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Properties

Label 2310.2.a.i.1.1
Level $2310$
Weight $2$
Character 2310.1
Self dual yes
Analytic conductor $18.445$
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] = [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: \(0\)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -1.00000 q^{22} +6.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -1.00000 q^{35} +1.00000 q^{36} +8.00000 q^{37} +4.00000 q^{38} -4.00000 q^{39} +1.00000 q^{40} +6.00000 q^{41} -1.00000 q^{42} +8.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} -6.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -4.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -1.00000 q^{55} -1.00000 q^{56} -4.00000 q^{57} -6.00000 q^{58} -1.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -1.00000 q^{66} +2.00000 q^{67} +6.00000 q^{69} +1.00000 q^{70} +12.0000 q^{71} -1.00000 q^{72} -10.0000 q^{73} -8.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +1.00000 q^{77} +4.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} +1.00000 q^{84} -8.00000 q^{86} +6.00000 q^{87} -1.00000 q^{88} +1.00000 q^{90} -4.00000 q^{91} +6.00000 q^{92} -4.00000 q^{93} +4.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) −1.00000 −0.213201
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 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
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 4.00000 0.648886
\(39\) −4.00000 −0.640513
\(40\) 1.00000 0.158114
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −1.00000 −0.154303
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 1.00000 0.150756
\(45\) −1.00000 −0.149071
\(46\) −6.00000 −0.884652
\(47\) 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
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) −4.00000 −0.529813
\(58\) −6.00000 −0.787839
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 4.00000 0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −1.00000 −0.123091
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 1.00000 0.119523
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −1.00000 −0.117851
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −8.00000 −0.929981
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 1.00000 0.113961
\(78\) 4.00000 0.452911
\(79\) −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
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 6.00000 0.643268
\(88\) −1.00000 −0.106600
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 1.00000 0.105409
\(91\) −4.00000 −0.419314
\(92\) 6.00000 0.625543
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 4.00000 0.392232
\(105\) −1.00000 −0.0975900
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) −4.00000 −0.383131 −0.191565 − 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 1.00000 0.0953463
\(111\) 8.00000 0.759326
\(112\) 1.00000 0.0944911
\(113\) −12.0000 −1.12887 −0.564433 − 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 4.00000 0.374634
\(115\) −6.00000 −0.559503
\(116\) 6.00000 0.557086
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) 1.00000 0.0909091
\(122\) 10.0000 0.905357
\(123\) 6.00000 0.541002
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) −4.00000 −0.350823
\(131\) −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
\(134\) −2.00000 −0.172774
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 − 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) −6.00000 −0.510754
\(139\) 8.00000 0.678551 0.339276 − 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) 10.0000 0.827606
\(147\) 1.00000 0.0824786
\(148\) 8.00000 0.657596
\(149\) −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
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 4.00000 0.321288
\(156\) −4.00000 −0.320256
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 4.00000 0.318223
\(159\) 6.00000 0.475831
\(160\) 1.00000 0.0790569
