Properties

Label 4650.2.a.bh.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4650,2,Mod(1,4650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4650.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.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: \(37.1304369399\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 186)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4650.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +5.00000 q^{11} -1.00000 q^{12} +7.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +7.00000 q^{19} -2.00000 q^{21} +5.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +7.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} -8.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} -5.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} +7.00000 q^{38} -7.00000 q^{39} -2.00000 q^{41} -2.00000 q^{42} +10.0000 q^{43} +5.00000 q^{44} -4.00000 q^{46} +1.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{51} +7.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +2.00000 q^{56} -7.00000 q^{57} -8.00000 q^{58} -10.0000 q^{59} +1.00000 q^{61} -1.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} +3.00000 q^{67} +1.00000 q^{68} +4.00000 q^{69} +3.00000 q^{71} +1.00000 q^{72} -14.0000 q^{73} +6.00000 q^{74} +7.00000 q^{76} +10.0000 q^{77} -7.00000 q^{78} -11.0000 q^{79} +1.00000 q^{81} -2.00000 q^{82} -7.00000 q^{83} -2.00000 q^{84} +10.0000 q^{86} +8.00000 q^{87} +5.00000 q^{88} -6.00000 q^{89} +14.0000 q^{91} -4.00000 q^{92} +1.00000 q^{93} +1.00000 q^{94} -1.00000 q^{96} +3.00000 q^{97} -3.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) −1.00000 −0.288675
\(13\) 7.00000 1.94145 0.970725 0.240192i \(-0.0772105\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 1.00000 0.235702
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 5.00000 1.06600
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 7.00000 1.37281
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 1.00000 0.176777
\(33\) −5.00000 −0.870388
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 7.00000 1.13555
\(39\) −7.00000 −1.12090
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −2.00000 −0.308607
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 5.00000 0.753778
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 1.00000 0.145865 0.0729325 0.997337i \(-0.476764\pi\)
0.0729325 + 0.997337i \(0.476764\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 7.00000 0.970725
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) −7.00000 −0.927173
\(58\) −8.00000 −1.05045
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) −1.00000 −0.127000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) 3.00000 0.366508 0.183254 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) 1.00000 0.121268
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 7.00000 0.802955
\(77\) 10.0000 1.13961
\(78\) −7.00000 −0.792594
\(79\) −11.0000 −1.23760 −0.618798 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −7.00000 −0.768350 −0.384175 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) 8.00000 0.857690
\(88\) 5.00000 0.533002
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 14.0000 1.46760
\(92\) −4.00000 −0.417029
\(93\) 1.00000 0.103695
\(94\) 1.00000 0.103142
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 3.00000 0.304604 0.152302 0.988334i \(-0.451331\pi\)
0.152302 + 0.988334i \(0.451331\pi\)
\(98\) −3.00000 −0.303046
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 7.00000 0.686406
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 2.00000 0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −7.00000 −0.655610
\(115\) 0 0
\(116\) −8.00000 −0.742781
\(117\) 7.00000 0.647150
\(118\) −10.0000 −0.920575
