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Properties

Label 5610.2.a.bd.1.1
Level $5610$
Weight $2$
Character 5610.1
Self dual yes
Analytic conductor $44.796$
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] = [5610,2,Mod(1,5610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5610, 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("5610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5610.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: \(44.7960755339\)
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 \(\chi\) \(=\) 5610.1

$q$-expansion

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

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
\(7\) 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
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 3.00000 0.801784
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 1.00000 0.223607
\(21\) −3.00000 −0.654654
\(22\) −1.00000 −0.213201
\(23\) 3.00000 0.625543 0.312772 − 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 3.00000 0.566947
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) −1.00000 −0.182574
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) 1.00000 0.171499
\(35\) 3.00000 0.507093
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) 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 \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.00000 0.149071
\(46\) 3.00000 0.442326
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) 1.00000 0.141421
\(51\) −1.00000 −0.140028
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 − 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) 3.00000 0.400892
\(57\) 8.00000 1.05963
\(58\) 3.00000 0.393919
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 3.00000 0.381000
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 1.00000 0.121268
\(69\) −3.00000 −0.361158
\(70\) 3.00000 0.358569
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 10.0000 1.16248
\(75\) −1.00000 −0.115470
\(76\) −8.00000 −0.917663
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −3.00000 −0.327327
\(85\) 1.00000 0.108465
\(86\) −1.00000 −0.107833
\(87\) −3.00000 −0.321634
\(88\) −1.00000 −0.106600
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 3.00000 0.312772
\(93\) −3.00000 −0.311086
\(94\) 4.00000 0.412568
\(95\) −8.00000 −0.820783
\(96\) −1.00000 −0.102062
\(97\) −1.00000 −0.101535 −0.0507673 − 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 2.00000 0.202031
\(99\) −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 \(0.403510\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −11.0000 −1.08386 −0.541931 − 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 8.00000 0.777029
\(107\) −5.00000 −0.483368 −0.241684 − 0.970355i \(-0.577700\pi\)
−0.241684 + 0.970355i \(0.577700\pi\)
\(108\) −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
\(111\) −10.0000 −0.949158
\(112\) 3.00000 0.283473
\(113\) 14.0000 1.31701 0.658505 − 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 8.00000 0.749269
\(115\) 3.00000 0.279751
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 3.00000 0.275010
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) 3.00000 0.269408
\(125\) 1.00000 0.0894427
\(126\) 3.00000 0.267261
\(127\) 10.0000 0.887357 0.443678 − 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 1.00000 0.0870388
\(133\) −24.0000 −2.08106
\(134\) 4.00000 0.345547
\(135\) −1.00000 −0.0860663
\(136\) 1.00000 0.0857493
\(137\) −11.0000 −0.939793 −0.469897 − 0.882721i \(-0.655709\pi\)
−0.469897 + 0.882721i \(0.655709\pi\)
\(138\) −3.00000 −0.255377
\(139\) −5.00000 −0.424094 −0.212047 − 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 3.00000 0.253546
\(141\) −4.00000 −0.336861
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 3.00000 0.249136
\(146\) 6.00000 0.496564
\(147\) −2.00000 −0.164957
\(148\) 10.0000 0.821995
\(149\) −20.0000 −1.63846 −0.819232 − 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 20.0000 1.62758 0.813788 − 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −8.00000 −0.648886
\(153\) 1.00000 0.0808452