\(161\) 6.00000 0.472866
\(162\) −1.00000 −0.0785674
\(163\) 2.00000 0.156652 0.0783260 − 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 6.00000 0.468521
\(165\) −1.00000 −0.0778499
\(166\) −12.0000 −0.931381
\(167\) −6.00000 −0.464294 −0.232147 − 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 8.00000 0.609994
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −6.00000 −0.454859
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 20.0000 1.48659 0.743294 − 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 4.00000 0.296500
\(183\) −10.0000 −0.739221
\(184\) −6.00000 −0.442326
\(185\) −8.00000 −0.588172
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) 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
\(193\) −22.0000 −1.58359 −0.791797 − 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −2.00000 −0.143592
\(195\) 4.00000 0.286446
\(196\) 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
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 2.00000 0.141069
\(202\) −6.00000 −0.422159
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) −8.00000 −0.557386
\(207\) 6.00000 0.417029
\(208\) −4.00000 −0.277350
\(209\) −4.00000 −0.276686
\(210\) 1.00000 0.0690066
\(211\) −10.0000 −0.688428 −0.344214 − 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 6.00000 0.412082
\(213\) 12.0000 0.822226
\(214\) −12.0000 −0.820303
\(215\) −8.00000 −0.545595
\(216\) −1.00000 −0.0680414
\(217\) −4.00000 −0.271538
\(218\) 4.00000 0.270914
\(219\) −10.0000 −0.675737
\(220\) −1.00000 −0.0674200
\(221\) 0 0
\(222\) −8.00000 −0.536925
\(223\) −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
\(226\) 12.0000 0.798228
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −4.00000 −0.264906
\(229\) 8.00000 0.528655 0.264327 − 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 6.00000 0.395628
\(231\) 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 \(0.437031\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −30.0000 −1.94054 −0.970269 − 0.242028i \(-0.922188\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) −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
\(244\) −10.0000 −0.640184
\(245\) −1.00000 −0.0638877
\(246\) −6.00000 −0.382546
\(247\) 16.0000 1.01806
\(248\) 4.00000 0.254000
\(249\) 12.0000 0.760469
\(250\) 1.00000 0.0632456
\(251\) 24.0000 1.51487 0.757433 − 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 1.00000 0.0629941
\(253\) 6.00000 0.377217
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −8.00000 −0.498058
\(259\) 8.00000 0.497096
\(260\) 4.00000 0.248069
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −6.00000 −0.368577
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) 2.00000 0.122169
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 1.00000 0.0608581
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) −12.0000 −0.724947
\(275\) 1.00000 0.0603023
\(276\) 6.00000 0.361158
\(277\) 14.0000 0.841178 0.420589 − 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) −8.00000 −0.479808
\(279\) −4.00000 −0.239474
\(280\) 1.00000 0.0597614
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 2.00000 0.118888 0.0594438 − 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) 12.0000 0.712069
\(285\) 4.00000 0.236940
\(286\) 4.00000 0.236525
\(287\) 6.00000 0.354169
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 6.00000 0.352332
\(291\) 2.00000 0.117242
\(292\) −10.0000 −0.585206
\(293\) 30.0000 1.75262 0.876309 − 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 1.00000 0.0580259
\(298\) 6.00000 0.347571
\(299\) −24.0000 −1.38796
\(300\) 1.00000 0.0577350
\(301\) 8.00000 0.461112
\(302\) −20.0000 −1.15087
\(303\) 6.00000 0.344691
\(304\) −4.00000 −0.229416
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 2.00000 0.114146 0.0570730 − 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 1.00000 0.0569803
\(309\) 8.00000 0.455104
\(310\) −4.00000 −0.227185