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 1.00000 0.0905357
\(123\) 2.00000 0.180334
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 1.00000 0.0883883
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) −5.00000 −0.435194
\(133\) 14.0000 1.21395
\(134\) 3.00000 0.259161
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 4.00000 0.340503
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) 3.00000 0.251754
\(143\) 35.0000 2.92685
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) 3.00000 0.247436
\(148\) 6.00000 0.493197
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) −1.00000 −0.0813788 −0.0406894 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) 7.00000 0.567775
\(153\) 1.00000 0.0808452
\(154\) 10.0000 0.805823
\(155\) 0 0
\(156\) −7.00000 −0.560449
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) −11.0000 −0.875113
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 1.00000 0.0785674
\(163\) 9.00000 0.704934 0.352467 0.935824i \(-0.385343\pi\)
0.352467 + 0.935824i \(0.385343\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −7.00000 −0.543305
\(167\) −10.0000 −0.773823 −0.386912 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) −2.00000 −0.154303
\(169\) 36.0000 2.76923
\(170\) 0 0
\(171\) 7.00000 0.535303
\(172\) 10.0000 0.762493
\(173\) −1.00000 −0.0760286 −0.0380143 0.999277i \(-0.512103\pi\)
−0.0380143 + 0.999277i \(0.512103\pi\)
\(174\) 8.00000 0.606478
\(175\) 0 0
\(176\) 5.00000 0.376889
\(177\) 10.0000 0.751646
\(178\) −6.00000 −0.449719
\(179\) 19.0000 1.42013 0.710063 0.704138i \(-0.248666\pi\)
0.710063 + 0.704138i \(0.248666\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 14.0000 1.03775
\(183\) −1.00000 −0.0739221
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 5.00000 0.365636
\(188\) 1.00000 0.0729325
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −19.0000 −1.36765 −0.683825 0.729646i \(-0.739685\pi\)
−0.683825 + 0.729646i \(0.739685\pi\)
\(194\) 3.00000 0.215387
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 5.00000 0.355335
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) −3.00000 −0.211604
\(202\) 6.00000 0.422159
\(203\) −16.0000 −1.12298
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) −10.0000 −0.696733
\(207\) −4.00000 −0.278019
\(208\) 7.00000 0.485363
\(209\) 35.0000 2.42100
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −6.00000 −0.412082
\(213\) −3.00000 −0.205557
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −2.00000 −0.135769
\(218\) −6.00000 −0.406371
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) 7.00000 0.470871
\(222\) −6.00000 −0.402694
\(223\) 5.00000 0.334825 0.167412 0.985887i \(-0.446459\pi\)
0.167412 + 0.985887i \(0.446459\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −22.0000 −1.46019 −0.730096 0.683345i \(-0.760525\pi\)
−0.730096 + 0.683345i \(0.760525\pi\)
\(228\) −7.00000 −0.463586
\(229\) 13.0000 0.859064 0.429532 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) 0 0
\(231\) −10.0000 −0.657952
\(232\) −8.00000 −0.525226
\(233\) −20.0000 −1.31024 −0.655122 0.755523i \(-0.727383\pi\)
−0.655122 + 0.755523i \(0.727383\pi\)
\(234\) 7.00000 0.457604
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) 11.0000 0.714527
\(238\) 2.00000 0.129641
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 14.0000 0.899954
\(243\) −1.00000 −0.0641500
\(244\) 1.00000 0.0640184
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 49.0000 3.11780
\(248\) −1.00000 −0.0635001
\(249\) 7.00000 0.443607
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 2.00000 0.125988
\(253\) −20.0000 −1.25739
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) −10.0000 −0.622573
\(259\) 12.0000 0.745644
\(260\) 0 0
