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Properties

Label 2310.2.a.n.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} -1.00000 q^{22} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} -1.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -6.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} +8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -1.00000 q^{55} -1.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} -4.00000 q^{59} -1.00000 q^{60} -2.00000 q^{61} -8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +1.00000 q^{66} -4.00000 q^{67} -6.00000 q^{68} -1.00000 q^{70} +1.00000 q^{72} +10.0000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +1.00000 q^{77} +2.00000 q^{78} -16.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} +1.00000 q^{84} -6.00000 q^{85} -4.00000 q^{86} +2.00000 q^{87} -1.00000 q^{88} -6.00000 q^{89} +1.00000 q^{90} +2.00000 q^{91} +8.00000 q^{93} +8.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) 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 \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −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
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −1.00000 −0.182574
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) −6.00000 −1.02899
\(35\) −1.00000 −0.169031
\(36\) 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
\(39\) 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
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 6.00000 0.840168
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) 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 \(0.583847\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −8.00000 −1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 1.00000 0.123091
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −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
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 6.00000 0.697486
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) 1.00000 0.113961
\(78\) 2.00000 0.226455
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 1.00000 0.109109
\(85\) −6.00000 −0.650791
\(86\) −4.00000 −0.431331
\(87\) 2.00000 0.214423
\(88\) −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
\(93\) 8.00000 0.829561
\(94\) 8.00000 0.825137
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 − 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 6.00000 0.594089
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −2.00000 −0.196116
\(105\) 1.00000 0.0975900
\(106\) −2.00000 −0.194257
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −6.00000 −0.569495
\(112\) −1.00000 −0.0944911
\(113\) 10.0000 0.940721 0.470360 − 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) −4.00000 −0.368230
\(119\) 6.00000 0.550019
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) 6.00000 0.541002
\(124\) −8.00000 −0.718421
\(125\) 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 \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 1.00000 0.0870388
\(133\) 4.00000 0.346844
\(134\) −4.00000 −0.345547
\(135\) −1.00000 −0.0860663
\(136\) −6.00000 −0.514496
\(137\) −14.0000 −1.19610 −0.598050 − 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 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
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 10.0000 0.827606
\(147\) −1.00000 −0.0824786
\(148\) 6.00000 0.493197
\(149\) 22.0000 1.80231 0.901155 − 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −4.00000 −0.324443
\(153\) −6.00000 −0.485071
\(154\) 1.00000 0.0805823
\(155\) −8.00000 −0.642575
\(156\) 2.00000 0.160128
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −16.0000 −1.27289
\(159\) 2.00000 0.158610
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −6.00000 −0.468521
\(165\) 1.00000 0.0778499
\(166\) 4.00000 0.310460
\(167\) −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
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 2.00000 0.151620
\(175\) −1.00000 −0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 4.00000 0.300658