\(311\) 6.00000 0.340229 0.170114 − 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 4.00000 0.226455
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −14.0000 −0.790066
\(315\) −1.00000 −0.0563436
\(316\) −4.00000 −0.225018
\(317\) 6.00000 0.336994 0.168497 − 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) −6.00000 −0.336463
\(319\) 6.00000 0.335936
\(320\) −1.00000 −0.0559017
\(321\) 12.0000 0.669775
\(322\) −6.00000 −0.334367
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) −2.00000 −0.110770
\(327\) −4.00000 −0.221201
\(328\) −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 \(0.314761\pi\)
\(332\) 12.0000 0.658586
\(333\) 8.00000 0.438397
\(334\) 6.00000 0.328305
\(335\) −2.00000 −0.109272
\(336\) 1.00000 0.0545545
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −3.00000 −0.163178
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 4.00000 0.216295
\(343\) 1.00000 0.0539949
\(344\) −8.00000 −0.431331
\(345\) −6.00000 −0.323029
\(346\) 6.00000 0.322562
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 6.00000 0.321634
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −4.00000 −0.213504
\(352\) −1.00000 −0.0533002
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −18.0000 −0.950004 −0.475002 − 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) −20.0000 −1.05118
\(363\) 1.00000 0.0524864
\(364\) −4.00000 −0.209657
\(365\) 10.0000 0.523424
\(366\) 10.0000 0.522708
\(367\) 8.00000 0.417597 0.208798 − 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 6.00000 0.312772
\(369\) 6.00000 0.312348
\(370\) 8.00000 0.415900
\(371\) 6.00000 0.311504
\(372\) −4.00000 −0.207390
\(373\) 2.00000 0.103556 0.0517780 − 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) −1.00000 −0.0514344
\(379\) −28.0000 −1.43826 −0.719132 − 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 4.00000 0.205196
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) −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
\(386\) 22.0000 1.11977
\(387\) 8.00000 0.406663
\(388\) 2.00000 0.101535
\(389\) 6.00000 0.304212 0.152106 − 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −4.00000 −0.202548
\(391\) 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 \(0.484018\pi\)
\(398\) 4.00000 0.200502
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 16.0000 0.797017
\(404\) 6.00000 0.298511
\(405\) −1.00000 −0.0496904
\(406\) −6.00000 −0.297775
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 6.00000 0.296319
\(411\) 12.0000 0.591916
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) −12.0000 −0.589057
\(416\) 4.00000 0.196116
\(417\) 8.00000 0.391762
\(418\) 4.00000 0.195646
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 14.0000 0.682318 0.341159 − 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 10.0000 0.486792
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) −10.0000 −0.483934
\(428\) 12.0000 0.580042
\(429\) −4.00000 −0.193122
\(430\) 8.00000 0.385794
\(431\) −30.0000 −1.44505 −0.722525 − 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 − 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 4.00000 0.192006
\(435\) −6.00000 −0.287678
\(436\) −4.00000 −0.191565
\(437\) −24.0000 −1.14808
\(438\) 10.0000 0.477818
\(439\) −16.0000 −0.763638 −0.381819 − 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 1.00000 0.0476731
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) −6.00000 −0.283790
\(448\) 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
\(452\) −12.0000 −0.564433
\(453\) 20.0000 0.939682
\(454\) −12.0000 −0.563188
\(455\) 4.00000 0.187523
\(456\) 4.00000 0.187317
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −8.00000 −0.373815
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) −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 \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −4.00000 −0.184900
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 4.00000 0.183726