\(261\) −8.00000 −0.495188
\(262\) 10.0000 0.617802
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −5.00000 −0.307729
\(265\) 0 0
\(266\) 14.0000 0.858395
\(267\) 6.00000 0.367194
\(268\) 3.00000 0.183254
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −19.0000 −1.15417 −0.577084 0.816685i \(-0.695809\pi\)
−0.577084 + 0.816685i \(0.695809\pi\)
\(272\) 1.00000 0.0606339
\(273\) −14.0000 −0.847319
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 7.00000 0.420589 0.210295 0.977638i \(-0.432558\pi\)
0.210295 + 0.977638i \(0.432558\pi\)
\(278\) −8.00000 −0.479808
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) −1.00000 −0.0595491
\(283\) −5.00000 −0.297219 −0.148610 0.988896i \(-0.547480\pi\)
−0.148610 + 0.988896i \(0.547480\pi\)
\(284\) 3.00000 0.178017
\(285\) 0 0
\(286\) 35.0000 2.06959
\(287\) −4.00000 −0.236113
\(288\) 1.00000 0.0589256
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −3.00000 −0.175863
\(292\) −14.0000 −0.819288
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) −5.00000 −0.290129
\(298\) 15.0000 0.868927
\(299\) −28.0000 −1.61928
\(300\) 0 0
\(301\) 20.0000 1.15278
\(302\) −1.00000 −0.0575435
\(303\) −6.00000 −0.344691
\(304\) 7.00000 0.401478
\(305\) 0 0
\(306\) 1.00000 0.0571662
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 10.0000 0.569803
\(309\) 10.0000 0.568880
\(310\) 0 0
\(311\) 21.0000 1.19080 0.595400 0.803429i \(-0.296993\pi\)
0.595400 + 0.803429i \(0.296993\pi\)
\(312\) −7.00000 −0.396297
\(313\) −2.00000 −0.113047 −0.0565233 0.998401i \(-0.518002\pi\)
−0.0565233 + 0.998401i \(0.518002\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −11.0000 −0.618798
\(317\) −5.00000 −0.280828 −0.140414 0.990093i \(-0.544843\pi\)
−0.140414 + 0.990093i \(0.544843\pi\)
\(318\) 6.00000 0.336463
\(319\) −40.0000 −2.23957
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) −8.00000 −0.445823
\(323\) 7.00000 0.389490
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 9.00000 0.498464
\(327\) 6.00000 0.331801
\(328\) −2.00000 −0.110432
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) −7.00000 −0.384175
\(333\) 6.00000 0.328798
\(334\) −10.0000 −0.547176
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 36.0000 1.95814
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −5.00000 −0.270765
\(342\) 7.00000 0.378517
\(343\) −20.0000 −1.07990
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) −1.00000 −0.0537603
\(347\) −17.0000 −0.912608 −0.456304 0.889824i \(-0.650827\pi\)
−0.456304 + 0.889824i \(0.650827\pi\)
\(348\) 8.00000 0.428845
\(349\) 24.0000 1.28469 0.642345 0.766415i \(-0.277962\pi\)
0.642345 + 0.766415i \(0.277962\pi\)
\(350\) 0 0
\(351\) −7.00000 −0.373632
\(352\) 5.00000 0.266501
\(353\) 11.0000 0.585471 0.292735 0.956193i \(-0.405434\pi\)
0.292735 + 0.956193i \(0.405434\pi\)
\(354\) 10.0000 0.531494
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −2.00000 −0.105851
\(358\) 19.0000 1.00418
\(359\) −17.0000 −0.897226 −0.448613 0.893726i \(-0.648082\pi\)
−0.448613 + 0.893726i \(0.648082\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 18.0000 0.946059
\(363\) −14.0000 −0.734809
\(364\) 14.0000 0.733799
\(365\) 0 0
\(366\) −1.00000 −0.0522708
\(367\) 11.0000 0.574195 0.287098 0.957901i \(-0.407310\pi\)
0.287098 + 0.957901i \(0.407310\pi\)
\(368\) −4.00000 −0.208514
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 1.00000 0.0518476
\(373\) 32.0000 1.65690 0.828449 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 5.00000 0.258544
\(375\) 0 0
\(376\) 1.00000 0.0515711
\(377\) −56.0000 −2.88415
\(378\) −2.00000 −0.102869
\(379\) 1.00000 0.0513665 0.0256833 0.999670i \(-0.491824\pi\)
0.0256833 + 0.999670i \(0.491824\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) 10.0000 0.510976 0.255488 0.966812i \(-0.417764\pi\)
