Properties

Label 1122.2.a.n.1.1
Level $1122$
Weight $2$
Character 1122.1
Self dual yes
Analytic conductor $8.959$
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] = [1122,2,Mod(1,1122)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1122, 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("1122.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1122.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: \(8.95921510679\)
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 \(\chi\) \(=\) 1122.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{20} -2.00000 q^{21} +1.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} +1.00000 q^{27} -2.00000 q^{28} -2.00000 q^{29} +4.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -1.00000 q^{34} -8.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{40} -6.00000 q^{41} -2.00000 q^{42} -4.00000 q^{43} +1.00000 q^{44} +4.00000 q^{45} -6.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +11.0000 q^{50} -1.00000 q^{51} +8.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} -2.00000 q^{56} -2.00000 q^{58} +8.00000 q^{59} +4.00000 q^{60} -8.00000 q^{61} +4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} -4.00000 q^{67} -1.00000 q^{68} -6.00000 q^{69} -8.00000 q^{70} -6.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} +2.00000 q^{74} +11.0000 q^{75} -2.00000 q^{77} -6.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} -2.00000 q^{84} -4.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} +1.00000 q^{88} -14.0000 q^{89} +4.00000 q^{90} -6.00000 q^{92} +4.00000 q^{93} -6.00000 q^{94} +1.00000 q^{96} +14.0000 q^{97} -3.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.00000 1.26491
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −2.00000 −0.534522
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 4.00000 0.894427
\(21\) −2.00000 −0.436436
\(22\) 1.00000 0.213201
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000 0.204124
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 4.00000 0.730297
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) −1.00000 −0.171499
\(35\) −8.00000 −1.35225
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 4.00000 0.632456
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −2.00000 −0.308607
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 1.00000 0.150756
\(45\) 4.00000 0.596285
\(46\) −6.00000 −0.884652
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 11.0000 1.55563
\(51\) −1.00000 −0.140028
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.00000 0.539360
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 4.00000 0.516398
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 4.00000 0.508001
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −1.00000 −0.121268
\(69\) −6.00000 −0.722315
\(70\) −8.00000 −0.956183
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 2.00000 0.232495
\(75\) 11.0000 1.27017
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −2.00000 −0.218218
\(85\) −4.00000 −0.433861
\(86\) −4.00000 −0.431331
\(87\) −2.00000 −0.214423
\(88\) 1.00000 0.106600
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 4.00000 0.421637
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) 4.00000 0.414781
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −3.00000 −0.303046
\(99\) 1.00000 0.100504
\(100\) 11.0000 1.10000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) 8.00000 0.777029
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 1.00000 0.0962250
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 4.00000 0.381385
\(111\) 2.00000 0.189832
\(112\) −2.00000 −0.188982
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) −24.0000 −2.23801
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 8.00000 0.736460
\(119\) 2.00000 0.183340
\(120\) 4.00000 0.365148
\(121\) 1.00000 0.0909091
\(122\) −8.00000 −0.724286
\(123\) −6.00000 −0.541002
\(124\) 4.00000 0.359211
\(125\) 24.0000 2.14663
\(126\) −2.00000 −0.178174
