Properties

Label 174.2.a.a.1.1
Level $174$
Weight $2$
Character 174.1
Self dual yes
Analytic conductor $1.389$
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] = [174,2,Mod(1,174)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(174, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("174.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 174 = 2 \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 174.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: \(1.38939699517\)
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 \(\chi\) \(=\) 174.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +6.00000 q^{11} -1.00000 q^{12} +3.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} +7.00000 q^{17} -1.00000 q^{18} +5.00000 q^{19} +3.00000 q^{20} +3.00000 q^{21} -6.00000 q^{22} -8.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} -1.00000 q^{27} -3.00000 q^{28} +1.00000 q^{29} +3.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -7.00000 q^{34} -9.00000 q^{35} +1.00000 q^{36} -3.00000 q^{37} -5.00000 q^{38} -3.00000 q^{40} -5.00000 q^{41} -3.00000 q^{42} +3.00000 q^{43} +6.00000 q^{44} +3.00000 q^{45} +8.00000 q^{46} +9.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -4.00000 q^{50} -7.00000 q^{51} -2.00000 q^{53} +1.00000 q^{54} +18.0000 q^{55} +3.00000 q^{56} -5.00000 q^{57} -1.00000 q^{58} -11.0000 q^{59} -3.00000 q^{60} -6.00000 q^{61} +8.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +6.00000 q^{66} +7.00000 q^{68} +8.00000 q^{69} +9.00000 q^{70} -1.00000 q^{72} -10.0000 q^{73} +3.00000 q^{74} -4.00000 q^{75} +5.00000 q^{76} -18.0000 q^{77} -2.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +5.00000 q^{82} +3.00000 q^{84} +21.0000 q^{85} -3.00000 q^{86} -1.00000 q^{87} -6.00000 q^{88} +10.0000 q^{89} -3.00000 q^{90} -8.00000 q^{92} +8.00000 q^{93} -9.00000 q^{94} +15.0000 q^{95} +1.00000 q^{96} -2.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 1.00000 0.408248
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 3.00000 0.801784
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 3.00000 0.670820
\(21\) 3.00000 0.654654
\(22\) −6.00000 −1.27920
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 1.00000 0.204124
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −3.00000 −0.566947
\(29\) 1.00000 0.185695
\(30\) 3.00000 0.547723
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) −6.00000 −1.04447
\(34\) −7.00000 −1.20049
\(35\) −9.00000 −1.52128
\(36\) 1.00000 0.166667
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) −5.00000 −0.811107
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) −3.00000 −0.462910
\(43\) 3.00000 0.457496 0.228748 0.973486i \(-0.426537\pi\)
0.228748 + 0.973486i \(0.426537\pi\)
\(44\) 6.00000 0.904534
\(45\) 3.00000 0.447214
\(46\) 8.00000 1.17954
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) −4.00000 −0.565685
\(51\) −7.00000 −0.980196
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 18.0000 2.42712
\(56\) 3.00000 0.400892
\(57\) −5.00000 −0.662266
\(58\) −1.00000 −0.131306
\(59\) −11.0000 −1.43208 −0.716039 0.698060i \(-0.754047\pi\)
−0.716039 + 0.698060i \(0.754047\pi\)
\(60\) −3.00000 −0.387298
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 8.00000 1.01600
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 7.00000 0.848875
\(69\) 8.00000 0.963087
\(70\) 9.00000 1.07571
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 3.00000 0.348743
\(75\) −4.00000 −0.461880
\(76\) 5.00000 0.573539
\(77\) −18.0000 −2.05129
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) 5.00000 0.552158
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 3.00000 0.327327
\(85\) 21.0000 2.27777
\(86\) −3.00000 −0.323498
\(87\) −1.00000 −0.107211
\(88\) −6.00000 −0.639602
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 8.00000 0.829561
\(94\) −9.00000 −0.928279
\(95\) 15.0000 1.53897
\(96\) 1.00000 0.102062
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −2.00000 −0.202031
\(99\) 6.00000 0.603023
\(100\) 4.00000 0.400000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 7.00000 0.693103