\(154\) −3.00000 −0.241747
\(155\) 3.00000 0.240966
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 4.00000 0.318223
\(159\) −8.00000 −0.634441
\(160\) 1.00000 0.0790569
\(161\) 9.00000 0.709299
\(162\) 1.00000 0.0785674
\(163\) 9.00000 0.704934 0.352467 − 0.935824i \(-0.385343\pi\)
0.352467 + 0.935824i \(0.385343\pi\)
\(164\) 0 0
\(165\) 1.00000 0.0778499
\(166\) 4.00000 0.310460
\(167\) 2.00000 0.154765 0.0773823 − 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) −3.00000 −0.231455
\(169\) −13.0000 −1.00000
\(170\) 1.00000 0.0766965
\(171\) −8.00000 −0.611775
\(172\) −1.00000 −0.0762493
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −3.00000 −0.227429
\(175\) 3.00000 0.226779
\(176\) −1.00000 −0.0753778
\(177\) 6.00000 0.450988
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 − 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 1.00000 0.0745356
\(181\) −25.0000 −1.85824 −0.929118 − 0.369784i \(-0.879432\pi\)
−0.929118 + 0.369784i \(0.879432\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 3.00000 0.221163
\(185\) 10.0000 0.735215
\(186\) −3.00000 −0.219971
\(187\) −1.00000 −0.0731272
\(188\) 4.00000 0.291730
\(189\) −3.00000 −0.218218
\(190\) −8.00000 −0.580381
\(191\) 11.0000 0.795932 0.397966 − 0.917400i \(-0.369716\pi\)
0.397966 + 0.917400i \(0.369716\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −1.00000 −0.0717958
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 12.0000 0.854965 0.427482 − 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −24.0000 −1.70131 −0.850657 − 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) 6.00000 0.422159
\(203\) 9.00000 0.631676
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) −11.0000 −0.766406
\(207\) 3.00000 0.208514
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) −3.00000 −0.207020
\(211\) 13.0000 0.894957 0.447478 − 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) 8.00000 0.549442
\(213\) 6.00000 0.411113
\(214\) −5.00000 −0.341793
\(215\) −1.00000 −0.0681994
\(216\) −1.00000 −0.0680414
\(217\) 9.00000 0.610960
\(218\) −4.00000 −0.270914
\(219\) −6.00000 −0.405442
\(220\) −1.00000 −0.0674200
\(221\) 0 0
\(222\) −10.0000 −0.671156
\(223\) −11.0000 −0.736614 −0.368307 − 0.929704i \(-0.620063\pi\)
−0.368307 + 0.929704i \(0.620063\pi\)
\(224\) 3.00000 0.200446
\(225\) 1.00000 0.0666667
\(226\) 14.0000 0.931266
\(227\) −13.0000 −0.862840 −0.431420 − 0.902151i \(-0.641987\pi\)
−0.431420 + 0.902151i \(0.641987\pi\)
\(228\) 8.00000 0.529813
\(229\) 4.00000 0.264327 0.132164 − 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 3.00000 0.197814
\(231\) 3.00000 0.197386
\(232\) 3.00000 0.196960
\(233\) 21.0000 1.37576 0.687878 − 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) −6.00000 −0.390567
\(237\) −4.00000 −0.259828
\(238\) 3.00000 0.194461
\(239\) 14.0000 0.905585 0.452792 − 0.891616i \(-0.350428\pi\)
0.452792 + 0.891616i \(0.350428\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 17.0000 1.09507 0.547533 − 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 0 0
\(248\) 3.00000 0.190500
\(249\) −4.00000 −0.253490
\(250\) 1.00000 0.0632456
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 3.00000 0.188982
\(253\) −3.00000 −0.188608
\(254\) 10.0000 0.627456
\(255\) −1.00000 −0.0626224
\(256\) 1.00000 0.0625000
\(257\) 27.0000 1.68421 0.842107 − 0.539311i \(-0.181315\pi\)
0.842107 + 0.539311i \(0.181315\pi\)
\(258\) 1.00000 0.0622573
\(259\) 30.0000 1.86411
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 12.0000 0.741362
\(263\) −15.0000 −0.924940 −0.462470 − 0.886635i \(-0.653037\pi\)
−0.462470 + 0.886635i \(0.653037\pi\)
\(264\) 1.00000 0.0615457
\(265\) 8.00000 0.491436
\(266\) −24.0000 −1.47153
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −7.00000 −0.425220 −0.212610 − 0.977137i \(-0.568196\pi\)
−0.212610 + 0.977137i \(0.568196\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) −11.0000 −0.664534
\(275\) −1.00000 −0.0603023
\(276\) −3.00000 −0.180579
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −5.00000 −0.299880
\(279\) 3.00000 0.179605