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 4.00000 0.344265
\(136\) −1.00000 −0.0857493
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −6.00000 −0.510754
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) −8.00000 −0.676123
\(141\) −6.00000 −0.505291
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −8.00000 −0.664364
\(146\) 10.0000 0.827606
\(147\) −3.00000 −0.247436
\(148\) 2.00000 0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 11.0000 0.898146
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) 0 0
\(153\) −1.00000 −0.0808452
\(154\) −2.00000 −0.161165
\(155\) 16.0000 1.28515
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −6.00000 −0.477334
\(159\) 8.00000 0.634441
\(160\) 4.00000 0.316228
\(161\) 12.0000 0.945732
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −6.00000 −0.468521
\(165\) 4.00000 0.311400
\(166\) −4.00000 −0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −2.00000 −0.154303
\(169\) −13.0000 −1.00000
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) −2.00000 −0.151620
\(175\) −22.0000 −1.66304
\(176\) 1.00000 0.0753778
\(177\) 8.00000 0.601317
\(178\) −14.0000 −1.04934
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 4.00000 0.298142
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) −6.00000 −0.442326
\(185\) 8.00000 0.588172
\(186\) 4.00000 0.293294
\(187\) −1.00000 −0.0731272
\(188\) −6.00000 −0.437595
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 1.00000 0.0721688
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 1.00000 0.0710669
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 11.0000 0.777817
\(201\) −4.00000 −0.282138
\(202\) 18.0000 1.26648
\(203\) 4.00000 0.280745
\(204\) −1.00000 −0.0700140
\(205\) −24.0000 −1.67623
\(206\) −12.0000 −0.836080
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) −8.00000 −0.552052
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 8.00000 0.549442
\(213\) −6.00000 −0.411113
\(214\) 4.00000 0.273434
\(215\) −16.0000 −1.09119
\(216\) 1.00000 0.0680414
\(217\) −8.00000 −0.543075
\(218\) −12.0000 −0.812743
\(219\) 10.0000 0.675737
\(220\) 4.00000 0.269680
\(221\) 0 0
\(222\) 2.00000 0.134231
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) −2.00000 −0.133631
\(225\) 11.0000 0.733333
\(226\) 18.0000 1.19734
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) −24.0000 −1.58251
\(231\) −2.00000 −0.131590
\(232\) −2.00000 −0.131306
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) 8.00000 0.520756
\(237\) −6.00000 −0.389742
\(238\) 2.00000 0.129641
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 4.00000 0.258199
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) −8.00000 −0.512148
\(245\) −12.0000 −0.766652
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) −4.00000 −0.253490
\(250\) 24.0000 1.51789
\(251\) −16.0000 −1.00991 −0.504956 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) −2.00000 −0.125988
\(253\) −6.00000 −0.377217
\(254\) −2.00000 −0.125491
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −4.00000 −0.249029
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) −4.00000 −0.247121
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 1.00000 0.0615457
\(265\) 32.0000 1.96574
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) −4.00000 −0.244339
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 4.00000 0.243432
\(271\) −14.0000 −0.850439 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 11.0000 0.663325
\(276\) −6.00000 −0.361158
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) −8.00000 −0.479808
\(279\) 4.00000 0.239474
\(280\) −8.00000 −0.478091
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −6.00000 −0.357295
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) −8.00000 −0.469776
\(291\) 14.0000 0.820695
\(292\) 10.0000 0.585206
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) −3.00000 −0.174964
\(295\) 32.0000 1.86311
\(296\) 2.00000 0.116248
\(297\) 1.00000 0.0580259
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 11.0000 0.635085
