Properties

Label 930.2.a.e.1.1
Level $930$
Weight $2$
Character 930.1
Self dual yes
Analytic conductor $7.426$
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] = [930,2,Mod(1,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.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: \(7.42608738798\)
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 \(\chi\) \(=\) 930.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +8.00000 q^{17} -1.00000 q^{18} -7.00000 q^{19} +1.00000 q^{20} -3.00000 q^{21} -3.00000 q^{22} +7.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} -8.00000 q^{29} +1.00000 q^{30} -1.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} -8.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} -4.00000 q^{37} +7.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} +3.00000 q^{42} +1.00000 q^{43} +3.00000 q^{44} +1.00000 q^{45} -7.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} -8.00000 q^{51} -2.00000 q^{52} +5.00000 q^{53} +1.00000 q^{54} +3.00000 q^{55} -3.00000 q^{56} +7.00000 q^{57} +8.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} +1.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +3.00000 q^{66} +10.0000 q^{67} +8.00000 q^{68} -7.00000 q^{69} -3.00000 q^{70} +9.00000 q^{71} -1.00000 q^{72} +1.00000 q^{73} +4.00000 q^{74} -1.00000 q^{75} -7.00000 q^{76} +9.00000 q^{77} -2.00000 q^{78} +13.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -16.0000 q^{83} -3.00000 q^{84} +8.00000 q^{85} -1.00000 q^{86} +8.00000 q^{87} -3.00000 q^{88} -3.00000 q^{89} -1.00000 q^{90} -6.00000 q^{91} +7.00000 q^{92} +1.00000 q^{93} -6.00000 q^{94} -7.00000 q^{95} +1.00000 q^{96} +6.00000 q^{97} -2.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −3.00000 −0.801784
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 8.00000 1.94029 0.970143 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) −1.00000 −0.235702
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 1.00000 0.223607
\(21\) −3.00000 −0.654654
\(22\) −3.00000 −0.639602
\(23\) 7.00000 1.45960 0.729800 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 3.00000 0.566947
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 1.00000 0.182574
\(31\) −1.00000 −0.179605
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) −8.00000 −1.37199
\(35\) 3.00000 0.507093
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 7.00000 1.13555
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 3.00000 0.462910
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 3.00000 0.452267
\(45\) 1.00000 0.149071
\(46\) −7.00000 −1.03209
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) −1.00000 −0.141421
\(51\) −8.00000 −1.12022
\(52\) −2.00000 −0.277350
\(53\) 5.00000 0.686803 0.343401 0.939189i \(-0.388421\pi\)
0.343401 + 0.939189i \(0.388421\pi\)
\(54\) 1.00000 0.136083
\(55\) 3.00000 0.404520
\(56\) −3.00000 −0.400892
\(57\) 7.00000 0.927173
\(58\) 8.00000 1.05045
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −1.00000 −0.129099
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 1.00000 0.127000
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 3.00000 0.369274
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 8.00000 0.970143
\(69\) −7.00000 −0.842701
\(70\) −3.00000 −0.358569
\(71\) 9.00000 1.06810 0.534052 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) −1.00000 −0.117851
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 4.00000 0.464991
\(75\) −1.00000 −0.115470
\(76\) −7.00000 −0.802955
\(77\) 9.00000 1.02565
\(78\) −2.00000 −0.226455
\(79\) 13.0000 1.46261 0.731307 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) −3.00000 −0.327327
\(85\) 8.00000 0.867722
\(86\) −1.00000 −0.107833
\(87\) 8.00000 0.857690
\(88\) −3.00000 −0.319801
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) −1.00000 −0.105409
\(91\) −6.00000 −0.628971
\(92\) 7.00000 0.729800
\(93\) 1.00000 0.103695
\(94\) −6.00000 −0.618853
\(95\) −7.00000 −0.718185
\(96\) 1.00000 0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) −2.00000 −0.202031
\(99\) 3.00000 0.301511
\(100\) 1.00000 0.100000
\(101\) 5.00000 0.497519 0.248759 0.968565i \(-0.419977\pi\)
0.248759 + 0.968565i \(0.419977\pi\)
\(102\) 8.00000 0.792118