\(280\) 3.00000 0.179284
\(281\) 17.0000 1.01413 0.507067 − 0.861906i \(-0.330729\pi\)
0.507067 + 0.861906i \(0.330729\pi\)
\(282\) −4.00000 −0.238197
\(283\) −26.0000 −1.54554 −0.772770 − 0.634686i \(-0.781129\pi\)
−0.772770 + 0.634686i \(0.781129\pi\)
\(284\) −6.00000 −0.356034
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 3.00000 0.176166
\(291\) 1.00000 0.0586210
\(292\) 6.00000 0.351123
\(293\) −5.00000 −0.292103 −0.146052 − 0.989277i \(-0.546657\pi\)
−0.146052 + 0.989277i \(0.546657\pi\)
\(294\) −2.00000 −0.116642
\(295\) −6.00000 −0.349334
\(296\) 10.0000 0.581238
\(297\) 1.00000 0.0580259
\(298\) −20.0000 −1.15857
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −3.00000 −0.172917
\(302\) 20.0000 1.15087
\(303\) −6.00000 −0.344691
\(304\) −8.00000 −0.458831
\(305\) 10.0000 0.572598
\(306\) 1.00000 0.0571662
\(307\) 28.0000 1.59804 0.799022 − 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) −3.00000 −0.170941
\(309\) 11.0000 0.625768
\(310\) 3.00000 0.170389
\(311\) 24.0000 1.36092 0.680458 − 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 9.00000 0.508710 0.254355 − 0.967111i \(-0.418137\pi\)
0.254355 + 0.967111i \(0.418137\pi\)
\(314\) −10.0000 −0.564333
\(315\) 3.00000 0.169031
\(316\) 4.00000 0.225018
\(317\) 1.00000 0.0561656 0.0280828 − 0.999606i \(-0.491060\pi\)
0.0280828 + 0.999606i \(0.491060\pi\)
\(318\) −8.00000 −0.448618
\(319\) −3.00000 −0.167968
\(320\) 1.00000 0.0559017
\(321\) 5.00000 0.279073
\(322\) 9.00000 0.501550
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 9.00000 0.498464
\(327\) 4.00000 0.221201
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 1.00000 0.0550482
\(331\) 1.00000 0.0549650 0.0274825 − 0.999622i \(-0.491251\pi\)
0.0274825 + 0.999622i \(0.491251\pi\)
\(332\) 4.00000 0.219529
\(333\) 10.0000 0.547997
\(334\) 2.00000 0.109435
\(335\) 4.00000 0.218543
\(336\) −3.00000 −0.163663
\(337\) −26.0000 −1.41631 −0.708155 − 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) −13.0000 −0.707107
\(339\) −14.0000 −0.760376
\(340\) 1.00000 0.0542326
\(341\) −3.00000 −0.162459
\(342\) −8.00000 −0.432590
\(343\) −15.0000 −0.809924
\(344\) −1.00000 −0.0539164
\(345\) −3.00000 −0.161515
\(346\) 6.00000 0.322562
\(347\) 20.0000 1.07366 0.536828 − 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) −3.00000 −0.160817
\(349\) 6.00000 0.321173 0.160586 − 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 3.00000 0.160357
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −11.0000 −0.585471 −0.292735 − 0.956193i \(-0.594566\pi\)
−0.292735 + 0.956193i \(0.594566\pi\)
\(354\) 6.00000 0.318896
\(355\) −6.00000 −0.318447
\(356\) 0 0
\(357\) −3.00000 −0.158777
\(358\) 6.00000 0.317110
\(359\) 18.0000 0.950004 0.475002 − 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 1.00000 0.0527046
\(361\) 45.0000 2.36842
\(362\) −25.0000 −1.31397
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) −10.0000 −0.522708
\(367\) −22.0000 −1.14839 −0.574195 − 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 3.00000 0.156386
\(369\) 0 0
\(370\) 10.0000 0.519875
\(371\) 24.0000 1.24602
\(372\) −3.00000 −0.155543
\(373\) −6.00000 −0.310668 −0.155334 − 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) −1.00000 −0.0517088
\(375\) −1.00000 −0.0516398
\(376\) 4.00000 0.206284
\(377\) 0 0
\(378\) −3.00000 −0.154303
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −8.00000 −0.410391
\(381\) −10.0000 −0.512316
\(382\) 11.0000 0.562809
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −3.00000 −0.152894
\(386\) −14.0000 −0.712581
\(387\) −1.00000 −0.0508329
\(388\) −1.00000 −0.0507673
\(389\) −26.0000 −1.31825 −0.659126 − 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 2.00000 0.101015
\(393\) −12.0000 −0.605320
\(394\) 12.0000 0.604551
\(395\) 4.00000 0.201262
\(396\) −1.00000 −0.0502519
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −24.0000 −1.20301
\(399\) 24.0000 1.20150
\(400\) 1.00000 0.0500000
\(401\) −17.0000 −0.848939 −0.424470 − 0.905442i \(-0.639539\pi\)
−0.424470 + 0.905442i \(0.639539\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 1.00000 0.0496904
