 0.192006
\(435\) 6.00000 0.287678
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) −2.00000 −0.0955637
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 2.00000 0.0949158
\(445\) 6.00000 0.284427
\(446\) 16.0000 0.757622
\(447\) −6.00000 −0.283790
\(448\) 1.00000 0.0472456
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 6.00000 0.282529
\(452\) −6.00000 −0.282216
\(453\) 20.0000 0.939682
\(454\) 0 0
\(455\) −2.00000 −0.0937614
\(456\) 4.00000 0.187317
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −26.0000 −1.21490
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 1.00000 0.0465242
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) −6.00000 −0.278543
\(465\) 4.00000 0.185496
\(466\) −6.00000 −0.277945
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 2.00000 0.0924500
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) −12.0000 −0.552345
\(473\) 4.00000 0.183920
\(474\) 4.00000 0.183726
\(475\) −4.00000 −0.183533
\(476\) −6.00000 −0.275010
\(477\) −6.00000 −0.274721
\(478\) −12.0000 −0.548867
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 1.00000 0.0456435
\(481\) 4.00000 0.182384
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 10.0000 0.454077
\(486\) −1.00000 −0.0453609
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) −14.0000 −0.633750
\(489\) 8.00000 0.361773
\(490\) 1.00000 0.0451754
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −6.00000 −0.270501
\(493\) 36.0000 1.62136
\(494\) 8.00000 0.359937
\(495\) 1.00000 0.0449467
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −24.0000 −1.07224
\(502\) 12.0000 0.535586
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) −16.0000 −0.709885
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) −6.00000 −0.265684
\(511\) 2.00000 0.0884748
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) −6.00000 −0.264649
\(515\) −8.00000 −0.352522
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) −2.00000 −0.0878750
\(519\) 18.0000 0.790112
\(520\) 2.00000 0.0877058
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 6.00000 0.262613
\(523\) 32.0000 1.39926 0.699631 0.714504i \(-0.253348\pi\)
0.699631 + 0.714504i \(0.253348\pi\)
\(524\) 12.0000 0.524222
\(525\) 1.00000 0.0436436
\(526\) 24.0000 1.04645
\(527\) 24.0000 1.04546
\(528\) −1.00000 −0.0435194
\(529\) −23.0000 −1.00000
\(530\) −6.00000 −0.260623
\(531\) 12.0000 0.520756
\(532\) −4.00000 −0.173422
\(533\) −12.0000 −0.519778
\(534\) 6.00000 0.259645
\(535\) 12.0000 0.518805
\(536\) 16.0000 0.691095
\(537\) −12.0000 −0.517838
\(538\) −18.0000 −0.776035
\(539\) −1.00000 −0.0430730
\(540\) −1.00000 −0.0430331
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 16.0000 0.687259
\(543\) −22.0000 −0.944110
\(544\) 6.00000 0.257248
\(545\) −2.00000 −0.0856706
\(546\) −2.00000 −0.0855921
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 18.0000 0.768922
\(549\) 14.0000 0.597505
\(550\) 1.00000 0.0426401
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 22.0000 0.934690
\(555\) −2.00000 −0.0848953
\(556\) −4.00000 −0.169638
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 4.00000 0.169334
\(559\) −8.00000 −0.338364
\(560\) −1.00000 −0.0422577
\(561\) 6.00000 0.253320
\(562\) 6.00000 0.253095
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 16.0000 0.672530
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) −4.00000 −0.167542
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) −19.0000 −0.790296
\(579\) −10.0000 −0.415586
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) 10.0000 0.414513
\(583\) 6.00000 0.248495
\(584\) −2.00000 −0.0827606
\(585\) −2.00000 −0.0826898
\(586\) 30.0000 1.23929
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 1.00000 0.0412393
\(589\) 16.0000 0.659269
\(590\) 12.0000 0.494032
\(591\) −6.00000 −0.246807
\(592\) 2.00000 0.0821995
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 1.00000 0.0410305
\(595\) 6.00000 0.245976
\(596\) −6.00000 −0.245770
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 4.00000 0.163028
\(603\) −16.0000 −0.651570
\(604\) 20.0000 0.813788
\(605\) −1.00000 −0.0406558
\(606\) 18.0000 0.731200
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 4.00000 0.162221
\(609\) −6.00000 −0.243132
\(610\) 14.0000 0.566843
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −8.00000 −0.322854
\(615\) 6.00000 0.241943
\(616\) 1.00000 0.0402911
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) −8.00000 −0.321807
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) −6.00000 −0.240385
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −14.0000 −0.559553