0.255488 + 0.966812i \(0.417764\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −19.0000 −0.967075
\(387\) 10.0000 0.508329
\(388\) 3.00000 0.152302
\(389\) −34.0000 −1.72387 −0.861934 0.507020i \(-0.830747\pi\)
−0.861934 + 0.507020i \(0.830747\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) −3.00000 −0.151523
\(393\) −10.0000 −0.504433
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) 5.00000 0.251259
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −11.0000 −0.551380
\(399\) −14.0000 −0.700877
\(400\) 0 0
\(401\) −23.0000 −1.14857 −0.574283 0.818657i \(-0.694719\pi\)
−0.574283 + 0.818657i \(0.694719\pi\)
\(402\) −3.00000 −0.149626
\(403\) −7.00000 −0.348695
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) −16.0000 −0.794067
\(407\) 30.0000 1.48704
\(408\) −1.00000 −0.0495074
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) −10.0000 −0.492665
\(413\) −20.0000 −0.984136
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 7.00000 0.343203
\(417\) 8.00000 0.391762
\(418\) 35.0000 1.71191
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) 1.00000 0.0486217
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −3.00000 −0.145350
\(427\) 2.00000 0.0967868
\(428\) −8.00000 −0.386695
\(429\) −35.0000 −1.68982
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −28.0000 −1.34559 −0.672797 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) −28.0000 −1.33942
\(438\) 14.0000 0.668946
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 7.00000 0.332956
\(443\) 14.0000 0.665160 0.332580 0.943075i \(-0.392081\pi\)
0.332580 + 0.943075i \(0.392081\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) 5.00000 0.236757
\(447\) −15.0000 −0.709476
\(448\) 2.00000 0.0944911
\(449\) −11.0000 −0.519122 −0.259561 0.965727i \(-0.583578\pi\)
−0.259561 + 0.965727i \(0.583578\pi\)
\(450\) 0 0
\(451\) −10.0000 −0.470882
\(452\) −6.00000 −0.282216
\(453\) 1.00000 0.0469841
\(454\) −22.0000 −1.03251
\(455\) 0 0
\(456\) −7.00000 −0.327805
\(457\) 42.0000 1.96468 0.982339 0.187112i \(-0.0599128\pi\)
0.982339 + 0.187112i \(0.0599128\pi\)
\(458\) 13.0000 0.607450
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) −10.0000 −0.465242
\(463\) −33.0000 −1.53364 −0.766820 0.641862i \(-0.778162\pi\)
−0.766820 + 0.641862i \(0.778162\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) −20.0000 −0.926482
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 7.00000 0.323575
\(469\) 6.00000 0.277054
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) −10.0000 −0.460287
\(473\) 50.0000 2.29900
\(474\) 11.0000 0.505247
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) −6.00000 −0.274721
\(478\) −4.00000 −0.182956
\(479\) 29.0000 1.32504 0.662522 0.749043i \(-0.269486\pi\)
0.662522 + 0.749043i \(0.269486\pi\)
\(480\) 0 0
\(481\) 42.0000 1.91504
\(482\) −14.0000 −0.637683
\(483\) 8.00000 0.364013
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 27.0000 1.22349 0.611743 0.791056i \(-0.290469\pi\)
0.611743 + 0.791056i \(0.290469\pi\)
\(488\) 1.00000 0.0452679
\(489\) −9.00000 −0.406994
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 2.00000 0.0901670
\(493\) −8.00000 −0.360302
\(494\) 49.0000 2.20461
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 6.00000 0.269137
\(498\) 7.00000 0.313678
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 0 0
\(501\) 10.0000 0.446767
\(502\) 20.0000 0.892644
\(503\) 33.0000 1.47140 0.735699 0.677309i \(-0.236854\pi\)
0.735699 + 0.677309i \(0.236854\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) −20.0000 −0.889108
\(507\) −36.0000 −1.59882
\(508\) −4.00000 −0.177471
\(509\) 40.0000 1.77297 0.886484 0.462758i \(-0.153140\pi\)
0.886484 + 0.462758i \(0.153140\pi\)
