Properties

Label 735.2.a.f.1.1
Level $735$
Weight $2$
Character 735.1
Self dual yes
Analytic conductor $5.869$
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] = [735,2,Mod(1,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, 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("735.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.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: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 735.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} +1.00000 q^{20} +8.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} +5.00000 q^{32} -2.00000 q^{34} -1.00000 q^{36} -2.00000 q^{37} +8.00000 q^{38} -6.00000 q^{39} +3.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} -1.00000 q^{45} +8.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} +10.0000 q^{53} -1.00000 q^{54} -8.00000 q^{57} -2.00000 q^{58} -4.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -4.00000 q^{62} +7.00000 q^{64} -6.00000 q^{65} +4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} -12.0000 q^{71} -3.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} -1.00000 q^{75} -8.00000 q^{76} -6.00000 q^{78} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} +2.00000 q^{85} +4.00000 q^{86} +2.00000 q^{87} +6.00000 q^{89} -1.00000 q^{90} -8.00000 q^{92} +4.00000 q^{93} -8.00000 q^{94} -8.00000 q^{95} -5.00000 q^{96} +18.0000 q^{97} +O(q^{100})\)

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 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 3.00000 0.612372
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 8.00000 1.29777
\(39\) −6.00000 −0.960769
\(40\) 3.00000 0.474342
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 8.00000 1.17954
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 2.00000 0.280056
\(52\) −6.00000 −0.832050
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) −1.00000 −0.129099
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 2.00000 0.242536
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −3.00000 −0.353553
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −2.00000 −0.232495
\(75\) −1.00000 −0.115470
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 4.00000 0.431331
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 4.00000 0.414781
\(94\) −8.00000 −0.825137
\(95\) −8.00000 −0.820783
\(96\) −5.00000 −0.510310
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 2.00000 0.198030
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −18.0000 −1.76505
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −8.00000 −0.749269
\(115\) −8.00000 −0.746004
\(116\) 2.00000 0.185695
\(117\) 6.00000 0.554700
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) −3.00000 −0.273861
\(121\) −11.0000 −1.00000
\(122\) 2.00000 0.181071
\(123\) −6.00000 −0.541002
\(124\) 4.00000 0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −3.00000 −0.265165
\(129\) −4.00000 −0.352180
\(130\) −6.00000 −0.526235
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 1.00000 0.0860663
\(136\) 6.00000 0.514496
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) −8.00000 −0.681005
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 2.00000 0.166091
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −24.0000 −1.94666
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 6.00000 0.480384
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 8.00000 0.636446
\(159\) −10.0000 −0.793052
\(160\) −5.00000 −0.395285
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 2.00000 0.153393
\(171\) 8.00000 0.611775
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 6.00000 0.449719
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 1.00000 0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) −24.0000 −1.76930
\(185\) 2.00000 0.147043
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) −7.00000 −0.505181
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 18.0000 1.29232
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) −3.00000 −0.212132
\(201\) −4.00000 −0.282138
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) −6.00000 −0.419058
\(206\) −8.00000 −0.557386
\(207\) 8.00000 0.556038
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −10.0000 −0.686803
\(213\) 12.0000 0.822226
\(214\) −12.0000 −0.820303
\(215\) −4.00000 −0.272798
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) −18.0000 −1.21911