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 1.00000 0.0745356
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 2.00000 0.148250
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) 8.00000 0.586588
\(187\) 6.00000 0.438763
\(188\) 8.00000 0.583460
\(189\) 1.00000 0.0727393
\(190\) −4.00000 −0.290191
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 10.0000 0.719816 0.359908 − 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 10.0000 0.717958
\(195\) 2.00000 0.143223
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) −10.0000 −0.703598
\(203\) 2.00000 0.140372
\(204\) 6.00000 0.420084
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 4.00000 0.276686
\(210\) 1.00000 0.0690066
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) 8.00000 0.543075
\(218\) −2.00000 −0.135457
\(219\) −10.0000 −0.675737
\(220\) −1.00000 −0.0674200
\(221\) 12.0000 0.807207
\(222\) −6.00000 −0.402694
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 10.0000 0.665190
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 4.00000 0.264906
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) −2.00000 −0.131306
\(233\) 26.0000 1.70332 0.851658 − 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) −2.00000 −0.130744
\(235\) 8.00000 0.521862
\(236\) −4.00000 −0.260378
\(237\) 16.0000 1.03931
\(238\) 6.00000 0.388922
\(239\) 8.00000 0.517477 0.258738 − 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) −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
\(244\) −2.00000 −0.128037
\(245\) 1.00000 0.0638877
\(246\) 6.00000 0.382546
\(247\) 8.00000 0.509028
\(248\) −8.00000 −0.508001
\(249\) −4.00000 −0.253490
\(250\) 1.00000 0.0632456
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 4.00000 0.249029
\(259\) −6.00000 −0.372822
\(260\) −2.00000 −0.124035
\(261\) −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 \(0.579330\pi\)
\(264\) 1.00000 0.0615457
\(265\) −2.00000 −0.122859
\(266\) 4.00000 0.245256
\(267\) 6.00000 0.367194
\(268\) −4.00000 −0.244339
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −8.00000 −0.485965 −0.242983 − 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −6.00000 −0.363803
\(273\) −2.00000 −0.121046
\(274\) −14.0000 −0.845771
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 30.0000 1.80253 0.901263 − 0.433273i \(-0.142641\pi\)
0.901263 + 0.433273i \(0.142641\pi\)
\(278\) 4.00000 0.239904
\(279\) −8.00000 −0.478947
\(280\) −1.00000 −0.0597614
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) −8.00000 −0.476393
\(283\) 12.0000 0.713326 0.356663 − 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 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
\(290\) −2.00000 −0.117444
\(291\) −10.0000 −0.586210
\(292\) 10.0000 0.585206
\(293\) 14.0000 0.817889 0.408944 − 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −4.00000 −0.232889
\(296\) 6.00000 0.348743
\(297\) 1.00000 0.0580259
\(298\) 22.0000 1.27443
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 4.00000 0.230556
\(302\) −8.00000 −0.460348
\(303\) 10.0000 0.574485
\(304\) −4.00000 −0.229416
\(305\) −2.00000 −0.114520
\(306\) −6.00000 −0.342997
\(307\) −28.0000 −1.59804 −0.799022 − 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) 24.0000 1.36092 0.680458 − 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 2.00000 0.113228
\(313\) −14.0000 −0.791327 −0.395663 − 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −10.0000 −0.564333
\(315\) −1.00000 −0.0563436
\(316\) −16.0000 −0.900070
\(317\) −10.0000 −0.561656 −0.280828 − 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 2.00000 0.112154
\(319\) 2.00000 0.111979
\(320\) 1.00000 0.0559017
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) 12.0000 0.664619
\(327\) 2.00000 0.110600
\(328\) −6.00000 −0.331295
\(329\) −8.00000 −0.441054
\(330\) 1.00000 0.0550482
\(331\) −20.0000 −1.09930 −0.549650 − 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 4.00000 0.219529
\(333\) 6.00000 0.328798
\(334\) −24.0000 −1.31322
\(335\) −4.00000 −0.218543
\(336\) 1.00000 0.0545545