\(301\) 8.00000 0.461112
\(302\) 6.00000 0.345261
\(303\) 18.0000 1.03407
\(304\) 0 0
\(305\) −32.0000 −1.83231
\(306\) −1.00000 −0.0571662
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) −2.00000 −0.113961
\(309\) −12.0000 −0.682656
\(310\) 16.0000 0.908739
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 10.0000 0.564333
\(315\) −8.00000 −0.450749
\(316\) −6.00000 −0.337526
\(317\) 8.00000 0.449325 0.224662 0.974437i \(-0.427872\pi\)
0.224662 + 0.974437i \(0.427872\pi\)
\(318\) 8.00000 0.448618
\(319\) −2.00000 −0.111979
\(320\) 4.00000 0.223607
\(321\) 4.00000 0.223258
\(322\) 12.0000 0.668734
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) −12.0000 −0.663602
\(328\) −6.00000 −0.331295
\(329\) 12.0000 0.661581
\(330\) 4.00000 0.220193
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −4.00000 −0.219529
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) −16.0000 −0.874173
\(336\) −2.00000 −0.109109
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) −13.0000 −0.707107
\(339\) 18.0000 0.977626
\(340\) −4.00000 −0.216930
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) −4.00000 −0.215666
\(345\) −24.0000 −1.29212
\(346\) −2.00000 −0.107521
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −2.00000 −0.107211
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) −22.0000 −1.17595
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 34.0000 1.80964 0.904819 0.425797i \(-0.140006\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) 8.00000 0.425195
\(355\) −24.0000 −1.27379
\(356\) −14.0000 −0.741999
\(357\) 2.00000 0.105851
\(358\) −24.0000 −1.26844
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 4.00000 0.210819
\(361\) −19.0000 −1.00000
\(362\) −18.0000 −0.946059
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 40.0000 2.09370
\(366\) −8.00000 −0.418167
\(367\) −12.0000 −0.626395 −0.313197 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) −6.00000 −0.312772
\(369\) −6.00000 −0.312348
\(370\) 8.00000 0.415900
\(371\) −16.0000 −0.830679
\(372\) 4.00000 0.207390
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) −1.00000 −0.0517088
\(375\) 24.0000 1.23935
\(376\) −6.00000 −0.309426
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) 18.0000 0.920960
\(383\) 30.0000 1.53293 0.766464 0.642287i \(-0.222014\pi\)
0.766464 + 0.642287i \(0.222014\pi\)
\(384\) 1.00000 0.0510310
\(385\) −8.00000 −0.407718
\(386\) 22.0000 1.11977
\(387\) −4.00000 −0.203331
\(388\) 14.0000 0.710742
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) −3.00000 −0.151523
\(393\) −4.00000 −0.201773
\(394\) −18.0000 −0.906827
\(395\) −24.0000 −1.20757
\(396\) 1.00000 0.0502519
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) 14.0000 0.699127 0.349563 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) 18.0000 0.895533
\(405\) 4.00000 0.198762
\(406\) 4.00000 0.198517
\(407\) 2.00000 0.0991363
\(408\) −1.00000 −0.0495074
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) −24.0000 −1.18528
\(411\) 6.00000 0.295958
\(412\) −12.0000 −0.591198
\(413\) −16.0000 −0.787309
\(414\) −6.00000 −0.294884
\(415\) −16.0000 −0.785409
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) −8.00000 −0.390360
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −12.0000 −0.584151
\(423\) −6.00000 −0.291730
\(424\) 8.00000 0.388514
\(425\) −11.0000 −0.533578
\(426\) −6.00000 −0.290701
\(427\) 16.0000 0.774294
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) −16.0000 −0.771589
\(431\) 20.0000 0.963366 0.481683 0.876346i \(-0.340026\pi\)
0.481683 + 0.876346i \(0.340026\pi\)
\(432\) 1.00000 0.0481125
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) −8.00000 −0.384012
\(435\) −8.00000 −0.383571
\(436\) −12.0000 −0.574696
\(437\) 0 0
\(438\) 10.0000 0.477818
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 4.00000 0.190693
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 2.00000 0.0949158
\(445\) −56.0000 −2.65465
\(446\) −24.0000 −1.13643
\(447\) 6.00000 0.283790
