Properties

Label 2366.2.a.p.1.1
Level $2366$
Weight $2$
Character 2366.1
Self dual yes
Analytic conductor $18.893$
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] = [2366,2,Mod(1,2366)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2366, 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("2366.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2366.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +3.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +3.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +3.00000 q^{10} -4.00000 q^{11} +3.00000 q^{12} -1.00000 q^{14} +9.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +6.00000 q^{18} +4.00000 q^{19} +3.00000 q^{20} -3.00000 q^{21} -4.00000 q^{22} -1.00000 q^{23} +3.00000 q^{24} +4.00000 q^{25} +9.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +9.00000 q^{30} -10.0000 q^{31} +1.00000 q^{32} -12.0000 q^{33} +2.00000 q^{34} -3.00000 q^{35} +6.00000 q^{36} -4.00000 q^{37} +4.00000 q^{38} +3.00000 q^{40} -12.0000 q^{41} -3.00000 q^{42} -4.00000 q^{44} +18.0000 q^{45} -1.00000 q^{46} +10.0000 q^{47} +3.00000 q^{48} +1.00000 q^{49} +4.00000 q^{50} +6.00000 q^{51} -4.00000 q^{53} +9.00000 q^{54} -12.0000 q^{55} -1.00000 q^{56} +12.0000 q^{57} -2.00000 q^{58} -9.00000 q^{59} +9.00000 q^{60} +13.0000 q^{61} -10.0000 q^{62} -6.00000 q^{63} +1.00000 q^{64} -12.0000 q^{66} -2.00000 q^{67} +2.00000 q^{68} -3.00000 q^{69} -3.00000 q^{70} +15.0000 q^{71} +6.00000 q^{72} +2.00000 q^{73} -4.00000 q^{74} +12.0000 q^{75} +4.00000 q^{76} +4.00000 q^{77} +4.00000 q^{79} +3.00000 q^{80} +9.00000 q^{81} -12.0000 q^{82} +8.00000 q^{83} -3.00000 q^{84} +6.00000 q^{85} -6.00000 q^{87} -4.00000 q^{88} -14.0000 q^{89} +18.0000 q^{90} -1.00000 q^{92} -30.0000 q^{93} +10.0000 q^{94} +12.0000 q^{95} +3.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} -24.0000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 3.00000 1.22474
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 6.00000 2.00000
\(10\) 3.00000 0.948683
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 3.00000 0.866025
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) 9.00000 2.32379
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 6.00000 1.41421
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 3.00000 0.670820
\(21\) −3.00000 −0.654654
\(22\) −4.00000 −0.852803
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 3.00000 0.612372
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 9.00000 1.64317
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 1.00000 0.176777
\(33\) −12.0000 −2.08893
\(34\) 2.00000 0.342997
\(35\) −3.00000 −0.507093
\(36\) 6.00000 1.00000
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) −3.00000 −0.462910
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −4.00000 −0.603023
\(45\) 18.0000 2.68328
\(46\) −1.00000 −0.147442
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 3.00000 0.433013
\(49\) 1.00000 0.142857
\(50\) 4.00000 0.565685
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 9.00000 1.22474
\(55\) −12.0000 −1.61808
\(56\) −1.00000 −0.133631
\(57\) 12.0000 1.58944
\(58\) −2.00000 −0.262613
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 9.00000 1.16190
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) −10.0000 −1.27000
\(63\) −6.00000 −0.755929
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −12.0000 −1.47710
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 2.00000 0.242536
\(69\) −3.00000 −0.361158
\(70\) −3.00000 −0.358569
\(71\) 15.0000 1.78017 0.890086 0.455792i \(-0.150644\pi\)
0.890086 + 0.455792i \(0.150644\pi\)
\(72\) 6.00000 0.707107
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −4.00000 −0.464991
\(75\) 12.0000 1.38564
\(76\) 4.00000 0.458831
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 3.00000 0.335410
\(81\) 9.00000 1.00000
\(82\) −12.0000 −1.32518
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) −3.00000 −0.327327
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) −4.00000 −0.426401
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 18.0000 1.89737
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −30.0000 −3.11086
\(94\) 10.0000 1.03142
\(95\) 12.0000 1.23117