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 2.00000 0.196116
\(105\) −3.00000 −0.292770
\(106\) −5.00000 −0.485643
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) −3.00000 −0.286039
\(111\) 4.00000 0.379663
\(112\) 3.00000 0.283473
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) −7.00000 −0.655610
\(115\) 7.00000 0.652753
\(116\) −8.00000 −0.742781
\(117\) −2.00000 −0.184900
\(118\) −6.00000 −0.552345
\(119\) 24.0000 2.20008
\(120\) 1.00000 0.0912871
\(121\) −2.00000 −0.181818
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) −1.00000 −0.0898027
\(125\) 1.00000 0.0894427
\(126\) −3.00000 −0.267261
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.00000 −0.0880451
\(130\) 2.00000 0.175412
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) −3.00000 −0.261116
\(133\) −21.0000 −1.82093
\(134\) −10.0000 −0.863868
\(135\) −1.00000 −0.0860663
\(136\) −8.00000 −0.685994
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 7.00000 0.595880
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 3.00000 0.253546
\(141\) −6.00000 −0.505291
\(142\) −9.00000 −0.755263
\(143\) −6.00000 −0.501745
\(144\) 1.00000 0.0833333
\(145\) −8.00000 −0.664364
\(146\) −1.00000 −0.0827606
\(147\) −2.00000 −0.164957
\(148\) −4.00000 −0.328798
\(149\) −9.00000 −0.737309 −0.368654 0.929567i \(-0.620181\pi\)
−0.368654 + 0.929567i \(0.620181\pi\)
\(150\) 1.00000 0.0816497
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 7.00000 0.567775
\(153\) 8.00000 0.646762
\(154\) −9.00000 −0.725241
\(155\) −1.00000 −0.0803219
\(156\) 2.00000 0.160128
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) −13.0000 −1.03422
\(159\) −5.00000 −0.396526
\(160\) −1.00000 −0.0790569
\(161\) 21.0000 1.65503
\(162\) −1.00000 −0.0785674
\(163\) 18.0000 1.40987 0.704934 0.709273i \(-0.250976\pi\)
0.704934 + 0.709273i \(0.250976\pi\)
\(164\) 0 0
\(165\) −3.00000 −0.233550
\(166\) 16.0000 1.24184
\(167\) 15.0000 1.16073 0.580367 0.814355i \(-0.302909\pi\)
0.580367 + 0.814355i \(0.302909\pi\)
\(168\) 3.00000 0.231455
\(169\) −9.00000 −0.692308
\(170\) −8.00000 −0.613572
\(171\) −7.00000 −0.535303
\(172\) 1.00000 0.0762493
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −8.00000 −0.606478
\(175\) 3.00000 0.226779
\(176\) 3.00000 0.226134
\(177\) −6.00000 −0.450988
\(178\) 3.00000 0.224860
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 1.00000 0.0745356
\(181\) −21.0000 −1.56092 −0.780459 0.625207i \(-0.785014\pi\)
−0.780459 + 0.625207i \(0.785014\pi\)
\(182\) 6.00000 0.444750
\(183\) −2.00000 −0.147844
\(184\) −7.00000 −0.516047
\(185\) −4.00000 −0.294086
\(186\) −1.00000 −0.0733236
\(187\) 24.0000 1.75505
\(188\) 6.00000 0.437595
\(189\) −3.00000 −0.218218
\(190\) 7.00000 0.507833
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −6.00000 −0.430775
\(195\) 2.00000 0.143223
\(196\) 2.00000 0.142857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −3.00000 −0.213201
\(199\) −3.00000 −0.212664 −0.106332 0.994331i \(-0.533911\pi\)
−0.106332 + 0.994331i \(0.533911\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −10.0000 −0.705346
\(202\) −5.00000 −0.351799
\(203\) −24.0000 −1.68447
\(204\) −8.00000 −0.560112
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 7.00000 0.486534
\(208\) −2.00000 −0.138675
\(209\) −21.0000 −1.45260
\(210\) 3.00000 0.207020
\(211\) 17.0000 1.17033 0.585164 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(212\) 5.00000 0.343401
\(213\) −9.00000 −0.616670
\(214\) 3.00000 0.205076
\(215\) 1.00000 0.0681994
\(216\) 1.00000 0.0680414
\(217\) −3.00000 −0.203653
\(218\) −4.00000 −0.270914
\(219\) −1.00000 −0.0675737
\(220\) 3.00000 0.202260
\(221\) −16.0000 −1.07628
\(222\) −4.00000 −0.268462
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) −3.00000 −0.200446
\(225\) 1.00000 0.0666667
\(226\) −1.00000 −0.0665190
\(227\) 25.0000 1.65931 0.829654 0.558278i \(-0.188538\pi\)
0.829654 + 0.558278i \(0.188538\pi\)
\(228\) 7.00000 0.463586
\(229\) −13.0000 −0.859064 −0.429532 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) −7.00000 −0.461566
\(231\) −9.00000 −0.592157
\(232\) 8.00000 0.525226
\(233\) −25.0000 −1.63780 −0.818902 0.573933i \(-0.805417\pi\)
−0.818902 + 0.573933i \(0.805417\pi\)
\(234\) 2.00000 0.130744
\(235\) 6.00000 0.391397