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 2.00000 0.134231
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 8.00000 0.529813
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 6.00000 0.392232
\(235\) 8.00000 0.521862
\(236\) 4.00000 0.260378
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) −11.0000 −0.707107
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 48.0000 3.05417
\(248\) 12.0000 0.762001
\(249\) −4.00000 −0.253490
\(250\) −1.00000 −0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) −2.00000 −0.125245
\(256\) −17.0000 −1.06250
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 6.00000 0.372104
\(261\) −2.00000 −0.123797
\(262\) −20.0000 −1.23560
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) −4.00000 −0.244339
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 1.00000 0.0608581
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 8.00000 0.476393
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 12.0000 0.712069
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) 0 0
\(288\) 5.00000 0.294628
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) −18.0000 −1.05518
\(292\) −2.00000 −0.117041
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 14.0000 0.810998
\(299\) 48.0000 2.77591
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) −10.0000 −0.574485
\(304\) −8.00000 −0.458831
\(305\) −2.00000 −0.114520
\(306\) −2.00000 −0.114332
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 4.00000 0.227185
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 18.0000 1.01905
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) −10.0000 −0.560772
\(319\) 0 0
\(320\) −7.00000 −0.391312
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) −1.00000 −0.0555556
\(325\) 6.00000 0.332820
\(326\) 12.0000 0.664619
\(327\) 18.0000 0.995402
\(328\) −18.0000 −0.993884
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −4.00000 −0.219529
\(333\) −2.00000 −0.109599
\(334\) −8.00000 −0.437741
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 23.0000 1.25104
\(339\) −6.00000 −0.325875
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 8.00000 0.432590
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) 8.00000 0.430706
\(346\) −6.00000 −0.322562
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) −2.00000 −0.107211
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 4.00000 0.212598
\(355\) 12.0000 0.636894
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −24.0000 −1.26844
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 3.00000 0.158114
\(361\) 45.0000 2.36842
\(362\) 2.00000 0.105118
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) −2.00000 −0.104542
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −8.00000 −0.417029
\(369\) 6.00000 0.312348
\(370\) 2.00000 0.103975
\(371\) 0 0
\(372\) −4.00000 −0.207390
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 24.0000 1.23771
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 8.00000 0.410391
\(381\) −8.00000 −0.409852
\(382\) 4.00000 0.204658
\(383\) 32.0000 1.63512 0.817562 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) 4.00000 0.203331
\(388\) −18.0000 −0.913812
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 6.00000 0.303822
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 20.0000 1.00887
\(394\) 18.0000 0.906827
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −4.00000 −0.199502
\(403\) −24.0000 −1.19553
\(404\) −10.0000 −0.497519
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) −6.00000 −0.297044
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) −6.00000 −0.296319
\(411\) 10.0000 0.493264
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) −4.00000 −0.196352
\(416\) 30.0000 1.47087
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −20.0000 −0.973585
\(423\) −8.00000 −0.388973
\(424\) −30.0000 −1.45693
\(425\) −2.00000 −0.0970143
\(426\) 12.0000 0.581402
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 28.0000 1.34871 0.674356 0.738406i \(-0.264421\pi\)
0.674356 + 0.738406i \(0.264421\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 18.0000 0.862044
\(437\) 64.0000 3.06154
\(438\) −2.00000 −0.0955637
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −6.00000 −0.284427
\(446\) 24.0000 1.13643
\(447\) −14.0000 −0.662177
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) −8.00000 −0.375873