\(96\) 3.00000 0.306186
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 1.00000 0.101015
\(99\) −24.0000 −2.41209
\(100\) 4.00000 0.400000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 6.00000 0.594089
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) −9.00000 −0.878310
\(106\) −4.00000 −0.388514
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 9.00000 0.866025
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) −12.0000 −1.14416
\(111\) −12.0000 −1.13899
\(112\) −1.00000 −0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 12.0000 1.12390
\(115\) −3.00000 −0.279751
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) −9.00000 −0.828517
\(119\) −2.00000 −0.183340
\(120\) 9.00000 0.821584
\(121\) 5.00000 0.454545
\(122\) 13.0000 1.17696
\(123\) −36.0000 −3.24601
\(124\) −10.0000 −0.898027
\(125\) −3.00000 −0.268328
\(126\) −6.00000 −0.534522
\(127\) 15.0000 1.33103 0.665517 0.746382i \(-0.268211\pi\)
0.665517 + 0.746382i \(0.268211\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 5.00000 0.436852 0.218426 0.975854i \(-0.429908\pi\)
0.218426 + 0.975854i \(0.429908\pi\)
\(132\) −12.0000 −1.04447
\(133\) −4.00000 −0.346844
\(134\) −2.00000 −0.172774
\(135\) 27.0000 2.32379
\(136\) 2.00000 0.171499
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) −3.00000 −0.255377
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) −3.00000 −0.253546
\(141\) 30.0000 2.52646
\(142\) 15.0000 1.25877
\(143\) 0 0
\(144\) 6.00000 0.500000
\(145\) −6.00000 −0.498273
\(146\) 2.00000 0.165521
\(147\) 3.00000 0.247436
\(148\) −4.00000 −0.328798
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 12.0000 0.979796
\(151\) −3.00000 −0.244137 −0.122068 0.992522i \(-0.538953\pi\)
−0.122068 + 0.992522i \(0.538953\pi\)
\(152\) 4.00000 0.324443
\(153\) 12.0000 0.970143
\(154\) 4.00000 0.322329
\(155\) −30.0000 −2.40966
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 4.00000 0.318223
\(159\) −12.0000 −0.951662
\(160\) 3.00000 0.237171
\(161\) 1.00000 0.0788110
\(162\) 9.00000 0.707107
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) −12.0000 −0.937043
\(165\) −36.0000 −2.80260
\(166\) 8.00000 0.620920
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) −3.00000 −0.231455
\(169\) 0 0
\(170\) 6.00000 0.460179
\(171\) 24.0000 1.83533
\(172\) 0 0
\(173\) −9.00000 −0.684257 −0.342129 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) −6.00000 −0.454859
\(175\) −4.00000 −0.302372
\(176\) −4.00000 −0.301511
\(177\) −27.0000 −2.02944
\(178\) −14.0000 −1.04934
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 18.0000 1.34164
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 0 0
\(183\) 39.0000 2.88296
\(184\) −1.00000 −0.0737210
\(185\) −12.0000 −0.882258
\(186\) −30.0000 −2.19971
\(187\) −8.00000 −0.585018
\(188\) 10.0000 0.729325
\(189\) −9.00000 −0.654654
\(190\) 12.0000 0.870572
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 3.00000 0.216506
\(193\) −13.0000 −0.935760 −0.467880 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) −24.0000 −1.70561
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 4.00000 0.282843
\(201\) −6.00000 −0.423207
\(202\) −18.0000 −1.26648
\(203\) 2.00000 0.140372
\(204\) 6.00000 0.420084
\(205\) −36.0000 −2.51435
\(206\) 6.00000 0.418040
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) −9.00000 −0.621059
\(211\) 28.0000 1.92760 0.963800 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) −4.00000 −0.274721
\(213\) 45.0000 3.08335
\(214\) 10.0000 0.683586
\(215\) 0 0
\(216\) 9.00000 0.612372
\(217\) 10.0000 0.678844
\(218\) −12.0000 −0.812743
\(219\) 6.00000 0.405442
\(220\) −12.0000 −0.809040
\(221\) 0 0
\(222\) −12.0000 −0.805387
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 24.0000 1.60000
\(226\) −2.00000 −0.133038
\(227\) 21.0000 1.39382 0.696909 0.717159i \(-0.254558\pi\)
0.696909 + 0.717159i \(0.254558\pi\)
\(228\) 12.0000 0.794719
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) −3.00000 −0.197814
\(231\) 12.0000 0.789542
\(232\) −2.00000 −0.131306
\(233\) −3.00000 −0.196537 −0.0982683 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) 0 0
\(235\) 30.0000 1.95698
\(236\) −9.00000 −0.585850
\(237\) 12.0000 0.779484