\(236\) 6.00000 0.390567
\(237\) −13.0000 −0.844441
\(238\) −24.0000 −1.55569
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 2.00000 0.128565
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 14.0000 0.890799
\(248\) 1.00000 0.0635001
\(249\) 16.0000 1.01396
\(250\) −1.00000 −0.0632456
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 3.00000 0.188982
\(253\) 21.0000 1.32026
\(254\) 12.0000 0.752947
\(255\) −8.00000 −0.500979
\(256\) 1.00000 0.0625000
\(257\) −23.0000 −1.43470 −0.717350 0.696713i \(-0.754645\pi\)
−0.717350 + 0.696713i \(0.754645\pi\)
\(258\) 1.00000 0.0622573
\(259\) −12.0000 −0.745644
\(260\) −2.00000 −0.124035
\(261\) −8.00000 −0.495188
\(262\) −6.00000 −0.370681
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 3.00000 0.184637
\(265\) 5.00000 0.307148
\(266\) 21.0000 1.28759
\(267\) 3.00000 0.183597
\(268\) 10.0000 0.610847
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 1.00000 0.0608581
\(271\) 7.00000 0.425220 0.212610 0.977137i \(-0.431804\pi\)
0.212610 + 0.977137i \(0.431804\pi\)
\(272\) 8.00000 0.485071
\(273\) 6.00000 0.363137
\(274\) −10.0000 −0.604122
\(275\) 3.00000 0.180907
\(276\) −7.00000 −0.421350
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 10.0000 0.599760
\(279\) −1.00000 −0.0598684
\(280\) −3.00000 −0.179284
\(281\) −28.0000 −1.67034 −0.835170 0.549992i \(-0.814631\pi\)
−0.835170 + 0.549992i \(0.814631\pi\)
\(282\) 6.00000 0.357295
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 9.00000 0.534052
\(285\) 7.00000 0.414644
\(286\) 6.00000 0.354787
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 47.0000 2.76471
\(290\) 8.00000 0.469776
\(291\) −6.00000 −0.351726
\(292\) 1.00000 0.0585206
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 2.00000 0.116642
\(295\) 6.00000 0.349334
\(296\) 4.00000 0.232495
\(297\) −3.00000 −0.174078
\(298\) 9.00000 0.521356
\(299\) −14.0000 −0.809641
\(300\) −1.00000 −0.0577350
\(301\) 3.00000 0.172917
\(302\) 8.00000 0.460348
\(303\) −5.00000 −0.287242
\(304\) −7.00000 −0.401478
\(305\) 2.00000 0.114520
\(306\) −8.00000 −0.457330
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 9.00000 0.512823
\(309\) −8.00000 −0.455104
\(310\) 1.00000 0.0567962
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) −2.00000 −0.113228
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 7.00000 0.395033
\(315\) 3.00000 0.169031
\(316\) 13.0000 0.731307
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 5.00000 0.280386
\(319\) −24.0000 −1.34374
\(320\) 1.00000 0.0559017
\(321\) 3.00000 0.167444
\(322\) −21.0000 −1.17028
\(323\) −56.0000 −3.11592
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −18.0000 −0.996928
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) 18.0000 0.992372
\(330\) 3.00000 0.165145
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) −16.0000 −0.878114
\(333\) −4.00000 −0.219199
\(334\) −15.0000 −0.820763
\(335\) 10.0000 0.546358
\(336\) −3.00000 −0.163663
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 9.00000 0.489535
\(339\) −1.00000 −0.0543125
\(340\) 8.00000 0.433861
\(341\) −3.00000 −0.162459
\(342\) 7.00000 0.378517
\(343\) −15.0000 −0.809924
\(344\) −1.00000 −0.0539164
\(345\) −7.00000 −0.376867
\(346\) 6.00000 0.322562
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 8.00000 0.428845
\(349\) 20.0000 1.07058 0.535288 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) −3.00000 −0.160357
\(351\) 2.00000 0.106752
\(352\) −3.00000 −0.159901
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 6.00000 0.318896
\(355\) 9.00000 0.477670
\(356\) −3.00000 −0.159000
\(357\) −24.0000 −1.27021
\(358\) −4.00000 −0.211407
\(359\) 1.00000 0.0527780 0.0263890 0.999652i \(-0.491599\pi\)
0.0263890 + 0.999652i \(0.491599\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 30.0000 1.57895
\(362\) 21.0000 1.10374
\(363\) 2.00000 0.104973
\(364\) −6.00000 −0.314485
\(365\) 1.00000 0.0523424
\(366\) 2.00000 0.104542
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) 7.00000 0.364900
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) 15.0000 0.778761
\(372\) 1.00000 0.0518476
\(373\) 13.0000 0.673114 0.336557 0.941663i \(-0.390737\pi\)
0.336557 + 0.941663i \(0.390737\pi\)
\(374\) −24.0000 −1.24101