\(627\) 4.00000 0.159745
\(628\) 2.00000 0.0798087
\(629\) −12.0000 −0.478471
\(630\) 1.00000 0.0398410
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 4.00000 0.159111
\(633\) −4.00000 −0.158986
\(634\) 30.0000 1.19145
\(635\) 16.0000 0.634941
\(636\) −6.00000 −0.237915
\(637\) 2.00000 0.0792429
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 12.0000 0.473602
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) −24.0000 −0.944267
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −12.0000 −0.471041
\(650\) −2.00000 −0.0784465
\(651\) −4.00000 −0.156772
\(652\) 8.00000 0.313304
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −12.0000 −0.468879
\(656\) −6.00000 −0.234261
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 1.00000 0.0389249
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) −20.0000 −0.777322
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) 4.00000 0.155113
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) −24.0000 −0.928588
\(669\) −16.0000 −0.618596
\(670\) −16.0000 −0.618134
\(671\) −14.0000 −0.540464
\(672\) −1.00000 −0.0385758
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −14.0000 −0.539260
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 6.00000 0.230429
\(679\) −10.0000 −0.383765
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) −4.00000 −0.153168
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) −4.00000 −0.152944
\(685\) −18.0000 −0.687745
\(686\) −1.00000 −0.0381802
\(687\) 26.0000 0.991962
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 18.0000 0.684257
\(693\) −1.00000 −0.0379869
\(694\) 12.0000 0.455514
\(695\) 4.00000 0.151729
\(696\) 6.00000 0.227429
\(697\) 36.0000 1.36360
\(698\) −14.0000 −0.529908
\(699\) 6.00000 0.226941
\(700\) 1.00000 0.0377964
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −8.00000 −0.301726
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −18.0000 −0.676960
\(708\) 12.0000 0.450988
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 6.00000 0.224544
\(715\) 2.00000 0.0747958
\(716\) −12.0000 −0.448461
\(717\) 12.0000 0.448148
\(718\) −36.0000 −1.34351
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 8.00000 0.297936
\(722\) 3.00000 0.111648
\(723\) 2.00000 0.0743808
\(724\) −22.0000 −0.817624
\(725\) −6.00000 −0.222834
\(726\) −1.00000 −0.0371135
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) 24.0000 0.887672
\(732\) 14.0000 0.517455
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −8.00000 −0.295285
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 6.00000 0.220863
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −8.00000 −0.293887
\(742\) 6.00000 0.220267
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 4.00000 0.146647
\(745\) 6.00000 0.219823
\(746\) 22.0000 0.805477
\(747\) 0 0
\(748\) 6.00000 0.219382
\(749\) −12.0000 −0.438470
\(750\) 1.00000 0.0365148
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 12.0000 0.437014
\(755\) −20.0000 −0.727875
\(756\) 1.00000 0.0363696
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 16.0000 0.579619
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 1.00000 0.0360844
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 6.00000 0.216085
\(772\) −10.0000 −0.359908
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 4.00000 0.143777
\(775\) −4.00000 −0.143684
\(776\) 10.0000 0.358979
\(777\) 2.00000 0.0717496
\(778\) −30.0000 −1.07555
\(779\) 24.0000 0.859889
\(780\) −2.00000 −0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 1.00000 0.0357143
\(785\) −2.00000 −0.0713831
\(786\) −12.0000 −0.428026
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) −6.00000 −0.213741
\(789\) −24.0000 −0.854423
\(790\) −4.00000 −0.142314
\(791\) −6.00000 −0.213335
\(792\) 1.00000 0.0355335
\(793\) 28.0000 0.994309
\(794\) −2.00000 −0.0709773
\(795\) 6.00000 0.212798
\(796\) 20.0000 0.708881
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 4.00000 0.141598
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) −18.0000 −0.635602
\(803\) −2.00000 −0.0705785
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 18.0000 0.633630
\(808\) 18.0000 0.633238
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 1.00000 0.0351364
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −6.00000 −0.210559
\(813\) −16.0000 −0.561144
\(814\) 2.00000 0.0701000
\(815\) −8.00000 −0.280228
\(816\) −6.00000 −0.210042
\(817\) 16.0000 0.559769
\(818\) 22.0000 0.769212
\(819\) 2.00000 0.0698857
\(820\) 6.00000 0.209529
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) −18.0000 −0.627822