\(238\) −2.00000 −0.129641
\(239\) −9.00000 −0.582162 −0.291081 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 9.00000 0.580948
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) 13.0000 0.832240
\(245\) 3.00000 0.191663
\(246\) −36.0000 −2.29528
\(247\) 0 0
\(248\) −10.0000 −0.635001
\(249\) 24.0000 1.52094
\(250\) −3.00000 −0.189737
\(251\) −7.00000 −0.441836 −0.220918 0.975292i \(-0.570905\pi\)
−0.220918 + 0.975292i \(0.570905\pi\)
\(252\) −6.00000 −0.377964
\(253\) 4.00000 0.251478
\(254\) 15.0000 0.941184
\(255\) 18.0000 1.12720
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −12.0000 −0.742781
\(262\) 5.00000 0.308901
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) −12.0000 −0.738549
\(265\) −12.0000 −0.737154
\(266\) −4.00000 −0.245256
\(267\) −42.0000 −2.57036
\(268\) −2.00000 −0.122169
\(269\) −17.0000 −1.03651 −0.518254 0.855227i \(-0.673418\pi\)
−0.518254 + 0.855227i \(0.673418\pi\)
\(270\) 27.0000 1.64317
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −3.00000 −0.181237
\(275\) −16.0000 −0.964836
\(276\) −3.00000 −0.180579
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) −8.00000 −0.479808
\(279\) −60.0000 −3.59211
\(280\) −3.00000 −0.179284
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 30.0000 1.78647
\(283\) 3.00000 0.178331 0.0891657 0.996017i \(-0.471580\pi\)
0.0891657 + 0.996017i \(0.471580\pi\)
\(284\) 15.0000 0.890086
\(285\) 36.0000 2.13246
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 6.00000 0.353553
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) −6.00000 −0.351726
\(292\) 2.00000 0.117041
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 3.00000 0.174964
\(295\) −27.0000 −1.57200
\(296\) −4.00000 −0.232495
\(297\) −36.0000 −2.08893
\(298\) −4.00000 −0.231714
\(299\) 0 0
\(300\) 12.0000 0.692820
\(301\) 0 0
\(302\) −3.00000 −0.172631
\(303\) −54.0000 −3.10222
\(304\) 4.00000 0.229416
\(305\) 39.0000 2.23313
\(306\) 12.0000 0.685994
\(307\) 7.00000 0.399511 0.199756 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 4.00000 0.227921
\(309\) 18.0000 1.02398
\(310\) −30.0000 −1.70389
\(311\) 2.00000 0.113410 0.0567048 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) −2.00000 −0.112867
\(315\) −18.0000 −1.01419
\(316\) 4.00000 0.225018
\(317\) 4.00000 0.224662 0.112331 0.993671i \(-0.464168\pi\)
0.112331 + 0.993671i \(0.464168\pi\)
\(318\) −12.0000 −0.672927
\(319\) 8.00000 0.447914
\(320\) 3.00000 0.167705
\(321\) 30.0000 1.67444
\(322\) 1.00000 0.0557278
\(323\) 8.00000 0.445132
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) 8.00000 0.443079
\(327\) −36.0000 −1.99080
\(328\) −12.0000 −0.662589
\(329\) −10.0000 −0.551318
\(330\) −36.0000 −1.98173
\(331\) −30.0000 −1.64895 −0.824475 0.565899i \(-0.808529\pi\)
−0.824475 + 0.565899i \(0.808529\pi\)
\(332\) 8.00000 0.439057
\(333\) −24.0000 −1.31519
\(334\) 16.0000 0.875481
\(335\) −6.00000 −0.327815
\(336\) −3.00000 −0.163663
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 6.00000 0.325396
\(341\) 40.0000 2.16612
\(342\) 24.0000 1.29777
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −9.00000 −0.484544
\(346\) −9.00000 −0.483843
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) −6.00000 −0.321634
\(349\) 13.0000 0.695874 0.347937 0.937518i \(-0.386882\pi\)
0.347937 + 0.937518i \(0.386882\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) −27.0000 −1.43503
\(355\) 45.0000 2.38835
\(356\) −14.0000 −0.741999
\(357\) −6.00000 −0.317554
\(358\) 10.0000 0.528516
\(359\) 13.0000 0.686114 0.343057 0.939315i \(-0.388538\pi\)
0.343057 + 0.939315i \(0.388538\pi\)
\(360\) 18.0000 0.948683
\(361\) −3.00000 −0.157895
\(362\) −5.00000 −0.262794
\(363\) 15.0000 0.787296
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 39.0000 2.03856
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −72.0000 −3.74817
\(370\) −12.0000 −0.623850
\(371\) 4.00000 0.207670
\(372\) −30.0000 −1.55543
\(373\) −8.00000 −0.414224 −0.207112 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(374\) −8.00000 −0.413670
\(375\) −9.00000 −0.464758
\(376\) 10.0000 0.515711
\(377\) 0 0
\(378\) −9.00000 −0.462910
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 12.0000 0.615587