\(375\) −1.00000 −0.0516398
\(376\) −6.00000 −0.309426
\(377\) 16.0000 0.824042
\(378\) 3.00000 0.154303
\(379\) 19.0000 0.975964 0.487982 0.872854i \(-0.337733\pi\)
0.487982 + 0.872854i \(0.337733\pi\)
\(380\) −7.00000 −0.359092
\(381\) 12.0000 0.614779
\(382\) 24.0000 1.22795
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 1.00000 0.0510310
\(385\) 9.00000 0.458682
\(386\) 14.0000 0.712581
\(387\) 1.00000 0.0508329
\(388\) 6.00000 0.304604
\(389\) −22.0000 −1.11544 −0.557722 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) −2.00000 −0.101274
\(391\) 56.0000 2.83204
\(392\) −2.00000 −0.101015
\(393\) −6.00000 −0.302660
\(394\) −6.00000 −0.302276
\(395\) 13.0000 0.654101
\(396\) 3.00000 0.150756
\(397\) 25.0000 1.25471 0.627357 0.778732i \(-0.284137\pi\)
0.627357 + 0.778732i \(0.284137\pi\)
\(398\) 3.00000 0.150376
\(399\) 21.0000 1.05131
\(400\) 1.00000 0.0500000
\(401\) −35.0000 −1.74782 −0.873908 0.486091i \(-0.838422\pi\)
−0.873908 + 0.486091i \(0.838422\pi\)
\(402\) 10.0000 0.498755
\(403\) 2.00000 0.0996271
\(404\) 5.00000 0.248759
\(405\) 1.00000 0.0496904
\(406\) 24.0000 1.19110
\(407\) −12.0000 −0.594818
\(408\) 8.00000 0.396059
\(409\) 12.0000 0.593362 0.296681 0.954977i \(-0.404120\pi\)
0.296681 + 0.954977i \(0.404120\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) 8.00000 0.394132
\(413\) 18.0000 0.885722
\(414\) −7.00000 −0.344031
\(415\) −16.0000 −0.785409
\(416\) 2.00000 0.0980581
\(417\) 10.0000 0.489702
\(418\) 21.0000 1.02714
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) −3.00000 −0.146385
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) −17.0000 −0.827547
\(423\) 6.00000 0.291730
\(424\) −5.00000 −0.242821
\(425\) 8.00000 0.388057
\(426\) 9.00000 0.436051
\(427\) 6.00000 0.290360
\(428\) −3.00000 −0.145010
\(429\) 6.00000 0.289683
\(430\) −1.00000 −0.0482243
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 19.0000 0.913082 0.456541 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) 3.00000 0.144005
\(435\) 8.00000 0.383571
\(436\) 4.00000 0.191565
\(437\) −49.0000 −2.34399
\(438\) 1.00000 0.0477818
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) −3.00000 −0.143019
\(441\) 2.00000 0.0952381
\(442\) 16.0000 0.761042
\(443\) 19.0000 0.902717 0.451359 0.892343i \(-0.350940\pi\)
0.451359 + 0.892343i \(0.350940\pi\)
\(444\) 4.00000 0.189832
\(445\) −3.00000 −0.142214
\(446\) 10.0000 0.473514
\(447\) 9.00000 0.425685
\(448\) 3.00000 0.141737
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) 1.00000 0.0470360
\(453\) 8.00000 0.375873
\(454\) −25.0000 −1.17331
\(455\) −6.00000 −0.281284
\(456\) −7.00000 −0.327805
\(457\) −38.0000 −1.77757 −0.888783 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) 13.0000 0.607450
\(459\) −8.00000 −0.373408
\(460\) 7.00000 0.326377
\(461\) −20.0000 −0.931493 −0.465746 0.884918i \(-0.654214\pi\)
−0.465746 + 0.884918i \(0.654214\pi\)
\(462\) 9.00000 0.418718
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) −8.00000 −0.371391
\(465\) 1.00000 0.0463739
\(466\) 25.0000 1.15810
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 30.0000 1.38527
\(470\) −6.00000 −0.276759
\(471\) 7.00000 0.322543
\(472\) −6.00000 −0.276172
\(473\) 3.00000 0.137940
\(474\) 13.0000 0.597110
\(475\) −7.00000 −0.321182
\(476\) 24.0000 1.10004
\(477\) 5.00000 0.228934
\(478\) −8.00000 −0.365911
\(479\) −27.0000 −1.23366 −0.616831 0.787096i \(-0.711584\pi\)
−0.616831 + 0.787096i \(0.711584\pi\)
\(480\) 1.00000 0.0456435
\(481\) 8.00000 0.364769
\(482\) 22.0000 1.00207
\(483\) −21.0000 −0.955533
\(484\) −2.00000 −0.0909091
\(485\) 6.00000 0.272446
\(486\) 1.00000 0.0453609
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −18.0000 −0.813988
\(490\) −2.00000 −0.0903508
\(491\) −29.0000 −1.30875 −0.654376 0.756169i \(-0.727069\pi\)
−0.654376 + 0.756169i \(0.727069\pi\)
\(492\) 0 0
\(493\) −64.0000 −2.88242
\(494\) −14.0000 −0.629890
\(495\) 3.00000 0.134840
\(496\) −1.00000 −0.0449013
\(497\) 27.0000 1.21112
\(498\) −16.0000 −0.716977
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 1.00000 0.0447214
\(501\) −15.0000 −0.670151
\(502\) −20.0000 −0.892644
\(503\) 32.0000 1.42681 0.713405 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) −3.00000 −0.133631