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 30.0000 1.37217
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 1.00000 0.0456435
\(481\) −32.0000 −1.45907
\(482\) −2.00000 −0.0910975
\(483\) 6.00000 0.273009
\(484\) 1.00000 0.0454545
\(485\) −2.00000 −0.0908153
\(486\) −1.00000 −0.0453609
\(487\) −16.0000 −0.725029 −0.362515 − 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 10.0000 0.452679
\(489\) 2.00000 0.0904431
\(490\) 1.00000 0.0451754
\(491\) −36.0000 −1.62466 −0.812329 − 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 6.00000 0.270501
\(493\) 0 0
\(494\) −16.0000 −0.719874
\(495\) −1.00000 −0.0449467
\(496\) −4.00000 −0.179605
\(497\) 12.0000 0.538274
\(498\) −12.0000 −0.537733
\(499\) −16.0000 −0.716258 −0.358129 − 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −6.00000 −0.268060
\(502\) −24.0000 −1.07117
\(503\) 42.0000 1.87269 0.936344 − 0.351085i \(-0.114187\pi\)
0.936344 + 0.351085i \(0.114187\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −6.00000 −0.266996
\(506\) −6.00000 −0.266733
\(507\) 3.00000 0.133235
\(508\) 8.00000 0.354943
\(509\) −18.0000 −0.797836 −0.398918 − 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) −18.0000 −0.793946
\(515\) −8.00000 −0.352522
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) −6.00000 −0.263371
\(520\) −4.00000 −0.175412
\(521\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −6.00000 −0.262613
\(523\) −34.0000 −1.48672 −0.743358 − 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) −12.0000 −0.524222
\(525\) 1.00000 0.0436436
\(526\) −12.0000 −0.523225
\(527\) 0 0
\(528\) 1.00000 0.0435194
\(529\) 13.0000 0.565217
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) −24.0000 −1.03956
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) −2.00000 −0.0863868
\(537\) −12.0000 −0.517838
\(538\) −6.00000 −0.258678
\(539\) 1.00000 0.0430730
\(540\) −1.00000 −0.0430331
\(541\) −40.0000 −1.71973 −0.859867 − 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) 16.0000 0.687259
\(543\) 20.0000 0.858282
\(544\) 0 0
\(545\) 4.00000 0.171341
\(546\) 4.00000 0.171184
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 12.0000 0.512615
\(549\) −10.0000 −0.426790
\(550\) −1.00000 −0.0426401
\(551\) −24.0000 −1.02243
\(552\) −6.00000 −0.255377
\(553\) −4.00000 −0.170097
\(554\) −14.0000 −0.594803
\(555\) −8.00000 −0.339581
\(556\) 8.00000 0.339276
\(557\) −42.0000 −1.77960 −0.889799 − 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 4.00000 0.169334
\(559\) −32.0000 −1.35346
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 0 0
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) −2.00000 −0.0840663
\(567\) 1.00000 0.0419961
\(568\) −12.0000 −0.503509
\(569\) −12.0000 −0.503066 −0.251533 − 0.967849i \(-0.580935\pi\)
−0.251533 + 0.967849i \(0.580935\pi\)
\(570\) −4.00000 −0.167542
\(571\) −22.0000 −0.920671 −0.460336 − 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 6.00000 0.250217
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 − 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 17.0000 0.707107
\(579\) −22.0000 −0.914289
\(580\) −6.00000 −0.249136
\(581\) 12.0000 0.497844
\(582\) −2.00000 −0.0829027
\(583\) 6.00000 0.248495
\(584\) 10.0000 0.413803
\(585\) 4.00000 0.165380
\(586\) −30.0000 −1.23929
\(587\) 36.0000 1.48588 0.742940 − 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 1.00000 0.0412393
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 8.00000 0.328798
\(593\) −36.0000 −1.47834 −0.739171 − 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −4.00000 −0.163709
\(598\) 24.0000 0.981433
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −8.00000 −0.326056
\(603\) 2.00000 0.0814463
\(604\) 20.0000 0.813788
\(605\) −1.00000 −0.0406558
\(606\) −6.00000 −0.243733
\(607\) −16.0000 −0.649420 −0.324710 − 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 4.00000 0.162221
\(609\) 6.00000 0.243132