\(510\) 0 0
\(511\) −28.0000 −1.23865
\(512\) 1.00000 0.0441942
\(513\) −7.00000 −0.309058
\(514\) 12.0000 0.529297
\(515\) 0 0
\(516\) −10.0000 −0.440225
\(517\) 5.00000 0.219900
\(518\) 12.0000 0.527250
\(519\) 1.00000 0.0438951
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) −8.00000 −0.350150
\(523\) −30.0000 −1.31181 −0.655904 0.754844i \(-0.727712\pi\)
−0.655904 + 0.754844i \(0.727712\pi\)
\(524\) 10.0000 0.436852
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) −1.00000 −0.0435607
\(528\) −5.00000 −0.217597
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 14.0000 0.606977
\(533\) −14.0000 −0.606407
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 3.00000 0.129580
\(537\) −19.0000 −0.819911
\(538\) 6.00000 0.258678
\(539\) −15.0000 −0.646096
\(540\) 0 0
\(541\) 44.0000 1.89171 0.945854 0.324593i \(-0.105227\pi\)
0.945854 + 0.324593i \(0.105227\pi\)
\(542\) −19.0000 −0.816120
\(543\) −18.0000 −0.772454
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) −14.0000 −0.599145
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) 18.0000 0.768922
\(549\) 1.00000 0.0426790
\(550\) 0 0
\(551\) −56.0000 −2.38568
\(552\) 4.00000 0.170251
\(553\) −22.0000 −0.935535
\(554\) 7.00000 0.297402
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) −1.00000 −0.0423334
\(559\) 70.0000 2.96068
\(560\) 0 0
\(561\) −5.00000 −0.211100
\(562\) −30.0000 −1.26547
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −1.00000 −0.0421076
\(565\) 0 0
\(566\) −5.00000 −0.210166
\(567\) 2.00000 0.0839921
\(568\) 3.00000 0.125877
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 35.0000 1.46342
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −27.0000 −1.12402 −0.562012 0.827129i \(-0.689973\pi\)
−0.562012 + 0.827129i \(0.689973\pi\)
\(578\) −16.0000 −0.665512
\(579\) 19.0000 0.789613
\(580\) 0 0
\(581\) −14.0000 −0.580818
\(582\) −3.00000 −0.124354
\(583\) −30.0000 −1.24247
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 17.0000 0.701665 0.350833 0.936438i \(-0.385899\pi\)
0.350833 + 0.936438i \(0.385899\pi\)
\(588\) 3.00000 0.123718
\(589\) −7.00000 −0.288430
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 6.00000 0.246598
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) −5.00000 −0.205152
\(595\) 0 0
\(596\) 15.0000 0.614424
\(597\) 11.0000 0.450200
\(598\) −28.0000 −1.14501
\(599\) −39.0000 −1.59350 −0.796748 0.604311i \(-0.793448\pi\)
−0.796748 + 0.604311i \(0.793448\pi\)
\(600\) 0 0
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 20.0000 0.815139
\(603\) 3.00000 0.122169
\(604\) −1.00000 −0.0406894
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) 7.00000 0.283887
\(609\) 16.0000 0.648353
\(610\) 0 0
\(611\) 7.00000 0.283190
\(612\) 1.00000 0.0404226
\(613\) −5.00000 −0.201948 −0.100974 0.994889i \(-0.532196\pi\)
−0.100974 + 0.994889i \(0.532196\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) 10.0000 0.402911
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 10.0000 0.402259
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 21.0000 0.842023
\(623\) −12.0000 −0.480770
\(624\) −7.00000 −0.280224
\(625\) 0 0
\(626\) −2.00000 −0.0799361
\(627\) −35.0000 −1.39777
\(628\) 4.00000 0.159617
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −11.0000 −0.437557
\(633\) 0 0
\(634\) −5.00000 −0.198575
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) −21.0000 −0.832050
\(638\) −40.0000 −1.58362
\(639\) 3.00000 0.118678
\(640\) 0 0
\(641\) 23.0000 0.908445 0.454223 0.890888i \(-0.349917\pi\)
0.454223 + 0.890888i \(0.349917\pi\)
\(642\) 8.00000 0.315735
\(643\) 32.0000 1.26196 0.630978 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 7.00000 0.275411
\(647\) −14.0000 −0.550397 −0.275198 0.961387i \(-0.588744\pi\)