\(505\) 5.00000 0.222497
\(506\) −21.0000 −0.933564
\(507\) 9.00000 0.399704
\(508\) −12.0000 −0.532414
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 8.00000 0.354246
\(511\) 3.00000 0.132712
\(512\) −1.00000 −0.0441942
\(513\) 7.00000 0.309058
\(514\) 23.0000 1.01449
\(515\) 8.00000 0.352522
\(516\) −1.00000 −0.0440225
\(517\) 18.0000 0.791639
\(518\) 12.0000 0.527250
\(519\) 6.00000 0.263371
\(520\) 2.00000 0.0877058
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 8.00000 0.350150
\(523\) 29.0000 1.26808 0.634041 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 6.00000 0.262111
\(525\) −3.00000 −0.130931
\(526\) −24.0000 −1.04645
\(527\) −8.00000 −0.348485
\(528\) −3.00000 −0.130558
\(529\) 26.0000 1.13043
\(530\) −5.00000 −0.217186
\(531\) 6.00000 0.260378
\(532\) −21.0000 −0.910465
\(533\) 0 0
\(534\) −3.00000 −0.129823
\(535\) −3.00000 −0.129701
\(536\) −10.0000 −0.431934
\(537\) −4.00000 −0.172613
\(538\) 18.0000 0.776035
\(539\) 6.00000 0.258438
\(540\) −1.00000 −0.0430331
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) −7.00000 −0.300676
\(543\) 21.0000 0.901196
\(544\) −8.00000 −0.342997
\(545\) 4.00000 0.171341
\(546\) −6.00000 −0.256776
\(547\) −14.0000 −0.598597 −0.299298 0.954160i \(-0.596753\pi\)
−0.299298 + 0.954160i \(0.596753\pi\)
\(548\) 10.0000 0.427179
\(549\) 2.00000 0.0853579
\(550\) −3.00000 −0.127920
\(551\) 56.0000 2.38568
\(552\) 7.00000 0.297940
\(553\) 39.0000 1.65845
\(554\) 8.00000 0.339887
\(555\) 4.00000 0.169791
\(556\) −10.0000 −0.424094
\(557\) 17.0000 0.720313 0.360157 0.932892i \(-0.382723\pi\)
0.360157 + 0.932892i \(0.382723\pi\)
\(558\) 1.00000 0.0423334
\(559\) −2.00000 −0.0845910
\(560\) 3.00000 0.126773
\(561\) −24.0000 −1.01328
\(562\) 28.0000 1.18111
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) −6.00000 −0.252646
\(565\) 1.00000 0.0420703
\(566\) 16.0000 0.672530
\(567\) 3.00000 0.125988
\(568\) −9.00000 −0.377632
\(569\) 37.0000 1.55112 0.775560 0.631273i \(-0.217467\pi\)
0.775560 + 0.631273i \(0.217467\pi\)
\(570\) −7.00000 −0.293198
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) −6.00000 −0.250873
\(573\) 24.0000 1.00261
\(574\) 0 0
\(575\) 7.00000 0.291920
\(576\) 1.00000 0.0416667
\(577\) −8.00000 −0.333044 −0.166522 0.986038i \(-0.553254\pi\)
−0.166522 + 0.986038i \(0.553254\pi\)
\(578\) −47.0000 −1.95494
\(579\) 14.0000 0.581820
\(580\) −8.00000 −0.332182
\(581\) −48.0000 −1.99138
\(582\) 6.00000 0.248708
\(583\) 15.0000 0.621237
\(584\) −1.00000 −0.0413803
\(585\) −2.00000 −0.0826898
\(586\) −14.0000 −0.578335
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 7.00000 0.288430
\(590\) −6.00000 −0.247016
\(591\) −6.00000 −0.246807
\(592\) −4.00000 −0.164399
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 3.00000 0.123091
\(595\) 24.0000 0.983904
\(596\) −9.00000 −0.368654
\(597\) 3.00000 0.122782
\(598\) 14.0000 0.572503
\(599\) 35.0000 1.43006 0.715031 0.699093i \(-0.246413\pi\)
0.715031 + 0.699093i \(0.246413\pi\)
\(600\) 1.00000 0.0408248
\(601\) −30.0000 −1.22373 −0.611863 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) −3.00000 −0.122271
\(603\) 10.0000 0.407231
\(604\) −8.00000 −0.325515
\(605\) −2.00000 −0.0813116
\(606\) 5.00000 0.203111
\(607\) −13.0000 −0.527654 −0.263827 0.964570i \(-0.584985\pi\)
−0.263827 + 0.964570i \(0.584985\pi\)
\(608\) 7.00000 0.283887
\(609\) 24.0000 0.972529
\(610\) −2.00000 −0.0809776
\(611\) −12.0000 −0.485468
\(612\) 8.00000 0.323381
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) −9.00000 −0.362620
\(617\) −3.00000 −0.120775 −0.0603877 0.998175i \(-0.519234\pi\)
−0.0603877 + 0.998175i \(0.519234\pi\)
\(618\) 8.00000 0.321807
\(619\) −18.0000 −0.723481 −0.361741 0.932279i \(-0.617817\pi\)
−0.361741 + 0.932279i \(0.617817\pi\)
\(620\) −1.00000 −0.0401610
\(621\) −7.00000 −0.280900
\(622\) 16.0000 0.641542
\(623\) −9.00000 −0.360577
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 21.0000 0.838659
\(628\) −7.00000 −0.279330
\(629\) −32.0000 −1.27592
\(630\) −3.00000 −0.119523
\(631\) −5.00000 −0.199047 −0.0995234 0.995035i \(-0.531732\pi\)
−0.0995234 + 0.995035i \(0.531732\pi\)
\(632\) −13.0000 −0.517112
\(633\) −17.0000 −0.675689
\(634\) −24.0000 −0.953162
\(635\) −12.0000 −0.476205