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 24.0000 1.12390
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −22.0000 −1.02799
\(459\) 2.00000 0.0933520
\(460\) 8.00000 0.373002
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 2.00000 0.0928477
\(465\) −4.00000 −0.185496
\(466\) −18.0000 −0.833834
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) −6.00000 −0.277350
\(469\) 0 0
\(470\) 8.00000 0.369012
\(471\) −14.0000 −0.645086
\(472\) 12.0000 0.552345
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) −4.00000 −0.182956
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 5.00000 0.228218
\(481\) −12.0000 −0.547153
\(482\) 6.00000 0.273293
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) −18.0000 −0.817338
\(486\) −1.00000 −0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −6.00000 −0.271607
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000 0.270501
\(493\) 4.00000 0.180151
\(494\) 48.0000 2.15962
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 1.00000 0.0447214
\(501\) 8.00000 0.357414
\(502\) 12.0000 0.535586
\(503\) 8.00000 0.356702 0.178351 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) −8.00000 −0.354943
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 0 0
\(512\) −11.0000 −0.486136
\(513\) −8.00000 −0.353209
\(514\) 6.00000 0.264649
\(515\) 8.00000 0.352522
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 18.0000 0.789352
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −10.0000 −0.434372
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) −6.00000 −0.259645
\(535\) 12.0000 0.518805
\(536\) −12.0000 −0.518321
\(537\) 24.0000 1.03568
\(538\) −14.0000 −0.603583
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −12.0000 −0.515444
\(543\) −2.00000 −0.0858282
\(544\) −10.0000 −0.428746
\(545\) 18.0000 0.771035
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 10.0000 0.427179
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −16.0000 −0.681623
\(552\) 24.0000 1.02151
\(553\) 0 0
\(554\) 14.0000 0.594803
\(555\) −2.00000 −0.0848953
\(556\) 0 0
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) −4.00000 −0.169334
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 18.0000 0.759284
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) −8.00000 −0.336861
\(565\) −6.00000 −0.252422
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 36.0000 1.51053
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 8.00000 0.335083
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 0 0
\(573\) −4.00000 −0.167102
\(574\) 0 0
\(575\) 8.00000 0.333623
\(576\) 7.00000 0.291667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −13.0000 −0.540729
\(579\) −18.0000 −0.748054
\(580\) −2.00000 −0.0830455
\(581\) 0 0
\(582\) −18.0000 −0.746124
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) −6.00000 −0.248069
\(586\) −14.0000 −0.578335
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 4.00000 0.164677
\(591\) −18.0000 −0.740421
\(592\) 2.00000 0.0821995
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) −4.00000 −0.163709
\(598\) 48.0000 1.96287
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 3.00000 0.122474
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) −8.00000 −0.325515
\(605\) 11.0000 0.447214
\(606\) −10.0000 −0.406222
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 40.0000 1.62221
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) −48.0000 −1.94187
\(612\) 2.00000 0.0808452
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) −12.0000 −0.484281
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 8.00000 0.321807
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) −4.00000 −0.160644
\(621\) −8.00000 −0.321029
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −24.0000 −0.954669
\(633\) 20.0000 0.794929
\(634\) 2.00000 0.0794301
\(635\) −8.00000 −0.317470
\(636\) 10.0000 0.396526
\(637\) 0 0
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 3.00000 0.118585
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 12.0000 0.473602
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) −16.0000 −0.629512
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) −3.00000 −0.117851
\(649\) 0 0
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) 18.0000 0.703856
\(655\) 20.0000 0.781465
\(656\) −6.00000 −0.234261