\(406\) 9.00000 0.446663
\(407\) −10.0000 −0.495682
\(408\) −1.00000 −0.0495074
\(409\) 4.00000 0.197787 0.0988936 − 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 0 0
\(411\) 11.0000 0.542590
\(412\) −11.0000 −0.541931
\(413\) −18.0000 −0.885722
\(414\) 3.00000 0.147442
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 5.00000 0.244851
\(418\) 8.00000 0.391293
\(419\) −15.0000 −0.732798 −0.366399 − 0.930458i \(-0.619409\pi\)
−0.366399 + 0.930458i \(0.619409\pi\)
\(420\) −3.00000 −0.146385
\(421\) −16.0000 −0.779792 −0.389896 − 0.920859i \(-0.627489\pi\)
−0.389896 + 0.920859i \(0.627489\pi\)
\(422\) 13.0000 0.632830
\(423\) 4.00000 0.194487
\(424\) 8.00000 0.388514
\(425\) 1.00000 0.0485071
\(426\) 6.00000 0.290701
\(427\) 30.0000 1.45180
\(428\) −5.00000 −0.241684
\(429\) 0 0
\(430\) −1.00000 −0.0482243
\(431\) −23.0000 −1.10787 −0.553936 − 0.832560i \(-0.686875\pi\)
−0.553936 + 0.832560i \(0.686875\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 16.0000 0.768911 0.384455 − 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 9.00000 0.432014
\(435\) −3.00000 −0.143839
\(436\) −4.00000 −0.191565
\(437\) −24.0000 −1.14808
\(438\) −6.00000 −0.286691
\(439\) −34.0000 −1.62273 −0.811366 − 0.584539i \(-0.801275\pi\)
−0.811366 + 0.584539i \(0.801275\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) −35.0000 −1.66290 −0.831450 − 0.555599i \(-0.812489\pi\)
−0.831450 + 0.555599i \(0.812489\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) −11.0000 −0.520865
\(447\) 20.0000 0.945968
\(448\) 3.00000 0.141737
\(449\) 21.0000 0.991051 0.495526 − 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 14.0000 0.658505
\(453\) −20.0000 −0.939682
\(454\) −13.0000 −0.610120
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) 22.0000 1.02912 0.514558 − 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 4.00000 0.186908
\(459\) −1.00000 −0.0466760
\(460\) 3.00000 0.139876
\(461\) 10.0000 0.465746 0.232873 − 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 3.00000 0.139573
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 3.00000 0.139272
\(465\) −3.00000 −0.139122
\(466\) 21.0000 0.972806
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) 4.00000 0.184506
\(471\) 10.0000 0.460776
\(472\) −6.00000 −0.276172
\(473\) 1.00000 0.0459800
\(474\) −4.00000 −0.183726
\(475\) −8.00000 −0.367065
\(476\) 3.00000 0.137505
\(477\) 8.00000 0.366295
\(478\) 14.0000 0.640345
\(479\) −9.00000 −0.411220 −0.205610 − 0.978634i \(-0.565918\pi\)
−0.205610 + 0.978634i \(0.565918\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 17.0000 0.774329
\(483\) −9.00000 −0.409514
\(484\) 1.00000 0.0454545
\(485\) −1.00000 −0.0454077
\(486\) −1.00000 −0.0453609
\(487\) 18.0000 0.815658 0.407829 − 0.913058i \(-0.366286\pi\)
0.407829 + 0.913058i \(0.366286\pi\)
\(488\) 10.0000 0.452679
\(489\) −9.00000 −0.406994
\(490\) 2.00000 0.0903508
\(491\) 8.00000 0.361035 0.180517 − 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) 3.00000 0.135113
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 3.00000 0.134704
\(497\) −18.0000 −0.807410
\(498\) −4.00000 −0.179244
\(499\) 16.0000 0.716258 0.358129 − 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 1.00000 0.0447214
\(501\) −2.00000 −0.0893534
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 − 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 3.00000 0.133631
\(505\) 6.00000 0.266996
\(506\) −3.00000 −0.133366
\(507\) 13.0000 0.577350
\(508\) 10.0000 0.443678
\(509\) −2.00000 −0.0886484 −0.0443242 − 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) −1.00000 −0.0442807
\(511\) 18.0000 0.796273
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) 27.0000 1.19092
\(515\) −11.0000 −0.484718
\(516\) 1.00000 0.0440225
\(517\) −4.00000 −0.175920
\(518\) 30.0000 1.31812
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 − 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 3.00000 0.131306
\(523\) −33.0000 −1.44299 −0.721495 − 0.692420i \(-0.756545\pi\)
−0.721495 + 0.692420i \(0.756545\pi\)