\(636\) −5.00000 −0.198263
\(637\) −4.00000 −0.158486
\(638\) 24.0000 0.950169
\(639\) 9.00000 0.356034
\(640\) −1.00000 −0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −3.00000 −0.118401
\(643\) 19.0000 0.749287 0.374643 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(644\) 21.0000 0.827516
\(645\) −1.00000 −0.0393750
\(646\) 56.0000 2.20329
\(647\) 49.0000 1.92639 0.963194 0.268806i \(-0.0866290\pi\)
0.963194 + 0.268806i \(0.0866290\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 18.0000 0.706562
\(650\) 2.00000 0.0784465
\(651\) 3.00000 0.117579
\(652\) 18.0000 0.704934
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 4.00000 0.156412
\(655\) 6.00000 0.234439
\(656\) 0 0
\(657\) 1.00000 0.0390137
\(658\) −18.0000 −0.701713
\(659\) −32.0000 −1.24654 −0.623272 0.782006i \(-0.714197\pi\)
−0.623272 + 0.782006i \(0.714197\pi\)
\(660\) −3.00000 −0.116775
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) −32.0000 −1.24372
\(663\) 16.0000 0.621389
\(664\) 16.0000 0.620920
\(665\) −21.0000 −0.814345
\(666\) 4.00000 0.154997
\(667\) −56.0000 −2.16833
\(668\) 15.0000 0.580367
\(669\) 10.0000 0.386622
\(670\) −10.0000 −0.386334
\(671\) 6.00000 0.231627
\(672\) 3.00000 0.115728
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 10.0000 0.385186
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −43.0000 −1.65262 −0.826312 0.563212i \(-0.809565\pi\)
−0.826312 + 0.563212i \(0.809565\pi\)
\(678\) 1.00000 0.0384048
\(679\) 18.0000 0.690777
\(680\) −8.00000 −0.306786
\(681\) −25.0000 −0.958002
\(682\) 3.00000 0.114876
\(683\) −15.0000 −0.573959 −0.286980 0.957937i \(-0.592651\pi\)
−0.286980 + 0.957937i \(0.592651\pi\)
\(684\) −7.00000 −0.267652
\(685\) 10.0000 0.382080
\(686\) 15.0000 0.572703
\(687\) 13.0000 0.495981
\(688\) 1.00000 0.0381246
\(689\) −10.0000 −0.380970
\(690\) 7.00000 0.266485
\(691\) 5.00000 0.190209 0.0951045 0.995467i \(-0.469681\pi\)
0.0951045 + 0.995467i \(0.469681\pi\)
\(692\) −6.00000 −0.228086
\(693\) 9.00000 0.341882
\(694\) 12.0000 0.455514
\(695\) −10.0000 −0.379322
\(696\) −8.00000 −0.303239
\(697\) 0 0
\(698\) −20.0000 −0.757011
\(699\) 25.0000 0.945587
\(700\) 3.00000 0.113389
\(701\) 33.0000 1.24639 0.623196 0.782065i \(-0.285834\pi\)
0.623196 + 0.782065i \(0.285834\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 28.0000 1.05604
\(704\) 3.00000 0.113067
\(705\) −6.00000 −0.225973
\(706\) 18.0000 0.677439
\(707\) 15.0000 0.564133
\(708\) −6.00000 −0.225494
\(709\) −15.0000 −0.563337 −0.281668 0.959512i \(-0.590888\pi\)
−0.281668 + 0.959512i \(0.590888\pi\)
\(710\) −9.00000 −0.337764
\(711\) 13.0000 0.487538
\(712\) 3.00000 0.112430
\(713\) −7.00000 −0.262152
\(714\) 24.0000 0.898177
\(715\) −6.00000 −0.224387
\(716\) 4.00000 0.149487
\(717\) −8.00000 −0.298765
\(718\) −1.00000 −0.0373197
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 1.00000 0.0372678
\(721\) 24.0000 0.893807
\(722\) −30.0000 −1.11648
\(723\) 22.0000 0.818189
\(724\) −21.0000 −0.780459
\(725\) −8.00000 −0.297113
\(726\) −2.00000 −0.0742270
\(727\) −29.0000 −1.07555 −0.537775 0.843088i \(-0.680735\pi\)
−0.537775 + 0.843088i \(0.680735\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) −1.00000 −0.0370117
\(731\) 8.00000 0.295891
\(732\) −2.00000 −0.0739221
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 18.0000 0.664392
\(735\) −2.00000 −0.0737711
\(736\) −7.00000 −0.258023
\(737\) 30.0000 1.10506
\(738\) 0 0
\(739\) −40.0000 −1.47142 −0.735712 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) −4.00000 −0.147043
\(741\) −14.0000 −0.514303
\(742\) −15.0000 −0.550667
\(743\) 1.00000 0.0366864 0.0183432 0.999832i \(-0.494161\pi\)
0.0183432 + 0.999832i \(0.494161\pi\)
\(744\) −1.00000 −0.0366618
\(745\) −9.00000 −0.329734
\(746\) −13.0000 −0.475964
\(747\) −16.0000 −0.585409
\(748\) 24.0000 0.877527
\(749\) −9.00000 −0.328853
\(750\) 1.00000 0.0365148
\(751\) 6.00000 0.218943 0.109472 0.993990i \(-0.465084\pi\)
0.109472 + 0.993990i \(0.465084\pi\)
\(752\) 6.00000 0.218797
\(753\) −20.0000 −0.728841
\(754\) −16.0000 −0.582686
\(755\) −8.00000 −0.291150
\(756\) −3.00000 −0.109109
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −19.0000 −0.690111
\(759\) −21.0000 −0.762252
\(760\) 7.00000 0.253917