−0.275198 + 0.961387i \(0.588744\pi\)
\(648\) 1.00000 0.0392837
\(649\) −50.0000 −1.96267
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 9.00000 0.352467
\(653\) −27.0000 −1.05659 −0.528296 0.849060i \(-0.677169\pi\)
−0.528296 + 0.849060i \(0.677169\pi\)
\(654\) 6.00000 0.234619
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) −14.0000 −0.546192
\(658\) 2.00000 0.0779681
\(659\) −26.0000 −1.01282 −0.506408 0.862294i \(-0.669027\pi\)
−0.506408 + 0.862294i \(0.669027\pi\)
\(660\) 0 0
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) 0 0
\(663\) −7.00000 −0.271857
\(664\) −7.00000 −0.271653
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 32.0000 1.23904
\(668\) −10.0000 −0.386912
\(669\) −5.00000 −0.193311
\(670\) 0 0
\(671\) 5.00000 0.193023
\(672\) −2.00000 −0.0771517
\(673\) −8.00000 −0.308377 −0.154189 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) 8.00000 0.308148
\(675\) 0 0
\(676\) 36.0000 1.38462
\(677\) −36.0000 −1.38359 −0.691796 0.722093i \(-0.743180\pi\)
−0.691796 + 0.722093i \(0.743180\pi\)
\(678\) 6.00000 0.230429
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 22.0000 0.843042
\(682\) −5.00000 −0.191460
\(683\) −16.0000 −0.612223 −0.306111 0.951996i \(-0.599028\pi\)
−0.306111 + 0.951996i \(0.599028\pi\)
\(684\) 7.00000 0.267652
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) −13.0000 −0.495981
\(688\) 10.0000 0.381246
\(689\) −42.0000 −1.60007
\(690\) 0 0
\(691\) 13.0000 0.494543 0.247272 0.968946i \(-0.420466\pi\)
0.247272 + 0.968946i \(0.420466\pi\)
\(692\) −1.00000 −0.0380143
\(693\) 10.0000 0.379869
\(694\) −17.0000 −0.645311
\(695\) 0 0
\(696\) 8.00000 0.303239
\(697\) −2.00000 −0.0757554
\(698\) 24.0000 0.908413
\(699\) 20.0000 0.756469
\(700\) 0 0
\(701\) 35.0000 1.32193 0.660966 0.750416i \(-0.270147\pi\)
0.660966 + 0.750416i \(0.270147\pi\)
\(702\) −7.00000 −0.264198
\(703\) 42.0000 1.58406
\(704\) 5.00000 0.188445
\(705\) 0 0
\(706\) 11.0000 0.413990
\(707\) 12.0000 0.451306
\(708\) 10.0000 0.375823
\(709\) −21.0000 −0.788672 −0.394336 0.918966i \(-0.629025\pi\)
−0.394336 + 0.918966i \(0.629025\pi\)
\(710\) 0 0
\(711\) −11.0000 −0.412532
\(712\) −6.00000 −0.224860
\(713\) 4.00000 0.149801
\(714\) −2.00000 −0.0748481
\(715\) 0 0
\(716\) 19.0000 0.710063
\(717\) 4.00000 0.149383
\(718\) −17.0000 −0.634434
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) −20.0000 −0.744839
\(722\) 30.0000 1.11648
\(723\) 14.0000 0.520666
\(724\) 18.0000 0.668965
\(725\) 0 0
\(726\) −14.0000 −0.519589
\(727\) −4.00000 −0.148352 −0.0741759 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) 14.0000 0.518875
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 10.0000 0.369863
\(732\) −1.00000 −0.0369611
\(733\) 30.0000 1.10808 0.554038 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 11.0000 0.406017
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 15.0000 0.552532
\(738\) −2.00000 −0.0736210
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) −49.0000 −1.80006
\(742\) −12.0000 −0.440534
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) −7.00000 −0.256117
\(748\) 5.00000 0.182818
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) −38.0000 −1.38664 −0.693320 0.720630i \(-0.743853\pi\)
−0.693320 + 0.720630i \(0.743853\pi\)
\(752\) 1.00000 0.0364662
\(753\) −20.0000 −0.728841
\(754\) −56.0000 −2.03940
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 1.00000 0.0363216
\(759\) 20.0000 0.725954
\(760\) 0 0
\(761\) 1.00000 0.0362500 0.0181250 0.999836i \(-0.494230\pi\)
0.0181250 + 0.999836i \(0.494230\pi\)
\(762\) 4.00000 0.144905
\(763\) −12.0000 −0.434429
\(764\) 0 0
\(765\) 0 0
\(766\) 10.0000 0.361315
\(767\) −70.0000 −2.52755
\(768\) −1.00000 −0.0360844