\(103\) −5.00000 −0.492665 −0.246332 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 0 0
\(105\) 9.00000 0.878310
\(106\) 2.00000 0.194257
\(107\) 9.00000 0.870063 0.435031 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) −18.0000 −1.71623
\(111\) 3.00000 0.284747
\(112\) −3.00000 −0.283473
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 5.00000 0.468293
\(115\) −24.0000 −2.23801
\(116\) 1.00000 0.0928477
\(117\) 0 0
\(118\) 11.0000 1.01263
\(119\) −21.0000 −1.92507
\(120\) 3.00000 0.273861
\(121\) 25.0000 2.27273
\(122\) 6.00000 0.543214
\(123\) 5.00000 0.450835
\(124\) −8.00000 −0.718421
\(125\) −3.00000 −0.268328
\(126\) 3.00000 0.267261
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −3.00000 −0.264135
\(130\) 0 0
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) −6.00000 −0.522233
\(133\) −15.0000 −1.30066
\(134\) 0 0
\(135\) −3.00000 −0.258199
\(136\) −7.00000 −0.600245
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −8.00000 −0.681005
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) −9.00000 −0.760639
\(141\) −9.00000 −0.757937
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 3.00000 0.249136
\(146\) 10.0000 0.827606
\(147\) −2.00000 −0.164957
\(148\) −3.00000 −0.246598
\(149\) 5.00000 0.409616 0.204808 0.978802i \(-0.434343\pi\)
0.204808 + 0.978802i \(0.434343\pi\)
\(150\) 4.00000 0.326599
\(151\) 1.00000 0.0813788 0.0406894 0.999172i \(-0.487045\pi\)
0.0406894 + 0.999172i \(0.487045\pi\)
\(152\) −5.00000 −0.405554
\(153\) 7.00000 0.565916
\(154\) 18.0000 1.45048
\(155\) −24.0000 −1.92773
\(156\) 0 0
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) 2.00000 0.159111
\(159\) 2.00000 0.158610
\(160\) −3.00000 −0.237171
\(161\) 24.0000 1.89146
\(162\) −1.00000 −0.0785674
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) −5.00000 −0.390434
\(165\) −18.0000 −1.40130
\(166\) 0 0
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) −3.00000 −0.231455
\(169\) −13.0000 −1.00000
\(170\) −21.0000 −1.61063
\(171\) 5.00000 0.382360
\(172\) 3.00000 0.228748
\(173\) −3.00000 −0.228086 −0.114043 0.993476i \(-0.536380\pi\)
−0.114043 + 0.993476i \(0.536380\pi\)
\(174\) 1.00000 0.0758098
\(175\) −12.0000 −0.907115
\(176\) 6.00000 0.452267
\(177\) 11.0000 0.826811
\(178\) −10.0000 −0.749532
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 3.00000 0.223607
\(181\) −24.0000 −1.78391 −0.891953 0.452128i \(-0.850665\pi\)
−0.891953 + 0.452128i \(0.850665\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 8.00000 0.589768
\(185\) −9.00000 −0.661693
\(186\) −8.00000 −0.586588
\(187\) 42.0000 3.07134
\(188\) 9.00000 0.656392
\(189\) 3.00000 0.218218
\(190\) −15.0000 −1.08821
\(191\) 23.0000 1.66422 0.832111 0.554609i \(-0.187132\pi\)
0.832111 + 0.554609i \(0.187132\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −20.0000 −1.43963 −0.719816 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 7.00000 0.498729 0.249365 0.968410i \(-0.419778\pi\)
0.249365 + 0.968410i \(0.419778\pi\)
\(198\) −6.00000 −0.426401
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) −3.00000 −0.210559
\(204\) −7.00000 −0.490098
\(205\) −15.0000 −1.04765
\(206\) 5.00000 0.348367
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) 30.0000 2.07514
\(210\) −9.00000 −0.621059
\(211\) 13.0000 0.894957 0.447478 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) −9.00000 −0.615227
\(215\) 9.00000 0.613795
\(216\) 1.00000 0.0680414
\(217\) 24.0000 1.62923
\(218\) 4.00000 0.270914
\(219\) 10.0000 0.675737
\(220\) 18.0000 1.21356
\(221\) 0 0
\(222\) −3.00000 −0.201347
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 3.00000 0.200446
\(225\) 4.00000 0.266667
\(226\) −1.00000 −0.0665190
\(227\) 5.00000 0.331862 0.165931 0.986137i \(-0.446937\pi\)
0.165931 + 0.986137i \(0.446937\pi\)
\(228\) −5.00000 −0.331133
\(229\) −5.00000 −0.330409 −0.165205 0.986259i \(-0.552828\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 24.0000 1.58251
\(231\) 18.0000 1.18431
\(232\) −1.00000 −0.0656532
\(233\) −8.00000 −0.524097 −0.262049 0.965055i \(-0.584398\pi\)
−0.262049 + 0.965055i \(0.584398\pi\)
\(234\) 0 0