\(761\) 49.0000 1.77625 0.888124 0.459603i \(-0.152008\pi\)
0.888124 + 0.459603i \(0.152008\pi\)
\(762\) −12.0000 −0.434714
\(763\) 12.0000 0.434429
\(764\) −24.0000 −0.868290
\(765\) 8.00000 0.289241
\(766\) 36.0000 1.30073
\(767\) −12.0000 −0.433295
\(768\) −1.00000 −0.0360844
\(769\) 9.00000 0.324548 0.162274 0.986746i \(-0.448117\pi\)
0.162274 + 0.986746i \(0.448117\pi\)
\(770\) −9.00000 −0.324337
\(771\) 23.0000 0.828325
\(772\) −14.0000 −0.503871
\(773\) −1.00000 −0.0359675 −0.0179838 0.999838i \(-0.505725\pi\)
−0.0179838 + 0.999838i \(0.505725\pi\)
\(774\) −1.00000 −0.0359443
\(775\) −1.00000 −0.0359211
\(776\) −6.00000 −0.215387
\(777\) 12.0000 0.430498
\(778\) 22.0000 0.788738
\(779\) 0 0
\(780\) 2.00000 0.0716115
\(781\) 27.0000 0.966136
\(782\) −56.0000 −2.00256
\(783\) 8.00000 0.285897
\(784\) 2.00000 0.0714286
\(785\) −7.00000 −0.249841
\(786\) 6.00000 0.214013
\(787\) −27.0000 −0.962446 −0.481223 0.876598i \(-0.659807\pi\)
−0.481223 + 0.876598i \(0.659807\pi\)
\(788\) 6.00000 0.213741
\(789\) −24.0000 −0.854423
\(790\) −13.0000 −0.462519
\(791\) 3.00000 0.106668
\(792\) −3.00000 −0.106600
\(793\) −4.00000 −0.142044
\(794\) −25.0000 −0.887217
\(795\) −5.00000 −0.177332
\(796\) −3.00000 −0.106332
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) −21.0000 −0.743392
\(799\) 48.0000 1.69812
\(800\) −1.00000 −0.0353553
\(801\) −3.00000 −0.106000
\(802\) 35.0000 1.23589
\(803\) 3.00000 0.105868
\(804\) −10.0000 −0.352673
\(805\) 21.0000 0.740153
\(806\) −2.00000 −0.0704470
\(807\) 18.0000 0.633630
\(808\) −5.00000 −0.175899
\(809\) 13.0000 0.457056 0.228528 0.973537i \(-0.426609\pi\)
0.228528 + 0.973537i \(0.426609\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −23.0000 −0.807639 −0.403820 0.914839i \(-0.632318\pi\)
−0.403820 + 0.914839i \(0.632318\pi\)
\(812\) −24.0000 −0.842235
\(813\) −7.00000 −0.245501
\(814\) 12.0000 0.420600
\(815\) 18.0000 0.630512
\(816\) −8.00000 −0.280056
\(817\) −7.00000 −0.244899
\(818\) −12.0000 −0.419570
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 10.0000 0.348790
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) −8.00000 −0.278693
\(825\) −3.00000 −0.104447
\(826\) −18.0000 −0.626300
\(827\) −44.0000 −1.53003 −0.765015 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 7.00000 0.243267
\(829\) −21.0000 −0.729360 −0.364680 0.931133i \(-0.618822\pi\)
−0.364680 + 0.931133i \(0.618822\pi\)
\(830\) 16.0000 0.555368
\(831\) 8.00000 0.277517
\(832\) −2.00000 −0.0693375
\(833\) 16.0000 0.554367
\(834\) −10.0000 −0.346272
\(835\) 15.0000 0.519096
\(836\) −21.0000 −0.726300
\(837\) 1.00000 0.0345651
\(838\) −16.0000 −0.552711
\(839\) −27.0000 −0.932144 −0.466072 0.884747i \(-0.654331\pi\)
−0.466072 + 0.884747i \(0.654331\pi\)
\(840\) 3.00000 0.103510
\(841\) 35.0000 1.20690
\(842\) −34.0000 −1.17172
\(843\) 28.0000 0.964371
\(844\) 17.0000 0.585164
\(845\) −9.00000 −0.309609
\(846\) −6.00000 −0.206284
\(847\) −6.00000 −0.206162
\(848\) 5.00000 0.171701
\(849\) 16.0000 0.549119
\(850\) −8.00000 −0.274398
\(851\) −28.0000 −0.959828
\(852\) −9.00000 −0.308335
\(853\) 45.0000 1.54077 0.770385 0.637579i \(-0.220064\pi\)
0.770385 + 0.637579i \(0.220064\pi\)
\(854\) −6.00000 −0.205316
\(855\) −7.00000 −0.239395
\(856\) 3.00000 0.102538
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) −6.00000 −0.204837
\(859\) −24.0000 −0.818869 −0.409435 0.912339i \(-0.634274\pi\)
−0.409435 + 0.912339i \(0.634274\pi\)
\(860\) 1.00000 0.0340997
\(861\) 0 0
\(862\) −32.0000 −1.08992
\(863\) −31.0000 −1.05525 −0.527626 0.849477i \(-0.676918\pi\)
−0.527626 + 0.849477i \(0.676918\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.00000 −0.204006
\(866\) −19.0000 −0.645646
\(867\) −47.0000 −1.59620
\(868\) −3.00000 −0.101827
\(869\) 39.0000 1.32298
\(870\) −8.00000 −0.271225
\(871\) −20.0000 −0.677674
\(872\) −4.00000 −0.135457
\(873\) 6.00000 0.203069
\(874\) 49.0000 1.65745
\(875\) 3.00000 0.101419
\(876\) −1.00000 −0.0337869
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 26.0000 0.877457
\(879\) −14.0000 −0.472208
\(880\) 3.00000 0.101130
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −13.0000 −0.437485 −0.218742 0.975783i \(-0.570195\pi\)
−0.218742 + 0.975783i \(0.570195\pi\)
\(884\) −16.0000 −0.538138