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) −19.0000 −0.683825
\(773\) 16.0000 0.575480 0.287740 0.957709i \(-0.407096\pi\)
0.287740 + 0.957709i \(0.407096\pi\)
\(774\) 10.0000 0.359443
\(775\) 0 0
\(776\) 3.00000 0.107694
\(777\) −12.0000 −0.430498
\(778\) −34.0000 −1.21896
\(779\) −14.0000 −0.501602
\(780\) 0 0
\(781\) 15.0000 0.536742
\(782\) −4.00000 −0.143040
\(783\) 8.00000 0.285897
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −10.0000 −0.356688
\(787\) −40.0000 −1.42585 −0.712923 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) 2.00000 0.0712470
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 5.00000 0.177667
\(793\) 7.00000 0.248577
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) −11.0000 −0.389885
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) −14.0000 −0.495595
\(799\) 1.00000 0.0353775
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) −23.0000 −0.812158
\(803\) −70.0000 −2.47025
\(804\) −3.00000 −0.105802
\(805\) 0 0
\(806\) −7.00000 −0.246564
\(807\) −6.00000 −0.211210
\(808\) 6.00000 0.211079
\(809\) −29.0000 −1.01959 −0.509793 0.860297i \(-0.670278\pi\)
−0.509793 + 0.860297i \(0.670278\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −16.0000 −0.561490
\(813\) 19.0000 0.666359
\(814\) 30.0000 1.05150
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) 70.0000 2.44899
\(818\) −30.0000 −1.04893
\(819\) 14.0000 0.489200
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) −18.0000 −0.627822
\(823\) −43.0000 −1.49889 −0.749443 0.662069i \(-0.769679\pi\)
−0.749443 + 0.662069i \(0.769679\pi\)
\(824\) −10.0000 −0.348367
\(825\) 0 0
\(826\) −20.0000 −0.695889
\(827\) −27.0000 −0.938882 −0.469441 0.882964i \(-0.655545\pi\)
−0.469441 + 0.882964i \(0.655545\pi\)
\(828\) −4.00000 −0.139010
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) −7.00000 −0.242827
\(832\) 7.00000 0.242681
\(833\) −3.00000 −0.103944
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) 35.0000 1.21050
\(837\) 1.00000 0.0345651
\(838\) −4.00000 −0.138178
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 2.00000 0.0689246
\(843\) 30.0000 1.03325
\(844\) 0 0
\(845\) 0 0
\(846\) 1.00000 0.0343807
\(847\) 28.0000 0.962091
\(848\) −6.00000 −0.206041
\(849\) 5.00000 0.171600
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) −3.00000 −0.102778
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) 8.00000 0.273275 0.136637 0.990621i \(-0.456370\pi\)
0.136637 + 0.990621i \(0.456370\pi\)
\(858\) −35.0000 −1.19488
\(859\) −26.0000 −0.887109 −0.443554 0.896248i \(-0.646283\pi\)
−0.443554 + 0.896248i \(0.646283\pi\)
\(860\) 0 0
\(861\) 4.00000 0.136320
\(862\) 24.0000 0.817443
\(863\) −10.0000 −0.340404 −0.170202 0.985409i \(-0.554442\pi\)
−0.170202 + 0.985409i \(0.554442\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −28.0000 −0.951479
\(867\) 16.0000 0.543388
\(868\) −2.00000 −0.0678844
\(869\) −55.0000 −1.86575
\(870\) 0 0
\(871\) 21.0000 0.711558
\(872\) −6.00000 −0.203186
\(873\) 3.00000 0.101535
\(874\) −28.0000 −0.947114
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) −28.0000 −0.945493 −0.472746 0.881199i \(-0.656737\pi\)
−0.472746 + 0.881199i \(0.656737\pi\)
\(878\) 32.0000 1.07995
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 51.0000 1.71823 0.859117 0.511780i \(-0.171014\pi\)
0.859117 + 0.511780i \(0.171014\pi\)
\(882\) −3.00000 −0.101015
\(883\) 46.0000 1.54802 0.774012 0.633171i \(-0.218247\pi\)
0.774012 + 0.633171i \(0.218247\pi\)
\(884\) 7.00000 0.235435
\(885\) 0 0
\(886\) 14.0000 0.470339
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) −6.00000 −0.201347
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 5.00000 0.167412
\(893\) 7.00000 0.234246
\(894\) −15.0000 −0.501675