\(235\) 27.0000 1.76129
\(236\) −11.0000 −0.716039
\(237\) 2.00000 0.129914
\(238\) 21.0000 1.36123
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) −3.00000 −0.193649
\(241\) −7.00000 −0.450910 −0.225455 0.974254i \(-0.572387\pi\)
−0.225455 + 0.974254i \(0.572387\pi\)
\(242\) −25.0000 −1.60706
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) 6.00000 0.383326
\(246\) −5.00000 −0.318788
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) 3.00000 0.189737
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) −3.00000 −0.188982
\(253\) −48.0000 −3.01773
\(254\) −6.00000 −0.376473
\(255\) −21.0000 −1.31507
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 3.00000 0.186772
\(259\) 9.00000 0.559233
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 20.0000 1.23560
\(263\) −7.00000 −0.431638 −0.215819 0.976433i \(-0.569242\pi\)
−0.215819 + 0.976433i \(0.569242\pi\)
\(264\) 6.00000 0.369274
\(265\) −6.00000 −0.368577
\(266\) 15.0000 0.919709
\(267\) −10.0000 −0.611990
\(268\) 0 0
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 3.00000 0.182574
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 7.00000 0.424437
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 24.0000 1.44725
\(276\) 8.00000 0.481543
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −14.0000 −0.839664
\(279\) −8.00000 −0.478947
\(280\) 9.00000 0.537853
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 9.00000 0.535942
\(283\) −22.0000 −1.30776 −0.653882 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) 0 0
\(285\) −15.0000 −0.888523
\(286\) 0 0
\(287\) 15.0000 0.885422
\(288\) −1.00000 −0.0589256
\(289\) 32.0000 1.88235
\(290\) −3.00000 −0.176166
\(291\) 0 0
\(292\) −10.0000 −0.585206
\(293\) 22.0000 1.28525 0.642627 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 2.00000 0.116642
\(295\) −33.0000 −1.92133
\(296\) 3.00000 0.174371
\(297\) −6.00000 −0.348155
\(298\) −5.00000 −0.289642
\(299\) 0 0
\(300\) −4.00000 −0.230940
\(301\) −9.00000 −0.518751
\(302\) −1.00000 −0.0575435
\(303\) −6.00000 −0.344691
\(304\) 5.00000 0.286770
\(305\) −18.0000 −1.03068
\(306\) −7.00000 −0.400163
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) −18.0000 −1.02565
\(309\) 5.00000 0.284440
\(310\) 24.0000 1.36311
\(311\) 1.00000 0.0567048 0.0283524 0.999598i \(-0.490974\pi\)
0.0283524 + 0.999598i \(0.490974\pi\)
\(312\) 0 0
\(313\) −13.0000 −0.734803 −0.367402 0.930062i \(-0.619753\pi\)
−0.367402 + 0.930062i \(0.619753\pi\)
\(314\) 7.00000 0.395033
\(315\) −9.00000 −0.507093
\(316\) −2.00000 −0.112509
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) −2.00000 −0.112154
\(319\) 6.00000 0.335936
\(320\) 3.00000 0.167705
\(321\) −9.00000 −0.502331
\(322\) −24.0000 −1.33747
\(323\) 35.0000 1.94745
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 1.00000 0.0553849
\(327\) 4.00000 0.221201
\(328\) 5.00000 0.276079
\(329\) −27.0000 −1.48856
\(330\) 18.0000 0.990867
\(331\) 27.0000 1.48405 0.742027 0.670370i \(-0.233865\pi\)
0.742027 + 0.670370i \(0.233865\pi\)
\(332\) 0 0
\(333\) −3.00000 −0.164399
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) 3.00000 0.163663
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 13.0000 0.707107
\(339\) −1.00000 −0.0543125
\(340\) 21.0000 1.13888
\(341\) −48.0000 −2.59935
\(342\) −5.00000 −0.270369
\(343\) 15.0000 0.809924
\(344\) −3.00000 −0.161749
\(345\) 24.0000 1.29212
\(346\) 3.00000 0.161281
\(347\) 27.0000 1.44944 0.724718 0.689046i \(-0.241970\pi\)
0.724718 + 0.689046i \(0.241970\pi\)
\(348\) −1.00000 −0.0536056
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 12.0000 0.641427
\(351\) 0 0
\(352\) −6.00000 −0.319801
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) −11.0000 −0.584643
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 21.0000 1.11144
\(358\) 16.0000 0.845626
\(359\) 21.0000 1.10834 0.554169 0.832404i \(-0.313036\pi\)
0.554169 + 0.832404i \(0.313036\pi\)
\(360\) −3.00000 −0.158114
\(361\) 6.00000 0.315789
\(362\) 24.0000 1.26141
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) −30.0000 −1.57027
\(366\) −6.00000 −0.313625