\(337\) −6.00000 −0.326841 −0.163420 − 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) −9.00000 −0.489535
\(339\) −10.0000 −0.543125
\(340\) −6.00000 −0.325396
\(341\) 8.00000 0.433224
\(342\) −4.00000 −0.216295
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −4.00000 −0.214731 −0.107366 − 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 2.00000 0.107211
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 2.00000 0.106752
\(352\) −1.00000 −0.0533002
\(353\) 34.0000 1.80964 0.904819 − 0.425797i \(-0.140006\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −6.00000 −0.317554
\(358\) 12.0000 0.634220
\(359\) −16.0000 −0.844448 −0.422224 − 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) −1.00000 −0.0524864
\(364\) 2.00000 0.104828
\(365\) 10.0000 0.523424
\(366\) 2.00000 0.104542
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 6.00000 0.311925
\(371\) 2.00000 0.103835
\(372\) 8.00000 0.414781
\(373\) 30.0000 1.55334 0.776671 − 0.629907i \(-0.216907\pi\)
0.776671 + 0.629907i \(0.216907\pi\)
\(374\) 6.00000 0.310253
\(375\) −1.00000 −0.0516398
\(376\) 8.00000 0.412568
\(377\) 4.00000 0.206010
\(378\) 1.00000 0.0514344
\(379\) 12.0000 0.616399 0.308199 − 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) −4.00000 −0.205196
\(381\) 16.0000 0.819705
\(382\) 24.0000 1.22795
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 1.00000 0.0509647
\(386\) 10.0000 0.508987
\(387\) −4.00000 −0.203331
\(388\) 10.0000 0.507673
\(389\) 38.0000 1.92668 0.963338 − 0.268290i \(-0.0864585\pi\)
0.963338 + 0.268290i \(0.0864585\pi\)
\(390\) 2.00000 0.101274
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 12.0000 0.605320
\(394\) 6.00000 0.302276
\(395\) −16.0000 −0.805047
\(396\) −1.00000 −0.0502519
\(397\) 22.0000 1.10415 0.552074 − 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0 0
\(399\) −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 \(0.351624\pi\)
\(402\) 4.00000 0.199502
\(403\) 16.0000 0.797017
\(404\) −10.0000 −0.497519
\(405\) 1.00000 0.0496904
\(406\) 2.00000 0.0992583
\(407\) −6.00000 −0.297409
\(408\) 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
\(411\) 14.0000 0.690569
\(412\) 0 0
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) −2.00000 −0.0980581
\(417\) −4.00000 −0.195881
\(418\) 4.00000 0.195646
\(419\) 4.00000 0.195413 0.0977064 − 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 1.00000 0.0487950
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −12.0000 −0.584151
\(423\) 8.00000 0.388973
\(424\) −2.00000 −0.0971286
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) −4.00000 −0.193347
\(429\) −2.00000 −0.0965609
\(430\) −4.00000 −0.192897
\(431\) 8.00000 0.385346 0.192673 − 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −6.00000 −0.288342 −0.144171 − 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 8.00000 0.384012
\(435\) 2.00000 0.0958927
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) −10.0000 −0.477818
\(439\) 16.0000 0.763638 0.381819 − 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) 20.0000 0.950229 0.475114 − 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) −6.00000 −0.284747
\(445\) −6.00000 −0.284427
\(446\) −8.00000 −0.378811
\(447\) −22.0000 −1.04056
\(448\) −1.00000 −0.0472456
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 1.00000 0.0471405
\(451\) 6.00000 0.282529
\(452\) 10.0000 0.470360
\(453\) 8.00000 0.375873
\(454\) −12.0000 −0.563188
\(455\) 2.00000 0.0937614
\(456\) 4.00000 0.187317
\(457\) 18.0000 0.842004 0.421002 − 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −10.0000 −0.467269
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 − 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) −1.00000 −0.0465242
\(463\) −8.00000 −0.371792 −0.185896 − 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 8.00000 0.370991
\(466\) 26.0000 1.20443
\(467\) −36.0000 −1.66588 −0.832941 − 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 4.00000 0.184703
\(470\) 8.00000 0.369012
\(471\) 10.0000 0.460776
\(472\) −4.00000 −0.184115