\(381\) 45.0000 2.30542
\(382\) 8.00000 0.409316
\(383\) 18.0000 0.919757 0.459879 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) 3.00000 0.153093
\(385\) 12.0000 0.611577
\(386\) −13.0000 −0.661683
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) −2.00000 −0.101144
\(392\) 1.00000 0.0505076
\(393\) 15.0000 0.756650
\(394\) 12.0000 0.604551
\(395\) 12.0000 0.603786
\(396\) −24.0000 −1.20605
\(397\) 9.00000 0.451697 0.225849 0.974162i \(-0.427485\pi\)
0.225849 + 0.974162i \(0.427485\pi\)
\(398\) −14.0000 −0.701757
\(399\) −12.0000 −0.600751
\(400\) 4.00000 0.200000
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) −6.00000 −0.299253
\(403\) 0 0
\(404\) −18.0000 −0.895533
\(405\) 27.0000 1.34164
\(406\) 2.00000 0.0992583
\(407\) 16.0000 0.793091
\(408\) 6.00000 0.297044
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) −36.0000 −1.77791
\(411\) −9.00000 −0.443937
\(412\) 6.00000 0.295599
\(413\) 9.00000 0.442861
\(414\) −6.00000 −0.294884
\(415\) 24.0000 1.17811
\(416\) 0 0
\(417\) −24.0000 −1.17529
\(418\) −16.0000 −0.782586
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) −9.00000 −0.439155
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 28.0000 1.36302
\(423\) 60.0000 2.91730
\(424\) −4.00000 −0.194257
\(425\) 8.00000 0.388057
\(426\) 45.0000 2.18026
\(427\) −13.0000 −0.629114
\(428\) 10.0000 0.483368
\(429\) 0 0
\(430\) 0 0
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 9.00000 0.433013
\(433\) 20.0000 0.961139 0.480569 0.876957i \(-0.340430\pi\)
0.480569 + 0.876957i \(0.340430\pi\)
\(434\) 10.0000 0.480015
\(435\) −18.0000 −0.863034
\(436\) −12.0000 −0.574696
\(437\) −4.00000 −0.191346
\(438\) 6.00000 0.286691
\(439\) −2.00000 −0.0954548 −0.0477274 0.998860i \(-0.515198\pi\)
−0.0477274 + 0.998860i \(0.515198\pi\)
\(440\) −12.0000 −0.572078
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) −12.0000 −0.569495
\(445\) −42.0000 −1.99099
\(446\) −12.0000 −0.568216
\(447\) −12.0000 −0.567581
\(448\) −1.00000 −0.0472456
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 24.0000 1.13137
\(451\) 48.0000 2.26023
\(452\) −2.00000 −0.0940721
\(453\) −9.00000 −0.422857
\(454\) 21.0000 0.985579
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) −5.00000 −0.233890 −0.116945 0.993138i \(-0.537310\pi\)
−0.116945 + 0.993138i \(0.537310\pi\)
\(458\) 26.0000 1.21490
\(459\) 18.0000 0.840168
\(460\) −3.00000 −0.139876
\(461\) −3.00000 −0.139724 −0.0698620 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 12.0000 0.558291
\(463\) 27.0000 1.25480 0.627398 0.778699i \(-0.284120\pi\)
0.627398 + 0.778699i \(0.284120\pi\)
\(464\) −2.00000 −0.0928477
\(465\) −90.0000 −4.17365
\(466\) −3.00000 −0.138972
\(467\) −33.0000 −1.52706 −0.763529 0.645774i \(-0.776535\pi\)
−0.763529 + 0.645774i \(0.776535\pi\)
\(468\) 0 0
\(469\) 2.00000 0.0923514
\(470\) 30.0000 1.38380
\(471\) −6.00000 −0.276465
\(472\) −9.00000 −0.414259
\(473\) 0 0
\(474\) 12.0000 0.551178
\(475\) 16.0000 0.734130
\(476\) −2.00000 −0.0916698
\(477\) −24.0000 −1.09888
\(478\) −9.00000 −0.411650
\(479\) −2.00000 −0.0913823 −0.0456912 0.998956i \(-0.514549\pi\)
−0.0456912 + 0.998956i \(0.514549\pi\)
\(480\) 9.00000 0.410792
\(481\) 0 0
\(482\) −12.0000 −0.546585
\(483\) 3.00000 0.136505
\(484\) 5.00000 0.227273
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) −7.00000 −0.317200 −0.158600 0.987343i \(-0.550698\pi\)
−0.158600 + 0.987343i \(0.550698\pi\)
\(488\) 13.0000 0.588482
\(489\) 24.0000 1.08532
\(490\) 3.00000 0.135526
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) −36.0000 −1.62301
\(493\) −4.00000 −0.180151
\(494\) 0 0
\(495\) −72.0000 −3.23616
\(496\) −10.0000 −0.449013
\(497\) −15.0000 −0.672842
\(498\) 24.0000 1.07547
\(499\) 2.00000 0.0895323 0.0447661 0.998997i \(-0.485746\pi\)
0.0447661 + 0.998997i \(0.485746\pi\)
\(500\) −3.00000 −0.134164
\(501\) 48.0000 2.14448
\(502\) −7.00000 −0.312425
\(503\) 2.00000 0.0891756 0.0445878 0.999005i \(-0.485803\pi\)
0.0445878 + 0.999005i \(0.485803\pi\)
\(504\) −6.00000 −0.267261
\(505\) −54.0000 −2.40297
\(506\) 4.00000 0.177822
\(507\) 0 0
\(508\) 15.0000 0.665517
\(509\) 9.00000 0.398918 0.199459 0.979906i \(-0.436082\pi\)
0.199459 + 0.979906i \(0.436082\pi\)