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −8.00000 −0.278693
\(825\) −1.00000 −0.0348155
\(826\) −12.0000 −0.417533
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −46.0000 −1.59765 −0.798823 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 2.00000 0.0693375
\(833\) −6.00000 −0.207888
\(834\) 4.00000 0.138509
\(835\) 24.0000 0.830554
\(836\) 4.00000 0.138343
\(837\) −4.00000 −0.138260
\(838\) −36.0000 −1.24360
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 1.00000 0.0345033
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) −6.00000 −0.206651
\(844\) −4.00000 −0.137686
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) −6.00000 −0.206041
\(849\) −16.0000 −0.549119
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) −14.0000 −0.479070
\(855\) 4.00000 0.136797
\(856\) 12.0000 0.410152
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 2.00000 0.0682789
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 4.00000 0.136399
\(861\) −6.00000 −0.204479
\(862\) −12.0000 −0.408722
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −18.0000 −0.612018
\(866\) −14.0000 −0.475739
\(867\) 19.0000 0.645274
\(868\) −4.00000 −0.135769
\(869\) 4.00000 0.135691
\(870\) −6.00000 −0.203419
\(871\) −32.0000 −1.08428
\(872\) −2.00000 −0.0677285
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 2.00000 0.0675737
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) −8.00000 −0.269987
\(879\) −30.0000 −1.01187
\(880\) 1.00000 0.0337100
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) −12.0000 −0.403604
\(885\) −12.0000 −0.403376
\(886\) 24.0000 0.806296
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −16.0000 −0.536623
\(890\) −6.00000 −0.201120
\(891\) −1.00000 −0.0335013
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 6.00000 0.200670
\(895\) 12.0000 0.401116
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 24.0000 0.800445
\(900\) 1.00000 0.0333333
\(901\) 36.0000 1.19933
\(902\) −6.00000 −0.199778
\(903\) −4.00000 −0.133112
\(904\) 6.00000 0.199557
\(905\) 22.0000 0.731305
\(906\) −20.0000 −0.664455
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) 0 0
\(909\) −18.0000 −0.597022
\(910\) 2.00000 0.0662994
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) −4.00000 −0.132453
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) −14.0000 −0.462826
\(916\) 26.0000 0.859064
\(917\) 12.0000 0.396275
\(918\) 6.00000 0.198030
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) 18.0000 0.592798
\(923\) 0 0
\(924\) −1.00000 −0.0328976
\(925\) 2.00000 0.0657596
\(926\) −32.0000 −1.05159
\(927\) 8.00000 0.262754
\(928\) 6.00000 0.196960
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) −4.00000 −0.131165
\(931\) −4.00000 −0.131095
\(932\) 6.00000 0.196537
\(933\) −12.0000 −0.392862
\(934\) −12.0000 −0.392652
\(935\) −6.00000 −0.196221
\(936\) −2.00000 −0.0653720
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 16.0000 0.522419
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) −1.00000 −0.0325300
\(946\) −4.00000 −0.130051
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) −4.00000 −0.129914
\(949\) 4.00000 0.129845
\(950\) 4.00000 0.129777
\(951\) −30.0000 −0.972817
\(952\) 6.00000 0.194461
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 6.00000 0.193952
\(958\) −24.0000 −0.775405
\(959\) 18.0000 0.581250
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) −4.00000 −0.128965
\(963\) −12.0000 −0.386695
\(964\) 2.00000 0.0644157
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 24.0000 0.770991
\(970\) −10.0000 −0.321081
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000 0.0320750
\(973\) −4.00000 −0.128234
\(974\) 40.0000 1.28168
\(975\) 2.00000 0.0640513
\(976\) 14.0000 0.448129
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) −8.00000 −0.255812
\(979\) 6.00000 0.191761
\(980\) −1.00000 −0.0319438
\(981\) 2.00000 0.0638551
\(982\) 12.0000 0.382935
\(983\) 48.0000 1.53096 0.765481 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 6.00000 0.191273
\(985\) 6.00000 0.191176
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 0 0
\(990\) −1.00000 −0.0317821
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 4.00000 0.127000
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) −20.0000 −0.634043
\(996\) 0 0
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) 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 2310.2.a.h.1.1 1
3.2 odd 2 6930.2.a.bl.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.h.1.1 1 1.1 even 1 trivial
6930.2.a.bl.1.1 1 3.2 odd 2