\(367\) −26.0000 −1.35719 −0.678594 0.734513i \(-0.737411\pi\)
−0.678594 + 0.734513i \(0.737411\pi\)
\(368\) −8.00000 −0.417029
\(369\) −5.00000 −0.260290
\(370\) 9.00000 0.467888
\(371\) 6.00000 0.311504
\(372\) 8.00000 0.414781
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) −42.0000 −2.17177
\(375\) 3.00000 0.154919
\(376\) −9.00000 −0.464140
\(377\) 0 0
\(378\) −3.00000 −0.154303
\(379\) 36.0000 1.84920 0.924598 0.380945i \(-0.124401\pi\)
0.924598 + 0.380945i \(0.124401\pi\)
\(380\) 15.0000 0.769484
\(381\) −6.00000 −0.307389
\(382\) −23.0000 −1.17678
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 1.00000 0.0510310
\(385\) −54.0000 −2.75209
\(386\) 20.0000 1.01797
\(387\) 3.00000 0.152499
\(388\) 0 0
\(389\) −32.0000 −1.62246 −0.811232 0.584724i \(-0.801203\pi\)
−0.811232 + 0.584724i \(0.801203\pi\)
\(390\) 0 0
\(391\) −56.0000 −2.83204
\(392\) −2.00000 −0.101015
\(393\) 20.0000 1.00887
\(394\) −7.00000 −0.352655
\(395\) −6.00000 −0.301893
\(396\) 6.00000 0.301511
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0 0
\(399\) 15.0000 0.750939
\(400\) 4.00000 0.200000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 3.00000 0.149071
\(406\) 3.00000 0.148888
\(407\) −18.0000 −0.892227
\(408\) 7.00000 0.346552
\(409\) −32.0000 −1.58230 −0.791149 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 15.0000 0.740797
\(411\) 6.00000 0.295958
\(412\) −5.00000 −0.246332
\(413\) 33.0000 1.62382
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) 0 0
\(417\) −14.0000 −0.685583
\(418\) −30.0000 −1.46735
\(419\) −29.0000 −1.41674 −0.708371 0.705840i \(-0.750570\pi\)
−0.708371 + 0.705840i \(0.750570\pi\)
\(420\) 9.00000 0.439155
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) −13.0000 −0.632830
\(423\) 9.00000 0.437595
\(424\) 2.00000 0.0971286
\(425\) 28.0000 1.35820
\(426\) 0 0
\(427\) 18.0000 0.871081
\(428\) 9.00000 0.435031
\(429\) 0 0
\(430\) −9.00000 −0.434019
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 22.0000 1.05725 0.528626 0.848855i \(-0.322707\pi\)
0.528626 + 0.848855i \(0.322707\pi\)
\(434\) −24.0000 −1.15204
\(435\) −3.00000 −0.143839
\(436\) −4.00000 −0.191565
\(437\) −40.0000 −1.91346
\(438\) −10.0000 −0.477818
\(439\) −1.00000 −0.0477274 −0.0238637 0.999715i \(-0.507597\pi\)
−0.0238637 + 0.999715i \(0.507597\pi\)
\(440\) −18.0000 −0.858116
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 3.00000 0.142374
\(445\) 30.0000 1.42214
\(446\) 8.00000 0.378811
\(447\) −5.00000 −0.236492
\(448\) −3.00000 −0.141737
\(449\) 3.00000 0.141579 0.0707894 0.997491i \(-0.477448\pi\)
0.0707894 + 0.997491i \(0.477448\pi\)
\(450\) −4.00000 −0.188562
\(451\) −30.0000 −1.41264
\(452\) 1.00000 0.0470360
\(453\) −1.00000 −0.0469841
\(454\) −5.00000 −0.234662
\(455\) 0 0
\(456\) 5.00000 0.234146
\(457\) 3.00000 0.140334 0.0701670 0.997535i \(-0.477647\pi\)
0.0701670 + 0.997535i \(0.477647\pi\)
\(458\) 5.00000 0.233635
\(459\) −7.00000 −0.326732
\(460\) −24.0000 −1.11901
\(461\) 40.0000 1.86299 0.931493 0.363760i \(-0.118507\pi\)
0.931493 + 0.363760i \(0.118507\pi\)
\(462\) −18.0000 −0.837436
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 1.00000 0.0464238
\(465\) 24.0000 1.11297
\(466\) 8.00000 0.370593
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −27.0000 −1.24542
\(471\) 7.00000 0.322543
\(472\) 11.0000 0.506316
\(473\) 18.0000 0.827641
\(474\) −2.00000 −0.0918630
\(475\) 20.0000 0.917663
\(476\) −21.0000 −0.962533
\(477\) −2.00000 −0.0915737
\(478\) 6.00000 0.274434
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 3.00000 0.136931
\(481\) 0 0
\(482\) 7.00000 0.318841
\(483\) −24.0000 −1.09204
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 23.0000 1.04223 0.521115 0.853487i \(-0.325516\pi\)
0.521115 + 0.853487i \(0.325516\pi\)
\(488\) 6.00000 0.271607
\(489\) 1.00000 0.0452216
\(490\) −6.00000 −0.271052
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 5.00000 0.225417
\(493\) 7.00000 0.315264
\(494\) 0 0
\(495\) 18.0000 0.809040
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) 22.0000 0.984855 0.492428 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) −3.00000 −0.134164