\(510\) 18.0000 0.797053
\(511\) −2.00000 −0.0884748
\(512\) 1.00000 0.0441942
\(513\) 36.0000 1.58944
\(514\) −14.0000 −0.617514
\(515\) 18.0000 0.793175
\(516\) 0 0
\(517\) −40.0000 −1.75920
\(518\) 4.00000 0.175750
\(519\) −27.0000 −1.18517
\(520\) 0 0
\(521\) 16.0000 0.700973 0.350486 0.936568i \(-0.386016\pi\)
0.350486 + 0.936568i \(0.386016\pi\)
\(522\) −12.0000 −0.525226
\(523\) −43.0000 −1.88026 −0.940129 0.340818i \(-0.889296\pi\)
−0.940129 + 0.340818i \(0.889296\pi\)
\(524\) 5.00000 0.218426
\(525\) −12.0000 −0.523723
\(526\) 9.00000 0.392419
\(527\) −20.0000 −0.871214
\(528\) −12.0000 −0.522233
\(529\) −22.0000 −0.956522
\(530\) −12.0000 −0.521247
\(531\) −54.0000 −2.34340
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) −42.0000 −1.81752
\(535\) 30.0000 1.29701
\(536\) −2.00000 −0.0863868
\(537\) 30.0000 1.29460
\(538\) −17.0000 −0.732922
\(539\) −4.00000 −0.172292
\(540\) 27.0000 1.16190
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) −2.00000 −0.0859074
\(543\) −15.0000 −0.643712
\(544\) 2.00000 0.0857493
\(545\) −36.0000 −1.54207
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) −3.00000 −0.128154
\(549\) 78.0000 3.32896
\(550\) −16.0000 −0.682242
\(551\) −8.00000 −0.340811
\(552\) −3.00000 −0.127688
\(553\) −4.00000 −0.170097
\(554\) 14.0000 0.594803
\(555\) −36.0000 −1.52811
\(556\) −8.00000 −0.339276
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) −60.0000 −2.54000
\(559\) 0 0
\(560\) −3.00000 −0.126773
\(561\) −24.0000 −1.01328
\(562\) −2.00000 −0.0843649
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 30.0000 1.26323
\(565\) −6.00000 −0.252422
\(566\) 3.00000 0.126099
\(567\) −9.00000 −0.377964
\(568\) 15.0000 0.629386
\(569\) 27.0000 1.13190 0.565949 0.824440i \(-0.308510\pi\)
0.565949 + 0.824440i \(0.308510\pi\)
\(570\) 36.0000 1.50787
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) 24.0000 1.00261
\(574\) 12.0000 0.500870
\(575\) −4.00000 −0.166812
\(576\) 6.00000 0.250000
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) −13.0000 −0.540729
\(579\) −39.0000 −1.62078
\(580\) −6.00000 −0.249136
\(581\) −8.00000 −0.331896
\(582\) −6.00000 −0.248708
\(583\) 16.0000 0.662652
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) −9.00000 −0.371470 −0.185735 0.982600i \(-0.559467\pi\)
−0.185735 + 0.982600i \(0.559467\pi\)
\(588\) 3.00000 0.123718
\(589\) −40.0000 −1.64817
\(590\) −27.0000 −1.11157
\(591\) 36.0000 1.48084
\(592\) −4.00000 −0.164399
\(593\) −20.0000 −0.821302 −0.410651 0.911793i \(-0.634698\pi\)
−0.410651 + 0.911793i \(0.634698\pi\)
\(594\) −36.0000 −1.47710
\(595\) −6.00000 −0.245976
\(596\) −4.00000 −0.163846
\(597\) −42.0000 −1.71895
\(598\) 0 0
\(599\) −9.00000 −0.367730 −0.183865 0.982952i \(-0.558861\pi\)
−0.183865 + 0.982952i \(0.558861\pi\)
\(600\) 12.0000 0.489898
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) −3.00000 −0.122068
\(605\) 15.0000 0.609837
\(606\) −54.0000 −2.19360
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) 4.00000 0.162221
\(609\) 6.00000 0.243132
\(610\) 39.0000 1.57906
\(611\) 0 0
\(612\) 12.0000 0.485071
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 7.00000 0.282497
\(615\) −108.000 −4.35498
\(616\) 4.00000 0.161165
\(617\) −45.0000 −1.81163 −0.905816 0.423672i \(-0.860741\pi\)
−0.905816 + 0.423672i \(0.860741\pi\)
\(618\) 18.0000 0.724066
\(619\) 39.0000 1.56754 0.783771 0.621050i \(-0.213294\pi\)
0.783771 + 0.621050i \(0.213294\pi\)
\(620\) −30.0000 −1.20483
\(621\) −9.00000 −0.361158
\(622\) 2.00000 0.0801927
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) −48.0000 −1.91694
\(628\) −2.00000 −0.0798087
\(629\) −8.00000 −0.318981
\(630\) −18.0000 −0.717137
\(631\) −1.00000 −0.0398094 −0.0199047 0.999802i \(-0.506336\pi\)
−0.0199047 + 0.999802i \(0.506336\pi\)
\(632\) 4.00000 0.159111
\(633\) 84.0000 3.33870
\(634\) 4.00000 0.158860
\(635\) 45.0000 1.78577
\(636\) −12.0000 −0.475831
\(637\) 0 0
\(638\) 8.00000 0.316723
\(639\) 90.0000 3.56034
\(640\) 3.00000 0.118585
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 30.0000 1.18401
\(643\) 45.0000 1.77463 0.887313 0.461167i \(-0.152569\pi\)
0.887313 + 0.461167i \(0.152569\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) 36.0000 1.41531 0.707653 0.706560i \(-0.249754\pi\)