\(524\) 12.0000 0.524222
\(525\) −3.00000 −0.130931
\(526\) −15.0000 −0.654031
\(527\) 3.00000 0.130682
\(528\) 1.00000 0.0435194
\(529\) −14.0000 −0.608696
\(530\) 8.00000 0.347498
\(531\) −6.00000 −0.260378
\(532\) −24.0000 −1.04053
\(533\) 0 0
\(534\) 0 0
\(535\) −5.00000 −0.216169
\(536\) 4.00000 0.172774
\(537\) −6.00000 −0.258919
\(538\) 6.00000 0.258678
\(539\) −2.00000 −0.0861461
\(540\) −1.00000 −0.0430331
\(541\) −36.0000 −1.54776 −0.773880 − 0.633332i \(-0.781687\pi\)
−0.773880 + 0.633332i \(0.781687\pi\)
\(542\) −7.00000 −0.300676
\(543\) 25.0000 1.07285
\(544\) 1.00000 0.0428746
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −42.0000 −1.79579 −0.897895 − 0.440209i \(-0.854904\pi\)
−0.897895 + 0.440209i \(0.854904\pi\)
\(548\) −11.0000 −0.469897
\(549\) 10.0000 0.426790
\(550\) −1.00000 −0.0426401
\(551\) −24.0000 −1.02243
\(552\) −3.00000 −0.127688
\(553\) 12.0000 0.510292
\(554\) 10.0000 0.424859
\(555\) −10.0000 −0.424476
\(556\) −5.00000 −0.212047
\(557\) −27.0000 −1.14403 −0.572013 − 0.820244i \(-0.693837\pi\)
−0.572013 + 0.820244i \(0.693837\pi\)
\(558\) 3.00000 0.127000
\(559\) 0 0
\(560\) 3.00000 0.126773
\(561\) 1.00000 0.0422200
\(562\) 17.0000 0.717102
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) −4.00000 −0.168430
\(565\) 14.0000 0.588984
\(566\) −26.0000 −1.09286
\(567\) 3.00000 0.125988
\(568\) −6.00000 −0.251754
\(569\) 38.0000 1.59304 0.796521 − 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 8.00000 0.335083
\(571\) 20.0000 0.836974 0.418487 − 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) −11.0000 −0.459532
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 − 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 1.00000 0.0415945
\(579\) 14.0000 0.581820
\(580\) 3.00000 0.124568
\(581\) 12.0000 0.497844
\(582\) 1.00000 0.0414513
\(583\) −8.00000 −0.331326
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −5.00000 −0.206548
\(587\) −21.0000 −0.866763 −0.433381 − 0.901211i \(-0.642680\pi\)
−0.433381 + 0.901211i \(0.642680\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −24.0000 −0.988903
\(590\) −6.00000 −0.247016
\(591\) −12.0000 −0.493614
\(592\) 10.0000 0.410997
\(593\) 26.0000 1.06769 0.533846 − 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(594\) 1.00000 0.0410305
\(595\) 3.00000 0.122988
\(596\) −20.0000 −0.819232
\(597\) 24.0000 0.982255
\(598\) 0 0
\(599\) 21.0000 0.858037 0.429018 − 0.903296i \(-0.358860\pi\)
0.429018 + 0.903296i \(0.358860\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −3.00000 −0.122271
\(603\) 4.00000 0.162893
\(604\) 20.0000 0.813788
\(605\) 1.00000 0.0406558
\(606\) −6.00000 −0.243733
\(607\) −24.0000 −0.974130 −0.487065 − 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) −8.00000 −0.324443
\(609\) −9.00000 −0.364698
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) 1.00000 0.0404226
\(613\) 10.0000 0.403896 0.201948 − 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) −26.0000 −1.04672 −0.523360 − 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 11.0000 0.442485
\(619\) 2.00000 0.0803868 0.0401934 − 0.999192i \(-0.487203\pi\)
0.0401934 + 0.999192i \(0.487203\pi\)
\(620\) 3.00000 0.120483
\(621\) −3.00000 −0.120386
\(622\) 24.0000 0.962312
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 9.00000 0.359712
\(627\) −8.00000 −0.319489
\(628\) −10.0000 −0.399043
\(629\) 10.0000 0.398726
\(630\) 3.00000 0.119523
\(631\) −26.0000 −1.03504 −0.517522 − 0.855670i \(-0.673145\pi\)
−0.517522 + 0.855670i \(0.673145\pi\)
\(632\) 4.00000 0.159111
\(633\) −13.0000 −0.516704
\(634\) 1.00000 0.0397151
\(635\) 10.0000 0.396838
\(636\) −8.00000 −0.317221
\(637\) 0 0
\(638\) −3.00000 −0.118771
\(639\) −6.00000 −0.237356
\(640\) 1.00000 0.0395285
\(641\) 9.00000 0.355479 0.177739 − 0.984078i \(-0.443122\pi\)
0.177739 + 0.984078i \(0.443122\pi\)
\(642\) 5.00000 0.197334
\(643\) −41.0000 −1.61688 −0.808441 − 0.588577i \(-0.799688\pi\)
−0.808441 + 0.588577i \(0.799688\pi\)
\(644\) 9.00000 0.354650
\(645\) 1.00000 0.0393750
\(646\) −8.00000 −0.314756
\(647\) 30.0000 1.17942 0.589711 − 0.807614i \(-0.299242\pi\)