\(501\) −18.0000 −0.804181
\(502\) 28.0000 1.24970
\(503\) 19.0000 0.847168 0.423584 0.905857i \(-0.360772\pi\)
0.423584 + 0.905857i \(0.360772\pi\)
\(504\) 3.00000 0.133631
\(505\) 18.0000 0.800989
\(506\) 48.0000 2.13386
\(507\) 13.0000 0.577350
\(508\) 6.00000 0.266207
\(509\) −11.0000 −0.487566 −0.243783 0.969830i \(-0.578389\pi\)
−0.243783 + 0.969830i \(0.578389\pi\)
\(510\) 21.0000 0.929896
\(511\) 30.0000 1.32712
\(512\) −1.00000 −0.0441942
\(513\) −5.00000 −0.220755
\(514\) −12.0000 −0.529297
\(515\) −15.0000 −0.660979
\(516\) −3.00000 −0.132068
\(517\) 54.0000 2.37492
\(518\) −9.00000 −0.395437
\(519\) 3.00000 0.131685
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −1.00000 −0.0437688
\(523\) −6.00000 −0.262362 −0.131181 0.991358i \(-0.541877\pi\)
−0.131181 + 0.991358i \(0.541877\pi\)
\(524\) −20.0000 −0.873704
\(525\) 12.0000 0.523723
\(526\) 7.00000 0.305215
\(527\) −56.0000 −2.43940
\(528\) −6.00000 −0.261116
\(529\) 41.0000 1.78261
\(530\) 6.00000 0.260623
\(531\) −11.0000 −0.477359
\(532\) −15.0000 −0.650332
\(533\) 0 0
\(534\) 10.0000 0.432742
\(535\) 27.0000 1.16731
\(536\) 0 0
\(537\) 16.0000 0.690451
\(538\) −14.0000 −0.603583
\(539\) 12.0000 0.516877
\(540\) −3.00000 −0.129099
\(541\) 45.0000 1.93470 0.967351 0.253442i \(-0.0815627\pi\)
0.967351 + 0.253442i \(0.0815627\pi\)
\(542\) −20.0000 −0.859074
\(543\) 24.0000 1.02994
\(544\) −7.00000 −0.300123
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) 26.0000 1.11168 0.555840 0.831289i \(-0.312397\pi\)
0.555840 + 0.831289i \(0.312397\pi\)
\(548\) −6.00000 −0.256307
\(549\) −6.00000 −0.256074
\(550\) −24.0000 −1.02336
\(551\) 5.00000 0.213007
\(552\) −8.00000 −0.340503
\(553\) 6.00000 0.255146
\(554\) −2.00000 −0.0849719
\(555\) 9.00000 0.382029
\(556\) 14.0000 0.593732
\(557\) −29.0000 −1.22877 −0.614385 0.789007i \(-0.710596\pi\)
−0.614385 + 0.789007i \(0.710596\pi\)
\(558\) 8.00000 0.338667
\(559\) 0 0
\(560\) −9.00000 −0.380319
\(561\) −42.0000 −1.77324
\(562\) −18.0000 −0.759284
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) −9.00000 −0.378968
\(565\) 3.00000 0.126211
\(566\) 22.0000 0.924729
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) −27.0000 −1.13190 −0.565949 0.824440i \(-0.691490\pi\)
−0.565949 + 0.824440i \(0.691490\pi\)
\(570\) 15.0000 0.628281
\(571\) −10.0000 −0.418487 −0.209243 0.977864i \(-0.567100\pi\)
−0.209243 + 0.977864i \(0.567100\pi\)
\(572\) 0 0
\(573\) −23.0000 −0.960839
\(574\) −15.0000 −0.626088
\(575\) −32.0000 −1.33449
\(576\) 1.00000 0.0416667
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) −32.0000 −1.33102
\(579\) 20.0000 0.831172
\(580\) 3.00000 0.124568
\(581\) 0 0
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) −22.0000 −0.908812
\(587\) 3.00000 0.123823 0.0619116 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −40.0000 −1.64817
\(590\) 33.0000 1.35859
\(591\) −7.00000 −0.287942
\(592\) −3.00000 −0.123299
\(593\) −20.0000 −0.821302 −0.410651 0.911793i \(-0.634698\pi\)
−0.410651 + 0.911793i \(0.634698\pi\)
\(594\) 6.00000 0.246183
\(595\) −63.0000 −2.58275
\(596\) 5.00000 0.204808
\(597\) 0 0
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 4.00000 0.163299
\(601\) 44.0000 1.79480 0.897399 0.441221i \(-0.145454\pi\)
0.897399 + 0.441221i \(0.145454\pi\)
\(602\) 9.00000 0.366813
\(603\) 0 0
\(604\) 1.00000 0.0406894
\(605\) 75.0000 3.04918
\(606\) 6.00000 0.243733
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −5.00000 −0.202777
\(609\) 3.00000 0.121566
\(610\) 18.0000 0.728799
\(611\) 0 0
\(612\) 7.00000 0.282958
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 20.0000 0.807134
\(615\) 15.0000 0.604858
\(616\) 18.0000 0.725241
\(617\) 37.0000 1.48956 0.744782 0.667308i \(-0.232553\pi\)
0.744782 + 0.667308i \(0.232553\pi\)
\(618\) −5.00000 −0.201129
\(619\) −5.00000 −0.200967 −0.100483 0.994939i \(-0.532039\pi\)
−0.100483 + 0.994939i \(0.532039\pi\)
\(620\) −24.0000 −0.963863
\(621\) 8.00000 0.321029
\(622\) −1.00000 −0.0400963
\(623\) −30.0000 −1.20192
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 13.0000 0.519584
\(627\) −30.0000 −1.19808