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 − 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) −2.00000 −0.0807134
\(615\) −6.00000 −0.241943
\(616\) −1.00000 −0.0402911
\(617\) 12.0000 0.483102 0.241551 − 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) −8.00000 −0.321807
\(619\) 26.0000 1.04503 0.522514 − 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 4.00000 0.160644
\(621\) 6.00000 0.240772
\(622\) −6.00000 −0.240578
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) −4.00000 −0.159745
\(628\) 14.0000 0.558661
\(629\) 0 0
\(630\) 1.00000 0.0398410
\(631\) −40.0000 −1.59237 −0.796187 − 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 4.00000 0.159111
\(633\) −10.0000 −0.397464
\(634\) −6.00000 −0.238290
\(635\) −8.00000 −0.317470
\(636\) 6.00000 0.237915
\(637\) −4.00000 −0.158486
\(638\) −6.00000 −0.237542
\(639\) 12.0000 0.474713
\(640\) 1.00000 0.0395285
\(641\) −6.00000 −0.236986 −0.118493 − 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) −12.0000 −0.473602
\(643\) −28.0000 −1.10421 −0.552106 − 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 6.00000 0.236433
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 4.00000 0.156893
\(651\) −4.00000 −0.156772
\(652\) 2.00000 0.0783260
\(653\) 42.0000 1.64359 0.821794 − 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 4.00000 0.156412
\(655\) 12.0000 0.468879
\(656\) 6.00000 0.234261
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 − 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) −1.00000 −0.0389249
\(661\) 20.0000 0.777910 0.388955 − 0.921257i \(-0.372836\pi\)
0.388955 + 0.921257i \(0.372836\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 4.00000 0.155113
\(666\) −8.00000 −0.309994
\(667\) 36.0000 1.39393
\(668\) −6.00000 −0.232147
\(669\) −16.0000 −0.618596
\(670\) 2.00000 0.0772667
\(671\) −10.0000 −0.386046
\(672\) −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
\(676\) 3.00000 0.115385
\(677\) 42.0000 1.61419 0.807096 − 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 12.0000 0.460857
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 4.00000 0.153168
\(683\) −36.0000 −1.37750 −0.688751 − 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −4.00000 −0.152944
\(685\) −12.0000 −0.458496
\(686\) −1.00000 −0.0381802
\(687\) 8.00000 0.305219
\(688\) 8.00000 0.304997
\(689\) −24.0000 −0.914327
\(690\) 6.00000 0.228416
\(691\) 2.00000 0.0760836 0.0380418 − 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) −6.00000 −0.228086
\(693\) 1.00000 0.0379869
\(694\) −12.0000 −0.455514
\(695\) −8.00000 −0.303457
\(696\) −6.00000 −0.227429
\(697\) 0 0
\(698\) 10.0000 0.378506
\(699\) 6.00000 0.226941
\(700\) 1.00000 0.0377964
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 4.00000 0.150970
\(703\) −32.0000 −1.20690
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 12.0000 0.450352
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) −12.0000 −0.448461
\(717\) −30.0000 −1.12037
\(718\) 18.0000 0.671754
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 8.00000 0.297936
\(722\) 3.00000 0.111648
\(723\) 2.00000 0.0743808
\(724\) 20.0000 0.743294
\(725\) 6.00000 0.222834
\(726\) −1.00000 −0.0371135
\(727\) 8.00000 0.296704 0.148352 − 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) −10.0000 −0.370117
\(731\) 0 0
\(732\) −10.0000 −0.369611
\(733\) −28.0000 −1.03420 −0.517102 − 0.855924i \(-0.672989\pi\)
−0.517102 + 0.855924i \(0.672989\pi\)
\(734\) −8.00000 −0.295285
\(735\) −1.00000 −0.0368856
\(736\) −6.00000 −0.221163
\(737\) 2.00000 0.0736709
\(738\) −6.00000 −0.220863
\(739\) 2.00000 0.0735712 0.0367856 − 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) −8.00000 −0.294086
\(741\) 16.0000 0.587775
\(742\) −6.00000 −0.220267
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 4.00000 0.146647
\(745\) 6.00000 0.219823
\(746\) −2.00000 −0.0732252
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 1.00000 0.0365148