\(448\) −2.00000 −0.0944911
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 11.0000 0.518545
\(451\) −6.00000 −0.282529
\(452\) 18.0000 0.846649
\(453\) 6.00000 0.281905
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) −34.0000 −1.59045 −0.795226 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) 2.00000 0.0934539
\(459\) −1.00000 −0.0466760
\(460\) −24.0000 −1.11901
\(461\) 38.0000 1.76984 0.884918 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(462\) −2.00000 −0.0930484
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 16.0000 0.741982
\(466\) −22.0000 −1.01913
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) −24.0000 −1.10704
\(471\) 10.0000 0.460776
\(472\) 8.00000 0.368230
\(473\) −4.00000 −0.183920
\(474\) −6.00000 −0.275589
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) 8.00000 0.366295
\(478\) −4.00000 −0.182956
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 4.00000 0.182574
\(481\) 0 0
\(482\) −26.0000 −1.18427
\(483\) 12.0000 0.546019
\(484\) 1.00000 0.0454545
\(485\) 56.0000 2.54283
\(486\) 1.00000 0.0453609
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) −8.00000 −0.362143
\(489\) 4.00000 0.180886
\(490\) −12.0000 −0.542105
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) −6.00000 −0.270501
\(493\) 2.00000 0.0900755
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 4.00000 0.179605
\(497\) 12.0000 0.538274
\(498\) −4.00000 −0.179244
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 24.0000 1.07331
\(501\) 0 0
\(502\) −16.0000 −0.714115
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 72.0000 3.20396
\(506\) −6.00000 −0.266733
\(507\) −13.0000 −0.577350
\(508\) −2.00000 −0.0887357
\(509\) −8.00000 −0.354594 −0.177297 0.984157i \(-0.556735\pi\)
−0.177297 + 0.984157i \(0.556735\pi\)
\(510\) −4.00000 −0.177123
\(511\) −20.0000 −0.884748
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) −48.0000 −2.11513
\(516\) −4.00000 −0.176090
\(517\) −6.00000 −0.263880
\(518\) −4.00000 −0.175750
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) −4.00000 −0.174741
\(525\) −22.0000 −0.960159
\(526\) 8.00000 0.348817
\(527\) −4.00000 −0.174243
\(528\) 1.00000 0.0435194
\(529\) 13.0000 0.565217
\(530\) 32.0000 1.38999
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 0 0
\(534\) −14.0000 −0.605839
\(535\) 16.0000 0.691740
\(536\) −4.00000 −0.172774
\(537\) −24.0000 −1.03568
\(538\) −12.0000 −0.517357
\(539\) −3.00000 −0.129219
\(540\) 4.00000 0.172133
\(541\) −12.0000 −0.515920 −0.257960 0.966156i \(-0.583050\pi\)
−0.257960 + 0.966156i \(0.583050\pi\)
\(542\) −14.0000 −0.601351
\(543\) −18.0000 −0.772454
\(544\) −1.00000 −0.0428746
\(545\) −48.0000 −2.05609
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 6.00000 0.256307
\(549\) −8.00000 −0.341432
\(550\) 11.0000 0.469042
\(551\) 0 0
\(552\) −6.00000 −0.255377
\(553\) 12.0000 0.510292
\(554\) −16.0000 −0.679775
\(555\) 8.00000 0.339581
\(556\) −8.00000 −0.339276
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 4.00000 0.169334
\(559\) 0 0
\(560\) −8.00000 −0.338062
\(561\) −1.00000 −0.0422200
\(562\) 6.00000 0.253095
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) −6.00000 −0.252646
\(565\) 72.0000 3.02906
\(566\) −4.00000 −0.168133
\(567\) −2.00000 −0.0839921
\(568\) −6.00000 −0.251754
\(569\) 38.0000 1.59304 0.796521 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 0 0
\(571\) −24.0000 −1.00437 −0.502184 0.864761i \(-0.667470\pi\)
−0.502184 + 0.864761i \(0.667470\pi\)
\(572\) 0 0
\(573\) 18.0000 0.751961
\(574\) 12.0000 0.500870
\(575\) −66.0000 −2.75239
\(576\) 1.00000 0.0416667
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 1.00000 0.0415945
\(579\) 22.0000 0.914289
\(580\) −8.00000 −0.332182
\(581\) 8.00000 0.331896
\(582\) 14.0000 0.580319
\(583\) 8.00000 0.331326
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −3.00000 −0.123718
\(589\) 0 0
\(590\) 32.0000 1.31742
\(591\) −18.0000 −0.740421
\(592\) 2.00000 0.0821995
\(593\) 46.0000 1.88899 0.944497 0.328521i \(-0.106550\pi\)