\(473\) 4.00000 0.183920
\(474\) 16.0000 0.734904
\(475\) −4.00000 −0.183533
\(476\) 6.00000 0.275010
\(477\) −2.00000 −0.0915737
\(478\) 8.00000 0.365911
\(479\) 16.0000 0.731059 0.365529 − 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −12.0000 −0.547153
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 10.0000 0.454077
\(486\) −1.00000 −0.0453609
\(487\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −12.0000 −0.542659
\(490\) 1.00000 0.0451754
\(491\) 20.0000 0.902587 0.451294 − 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 6.00000 0.270501
\(493\) 12.0000 0.540453
\(494\) 8.00000 0.359937
\(495\) −1.00000 −0.0449467
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) −28.0000 −1.25345 −0.626726 − 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 1.00000 0.0447214
\(501\) 24.0000 1.07224
\(502\) 12.0000 0.535586
\(503\) 8.00000 0.356702 0.178351 − 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −10.0000 −0.444994
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) −16.0000 −0.709885
\(509\) 30.0000 1.32973 0.664863 − 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 6.00000 0.265684
\(511\) −10.0000 −0.442374
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) −8.00000 −0.351840
\(518\) −6.00000 −0.263625
\(519\) −6.00000 −0.263371
\(520\) −2.00000 −0.0877058
\(521\) 10.0000 0.438108 0.219054 − 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 28.0000 1.22435 0.612177 − 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) −12.0000 −0.524222
\(525\) 1.00000 0.0436436
\(526\) −8.00000 −0.348817
\(527\) 48.0000 2.09091
\(528\) 1.00000 0.0435194
\(529\) −23.0000 −1.00000
\(530\) −2.00000 −0.0868744
\(531\) −4.00000 −0.173585
\(532\) 4.00000 0.173422
\(533\) 12.0000 0.519778
\(534\) 6.00000 0.259645
\(535\) −4.00000 −0.172935
\(536\) −4.00000 −0.172774
\(537\) −12.0000 −0.517838
\(538\) −18.0000 −0.776035
\(539\) −1.00000 −0.0430730
\(540\) −1.00000 −0.0430331
\(541\) −18.0000 −0.773880 −0.386940 − 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −8.00000 −0.343629
\(543\) 10.0000 0.429141
\(544\) −6.00000 −0.257248
\(545\) −2.00000 −0.0856706
\(546\) −2.00000 −0.0855921
\(547\) 4.00000 0.171028 0.0855138 − 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −14.0000 −0.598050
\(549\) −2.00000 −0.0853579
\(550\) −1.00000 −0.0426401
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 30.0000 1.27458
\(555\) −6.00000 −0.254686
\(556\) 4.00000 0.169638
\(557\) −2.00000 −0.0847427 −0.0423714 − 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) −8.00000 −0.338667
\(559\) 8.00000 0.338364
\(560\) −1.00000 −0.0422577
\(561\) −6.00000 −0.253320
\(562\) 10.0000 0.421825
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) −8.00000 −0.336861
\(565\) 10.0000 0.420703
\(566\) 12.0000 0.504398
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 − 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 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
\(573\) −24.0000 −1.00261
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 − 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 19.0000 0.790296
\(579\) −10.0000 −0.415586
\(580\) −2.00000 −0.0830455
\(581\) −4.00000 −0.165948
\(582\) −10.0000 −0.414513
\(583\) 2.00000 0.0828315
\(584\) 10.0000 0.413803
\(585\) −2.00000 −0.0826898
\(586\) 14.0000 0.578335
\(587\) 36.0000 1.48588 0.742940 − 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 32.0000 1.31854
\(590\) −4.00000 −0.164677
\(591\) −6.00000 −0.246807
\(592\) 6.00000 0.246598
\(593\) −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
\(596\) 22.0000 0.901155
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 10.0000 0.407909 0.203954 − 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 4.00000 0.163028
\(603\) −4.00000 −0.162893
\(604\) −8.00000 −0.325515
\(605\) 1.00000 0.0406558
\(606\) 10.0000 0.406222
\(607\) −16.0000 −0.649420 −0.324710 − 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −4.00000 −0.162221
\(609\) −2.00000 −0.0810441
\(610\) −2.00000 −0.0809776
\(611\) −16.0000 −0.647291