\(628\) −7.00000 −0.279330
\(629\) −21.0000 −0.837325
\(630\) 9.00000 0.358569
\(631\) −5.00000 −0.199047 −0.0995234 0.995035i \(-0.531732\pi\)
−0.0995234 + 0.995035i \(0.531732\pi\)
\(632\) 2.00000 0.0795557
\(633\) −13.0000 −0.516704
\(634\) 12.0000 0.476581
\(635\) 18.0000 0.714308
\(636\) 2.00000 0.0793052
\(637\) 0 0
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) 25.0000 0.987441 0.493720 0.869621i \(-0.335637\pi\)
0.493720 + 0.869621i \(0.335637\pi\)
\(642\) 9.00000 0.355202
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 24.0000 0.945732
\(645\) −9.00000 −0.354375
\(646\) −35.0000 −1.37706
\(647\) −14.0000 −0.550397 −0.275198 0.961387i \(-0.588744\pi\)
−0.275198 + 0.961387i \(0.588744\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −66.0000 −2.59073
\(650\) 0 0
\(651\) −24.0000 −0.940634
\(652\) −1.00000 −0.0391630
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) −4.00000 −0.156412
\(655\) −60.0000 −2.34439
\(656\) −5.00000 −0.195217
\(657\) −10.0000 −0.390137
\(658\) 27.0000 1.05257
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) −18.0000 −0.700649
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) −27.0000 −1.04938
\(663\) 0 0
\(664\) 0 0
\(665\) −45.0000 −1.74503
\(666\) 3.00000 0.116248
\(667\) −8.00000 −0.309761
\(668\) 18.0000 0.696441
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) −36.0000 −1.38976
\(672\) −3.00000 −0.115728
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) −22.0000 −0.847408
\(675\) −4.00000 −0.153960
\(676\) −13.0000 −0.500000
\(677\) −34.0000 −1.30673 −0.653363 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) 1.00000 0.0384048
\(679\) 0 0
\(680\) −21.0000 −0.805313
\(681\) −5.00000 −0.191600
\(682\) 48.0000 1.83801
\(683\) 1.00000 0.0382639 0.0191320 0.999817i \(-0.493910\pi\)
0.0191320 + 0.999817i \(0.493910\pi\)
\(684\) 5.00000 0.191180
\(685\) −18.0000 −0.687745
\(686\) −15.0000 −0.572703
\(687\) 5.00000 0.190762
\(688\) 3.00000 0.114374
\(689\) 0 0
\(690\) −24.0000 −0.913664
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) −3.00000 −0.114043
\(693\) −18.0000 −0.683763
\(694\) −27.0000 −1.02491
\(695\) 42.0000 1.59315
\(696\) 1.00000 0.0379049
\(697\) −35.0000 −1.32572
\(698\) 22.0000 0.832712
\(699\) 8.00000 0.302588
\(700\) −12.0000 −0.453557
\(701\) 27.0000 1.01978 0.509888 0.860241i \(-0.329687\pi\)
0.509888 + 0.860241i \(0.329687\pi\)
\(702\) 0 0
\(703\) −15.0000 −0.565736
\(704\) 6.00000 0.226134
\(705\) −27.0000 −1.01688
\(706\) 2.00000 0.0752710
\(707\) −18.0000 −0.676960
\(708\) 11.0000 0.413405
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) −2.00000 −0.0750059
\(712\) −10.0000 −0.374766
\(713\) 64.0000 2.39682
\(714\) −21.0000 −0.785905
\(715\) 0 0
\(716\) −16.0000 −0.597948
\(717\) 6.00000 0.224074
\(718\) −21.0000 −0.783713
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 3.00000 0.111803
\(721\) 15.0000 0.558629
\(722\) −6.00000 −0.223297
\(723\) 7.00000 0.260333
\(724\) −24.0000 −0.891953
\(725\) 4.00000 0.148556
\(726\) 25.0000 0.927837
\(727\) 10.0000 0.370879 0.185440 0.982656i \(-0.440629\pi\)
0.185440 + 0.982656i \(0.440629\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 30.0000 1.11035
\(731\) 21.0000 0.776713
\(732\) 6.00000 0.221766
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 26.0000 0.959678
\(735\) −6.00000 −0.221313
\(736\) 8.00000 0.294884
\(737\) 0 0
\(738\) 5.00000 0.184053
\(739\) −32.0000 −1.17714 −0.588570 0.808447i \(-0.700309\pi\)
−0.588570 + 0.808447i \(0.700309\pi\)
\(740\) −9.00000 −0.330847
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) 47.0000 1.72426 0.862131 0.506685i \(-0.169129\pi\)
0.862131 + 0.506685i \(0.169129\pi\)
\(744\) −8.00000 −0.293294
\(745\) 15.0000 0.549557
\(746\) −4.00000 −0.146450
\(747\) 0 0
\(748\) 42.0000 1.53567
\(749\) −27.0000 −0.986559
\(750\) −3.00000 −0.109545
\(751\) −46.0000 −1.67856 −0.839282 0.543696i \(-0.817024\pi\)
−0.839282 + 0.543696i \(0.817024\pi\)
\(752\) 9.00000 0.328196
\(753\) 28.0000 1.02038
\(754\) 0 0
\(755\) 3.00000 0.109181
\(756\) 3.00000 0.109109
\(757\) −13.0000 −0.472493 −0.236247 0.971693i \(-0.575917\pi\)