\(751\) 32.0000 1.16770 0.583848 − 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) 24.0000 0.874609
\(754\) 24.0000 0.874028
\(755\) −20.0000 −0.727875
\(756\) 1.00000 0.0363696
\(757\) 8.00000 0.290765 0.145382 − 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) 28.0000 1.01701
\(759\) 6.00000 0.217786
\(760\) −4.00000 −0.145095
\(761\) 6.00000 0.217500 0.108750 − 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) −8.00000 −0.289809
\(763\) −4.00000 −0.144810
\(764\) 0 0
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 14.0000 0.504853 0.252426 − 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 1.00000 0.0360375
\(771\) 18.0000 0.648254
\(772\) −22.0000 −0.791797
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −8.00000 −0.287554
\(775\) −4.00000 −0.143684
\(776\) −2.00000 −0.0717958
\(777\) 8.00000 0.286998
\(778\) −6.00000 −0.215110
\(779\) −24.0000 −0.859889
\(780\) 4.00000 0.143223
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) −14.0000 −0.499681
\(786\) 12.0000 0.428026
\(787\) −46.0000 −1.63972 −0.819861 − 0.572562i \(-0.805950\pi\)
−0.819861 + 0.572562i \(0.805950\pi\)
\(788\) 6.00000 0.213741
\(789\) 12.0000 0.427211
\(790\) −4.00000 −0.142314
\(791\) −12.0000 −0.426671
\(792\) −1.00000 −0.0355335
\(793\) 40.0000 1.42044
\(794\) −2.00000 −0.0709773
\(795\) −6.00000 −0.212798
\(796\) −4.00000 −0.141776
\(797\) 54.0000 1.91278 0.956389 − 0.292096i \(-0.0943526\pi\)
0.956389 + 0.292096i \(0.0943526\pi\)
\(798\) 4.00000 0.141598
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 6.00000 0.211867
\(803\) −10.0000 −0.352892
\(804\) 2.00000 0.0705346
\(805\) −6.00000 −0.211472
\(806\) −16.0000 −0.563576
\(807\) 6.00000 0.211210
\(808\) −6.00000 −0.211079
\(809\) −36.0000 −1.26569 −0.632846 − 0.774277i \(-0.718114\pi\)
−0.632846 + 0.774277i \(0.718114\pi\)
\(810\) 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
\(814\) −8.00000 −0.280400
\(815\) −2.00000 −0.0700569
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 10.0000 0.349642
\(819\) −4.00000 −0.139771
\(820\) −6.00000 −0.209529
\(821\) −18.0000 −0.628204 −0.314102 − 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) −12.0000 −0.418548
\(823\) 44.0000 1.53374 0.766872 − 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) −8.00000 −0.278693
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 − 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 6.00000 0.208514
\(829\) −28.0000 −0.972480 −0.486240 − 0.873825i \(-0.661632\pi\)
−0.486240 + 0.873825i \(0.661632\pi\)
\(830\) 12.0000 0.416526
\(831\) 14.0000 0.485655
\(832\) −4.00000 −0.138675
\(833\) 0 0
\(834\) −8.00000 −0.277017
\(835\) 6.00000 0.207639
\(836\) −4.00000 −0.138343
\(837\) −4.00000 −0.138260
\(838\) −12.0000 −0.414533
\(839\) −42.0000 −1.45000 −0.725001 − 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 1.00000 0.0345033
\(841\) 7.00000 0.241379
\(842\) −14.0000 −0.482472
\(843\) 0 0
\(844\) −10.0000 −0.344214
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 6.00000 0.206041
\(849\) 2.00000 0.0686398
\(850\) 0 0
\(851\) 48.0000 1.64542
\(852\) 12.0000 0.411113
\(853\) 44.0000 1.50653 0.753266 − 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) 10.0000 0.342193
\(855\) 4.00000 0.136797
\(856\) −12.0000 −0.410152
\(857\) 36.0000 1.22974 0.614868 − 0.788630i \(-0.289209\pi\)
0.614868 + 0.788630i \(0.289209\pi\)
\(858\) 4.00000 0.136558
\(859\) 14.0000 0.477674 0.238837 − 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) −8.00000 −0.272798
\(861\) 6.00000 0.204479
\(862\) 30.0000 1.02180
\(863\) −18.0000 −0.612727 −0.306364 − 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) −2.00000 −0.0679628
\(867\) −17.0000 −0.577350
\(868\) −4.00000 −0.135769
\(869\) −4.00000 −0.135691
\(870\) 6.00000 0.203419
\(871\) −8.00000 −0.271070
\(872\) 4.00000 0.135457
\(873\) 2.00000 0.0676897
\(874\) 24.0000 0.811812
\(875\) −1.00000 −0.0338062