\(612\) −6.00000 −0.242536
\(613\) −2.00000 −0.0807792 −0.0403896 − 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −28.0000 −1.12999
\(615\) 6.00000 0.241943
\(616\) 1.00000 0.0402911
\(617\) −30.0000 −1.20775 −0.603877 − 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 36.0000 1.44696 0.723481 − 0.690344i \(-0.242541\pi\)
0.723481 + 0.690344i \(0.242541\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) 6.00000 0.240385
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −14.0000 −0.559553
\(627\) −4.00000 −0.159745
\(628\) −10.0000 −0.399043
\(629\) −36.0000 −1.43541
\(630\) −1.00000 −0.0398410
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −16.0000 −0.636446
\(633\) 12.0000 0.476957
\(634\) −10.0000 −0.397151
\(635\) −16.0000 −0.634941
\(636\) 2.00000 0.0793052
\(637\) −2.00000 −0.0792429
\(638\) 2.00000 0.0791808
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 4.00000 0.157867
\(643\) 12.0000 0.473234 0.236617 − 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 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
\(649\) 4.00000 0.157014
\(650\) −2.00000 −0.0784465
\(651\) −8.00000 −0.313545
\(652\) 12.0000 0.469956
\(653\) 38.0000 1.48705 0.743527 − 0.668705i \(-0.233151\pi\)
0.743527 + 0.668705i \(0.233151\pi\)
\(654\) 2.00000 0.0782062
\(655\) −12.0000 −0.468879
\(656\) −6.00000 −0.234261
\(657\) 10.0000 0.390137
\(658\) −8.00000 −0.311872
\(659\) −36.0000 −1.40236 −0.701180 − 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 1.00000 0.0389249
\(661\) −26.0000 −1.01128 −0.505641 − 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) −20.0000 −0.777322
\(663\) −12.0000 −0.466041
\(664\) 4.00000 0.155230
\(665\) 4.00000 0.155113
\(666\) 6.00000 0.232495
\(667\) 0 0
\(668\) −24.0000 −0.928588
\(669\) 8.00000 0.309298
\(670\) −4.00000 −0.154533
\(671\) 2.00000 0.0772091
\(672\) 1.00000 0.0385758
\(673\) −38.0000 −1.46479 −0.732396 − 0.680879i \(-0.761598\pi\)
−0.732396 + 0.680879i \(0.761598\pi\)
\(674\) −6.00000 −0.231111
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) −10.0000 −0.384048
\(679\) −10.0000 −0.383765
\(680\) −6.00000 −0.230089
\(681\) 12.0000 0.459841
\(682\) 8.00000 0.306336
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −4.00000 −0.152944
\(685\) −14.0000 −0.534913
\(686\) −1.00000 −0.0381802
\(687\) 10.0000 0.381524
\(688\) −4.00000 −0.152499
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 − 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 6.00000 0.228086
\(693\) 1.00000 0.0379869
\(694\) −4.00000 −0.151838
\(695\) 4.00000 0.151729
\(696\) 2.00000 0.0758098
\(697\) 36.0000 1.36360
\(698\) −2.00000 −0.0757011
\(699\) −26.0000 −0.983410
\(700\) −1.00000 −0.0377964
\(701\) −2.00000 −0.0755390 −0.0377695 − 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 2.00000 0.0754851
\(703\) −24.0000 −0.905177
\(704\) −1.00000 −0.0376889
\(705\) −8.00000 −0.301297
\(706\) 34.0000 1.27961
\(707\) 10.0000 0.376089
\(708\) 4.00000 0.150329
\(709\) 6.00000 0.225335 0.112667 − 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) −6.00000 −0.224544
\(715\) 2.00000 0.0747958
\(716\) 12.0000 0.448461
\(717\) −8.00000 −0.298765
\(718\) −16.0000 −0.597115
\(719\) −32.0000 −1.19340 −0.596699 − 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) −2.00000 −0.0743808
\(724\) −10.0000 −0.371647
\(725\) −2.00000 −0.0742781
\(726\) −1.00000 −0.0371135
\(727\) −48.0000 −1.78022 −0.890111 − 0.455744i \(-0.849373\pi\)
−0.890111 + 0.455744i \(0.849373\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) 24.0000 0.887672
\(732\) 2.00000 0.0739221
\(733\) −34.0000 −1.25582 −0.627909 − 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) −8.00000 −0.295285
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) −6.00000 −0.220863
\(739\) −12.0000 −0.441427 −0.220714 − 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 6.00000 0.220564
\(741\) −8.00000 −0.293887
\(742\) 2.00000 0.0734223
\(743\) 24.0000 0.880475 0.440237 − 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 8.00000 0.293294
\(745\) 22.0000 0.806018