−0.236247 + 0.971693i \(0.575917\pi\)
\(758\) −36.0000 −1.30758
\(759\) 48.0000 1.74229
\(760\) −15.0000 −0.544107
\(761\) −4.00000 −0.145000 −0.0724999 0.997368i \(-0.523098\pi\)
−0.0724999 + 0.997368i \(0.523098\pi\)
\(762\) 6.00000 0.217357
\(763\) 12.0000 0.434429
\(764\) 23.0000 0.832111
\(765\) 21.0000 0.759257
\(766\) 14.0000 0.505841
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 54.0000 1.94602
\(771\) −12.0000 −0.432169
\(772\) −20.0000 −0.719816
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) −3.00000 −0.107833
\(775\) −32.0000 −1.14947
\(776\) 0 0
\(777\) −9.00000 −0.322873
\(778\) 32.0000 1.14726
\(779\) −25.0000 −0.895718
\(780\) 0 0
\(781\) 0 0
\(782\) 56.0000 2.00256
\(783\) −1.00000 −0.0357371
\(784\) 2.00000 0.0714286
\(785\) −21.0000 −0.749522
\(786\) −20.0000 −0.713376
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) 7.00000 0.249365
\(789\) 7.00000 0.249207
\(790\) 6.00000 0.213470
\(791\) −3.00000 −0.106668
\(792\) −6.00000 −0.213201
\(793\) 0 0
\(794\) −22.0000 −0.780751
\(795\) 6.00000 0.212798
\(796\) 0 0
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) −15.0000 −0.530994
\(799\) 63.0000 2.22878
\(800\) −4.00000 −0.141421
\(801\) 10.0000 0.353333
\(802\) 18.0000 0.635602
\(803\) −60.0000 −2.11735
\(804\) 0 0
\(805\) 72.0000 2.53767
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) −6.00000 −0.211079
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) −3.00000 −0.105409
\(811\) −32.0000 −1.12367 −0.561836 0.827249i \(-0.689905\pi\)
−0.561836 + 0.827249i \(0.689905\pi\)
\(812\) −3.00000 −0.105279
\(813\) −20.0000 −0.701431
\(814\) 18.0000 0.630900
\(815\) −3.00000 −0.105085
\(816\) −7.00000 −0.245049
\(817\) 15.0000 0.524784
\(818\) 32.0000 1.11885
\(819\) 0 0
\(820\) −15.0000 −0.523823
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) −6.00000 −0.209274
\(823\) −10.0000 −0.348578 −0.174289 0.984695i \(-0.555763\pi\)
−0.174289 + 0.984695i \(0.555763\pi\)
\(824\) 5.00000 0.174183
\(825\) −24.0000 −0.835573
\(826\) −33.0000 −1.14822
\(827\) 8.00000 0.278187 0.139094 0.990279i \(-0.455581\pi\)
0.139094 + 0.990279i \(0.455581\pi\)
\(828\) −8.00000 −0.278019
\(829\) 37.0000 1.28506 0.642532 0.766259i \(-0.277884\pi\)
0.642532 + 0.766259i \(0.277884\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) 14.0000 0.485071
\(834\) 14.0000 0.484780
\(835\) 54.0000 1.86875
\(836\) 30.0000 1.03757
\(837\) 8.00000 0.276520
\(838\) 29.0000 1.00179
\(839\) −15.0000 −0.517858 −0.258929 0.965896i \(-0.583369\pi\)
−0.258929 + 0.965896i \(0.583369\pi\)
\(840\) −9.00000 −0.310530
\(841\) 1.00000 0.0344828
\(842\) 6.00000 0.206774
\(843\) −18.0000 −0.619953
\(844\) 13.0000 0.447478
\(845\) −39.0000 −1.34164
\(846\) −9.00000 −0.309426
\(847\) −75.0000 −2.57703
\(848\) −2.00000 −0.0686803
\(849\) 22.0000 0.755038
\(850\) −28.0000 −0.960392
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) −17.0000 −0.582069 −0.291034 0.956713i \(-0.593999\pi\)
−0.291034 + 0.956713i \(0.593999\pi\)
\(854\) −18.0000 −0.615947
\(855\) 15.0000 0.512989
\(856\) −9.00000 −0.307614
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 7.00000 0.238837 0.119418 0.992844i \(-0.461897\pi\)
0.119418 + 0.992844i \(0.461897\pi\)
\(860\) 9.00000 0.306897
\(861\) −15.0000 −0.511199
\(862\) 12.0000 0.408722
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 1.00000 0.0340207
\(865\) −9.00000 −0.306009
\(866\) −22.0000 −0.747590
\(867\) −32.0000 −1.08678
\(868\) 24.0000 0.814613
\(869\) −12.0000 −0.407072
\(870\) 3.00000 0.101710
\(871\) 0 0
\(872\) 4.00000 0.135457
\(873\) 0 0
\(874\) 40.0000 1.35302
\(875\) 9.00000 0.304256
\(876\) 10.0000 0.337869
\(877\) −24.0000 −0.810422 −0.405211 0.914223i \(-0.632802\pi\)
−0.405211 + 0.914223i \(0.632802\pi\)
\(878\) 1.00000 0.0337484
\(879\) −22.0000 −0.742042
\(880\) 18.0000 0.606780
\(881\) 54.0000 1.81931 0.909653 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 0 0
\(885\) 33.0000 1.10928
\(886\) −16.0000 −0.537531
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) −3.00000 −0.100673
\(889\) −18.0000 −0.603701
\(890\) −30.0000 −1.00560