0.589711 + 0.807614i \(0.299242\pi\)
\(648\) 1.00000 0.0392837
\(649\) 6.00000 0.235521
\(650\) 0 0
\(651\) −9.00000 −0.352738
\(652\) 9.00000 0.352467
\(653\) −39.0000 −1.52619 −0.763094 − 0.646288i \(-0.776321\pi\)
−0.763094 + 0.646288i \(0.776321\pi\)
\(654\) 4.00000 0.156412
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 12.0000 0.467809
\(659\) −41.0000 −1.59713 −0.798567 − 0.601906i \(-0.794408\pi\)
−0.798567 + 0.601906i \(0.794408\pi\)
\(660\) 1.00000 0.0389249
\(661\) −32.0000 −1.24466 −0.622328 − 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 1.00000 0.0388661
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) −24.0000 −0.930680
\(666\) 10.0000 0.387492
\(667\) 9.00000 0.348481
\(668\) 2.00000 0.0773823
\(669\) 11.0000 0.425285
\(670\) 4.00000 0.154533
\(671\) −10.0000 −0.386046
\(672\) −3.00000 −0.115728
\(673\) 44.0000 1.69608 0.848038 − 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) −26.0000 −1.00148
\(675\) −1.00000 −0.0384900
\(676\) −13.0000 −0.500000
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −14.0000 −0.537667
\(679\) −3.00000 −0.115129
\(680\) 1.00000 0.0383482
\(681\) 13.0000 0.498161
\(682\) −3.00000 −0.114876
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −8.00000 −0.305888
\(685\) −11.0000 −0.420288
\(686\) −15.0000 −0.572703
\(687\) −4.00000 −0.152610
\(688\) −1.00000 −0.0381246
\(689\) 0 0
\(690\) −3.00000 −0.114208
\(691\) −8.00000 −0.304334 −0.152167 − 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 6.00000 0.228086
\(693\) −3.00000 −0.113961
\(694\) 20.0000 0.759190
\(695\) −5.00000 −0.189661
\(696\) −3.00000 −0.113715
\(697\) 0 0
\(698\) 6.00000 0.227103
\(699\) −21.0000 −0.794293
\(700\) 3.00000 0.113389
\(701\) 22.0000 0.830929 0.415464 − 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) −80.0000 −3.01726
\(704\) −1.00000 −0.0376889
\(705\) −4.00000 −0.150649
\(706\) −11.0000 −0.413990
\(707\) 18.0000 0.676960
\(708\) 6.00000 0.225494
\(709\) −6.00000 −0.225335 −0.112667 − 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) −6.00000 −0.225176
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) 9.00000 0.337053
\(714\) −3.00000 −0.112272
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) −14.0000 −0.522840
\(718\) 18.0000 0.671754
\(719\) 34.0000 1.26799 0.633993 − 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(720\) 1.00000 0.0372678
\(721\) −33.0000 −1.22898
\(722\) 45.0000 1.67473
\(723\) −17.0000 −0.632237
\(724\) −25.0000 −0.929118
\(725\) 3.00000 0.111417
\(726\) −1.00000 −0.0371135
\(727\) 5.00000 0.185440 0.0927199 − 0.995692i \(-0.470444\pi\)
0.0927199 + 0.995692i \(0.470444\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.00000 0.222070
\(731\) −1.00000 −0.0369863
\(732\) −10.0000 −0.369611
\(733\) −4.00000 −0.147743 −0.0738717 − 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) −22.0000 −0.812035
\(735\) −2.00000 −0.0737711
\(736\) 3.00000 0.110581
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) −6.00000 −0.220714 −0.110357 − 0.993892i \(-0.535199\pi\)
−0.110357 + 0.993892i \(0.535199\pi\)
\(740\) 10.0000 0.367607
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) −3.00000 −0.109985
\(745\) −20.0000 −0.732743
\(746\) −6.00000 −0.219676
\(747\) 4.00000 0.146352
\(748\) −1.00000 −0.0365636
\(749\) −15.0000 −0.548088
\(750\) −1.00000 −0.0365148
\(751\) −27.0000 −0.985244 −0.492622 − 0.870243i \(-0.663961\pi\)
−0.492622 + 0.870243i \(0.663961\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) 0 0
\(755\) 20.0000 0.727875
\(756\) −3.00000 −0.109109
\(757\) −19.0000 −0.690567 −0.345283 − 0.938498i \(-0.612217\pi\)
−0.345283 + 0.938498i \(0.612217\pi\)
\(758\) −16.0000 −0.581146
\(759\) 3.00000 0.108893
\(760\) −8.00000 −0.290191
\(761\) −17.0000 −0.616250 −0.308125 − 0.951346i \(-0.599701\pi\)
−0.308125 + 0.951346i \(0.599701\pi\)
\(762\) −10.0000 −0.362262
\(763\) −12.0000 −0.434429
\(764\) 11.0000 0.397966
\(765\) 1.00000 0.0361551
\(766\) 0 0
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 10.0000 0.360609 0.180305 − 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) −3.00000 −0.108112