\(746\) 30.0000 1.09838
\(747\) 4.00000 0.146352
\(748\) 6.00000 0.219382
\(749\) 4.00000 0.146157
\(750\) −1.00000 −0.0365148
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 8.00000 0.291730
\(753\) −12.0000 −0.437304
\(754\) 4.00000 0.145671
\(755\) −8.00000 −0.291150
\(756\) 1.00000 0.0363696
\(757\) −42.0000 −1.52652 −0.763258 − 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 12.0000 0.435860
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) −38.0000 −1.37750 −0.688749 − 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 16.0000 0.579619
\(763\) 2.00000 0.0724049
\(764\) 24.0000 0.868290
\(765\) −6.00000 −0.216930
\(766\) −24.0000 −0.867155
\(767\) 8.00000 0.288863
\(768\) −1.00000 −0.0360844
\(769\) −46.0000 −1.65880 −0.829401 − 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 1.00000 0.0360375
\(771\) −18.0000 −0.648254
\(772\) 10.0000 0.359908
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −4.00000 −0.143777
\(775\) −8.00000 −0.287368
\(776\) 10.0000 0.358979
\(777\) 6.00000 0.215249
\(778\) 38.0000 1.36237
\(779\) 24.0000 0.859889
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 1.00000 0.0357143
\(785\) −10.0000 −0.356915
\(786\) 12.0000 0.428026
\(787\) 4.00000 0.142585 0.0712923 − 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 6.00000 0.213741
\(789\) 8.00000 0.284808
\(790\) −16.0000 −0.569254
\(791\) −10.0000 −0.355559
\(792\) −1.00000 −0.0355335
\(793\) 4.00000 0.142044
\(794\) 22.0000 0.780751
\(795\) 2.00000 0.0709327
\(796\) 0 0
\(797\) −18.0000 −0.637593 −0.318796 − 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) −4.00000 −0.141598
\(799\) −48.0000 −1.69812
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) 18.0000 0.635602
\(803\) −10.0000 −0.352892
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 18.0000 0.633630
\(808\) −10.0000 −0.351799
\(809\) −38.0000 −1.33601 −0.668004 − 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 1.00000 0.0351364
\(811\) −12.0000 −0.421377 −0.210688 − 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 2.00000 0.0701862
\(813\) 8.00000 0.280572
\(814\) −6.00000 −0.210300
\(815\) 12.0000 0.420342
\(816\) 6.00000 0.210042
\(817\) 16.0000 0.559769
\(818\) −22.0000 −0.769212
\(819\) 2.00000 0.0698857
\(820\) −6.00000 −0.209529
\(821\) −10.0000 −0.349002 −0.174501 − 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 14.0000 0.488306
\(823\) −48.0000 −1.67317 −0.836587 − 0.547833i \(-0.815453\pi\)
−0.836587 + 0.547833i \(0.815453\pi\)
\(824\) 0 0
\(825\) 1.00000 0.0348155
\(826\) 4.00000 0.139178
\(827\) −52.0000 −1.80822 −0.904109 − 0.427303i \(-0.859464\pi\)
−0.904109 + 0.427303i \(0.859464\pi\)
\(828\) 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
\(831\) −30.0000 −1.04069
\(832\) −2.00000 −0.0693375
\(833\) −6.00000 −0.207888
\(834\) −4.00000 −0.138509
\(835\) −24.0000 −0.830554
\(836\) 4.00000 0.138343
\(837\) 8.00000 0.276520
\(838\) 4.00000 0.138178
\(839\) −24.0000 −0.828572 −0.414286 − 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 1.00000 0.0345033
\(841\) −25.0000 −0.862069
\(842\) −26.0000 −0.896019
\(843\) −10.0000 −0.344418
\(844\) −12.0000 −0.413057
\(845\) −9.00000 −0.309609
\(846\) 8.00000 0.275046
\(847\) −1.00000 −0.0343604
\(848\) −2.00000 −0.0686803
\(849\) −12.0000 −0.411839
\(850\) −6.00000 −0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) 54.0000 1.84892 0.924462 − 0.381273i \(-0.124514\pi\)
0.924462 + 0.381273i \(0.124514\pi\)
\(854\) 2.00000 0.0684386
\(855\) −4.00000 −0.136797
\(856\) −4.00000 −0.136717
\(857\) 50.0000 1.70797 0.853984 − 0.520300i \(-0.174180\pi\)
0.853984 + 0.520300i \(0.174180\pi\)
\(858\) −2.00000 −0.0682789
\(859\) 4.00000 0.136478 0.0682391 − 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) −4.00000 −0.136399
\(861\) −6.00000 −0.204479
\(862\) 8.00000 0.272481
\(863\) −8.00000 −0.272323 −0.136162 − 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) −6.00000 −0.203888
\(867\) −19.0000 −0.645274
\(868\) 8.00000 0.271538
\(869\) 16.0000 0.542763
\(870\) 2.00000 0.0678064
\(871\) 8.00000 0.271070