0.944497 + 0.328521i \(0.106550\pi\)
\(594\) 1.00000 0.0410305
\(595\) 8.00000 0.327968
\(596\) 6.00000 0.245770
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 14.0000 0.572024 0.286012 0.958226i \(-0.407670\pi\)
0.286012 + 0.958226i \(0.407670\pi\)
\(600\) 11.0000 0.449073
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 8.00000 0.326056
\(603\) −4.00000 −0.162893
\(604\) 6.00000 0.244137
\(605\) 4.00000 0.162623
\(606\) 18.0000 0.731200
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) −32.0000 −1.29564
\(611\) 0 0
\(612\) −1.00000 −0.0404226
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) 8.00000 0.322854
\(615\) −24.0000 −0.967773
\(616\) −2.00000 −0.0805823
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −12.0000 −0.482711
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 16.0000 0.642575
\(621\) −6.00000 −0.240772
\(622\) 18.0000 0.721734
\(623\) 28.0000 1.12180
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) −2.00000 −0.0797452
\(630\) −8.00000 −0.318728
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −6.00000 −0.238667
\(633\) −12.0000 −0.476957
\(634\) 8.00000 0.317721
\(635\) −8.00000 −0.317470
\(636\) 8.00000 0.317221
\(637\) 0 0
\(638\) −2.00000 −0.0791808
\(639\) −6.00000 −0.237356
\(640\) 4.00000 0.158114
\(641\) −26.0000 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) 4.00000 0.157867
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 12.0000 0.472866
\(645\) −16.0000 −0.629999
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 1.00000 0.0392837
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 4.00000 0.156652
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) −12.0000 −0.469237
\(655\) −16.0000 −0.625172
\(656\) −6.00000 −0.234261
\(657\) 10.0000 0.390137
\(658\) 12.0000 0.467809
\(659\) 44.0000 1.71400 0.856998 0.515319i \(-0.172327\pi\)
0.856998 + 0.515319i \(0.172327\pi\)
\(660\) 4.00000 0.155700
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 12.0000 0.464642
\(668\) 0 0
\(669\) −24.0000 −0.927894
\(670\) −16.0000 −0.618134
\(671\) −8.00000 −0.308837
\(672\) −2.00000 −0.0771517
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 34.0000 1.30963
\(675\) 11.0000 0.423390
\(676\) −13.0000 −0.500000
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 18.0000 0.691286
\(679\) −28.0000 −1.07454
\(680\) −4.00000 −0.153393
\(681\) 12.0000 0.459841
\(682\) 4.00000 0.153168
\(683\) −16.0000 −0.612223 −0.306111 0.951996i \(-0.599028\pi\)
−0.306111 + 0.951996i \(0.599028\pi\)
\(684\) 0 0
\(685\) 24.0000 0.916993
\(686\) 20.0000 0.763604
\(687\) 2.00000 0.0763048
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) −24.0000 −0.913664
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −2.00000 −0.0760286
\(693\) −2.00000 −0.0759737
\(694\) 12.0000 0.455514
\(695\) −32.0000 −1.21383
\(696\) −2.00000 −0.0758098
\(697\) 6.00000 0.227266
\(698\) 8.00000 0.302804
\(699\) −22.0000 −0.832116
\(700\) −22.0000 −0.831522
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) −24.0000 −0.903892
\(706\) 34.0000 1.27961
\(707\) −36.0000 −1.35392
\(708\) 8.00000 0.300658
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) −24.0000 −0.900704
\(711\) −6.00000 −0.225018
\(712\) −14.0000 −0.524672
\(713\) −24.0000 −0.898807
\(714\) 2.00000 0.0748481
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) −4.00000 −0.149383
\(718\) 36.0000 1.34351
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 4.00000 0.149071
\(721\) 24.0000 0.893807
\(722\) −19.0000 −0.707107
\(723\) −26.0000 −0.966950
\(724\) −18.0000 −0.668965
\(725\) −22.0000 −0.817059
\(726\) 1.00000 0.0371135
\(727\) 20.0000 0.741759 0.370879 0.928681i \(-0.379056\pi\)
0.370879 + 0.928681i \(0.379056\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 40.0000 1.48047
\(731\) 4.00000 0.147945
\(732\) −8.00000 −0.295689
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) −12.0000 −0.442928
\(735\) −12.0000 −0.442627
\(736\) −6.00000 −0.221163
\(737\) −4.00000 −0.147342