\(876\) −10.0000 −0.337869
\(877\) −10.0000 −0.337676 −0.168838 − 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 16.0000 0.539974
\(879\) 30.0000 1.01187
\(880\) −1.00000 −0.0337100
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −46.0000 −1.54802 −0.774012 − 0.633171i \(-0.781753\pi\)
−0.774012 + 0.633171i \(0.781753\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) −42.0000 −1.41022 −0.705111 − 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) −8.00000 −0.268462
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 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
\(897\) −24.0000 −0.801337
\(898\) 30.0000 1.00111
\(899\) −24.0000 −0.800445
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) −6.00000 −0.199778
\(903\) 8.00000 0.266223
\(904\) 12.0000 0.399114
\(905\) −20.0000 −0.664822
\(906\) −20.0000 −0.664455
\(907\) −34.0000 −1.12895 −0.564476 − 0.825450i \(-0.690922\pi\)
−0.564476 + 0.825450i \(0.690922\pi\)
\(908\) 12.0000 0.398234
\(909\) 6.00000 0.199007
\(910\) −4.00000 −0.132599
\(911\) 36.0000 1.19273 0.596367 − 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) −4.00000 −0.132453
\(913\) 12.0000 0.397142
\(914\) 22.0000 0.727695
\(915\) 10.0000 0.330590
\(916\) 8.00000 0.264327
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) 56.0000 1.84727 0.923635 − 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 6.00000 0.197814
\(921\) 2.00000 0.0659022
\(922\) −18.0000 −0.592798
\(923\) −48.0000 −1.57994
\(924\) 1.00000 0.0328976
\(925\) 8.00000 0.263038
\(926\) −32.0000 −1.05159
\(927\) 8.00000 0.262754
\(928\) −6.00000 −0.196960
\(929\) 36.0000 1.18112 0.590561 − 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) −4.00000 −0.131165
\(931\) −4.00000 −0.131095
\(932\) 6.00000 0.196537
\(933\) 6.00000 0.196431
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −2.00000 −0.0653023
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 54.0000 1.76035 0.880175 − 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) −14.0000 −0.456145
\(943\) 36.0000 1.17232
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) −8.00000 −0.260102
\(947\) 24.0000 0.779895 0.389948 − 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) −4.00000 −0.129914
\(949\) 40.0000 1.29845
\(950\) 4.00000 0.129777
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 − 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −30.0000 −0.970269
\(957\) 6.00000 0.193952
\(958\) 0 0
\(959\) 12.0000 0.387500
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) 32.0000 1.03172
\(963\) 12.0000 0.386695
\(964\) 2.00000 0.0644157
\(965\) 22.0000 0.708205
\(966\) −6.00000 −0.193047
\(967\) 32.0000 1.02905 0.514525 − 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) −12.0000 −0.385098 −0.192549 − 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 1.00000 0.0320750
\(973\) 8.00000 0.256468
\(974\) 16.0000 0.512673
\(975\) −4.00000 −0.128103
\(976\) −10.0000 −0.320092
\(977\) −24.0000 −0.767828 −0.383914 − 0.923369i \(-0.625424\pi\)
−0.383914 + 0.923369i \(0.625424\pi\)
\(978\) −2.00000 −0.0639529
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −4.00000 −0.127710
\(982\) 36.0000 1.14881
\(983\) −24.0000 −0.765481 −0.382741 − 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −6.00000 −0.191273
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) 48.0000 1.52631
\(990\) 1.00000 0.0317821
\(991\) 56.0000 1.77890 0.889449 − 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 4.00000 0.127000
\(993\) 20.0000 0.634681
\(994\) −12.0000 −0.380617
\(995\) 4.00000 0.126809
\(996\) 12.0000 0.380235
\(997\) −40.0000 −1.26681 −0.633406 − 0.773819i \(-0.718344\pi\)
−0.633406 + 0.773819i \(0.718344\pi\)
\(998\) 16.0000 0.506471
\(999\) 8.00000 0.253109
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.i.1.1 ✓ 1
3.2 odd 2 6930.2.a.bg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.i.1.1 ✓ 1 1.1 even 1 trivial
6930.2.a.bg.1.1 1 3.2 odd 2