\(872\) −2.00000 −0.0677285
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) −10.0000 −0.337869
\(877\) 6.00000 0.202606 0.101303 − 0.994856i \(-0.467699\pi\)
0.101303 + 0.994856i \(0.467699\pi\)
\(878\) 16.0000 0.539974
\(879\) −14.0000 −0.472208
\(880\) −1.00000 −0.0337100
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 1.00000 0.0336718
\(883\) −52.0000 −1.74994 −0.874970 − 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 12.0000 0.403604
\(885\) 4.00000 0.134459
\(886\) 20.0000 0.671913
\(887\) 8.00000 0.268614 0.134307 − 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) −6.00000 −0.201347
\(889\) 16.0000 0.536623
\(890\) −6.00000 −0.201120
\(891\) −1.00000 −0.0335013
\(892\) −8.00000 −0.267860
\(893\) −32.0000 −1.07084
\(894\) −22.0000 −0.735790
\(895\) 12.0000 0.401116
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) 16.0000 0.533630
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) 6.00000 0.199778
\(903\) −4.00000 −0.133112
\(904\) 10.0000 0.332595
\(905\) −10.0000 −0.332411
\(906\) 8.00000 0.265782
\(907\) 4.00000 0.132818 0.0664089 − 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) −12.0000 −0.398234
\(909\) −10.0000 −0.331679
\(910\) 2.00000 0.0662994
\(911\) 8.00000 0.265052 0.132526 − 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 4.00000 0.132453
\(913\) −4.00000 −0.132381
\(914\) 18.0000 0.595387
\(915\) 2.00000 0.0661180
\(916\) −10.0000 −0.330409
\(917\) 12.0000 0.396275
\(918\) 6.00000 0.198030
\(919\) −24.0000 −0.791687 −0.395843 − 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) −2.00000 −0.0658665
\(923\) 0 0
\(924\) −1.00000 −0.0328976
\(925\) 6.00000 0.197279
\(926\) −8.00000 −0.262896
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) −30.0000 −0.984268 −0.492134 − 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 8.00000 0.262330
\(931\) −4.00000 −0.131095
\(932\) 26.0000 0.851658
\(933\) −24.0000 −0.785725
\(934\) −36.0000 −1.17796
\(935\) 6.00000 0.196221
\(936\) −2.00000 −0.0653720
\(937\) −6.00000 −0.196011 −0.0980057 − 0.995186i \(-0.531246\pi\)
−0.0980057 + 0.995186i \(0.531246\pi\)
\(938\) 4.00000 0.130605
\(939\) 14.0000 0.456873
\(940\) 8.00000 0.260931
\(941\) 30.0000 0.977972 0.488986 − 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 10.0000 0.325818
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 1.00000 0.0325300
\(946\) 4.00000 0.130051
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 16.0000 0.519656
\(949\) −20.0000 −0.649227
\(950\) −4.00000 −0.129777
\(951\) 10.0000 0.324272
\(952\) 6.00000 0.194461
\(953\) 42.0000 1.36051 0.680257 − 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 24.0000 0.776622
\(956\) 8.00000 0.258738
\(957\) −2.00000 −0.0646508
\(958\) 16.0000 0.516937
\(959\) 14.0000 0.452084
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) −12.0000 −0.386896
\(963\) −4.00000 −0.128898
\(964\) 2.00000 0.0644157
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 1.00000 0.0321412
\(969\) −24.0000 −0.770991
\(970\) 10.0000 0.321081
\(971\) 12.0000 0.385098 0.192549 − 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −4.00000 −0.128234
\(974\) 0 0
\(975\) 2.00000 0.0640513
\(976\) −2.00000 −0.0640184
\(977\) −54.0000 −1.72761 −0.863807 − 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) −12.0000 −0.383718
\(979\) 6.00000 0.191761
\(980\) 1.00000 0.0319438
\(981\) −2.00000 −0.0638551
\(982\) 20.0000 0.638226
\(983\) 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
\(986\) 12.0000 0.382158
\(987\) 8.00000 0.254643
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) −1.00000 −0.0317821
\(991\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) −8.00000 −0.254000
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) 38.0000 1.20347 0.601736 − 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) −28.0000 −0.886325
\(999\) −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 2310.2.a.n.1.1 ✓ 1
3.2 odd 2 6930.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.n.1.1 ✓ 1 1.1 even 1 trivial
6930.2.a.a.1.1 1 3.2 odd 2