0.707653 + 0.706560i \(0.249754\pi\)
\(648\) 9.00000 0.353553
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 30.0000 1.17579
\(652\) 8.00000 0.313304
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) −36.0000 −1.40771
\(655\) 15.0000 0.586098
\(656\) −12.0000 −0.468521
\(657\) 12.0000 0.468165
\(658\) −10.0000 −0.389841
\(659\) −10.0000 −0.389545 −0.194772 0.980848i \(-0.562397\pi\)
−0.194772 + 0.980848i \(0.562397\pi\)
\(660\) −36.0000 −1.40130
\(661\) −25.0000 −0.972387 −0.486194 0.873851i \(-0.661615\pi\)
−0.486194 + 0.873851i \(0.661615\pi\)
\(662\) −30.0000 −1.16598
\(663\) 0 0
\(664\) 8.00000 0.310460
\(665\) −12.0000 −0.465340
\(666\) −24.0000 −0.929981
\(667\) 2.00000 0.0774403
\(668\) 16.0000 0.619059
\(669\) −36.0000 −1.39184
\(670\) −6.00000 −0.231800
\(671\) −52.0000 −2.00744
\(672\) −3.00000 −0.115728
\(673\) 30.0000 1.15642 0.578208 0.815890i \(-0.303752\pi\)
0.578208 + 0.815890i \(0.303752\pi\)
\(674\) −18.0000 −0.693334
\(675\) 36.0000 1.38564
\(676\) 0 0
\(677\) 51.0000 1.96009 0.980045 0.198778i \(-0.0636972\pi\)
0.980045 + 0.198778i \(0.0636972\pi\)
\(678\) −6.00000 −0.230429
\(679\) 2.00000 0.0767530
\(680\) 6.00000 0.230089
\(681\) 63.0000 2.41417
\(682\) 40.0000 1.53168
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 24.0000 0.917663
\(685\) −9.00000 −0.343872
\(686\) −1.00000 −0.0381802
\(687\) 78.0000 2.97589
\(688\) 0 0
\(689\) 0 0
\(690\) −9.00000 −0.342624
\(691\) 27.0000 1.02713 0.513564 0.858051i \(-0.328325\pi\)
0.513564 + 0.858051i \(0.328325\pi\)
\(692\) −9.00000 −0.342129
\(693\) 24.0000 0.911685
\(694\) 4.00000 0.151838
\(695\) −24.0000 −0.910372
\(696\) −6.00000 −0.227429
\(697\) −24.0000 −0.909065
\(698\) 13.0000 0.492057
\(699\) −9.00000 −0.340411
\(700\) −4.00000 −0.151186
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −16.0000 −0.603451
\(704\) −4.00000 −0.150756
\(705\) 90.0000 3.38960
\(706\) 18.0000 0.677439
\(707\) 18.0000 0.676960
\(708\) −27.0000 −1.01472
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 45.0000 1.68882
\(711\) 24.0000 0.900070
\(712\) −14.0000 −0.524672
\(713\) 10.0000 0.374503
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) −27.0000 −1.00833
\(718\) 13.0000 0.485156
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 18.0000 0.670820
\(721\) −6.00000 −0.223452
\(722\) −3.00000 −0.111648
\(723\) −36.0000 −1.33885
\(724\) −5.00000 −0.185824
\(725\) −8.00000 −0.297113
\(726\) 15.0000 0.556702
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 6.00000 0.222070
\(731\) 0 0
\(732\) 39.0000 1.44148
\(733\) 43.0000 1.58824 0.794121 0.607760i \(-0.207932\pi\)
0.794121 + 0.607760i \(0.207932\pi\)
\(734\) −10.0000 −0.369107
\(735\) 9.00000 0.331970
\(736\) −1.00000 −0.0368605
\(737\) 8.00000 0.294684
\(738\) −72.0000 −2.65036
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) 4.00000 0.146845
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −30.0000 −1.09985
\(745\) −12.0000 −0.439646
\(746\) −8.00000 −0.292901
\(747\) 48.0000 1.75623
\(748\) −8.00000 −0.292509
\(749\) −10.0000 −0.365392
\(750\) −9.00000 −0.328634
\(751\) −11.0000 −0.401396 −0.200698 0.979653i \(-0.564321\pi\)
−0.200698 + 0.979653i \(0.564321\pi\)
\(752\) 10.0000 0.364662
\(753\) −21.0000 −0.765283
\(754\) 0 0
\(755\) −9.00000 −0.327544
\(756\) −9.00000 −0.327327
\(757\) −48.0000 −1.74459 −0.872295 0.488980i \(-0.837369\pi\)
−0.872295 + 0.488980i \(0.837369\pi\)
\(758\) 4.00000 0.145287
\(759\) 12.0000 0.435572
\(760\) 12.0000 0.435286
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 45.0000 1.63018
\(763\) 12.0000 0.434429
\(764\) 8.00000 0.289430
\(765\) 36.0000 1.30158
\(766\) 18.0000 0.650366
\(767\) 0 0
\(768\) 3.00000 0.108253
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 12.0000 0.432450
\(771\) −42.0000 −1.51259
\(772\) −13.0000 −0.467880
\(773\) −46.0000 −1.65451 −0.827253 0.561830i \(-0.810097\pi\)
−0.827253 + 0.561830i \(0.810097\pi\)
\(774\) 0 0
\(775\) −40.0000 −1.43684
\(776\) −2.00000 −0.0717958
\(777\) 12.0000 0.430498
\(778\) 2.00000 0.0717035
\(779\) −48.0000 −1.71978
\(780\) 0 0
\(781\) −60.0000 −2.14697
\(782\) −2.00000 −0.0715199
\(783\) −18.0000 −0.643268
\(784\) 1.00000 0.0357143