\(771\) −27.0000 −0.972381
\(772\) −14.0000 −0.503871
\(773\) 8.00000 0.287740 0.143870 − 0.989597i \(-0.454045\pi\)
0.143870 + 0.989597i \(0.454045\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 3.00000 0.107763
\(776\) −1.00000 −0.0358979
\(777\) −30.0000 −1.07624
\(778\) −26.0000 −0.932145
\(779\) 0 0
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) 3.00000 0.107280
\(783\) −3.00000 −0.107211
\(784\) 2.00000 0.0714286
\(785\) −10.0000 −0.356915
\(786\) −12.0000 −0.428026
\(787\) −28.0000 −0.998092 −0.499046 − 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 12.0000 0.427482
\(789\) 15.0000 0.534014
\(790\) 4.00000 0.142314
\(791\) 42.0000 1.49335
\(792\) −1.00000 −0.0355335
\(793\) 0 0
\(794\) 14.0000 0.496841
\(795\) −8.00000 −0.283731
\(796\) −24.0000 −0.850657
\(797\) 24.0000 0.850124 0.425062 − 0.905164i \(-0.360252\pi\)
0.425062 + 0.905164i \(0.360252\pi\)
\(798\) 24.0000 0.849591
\(799\) 4.00000 0.141510
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −17.0000 −0.600291
\(803\) −6.00000 −0.211735
\(804\) −4.00000 −0.141069
\(805\) 9.00000 0.317208
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 6.00000 0.211079
\(809\) 26.0000 0.914111 0.457056 − 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 1.00000 0.0351364
\(811\) −16.0000 −0.561836 −0.280918 − 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 9.00000 0.315838
\(813\) 7.00000 0.245501
\(814\) −10.0000 −0.350500
\(815\) 9.00000 0.315256
\(816\) −1.00000 −0.0350070
\(817\) 8.00000 0.279885
\(818\) 4.00000 0.139857
\(819\) 0 0
\(820\) 0 0
\(821\) −15.0000 −0.523504 −0.261752 − 0.965135i \(-0.584300\pi\)
−0.261752 + 0.965135i \(0.584300\pi\)
\(822\) 11.0000 0.383669
\(823\) 22.0000 0.766872 0.383436 − 0.923567i \(-0.374741\pi\)
0.383436 + 0.923567i \(0.374741\pi\)
\(824\) −11.0000 −0.383203
\(825\) 1.00000 0.0348155
\(826\) −18.0000 −0.626300
\(827\) −3.00000 −0.104320 −0.0521601 − 0.998639i \(-0.516611\pi\)
−0.0521601 + 0.998639i \(0.516611\pi\)
\(828\) 3.00000 0.104257
\(829\) −26.0000 −0.903017 −0.451509 − 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 4.00000 0.138842
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 5.00000 0.173136
\(835\) 2.00000 0.0692129
\(836\) 8.00000 0.276686
\(837\) −3.00000 −0.103695
\(838\) −15.0000 −0.518166
\(839\) 34.0000 1.17381 0.586905 − 0.809656i \(-0.300346\pi\)
0.586905 + 0.809656i \(0.300346\pi\)
\(840\) −3.00000 −0.103510
\(841\) −20.0000 −0.689655
\(842\) −16.0000 −0.551396
\(843\) −17.0000 −0.585511
\(844\) 13.0000 0.447478
\(845\) −13.0000 −0.447214
\(846\) 4.00000 0.137523
\(847\) 3.00000 0.103081
\(848\) 8.00000 0.274721
\(849\) 26.0000 0.892318
\(850\) 1.00000 0.0342997
\(851\) 30.0000 1.02839
\(852\) 6.00000 0.205557
\(853\) −17.0000 −0.582069 −0.291034 − 0.956713i \(-0.593999\pi\)
−0.291034 + 0.956713i \(0.593999\pi\)
\(854\) 30.0000 1.02658
\(855\) −8.00000 −0.273594
\(856\) −5.00000 −0.170896
\(857\) −13.0000 −0.444072 −0.222036 − 0.975039i \(-0.571270\pi\)
−0.222036 + 0.975039i \(0.571270\pi\)
\(858\) 0 0
\(859\) −25.0000 −0.852989 −0.426494 − 0.904490i \(-0.640252\pi\)
−0.426494 + 0.904490i \(0.640252\pi\)
\(860\) −1.00000 −0.0340997
\(861\) 0 0
\(862\) −23.0000 −0.783383
\(863\) −52.0000 −1.77010 −0.885050 − 0.465495i \(-0.845876\pi\)
−0.885050 + 0.465495i \(0.845876\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) 16.0000 0.543702
\(867\) −1.00000 −0.0339618
\(868\) 9.00000 0.305480
\(869\) −4.00000 −0.135691
\(870\) −3.00000 −0.101710
\(871\) 0 0
\(872\) −4.00000 −0.135457
\(873\) −1.00000 −0.0338449
\(874\) −24.0000 −0.811812
\(875\) 3.00000 0.101419
\(876\) −6.00000 −0.202721
\(877\) −17.0000 −0.574049 −0.287025 − 0.957923i \(-0.592666\pi\)
−0.287025 + 0.957923i \(0.592666\pi\)
\(878\) −34.0000 −1.14744
\(879\) 5.00000 0.168646
\(880\) −1.00000 −0.0337100
\(881\) −57.0000 −1.92038 −0.960189 − 0.279350i \(-0.909881\pi\)
−0.960189 + 0.279350i \(0.909881\pi\)
\(882\) 2.00000 0.0673435
\(883\) 24.0000 0.807664 0.403832 − 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(884\) 0 0
\(885\) 6.00000 0.201688