\(885\) −6.00000 −0.201688
\(886\) −19.0000 −0.638317
\(887\) 22.0000 0.738688 0.369344 0.929293i \(-0.379582\pi\)
0.369344 + 0.929293i \(0.379582\pi\)
\(888\) −4.00000 −0.134231
\(889\) −36.0000 −1.20740
\(890\) 3.00000 0.100560
\(891\) 3.00000 0.100504
\(892\) −10.0000 −0.334825
\(893\) −42.0000 −1.40548
\(894\) −9.00000 −0.301005
\(895\) 4.00000 0.133705
\(896\) −3.00000 −0.100223
\(897\) 14.0000 0.467446
\(898\) −34.0000 −1.13459
\(899\) 8.00000 0.266815
\(900\) 1.00000 0.0333333
\(901\) 40.0000 1.33259
\(902\) 0 0
\(903\) −3.00000 −0.0998337
\(904\) −1.00000 −0.0332595
\(905\) −21.0000 −0.698064
\(906\) −8.00000 −0.265782
\(907\) −40.0000 −1.32818 −0.664089 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) 25.0000 0.829654
\(909\) 5.00000 0.165840
\(910\) 6.00000 0.198898
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 7.00000 0.231793
\(913\) −48.0000 −1.58857
\(914\) 38.0000 1.25693
\(915\) −2.00000 −0.0661180
\(916\) −13.0000 −0.429532
\(917\) 18.0000 0.594412
\(918\) 8.00000 0.264039
\(919\) −10.0000 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) −7.00000 −0.230783
\(921\) 12.0000 0.395413
\(922\) 20.0000 0.658665
\(923\) −18.0000 −0.592477
\(924\) −9.00000 −0.296078
\(925\) −4.00000 −0.131519
\(926\) 40.0000 1.31448
\(927\) 8.00000 0.262754
\(928\) 8.00000 0.262613
\(929\) 3.00000 0.0984268 0.0492134 0.998788i \(-0.484329\pi\)
0.0492134 + 0.998788i \(0.484329\pi\)
\(930\) −1.00000 −0.0327913
\(931\) −14.0000 −0.458831
\(932\) −25.0000 −0.818902
\(933\) 16.0000 0.523816
\(934\) 0 0
\(935\) 24.0000 0.784884
\(936\) 2.00000 0.0653720
\(937\) −16.0000 −0.522697 −0.261349 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) −30.0000 −0.979535
\(939\) 10.0000 0.326338
\(940\) 6.00000 0.195698
\(941\) −28.0000 −0.912774 −0.456387 0.889781i \(-0.650857\pi\)
−0.456387 + 0.889781i \(0.650857\pi\)
\(942\) −7.00000 −0.228072
\(943\) 0 0
\(944\) 6.00000 0.195283
\(945\) −3.00000 −0.0975900
\(946\) −3.00000 −0.0975384
\(947\) 38.0000 1.23483 0.617417 0.786636i \(-0.288179\pi\)
0.617417 + 0.786636i \(0.288179\pi\)
\(948\) −13.0000 −0.422220
\(949\) −2.00000 −0.0649227
\(950\) 7.00000 0.227110
\(951\) −24.0000 −0.778253
\(952\) −24.0000 −0.777844
\(953\) 12.0000 0.388718 0.194359 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) −5.00000 −0.161881
\(955\) −24.0000 −0.776622
\(956\) 8.00000 0.258738
\(957\) 24.0000 0.775810
\(958\) 27.0000 0.872330
\(959\) 30.0000 0.968751
\(960\) −1.00000 −0.0322749
\(961\) 1.00000 0.0322581
\(962\) −8.00000 −0.257930
\(963\) −3.00000 −0.0966736
\(964\) −22.0000 −0.708572
\(965\) −14.0000 −0.450676
\(966\) 21.0000 0.675664
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 2.00000 0.0642824
\(969\) 56.0000 1.79898
\(970\) −6.00000 −0.192648
\(971\) −54.0000 −1.73294 −0.866471 0.499227i \(-0.833617\pi\)
−0.866471 + 0.499227i \(0.833617\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −30.0000 −0.961756
\(974\) 20.0000 0.640841
\(975\) 2.00000 0.0640513
\(976\) 2.00000 0.0640184
\(977\) 46.0000 1.47167 0.735835 0.677161i \(-0.236790\pi\)
0.735835 + 0.677161i \(0.236790\pi\)
\(978\) 18.0000 0.575577
\(979\) −9.00000 −0.287641
\(980\) 2.00000 0.0638877
\(981\) 4.00000 0.127710
\(982\) 29.0000 0.925427
\(983\) 48.0000 1.53096 0.765481 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 64.0000 2.03818
\(987\) −18.0000 −0.572946
\(988\) 14.0000 0.445399
\(989\) 7.00000 0.222587
\(990\) −3.00000 −0.0953463
\(991\) 5.00000 0.158830 0.0794151 0.996842i \(-0.474695\pi\)
0.0794151 + 0.996842i \(0.474695\pi\)
\(992\) 1.00000 0.0317500
\(993\) −32.0000 −1.01549
\(994\) −27.0000 −0.856388
\(995\) −3.00000 −0.0951064
\(996\) 16.0000 0.506979
\(997\) −6.00000 −0.190022 −0.0950110 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) 20.0000 0.633089
\(999\) 4.00000 0.126554
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 930.2.a.e.1.1 1
3.2 odd 2 2790.2.a.u.1.1 1
4.3 odd 2 7440.2.a.x.1.1 1
5.2 odd 4 4650.2.d.ba.3349.1 2
5.3 odd 4 4650.2.d.ba.3349.2 2
5.4 even 2 4650.2.a.bk.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.e.1.1 1 1.1 even 1 trivial
2790.2.a.u.1.1 1 3.2 odd 2
4650.2.a.bk.1.1 1 5.4 even 2
4650.2.d.ba.3349.1 2 5.2 odd 4
4650.2.d.ba.3349.2 2 5.3 odd 4
7440.2.a.x.1.1 1 4.3 odd 2