\(785\) −6.00000 −0.214149
\(786\) 15.0000 0.535032
\(787\) 25.0000 0.891154 0.445577 0.895244i \(-0.352999\pi\)
0.445577 + 0.895244i \(0.352999\pi\)
\(788\) 12.0000 0.427482
\(789\) 27.0000 0.961225
\(790\) 12.0000 0.426941
\(791\) 2.00000 0.0711118
\(792\) −24.0000 −0.852803
\(793\) 0 0
\(794\) 9.00000 0.319398
\(795\) −36.0000 −1.27679
\(796\) −14.0000 −0.496217
\(797\) −9.00000 −0.318796 −0.159398 0.987214i \(-0.550955\pi\)
−0.159398 + 0.987214i \(0.550955\pi\)
\(798\) −12.0000 −0.424795
\(799\) 20.0000 0.707549
\(800\) 4.00000 0.141421
\(801\) −84.0000 −2.96799
\(802\) 10.0000 0.353112
\(803\) −8.00000 −0.282314
\(804\) −6.00000 −0.211604
\(805\) 3.00000 0.105736
\(806\) 0 0
\(807\) −51.0000 −1.79529
\(808\) −18.0000 −0.633238
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 27.0000 0.948683
\(811\) −31.0000 −1.08856 −0.544279 0.838905i \(-0.683197\pi\)
−0.544279 + 0.838905i \(0.683197\pi\)
\(812\) 2.00000 0.0701862
\(813\) −6.00000 −0.210429
\(814\) 16.0000 0.560800
\(815\) 24.0000 0.840683
\(816\) 6.00000 0.210042
\(817\) 0 0
\(818\) 30.0000 1.04893
\(819\) 0 0
\(820\) −36.0000 −1.25717
\(821\) −48.0000 −1.67521 −0.837606 0.546275i \(-0.816045\pi\)
−0.837606 + 0.546275i \(0.816045\pi\)
\(822\) −9.00000 −0.313911
\(823\) −9.00000 −0.313720 −0.156860 0.987621i \(-0.550137\pi\)
−0.156860 + 0.987621i \(0.550137\pi\)
\(824\) 6.00000 0.209020
\(825\) −48.0000 −1.67115
\(826\) 9.00000 0.313150
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −6.00000 −0.208514
\(829\) 33.0000 1.14614 0.573069 0.819507i \(-0.305753\pi\)
0.573069 + 0.819507i \(0.305753\pi\)
\(830\) 24.0000 0.833052
\(831\) 42.0000 1.45696
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) −24.0000 −0.831052
\(835\) 48.0000 1.66111
\(836\) −16.0000 −0.553372
\(837\) −90.0000 −3.11086
\(838\) −24.0000 −0.829066
\(839\) −2.00000 −0.0690477 −0.0345238 0.999404i \(-0.510991\pi\)
−0.0345238 + 0.999404i \(0.510991\pi\)
\(840\) −9.00000 −0.310530
\(841\) −25.0000 −0.862069
\(842\) −10.0000 −0.344623
\(843\) −6.00000 −0.206651
\(844\) 28.0000 0.963800
\(845\) 0 0
\(846\) 60.0000 2.06284
\(847\) −5.00000 −0.171802
\(848\) −4.00000 −0.137361
\(849\) 9.00000 0.308879
\(850\) 8.00000 0.274398
\(851\) 4.00000 0.137118
\(852\) 45.0000 1.54167
\(853\) 49.0000 1.67773 0.838864 0.544341i \(-0.183220\pi\)
0.838864 + 0.544341i \(0.183220\pi\)
\(854\) −13.0000 −0.444851
\(855\) 72.0000 2.46235
\(856\) 10.0000 0.341793
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) 36.0000 1.22688
\(862\) −3.00000 −0.102180
\(863\) 21.0000 0.714848 0.357424 0.933942i \(-0.383655\pi\)
0.357424 + 0.933942i \(0.383655\pi\)
\(864\) 9.00000 0.306186
\(865\) −27.0000 −0.918028
\(866\) 20.0000 0.679628
\(867\) −39.0000 −1.32451
\(868\) 10.0000 0.339422
\(869\) −16.0000 −0.542763
\(870\) −18.0000 −0.610257
\(871\) 0 0
\(872\) −12.0000 −0.406371
\(873\) −12.0000 −0.406138
\(874\) −4.00000 −0.135302
\(875\) 3.00000 0.101419
\(876\) 6.00000 0.202721
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −2.00000 −0.0674967
\(879\) 54.0000 1.82137
\(880\) −12.0000 −0.404520
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 6.00000 0.202031
\(883\) 48.0000 1.61533 0.807664 0.589643i \(-0.200731\pi\)
0.807664 + 0.589643i \(0.200731\pi\)
\(884\) 0 0
\(885\) −81.0000 −2.72279
\(886\) 6.00000 0.201574
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) −12.0000 −0.402694
\(889\) −15.0000 −0.503084
\(890\) −42.0000 −1.40784
\(891\) −36.0000 −1.20605
\(892\) −12.0000 −0.401790
\(893\) 40.0000 1.33855
\(894\) −12.0000 −0.401340
\(895\) 30.0000 1.00279
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 15.0000 0.500556
\(899\) 20.0000 0.667037
\(900\) 24.0000 0.800000
\(901\) −8.00000 −0.266519
\(902\) 48.0000 1.59823
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) −15.0000 −0.498617
\(906\) −9.00000 −0.299005
\(907\) 14.0000 0.464862 0.232431 0.972613i \(-0.425332\pi\)
0.232431 + 0.972613i \(0.425332\pi\)
\(908\) 21.0000 0.696909
\(909\) −108.000 −3.58213
\(910\) 0 0
\(911\) −23.0000 −0.762024 −0.381012 0.924570i \(-0.624424\pi\)
−0.381012 + 0.924570i \(0.624424\pi\)
\(912\) 12.0000 0.397360
\(913\) −32.0000 −1.05905
\(914\) −5.00000 −0.165385
\(915\) 117.000 3.86790