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 28.0000 0.934893
\(898\) −11.0000 −0.367075
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) −6.00000 −0.199889
\(902\) −10.0000 −0.332964
\(903\) −20.0000 −0.665558
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 1.00000 0.0332228
\(907\) 1.00000 0.0332045 0.0166022 0.999862i \(-0.494715\pi\)
0.0166022 + 0.999862i \(0.494715\pi\)
\(908\) −22.0000 −0.730096
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) −7.00000 −0.231793
\(913\) −35.0000 −1.15833
\(914\) 42.0000 1.38924
\(915\) 0 0
\(916\) 13.0000 0.429532
\(917\) 20.0000 0.660458
\(918\) −1.00000 −0.0330049
\(919\) 28.0000 0.923635 0.461817 0.886975i \(-0.347198\pi\)
0.461817 + 0.886975i \(0.347198\pi\)
\(920\) 0 0
\(921\) 16.0000 0.527218
\(922\) −34.0000 −1.11973
\(923\) 21.0000 0.691223
\(924\) −10.0000 −0.328976
\(925\) 0 0
\(926\) −33.0000 −1.08445
\(927\) −10.0000 −0.328443
\(928\) −8.00000 −0.262613
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) −21.0000 −0.688247
\(932\) −20.0000 −0.655122
\(933\) −21.0000 −0.687509
\(934\) −24.0000 −0.785304
\(935\) 0 0
\(936\) 7.00000 0.228802
\(937\) −1.00000 −0.0326686 −0.0163343 0.999867i \(-0.505200\pi\)
−0.0163343 + 0.999867i \(0.505200\pi\)
\(938\) 6.00000 0.195907
\(939\) 2.00000 0.0652675
\(940\) 0 0
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) −4.00000 −0.130327
\(943\) 8.00000 0.260516
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) 50.0000 1.62564
\(947\) 5.00000 0.162478 0.0812391 0.996695i \(-0.474112\pi\)
0.0812391 + 0.996695i \(0.474112\pi\)
\(948\) 11.0000 0.357263
\(949\) −98.0000 −3.18121
\(950\) 0 0
\(951\) 5.00000 0.162136
\(952\) 2.00000 0.0648204
\(953\) 5.00000 0.161966 0.0809829 0.996715i \(-0.474194\pi\)
0.0809829 + 0.996715i \(0.474194\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −4.00000 −0.129369
\(957\) 40.0000 1.29302
\(958\) 29.0000 0.936947
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 42.0000 1.35413
\(963\) −8.00000 −0.257796
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) −33.0000 −1.06121 −0.530604 0.847620i \(-0.678035\pi\)
−0.530604 + 0.847620i \(0.678035\pi\)
\(968\) 14.0000 0.449977
\(969\) −7.00000 −0.224872
\(970\) 0 0
\(971\) 34.0000 1.09111 0.545556 0.838074i \(-0.316319\pi\)
0.545556 + 0.838074i \(0.316319\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −16.0000 −0.512936
\(974\) 27.0000 0.865136
\(975\) 0 0
\(976\) 1.00000 0.0320092
\(977\) 22.0000 0.703842 0.351921 0.936030i \(-0.385529\pi\)
0.351921 + 0.936030i \(0.385529\pi\)
\(978\) −9.00000 −0.287788
\(979\) −30.0000 −0.958804
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) −20.0000 −0.638226
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) −2.00000 −0.0636607
\(988\) 49.0000 1.55890
\(989\) −40.0000 −1.27193
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 0 0
\(994\) 6.00000 0.190308
\(995\) 0 0
\(996\) 7.00000 0.221803
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) −24.0000 −0.759707
\(999\) −6.00000 −0.189832
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 4650.2.a.bh.1.1 1
5.2 odd 4 4650.2.d.k.3349.2 2
5.3 odd 4 4650.2.d.k.3349.1 2
5.4 even 2 186.2.a.b.1.1 1
15.14 odd 2 558.2.a.f.1.1 1
20.19 odd 2 1488.2.a.h.1.1 1
35.34 odd 2 9114.2.a.b.1.1 1
40.19 odd 2 5952.2.a.s.1.1 1
40.29 even 2 5952.2.a.b.1.1 1
60.59 even 2 4464.2.a.c.1.1 1
155.154 odd 2 5766.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
186.2.a.b.1.1 1 5.4 even 2
558.2.a.f.1.1 1 15.14 odd 2
1488.2.a.h.1.1 1 20.19 odd 2
4464.2.a.c.1.1 1 60.59 even 2
4650.2.a.bh.1.1 1 1.1 even 1 trivial
4650.2.d.k.3349.1 2 5.3 odd 4
4650.2.d.k.3349.2 2 5.2 odd 4
5766.2.a.c.1.1 1 155.154 odd 2
5952.2.a.b.1.1 1 40.29 even 2
5952.2.a.s.1.1 1 40.19 odd 2
9114.2.a.b.1.1 1 35.34 odd 2