\(738\) −6.00000 −0.220863
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) −16.0000 −0.587378
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 4.00000 0.146647
\(745\) 24.0000 0.879292
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) −1.00000 −0.0365636
\(749\) −8.00000 −0.292314
\(750\) 24.0000 0.876356
\(751\) −36.0000 −1.31366 −0.656829 0.754039i \(-0.728103\pi\)
−0.656829 + 0.754039i \(0.728103\pi\)
\(752\) −6.00000 −0.218797
\(753\) −16.0000 −0.583072
\(754\) 0 0
\(755\) 24.0000 0.873449
\(756\) −2.00000 −0.0727393
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) −20.0000 −0.726433
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) −2.00000 −0.0724524
\(763\) 24.0000 0.868858
\(764\) 18.0000 0.651217
\(765\) −4.00000 −0.144620
\(766\) 30.0000 1.08394
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 54.0000 1.94729 0.973645 0.228069i \(-0.0732413\pi\)
0.973645 + 0.228069i \(0.0732413\pi\)
\(770\) −8.00000 −0.288300
\(771\) 18.0000 0.648254
\(772\) 22.0000 0.791797
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) −4.00000 −0.143777
\(775\) 44.0000 1.58053
\(776\) 14.0000 0.502571
\(777\) −4.00000 −0.143499
\(778\) −8.00000 −0.286814
\(779\) 0 0
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 6.00000 0.214560
\(783\) −2.00000 −0.0714742
\(784\) −3.00000 −0.107143
\(785\) 40.0000 1.42766
\(786\) −4.00000 −0.142675
\(787\) 36.0000 1.28326 0.641631 0.767014i \(-0.278258\pi\)
0.641631 + 0.767014i \(0.278258\pi\)
\(788\) −18.0000 −0.641223
\(789\) 8.00000 0.284808
\(790\) −24.0000 −0.853882
\(791\) −36.0000 −1.28001
\(792\) 1.00000 0.0355335
\(793\) 0 0
\(794\) −2.00000 −0.0709773
\(795\) 32.0000 1.13492
\(796\) 16.0000 0.567105
\(797\) −48.0000 −1.70025 −0.850124 0.526583i \(-0.823473\pi\)
−0.850124 + 0.526583i \(0.823473\pi\)
\(798\) 0 0
\(799\) 6.00000 0.212265
\(800\) 11.0000 0.388909
\(801\) −14.0000 −0.494666
\(802\) 14.0000 0.494357
\(803\) 10.0000 0.352892
\(804\) −4.00000 −0.141069
\(805\) 48.0000 1.69178
\(806\) 0 0
\(807\) −12.0000 −0.422420
\(808\) 18.0000 0.633238
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 4.00000 0.140546
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 4.00000 0.140372
\(813\) −14.0000 −0.491001
\(814\) 2.00000 0.0701000
\(815\) 16.0000 0.560456
\(816\) −1.00000 −0.0350070
\(817\) 0 0
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) −24.0000 −0.838116
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 6.00000 0.209274
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) −12.0000 −0.418040
\(825\) 11.0000 0.382971
\(826\) −16.0000 −0.556711
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −6.00000 −0.208514
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) −16.0000 −0.555368
\(831\) −16.0000 −0.555034
\(832\) 0 0
\(833\) 3.00000 0.103944
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) −16.0000 −0.552711
\(839\) −34.0000 −1.17381 −0.586905 0.809656i \(-0.699654\pi\)
−0.586905 + 0.809656i \(0.699654\pi\)
\(840\) −8.00000 −0.276026
\(841\) −25.0000 −0.862069
\(842\) 22.0000 0.758170
\(843\) 6.00000 0.206651
\(844\) −12.0000 −0.413057
\(845\) −52.0000 −1.78885
\(846\) −6.00000 −0.206284
\(847\) −2.00000 −0.0687208
\(848\) 8.00000 0.274721
\(849\) −4.00000 −0.137280
\(850\) −11.0000 −0.377297
\(851\) −12.0000 −0.411355
\(852\) −6.00000 −0.205557
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 16.0000 0.547509
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) −30.0000 −1.02478 −0.512390 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) −16.0000 −0.545595
\(861\) 12.0000 0.408959
\(862\) 20.0000 0.681203
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 1.00000 0.0340207
\(865\) −8.00000 −0.272008
\(866\) 10.0000 0.339814
\(867\) 1.00000 0.0339618
\(868\) −8.00000 −0.271538
\(869\) −6.00000 −0.203536
\(870\) −8.00000 −0.271225
\(871\) 0 0
\(872\) −12.0000 −0.406371
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) −48.0000 −1.62270
\(876\) 10.0000 0.337869
\(877\) −52.0000 −1.75592 −0.877958 0.478738i \(-0.841094\pi\)