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) −12.0000 −0.466393
\(663\) 12.0000 0.466041
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −16.0000 −0.619522
\(668\) 8.00000 0.309529
\(669\) −24.0000 −0.927894
\(670\) −4.00000 −0.154533
\(671\) 0 0
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −14.0000 −0.539260
\(675\) −1.00000 −0.0384900
\(676\) −23.0000 −0.884615
\(677\) −46.0000 −1.76792 −0.883962 0.467559i \(-0.845134\pi\)
−0.883962 + 0.467559i \(0.845134\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) −6.00000 −0.230089
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −8.00000 −0.305888
\(685\) 10.0000 0.382080
\(686\) 0 0
\(687\) 22.0000 0.839352
\(688\) −4.00000 −0.152499
\(689\) 60.0000 2.28582
\(690\) 8.00000 0.304555
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) −12.0000 −0.454532
\(698\) −14.0000 −0.529908
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) −6.00000 −0.226455
\(703\) −16.0000 −0.603451
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 12.0000 0.450352
\(711\) 8.00000 0.300023
\(712\) −18.0000 −0.674579
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) 4.00000 0.149383
\(718\) −36.0000 −1.34351
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) 45.0000 1.67473
\(723\) −6.00000 −0.223142
\(724\) −2.00000 −0.0743294
\(725\) −2.00000 −0.0742781
\(726\) 11.0000 0.408248
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) −8.00000 −0.295891
\(732\) 2.00000 0.0739221
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 40.0000 1.47442
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −48.0000 −1.76332
\(742\) 0 0
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) −12.0000 −0.439941
\(745\) −14.0000 −0.512920
\(746\) −10.0000 −0.366126
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 1.00000 0.0365148
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 8.00000 0.291730
\(753\) −12.0000 −0.437304
\(754\) −12.0000 −0.437014
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) 54.0000 1.95750 0.978749 0.205061i \(-0.0657392\pi\)
0.978749 + 0.205061i \(0.0657392\pi\)
\(762\) −8.00000 −0.289809
\(763\) 0 0
\(764\) −4.00000 −0.144715
\(765\) 2.00000 0.0723102
\(766\) 32.0000 1.15621
\(767\) −24.0000 −0.866590
\(768\) 17.0000 0.613435
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) −18.0000 −0.647834
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 4.00000 0.143777
\(775\) −4.00000 −0.143684
\(776\) −54.0000 −1.93849
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) 48.0000 1.71978
\(780\) −6.00000 −0.214834
\(781\) 0 0
\(782\) −16.0000 −0.572159
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) −14.0000 −0.499681
\(786\) 20.0000 0.713376
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −18.0000 −0.641223
\(789\) −16.0000 −0.569615
\(790\) −8.00000 −0.284627
\(791\) 0 0
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 22.0000 0.780751
\(795\) 10.0000 0.354663
\(796\) −4.00000 −0.141776
\(797\) −54.0000 −1.91278 −0.956389 0.292096i \(-0.905647\pi\)
−0.956389 + 0.292096i \(0.905647\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 5.00000 0.176777
\(801\) 6.00000 0.212000
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −24.0000 −0.845364
\(807\) 14.0000 0.492823
\(808\) −30.0000 −1.05540
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 12.0000 0.420858
\(814\) 0 0
\(815\) −12.0000 −0.420342
\(816\) −2.00000 −0.0700140
\(817\) 32.0000 1.11954
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 10.0000 0.348790
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −8.00000 −0.278019
\(829\) 42.0000 1.45872 0.729360 0.684130i \(-0.239818\pi\)
0.729360 + 0.684130i \(0.239818\pi\)
\(830\) −4.00000 −0.138842
\(831\) −14.0000 −0.485655
\(832\) 42.0000 1.45609
\(833\) 0 0
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) 12.0000 0.414533
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −26.0000 −0.896019
\(843\) −18.0000 −0.619953
\(844\) 20.0000 0.688428
\(845\) −23.0000 −0.791224
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) −10.0000 −0.343401
\(849\) −4.00000 −0.137280
\(850\) −2.00000 −0.0685994
\(851\) −16.0000 −0.548473
\(852\) −12.0000 −0.411113
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 36.0000 1.23045
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 28.0000 0.953684
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −5.00000 −0.170103