\(891\) 6.00000 0.201008
\(892\) −8.00000 −0.267860
\(893\) 45.0000 1.50587
\(894\) 5.00000 0.167225
\(895\) −48.0000 −1.60446
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) −3.00000 −0.100111
\(899\) −8.00000 −0.266815
\(900\) 4.00000 0.133333
\(901\) −14.0000 −0.466408
\(902\) 30.0000 0.998891
\(903\) 9.00000 0.299501
\(904\) −1.00000 −0.0332595
\(905\) −72.0000 −2.39336
\(906\) 1.00000 0.0332228
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 5.00000 0.165931
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −5.00000 −0.165567
\(913\) 0 0
\(914\) −3.00000 −0.0992312
\(915\) 18.0000 0.595062
\(916\) −5.00000 −0.165205
\(917\) 60.0000 1.98137
\(918\) 7.00000 0.231034
\(919\) −3.00000 −0.0989609 −0.0494804 0.998775i \(-0.515757\pi\)
−0.0494804 + 0.998775i \(0.515757\pi\)
\(920\) 24.0000 0.791257
\(921\) 20.0000 0.659022
\(922\) −40.0000 −1.31733
\(923\) 0 0
\(924\) 18.0000 0.592157
\(925\) −12.0000 −0.394558
\(926\) −8.00000 −0.262896
\(927\) −5.00000 −0.164222
\(928\) −1.00000 −0.0328266
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) −24.0000 −0.786991
\(931\) 10.0000 0.327737
\(932\) −8.00000 −0.262049
\(933\) −1.00000 −0.0327385
\(934\) 0 0
\(935\) 126.000 4.12064
\(936\) 0 0
\(937\) −49.0000 −1.60076 −0.800380 0.599493i \(-0.795369\pi\)
−0.800380 + 0.599493i \(0.795369\pi\)
\(938\) 0 0
\(939\) 13.0000 0.424239
\(940\) 27.0000 0.880643
\(941\) 2.00000 0.0651981 0.0325991 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) −7.00000 −0.228072
\(943\) 40.0000 1.30258
\(944\) −11.0000 −0.358020
\(945\) 9.00000 0.292770
\(946\) −18.0000 −0.585230
\(947\) 32.0000 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 2.00000 0.0649570
\(949\) 0 0
\(950\) −20.0000 −0.648886
\(951\) 12.0000 0.389127
\(952\) 21.0000 0.680614
\(953\) −12.0000 −0.388718 −0.194359 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(954\) 2.00000 0.0647524
\(955\) 69.0000 2.23279
\(956\) −6.00000 −0.194054
\(957\) −6.00000 −0.193952
\(958\) −16.0000 −0.516937
\(959\) 18.0000 0.581250
\(960\) −3.00000 −0.0968246
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 9.00000 0.290021
\(964\) −7.00000 −0.225455
\(965\) −60.0000 −1.93147
\(966\) 24.0000 0.772187
\(967\) 52.0000 1.67221 0.836104 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) −25.0000 −0.803530
\(969\) −35.0000 −1.12436
\(970\) 0 0
\(971\) −42.0000 −1.34784 −0.673922 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −42.0000 −1.34646
\(974\) −23.0000 −0.736968
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) 60.0000 1.91957 0.959785 0.280736i \(-0.0905785\pi\)
0.959785 + 0.280736i \(0.0905785\pi\)
\(978\) −1.00000 −0.0319765
\(979\) 60.0000 1.91761
\(980\) 6.00000 0.191663
\(981\) −4.00000 −0.127710
\(982\) 6.00000 0.191468
\(983\) −40.0000 −1.27580 −0.637901 0.770118i \(-0.720197\pi\)
−0.637901 + 0.770118i \(0.720197\pi\)
\(984\) −5.00000 −0.159394
\(985\) 21.0000 0.669116
\(986\) −7.00000 −0.222925
\(987\) 27.0000 0.859419
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) −18.0000 −0.572078
\(991\) −17.0000 −0.540023 −0.270011 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(992\) 8.00000 0.254000
\(993\) −27.0000 −0.856819
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −41.0000 −1.29848 −0.649242 0.760582i \(-0.724914\pi\)
−0.649242 + 0.760582i \(0.724914\pi\)
\(998\) −22.0000 −0.696398
\(999\) 3.00000 0.0949158
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 174.2.a.a.1.1 1
3.2 odd 2 522.2.a.h.1.1 1
4.3 odd 2 1392.2.a.o.1.1 1
5.2 odd 4 4350.2.e.q.349.1 2
5.3 odd 4 4350.2.e.q.349.2 2
5.4 even 2 4350.2.a.y.1.1 1
7.6 odd 2 8526.2.a.g.1.1 1
8.3 odd 2 5568.2.a.b.1.1 1
8.5 even 2 5568.2.a.t.1.1 1
12.11 even 2 4176.2.a.e.1.1 1
29.28 even 2 5046.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
174.2.a.a.1.1 1 1.1 even 1 trivial
522.2.a.h.1.1 1 3.2 odd 2
1392.2.a.o.1.1 1 4.3 odd 2
4176.2.a.e.1.1 1 12.11 even 2
4350.2.a.y.1.1 1 5.4 even 2
4350.2.e.q.349.1 2 5.2 odd 4
4350.2.e.q.349.2 2 5.3 odd 4
5046.2.a.o.1.1 1 29.28 even 2
5568.2.a.b.1.1 1 8.3 odd 2
5568.2.a.t.1.1 1 8.5 even 2
8526.2.a.g.1.1 1 7.6 odd 2