−0.877958 + 0.478738i \(0.841094\pi\)
\(878\) −14.0000 −0.472477
\(879\) −18.0000 −0.607125
\(880\) 4.00000 0.134840
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) −3.00000 −0.101015
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 0 0
\(885\) 32.0000 1.07567
\(886\) 16.0000 0.537531
\(887\) −40.0000 −1.34307 −0.671534 0.740973i \(-0.734364\pi\)
−0.671534 + 0.740973i \(0.734364\pi\)
\(888\) 2.00000 0.0671156
\(889\) 4.00000 0.134156
\(890\) −56.0000 −1.87712
\(891\) 1.00000 0.0335013
\(892\) −24.0000 −0.803579
\(893\) 0 0
\(894\) 6.00000 0.200670
\(895\) −96.0000 −3.20893
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) −8.00000 −0.266815
\(900\) 11.0000 0.366667
\(901\) −8.00000 −0.266519
\(902\) −6.00000 −0.199778
\(903\) 8.00000 0.266223
\(904\) 18.0000 0.598671
\(905\) −72.0000 −2.39336
\(906\) 6.00000 0.199337
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 12.0000 0.398234
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) −22.0000 −0.728893 −0.364446 0.931224i \(-0.618742\pi\)
−0.364446 + 0.931224i \(0.618742\pi\)
\(912\) 0 0
\(913\) −4.00000 −0.132381
\(914\) −34.0000 −1.12462
\(915\) −32.0000 −1.05789
\(916\) 2.00000 0.0660819
\(917\) 8.00000 0.264183
\(918\) −1.00000 −0.0330049
\(919\) −46.0000 −1.51740 −0.758700 0.651440i \(-0.774165\pi\)
−0.758700 + 0.651440i \(0.774165\pi\)
\(920\) −24.0000 −0.791257
\(921\) 8.00000 0.263609
\(922\) 38.0000 1.25146
\(923\) 0 0
\(924\) −2.00000 −0.0657952
\(925\) 22.0000 0.723356
\(926\) 36.0000 1.18303
\(927\) −12.0000 −0.394132
\(928\) −2.00000 −0.0656532
\(929\) 38.0000 1.24674 0.623370 0.781927i \(-0.285763\pi\)
0.623370 + 0.781927i \(0.285763\pi\)
\(930\) 16.0000 0.524661
\(931\) 0 0
\(932\) −22.0000 −0.720634
\(933\) 18.0000 0.589294
\(934\) −36.0000 −1.17796
\(935\) −4.00000 −0.130814
\(936\) 0 0
\(937\) 18.0000 0.588034 0.294017 0.955800i \(-0.405008\pi\)
0.294017 + 0.955800i \(0.405008\pi\)
\(938\) 8.00000 0.261209
\(939\) 2.00000 0.0652675
\(940\) −24.0000 −0.782794
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 10.0000 0.325818
\(943\) 36.0000 1.17232
\(944\) 8.00000 0.260378
\(945\) −8.00000 −0.260240
\(946\) −4.00000 −0.130051
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) −6.00000 −0.194871
\(949\) 0 0
\(950\) 0 0
\(951\) 8.00000 0.259418
\(952\) 2.00000 0.0648204
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 8.00000 0.259010
\(955\) 72.0000 2.32987
\(956\) −4.00000 −0.129369
\(957\) −2.00000 −0.0646508
\(958\) −12.0000 −0.387702
\(959\) −12.0000 −0.387500
\(960\) 4.00000 0.129099
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) −26.0000 −0.837404
\(965\) 88.0000 2.83282
\(966\) 12.0000 0.386094
\(967\) 6.00000 0.192947 0.0964735 0.995336i \(-0.469244\pi\)
0.0964735 + 0.995336i \(0.469244\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 56.0000 1.79805
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 1.00000 0.0320750
\(973\) 16.0000 0.512936
\(974\) 0 0
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 4.00000 0.127906
\(979\) −14.0000 −0.447442
\(980\) −12.0000 −0.383326
\(981\) −12.0000 −0.383131
\(982\) 36.0000 1.14881
\(983\) −54.0000 −1.72233 −0.861166 0.508323i \(-0.830265\pi\)
−0.861166 + 0.508323i \(0.830265\pi\)
\(984\) −6.00000 −0.191273
\(985\) −72.0000 −2.29411
\(986\) 2.00000 0.0636930
\(987\) 12.0000 0.381964
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 4.00000 0.127128
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) 4.00000 0.127000
\(993\) 28.0000 0.888553
\(994\) 12.0000 0.380617
\(995\) 64.0000 2.02894
\(996\) −4.00000 −0.126745
\(997\) 4.00000 0.126681 0.0633406 0.997992i \(-0.479825\pi\)
0.0633406 + 0.997992i \(0.479825\pi\)
\(998\) −28.0000 −0.886325
\(999\) 2.00000 0.0632772
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 1122.2.a.n.1.1 1
3.2 odd 2 3366.2.a.a.1.1 1
4.3 odd 2 8976.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1122.2.a.n.1.1 1 1.1 even 1 trivial
3366.2.a.a.1.1 1 3.2 odd 2
8976.2.a.s.1.1 1 4.3 odd 2