\(886\) −35.0000 −1.17585
\(887\) −20.0000 −0.671534 −0.335767 − 0.941945i \(-0.608996\pi\)
−0.335767 + 0.941945i \(0.608996\pi\)
\(888\) −10.0000 −0.335578
\(889\) 30.0000 1.00617
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) −11.0000 −0.368307
\(893\) −32.0000 −1.07084
\(894\) 20.0000 0.668900
\(895\) 6.00000 0.200558
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) 21.0000 0.700779
\(899\) 9.00000 0.300167
\(900\) 1.00000 0.0333333
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) 3.00000 0.0998337
\(904\) 14.0000 0.465633
\(905\) −25.0000 −0.831028
\(906\) −20.0000 −0.664455
\(907\) −41.0000 −1.36138 −0.680691 − 0.732570i \(-0.738320\pi\)
−0.680691 + 0.732570i \(0.738320\pi\)
\(908\) −13.0000 −0.431420
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 52.0000 1.72284 0.861418 − 0.507896i \(-0.169577\pi\)
0.861418 + 0.507896i \(0.169577\pi\)
\(912\) 8.00000 0.264906
\(913\) −4.00000 −0.132381
\(914\) 22.0000 0.727695
\(915\) −10.0000 −0.330590
\(916\) 4.00000 0.132164
\(917\) 36.0000 1.18882
\(918\) −1.00000 −0.0330049
\(919\) 47.0000 1.55039 0.775193 − 0.631724i \(-0.217652\pi\)
0.775193 + 0.631724i \(0.217652\pi\)
\(920\) 3.00000 0.0989071
\(921\) −28.0000 −0.922631
\(922\) 10.0000 0.329332
\(923\) 0 0
\(924\) 3.00000 0.0986928
\(925\) 10.0000 0.328798
\(926\) 16.0000 0.525793
\(927\) −11.0000 −0.361287
\(928\) 3.00000 0.0984798
\(929\) −11.0000 −0.360898 −0.180449 − 0.983584i \(-0.557755\pi\)
−0.180449 + 0.983584i \(0.557755\pi\)
\(930\) −3.00000 −0.0983739
\(931\) −16.0000 −0.524379
\(932\) 21.0000 0.687878
\(933\) −24.0000 −0.785725
\(934\) −12.0000 −0.392652
\(935\) −1.00000 −0.0327035
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 − 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 12.0000 0.391814
\(939\) −9.00000 −0.293704
\(940\) 4.00000 0.130466
\(941\) −30.0000 −0.977972 −0.488986 − 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 10.0000 0.325818
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) −3.00000 −0.0975900
\(946\) 1.00000 0.0325128
\(947\) −18.0000 −0.584921 −0.292461 − 0.956278i \(-0.594474\pi\)
−0.292461 + 0.956278i \(0.594474\pi\)
\(948\) −4.00000 −0.129914
\(949\) 0 0
\(950\) −8.00000 −0.259554
\(951\) −1.00000 −0.0324272
\(952\) 3.00000 0.0972306
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 8.00000 0.259010
\(955\) 11.0000 0.355952
\(956\) 14.0000 0.452792
\(957\) 3.00000 0.0969762
\(958\) −9.00000 −0.290777
\(959\) −33.0000 −1.06563
\(960\) −1.00000 −0.0322749
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) −5.00000 −0.161123
\(964\) 17.0000 0.547533
\(965\) −14.0000 −0.450676
\(966\) −9.00000 −0.289570
\(967\) −34.0000 −1.09337 −0.546683 − 0.837340i \(-0.684110\pi\)
−0.546683 + 0.837340i \(0.684110\pi\)
\(968\) 1.00000 0.0321412
\(969\) 8.00000 0.256997
\(970\) −1.00000 −0.0321081
\(971\) −8.00000 −0.256732 −0.128366 − 0.991727i \(-0.540973\pi\)
−0.128366 + 0.991727i \(0.540973\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −15.0000 −0.480878
\(974\) 18.0000 0.576757
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 18.0000 0.575871 0.287936 − 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −9.00000 −0.287788
\(979\) 0 0
\(980\) 2.00000 0.0638877
\(981\) −4.00000 −0.127710
\(982\) 8.00000 0.255290
\(983\) −21.0000 −0.669796 −0.334898 − 0.942254i \(-0.608702\pi\)
−0.334898 + 0.942254i \(0.608702\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 3.00000 0.0955395
\(987\) −12.0000 −0.381964
\(988\) 0 0
\(989\) −3.00000 −0.0953945
\(990\) −1.00000 −0.0317821
\(991\) −7.00000 −0.222362 −0.111181 − 0.993800i \(-0.535463\pi\)
−0.111181 + 0.993800i \(0.535463\pi\)
\(992\) 3.00000 0.0952501
\(993\) −1.00000 −0.0317340
\(994\) −18.0000 −0.570925
\(995\) −24.0000 −0.760851
\(996\) −4.00000 −0.126745
\(997\) 45.0000 1.42516 0.712582 − 0.701589i \(-0.247526\pi\)
0.712582 + 0.701589i \(0.247526\pi\)
\(998\) 16.0000 0.506471
\(999\) −10.0000 −0.316386
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 5610.2.a.bd.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.bd.1.1 ✓ 1 1.1 even 1 trivial