\(916\) 26.0000 0.859064
\(917\) −5.00000 −0.165115
\(918\) 18.0000 0.594089
\(919\) −11.0000 −0.362857 −0.181428 0.983404i \(-0.558072\pi\)
−0.181428 + 0.983404i \(0.558072\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 21.0000 0.691974
\(922\) −3.00000 −0.0987997
\(923\) 0 0
\(924\) 12.0000 0.394771
\(925\) −16.0000 −0.526077
\(926\) 27.0000 0.887275
\(927\) 36.0000 1.18240
\(928\) −2.00000 −0.0656532
\(929\) 2.00000 0.0656179 0.0328089 0.999462i \(-0.489555\pi\)
0.0328089 + 0.999462i \(0.489555\pi\)
\(930\) −90.0000 −2.95122
\(931\) 4.00000 0.131095
\(932\) −3.00000 −0.0982683
\(933\) 6.00000 0.196431
\(934\) −33.0000 −1.07979
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) 16.0000 0.522697 0.261349 0.965244i \(-0.415833\pi\)
0.261349 + 0.965244i \(0.415833\pi\)
\(938\) 2.00000 0.0653023
\(939\) 0 0
\(940\) 30.0000 0.978492
\(941\) 15.0000 0.488986 0.244493 0.969651i \(-0.421378\pi\)
0.244493 + 0.969651i \(0.421378\pi\)
\(942\) −6.00000 −0.195491
\(943\) 12.0000 0.390774
\(944\) −9.00000 −0.292925
\(945\) −27.0000 −0.878310
\(946\) 0 0
\(947\) −38.0000 −1.23483 −0.617417 0.786636i \(-0.711821\pi\)
−0.617417 + 0.786636i \(0.711821\pi\)
\(948\) 12.0000 0.389742
\(949\) 0 0
\(950\) 16.0000 0.519109
\(951\) 12.0000 0.389127
\(952\) −2.00000 −0.0648204
\(953\) −39.0000 −1.26333 −0.631667 0.775240i \(-0.717629\pi\)
−0.631667 + 0.775240i \(0.717629\pi\)
\(954\) −24.0000 −0.777029
\(955\) 24.0000 0.776622
\(956\) −9.00000 −0.291081
\(957\) 24.0000 0.775810
\(958\) −2.00000 −0.0646171
\(959\) 3.00000 0.0968751
\(960\) 9.00000 0.290474
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 60.0000 1.93347
\(964\) −12.0000 −0.386494
\(965\) −39.0000 −1.25545
\(966\) 3.00000 0.0965234
\(967\) −56.0000 −1.80084 −0.900419 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) 5.00000 0.160706
\(969\) 24.0000 0.770991
\(970\) −6.00000 −0.192648
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) −7.00000 −0.224294
\(975\) 0 0
\(976\) 13.0000 0.416120
\(977\) −31.0000 −0.991778 −0.495889 0.868386i \(-0.665158\pi\)
−0.495889 + 0.868386i \(0.665158\pi\)
\(978\) 24.0000 0.767435
\(979\) 56.0000 1.78977
\(980\) 3.00000 0.0958315
\(981\) −72.0000 −2.29878
\(982\) −20.0000 −0.638226
\(983\) 14.0000 0.446531 0.223265 0.974758i \(-0.428328\pi\)
0.223265 + 0.974758i \(0.428328\pi\)
\(984\) −36.0000 −1.14764
\(985\) 36.0000 1.14706
\(986\) −4.00000 −0.127386
\(987\) −30.0000 −0.954911
\(988\) 0 0
\(989\) 0 0
\(990\) −72.0000 −2.28831
\(991\) 5.00000 0.158830 0.0794151 0.996842i \(-0.474695\pi\)
0.0794151 + 0.996842i \(0.474695\pi\)
\(992\) −10.0000 −0.317500
\(993\) −90.0000 −2.85606
\(994\) −15.0000 −0.475771
\(995\) −42.0000 −1.33149
\(996\) 24.0000 0.760469
\(997\) −27.0000 −0.855099 −0.427549 0.903992i \(-0.640623\pi\)
−0.427549 + 0.903992i \(0.640623\pi\)
\(998\) 2.00000 0.0633089
\(999\) −36.0000 −1.13899
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 2366.2.a.p.1.1 1
13.4 even 6 182.2.g.a.29.1 2
13.5 odd 4 2366.2.d.h.337.1 2
13.8 odd 4 2366.2.d.h.337.2 2
13.10 even 6 182.2.g.a.113.1 yes 2
13.12 even 2 2366.2.a.g.1.1 1
39.17 odd 6 1638.2.r.n.757.1 2
39.23 odd 6 1638.2.r.n.1387.1 2
52.23 odd 6 1456.2.s.g.113.1 2
52.43 odd 6 1456.2.s.g.1121.1 2
91.4 even 6 1274.2.e.a.471.1 2
91.10 odd 6 1274.2.h.c.373.1 2
91.17 odd 6 1274.2.e.l.471.1 2
91.23 even 6 1274.2.e.a.165.1 2
91.30 even 6 1274.2.h.n.263.1 2
91.62 odd 6 1274.2.g.k.295.1 2
91.69 odd 6 1274.2.g.k.393.1 2
91.75 odd 6 1274.2.e.l.165.1 2
91.82 odd 6 1274.2.h.c.263.1 2
91.88 even 6 1274.2.h.n.373.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.g.a.29.1 2 13.4 even 6
182.2.g.a.113.1 yes 2 13.10 even 6
1274.2.e.a.165.1 2 91.23 even 6
1274.2.e.a.471.1 2 91.4 even 6
1274.2.e.l.165.1 2 91.75 odd 6
1274.2.e.l.471.1 2 91.17 odd 6
1274.2.g.k.295.1 2 91.62 odd 6
1274.2.g.k.393.1 2 91.69 odd 6
1274.2.h.c.263.1 2 91.82 odd 6
1274.2.h.c.373.1 2 91.10 odd 6
1274.2.h.n.263.1 2 91.30 even 6
1274.2.h.n.373.1 2 91.88 even 6
1456.2.s.g.113.1 2 52.23 odd 6
1456.2.s.g.1121.1 2 52.43 odd 6
1638.2.r.n.757.1 2 39.17 odd 6
1638.2.r.n.1387.1 2 39.23 odd 6
2366.2.a.g.1.1 1 13.12 even 2
2366.2.a.p.1.1 1 1.1 even 1 trivial
2366.2.d.h.337.1 2 13.5 odd 4
2366.2.d.h.337.2 2 13.8 odd 4