\(865\) 6.00000 0.204006
\(866\) 2.00000 0.0679628
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) −2.00000 −0.0678064
\(871\) 24.0000 0.813209
\(872\) 54.0000 1.82867
\(873\) 18.0000 0.609208
\(874\) 64.0000 2.16483
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) −58.0000 −1.95852 −0.979260 0.202606i \(-0.935059\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) −28.0000 −0.944954
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 12.0000 0.403604
\(885\) −4.00000 −0.134459
\(886\) 12.0000 0.403148
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) −24.0000 −0.803579
\(893\) −64.0000 −2.14168
\(894\) −14.0000 −0.468230
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) −48.0000 −1.60267
\(898\) −30.0000 −1.00111
\(899\) 8.00000 0.266815
\(900\) −1.00000 −0.0333333
\(901\) −20.0000 −0.666297
\(902\) 0 0
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) −2.00000 −0.0664822
\(906\) −8.00000 −0.265782
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −4.00000 −0.132745
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) 18.0000 0.595387
\(915\) 2.00000 0.0661180
\(916\) 22.0000 0.726900
\(917\) 0 0
\(918\) 2.00000 0.0660098
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 24.0000 0.791257
\(921\) 12.0000 0.395413
\(922\) 2.00000 0.0658665
\(923\) −72.0000 −2.36991
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −24.0000 −0.788689
\(927\) −8.00000 −0.262754
\(928\) −10.0000 −0.328266
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) −4.00000 −0.131165
\(931\) 0 0
\(932\) 18.0000 0.589610
\(933\) 24.0000 0.785725
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) −18.0000 −0.588348
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) −8.00000 −0.260931
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −14.0000 −0.456145
\(943\) 48.0000 1.56310
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) 20.0000 0.649913 0.324956 0.945729i \(-0.394650\pi\)
0.324956 + 0.945729i \(0.394650\pi\)
\(948\) 8.00000 0.259828
\(949\) 12.0000 0.389536
\(950\) 8.00000 0.259554
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) −2.00000 −0.0647864 −0.0323932 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 10.0000 0.323762
\(955\) −4.00000 −0.129437
\(956\) 4.00000 0.129369
\(957\) 0 0
\(958\) −32.0000 −1.03387
\(959\) 0 0
\(960\) 7.00000 0.225924
\(961\) −15.0000 −0.483871
\(962\) −12.0000 −0.386896
\(963\) −12.0000 −0.386695
\(964\) −6.00000 −0.193247
\(965\) −18.0000 −0.579441
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 33.0000 1.06066
\(969\) 16.0000 0.513994
\(970\) −18.0000 −0.577945
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) −6.00000 −0.192154
\(976\) −2.00000 −0.0640184
\(977\) −10.0000 −0.319928 −0.159964 0.987123i \(-0.551138\pi\)
−0.159964 + 0.987123i \(0.551138\pi\)
\(978\) −12.0000 −0.383718
\(979\) 0 0
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 0 0
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 18.0000 0.573819
\(985\) −18.0000 −0.573528
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) −48.0000 −1.52708
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) −24.0000 −0.762385 −0.381193 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) −20.0000 −0.635001
\(993\) 12.0000 0.380808
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) 4.00000 0.126745
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) 20.0000 0.633089
\(999\) 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 735.2.a.f.1.1 1
3.2 odd 2 2205.2.a.b.1.1 1
5.4 even 2 3675.2.a.f.1.1 1
7.2 even 3 735.2.i.b.361.1 2
7.3 odd 6 735.2.i.a.226.1 2
7.4 even 3 735.2.i.b.226.1 2
7.5 odd 6 735.2.i.a.361.1 2
7.6 odd 2 105.2.a.a.1.1 1
21.20 even 2 315.2.a.a.1.1 1
28.27 even 2 1680.2.a.f.1.1 1
35.13 even 4 525.2.d.b.274.1 2
35.27 even 4 525.2.d.b.274.2 2
35.34 odd 2 525.2.a.a.1.1 1
56.13 odd 2 6720.2.a.p.1.1 1
56.27 even 2 6720.2.a.bk.1.1 1
84.83 odd 2 5040.2.a.d.1.1 1
105.62 odd 4 1575.2.d.b.1324.1 2
105.83 odd 4 1575.2.d.b.1324.2 2
105.104 even 2 1575.2.a.h.1.1 1
140.139 even 2 8400.2.a.co.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.a.a.1.1 1 7.6 odd 2
315.2.a.a.1.1 1 21.20 even 2
525.2.a.a.1.1 1 35.34 odd 2
525.2.d.b.274.1 2 35.13 even 4
525.2.d.b.274.2 2 35.27 even 4
735.2.a.f.1.1 1 1.1 even 1 trivial
735.2.i.a.226.1 2 7.3 odd 6
735.2.i.a.361.1 2 7.5 odd 6
735.2.i.b.226.1 2 7.4 even 3
735.2.i.b.361.1 2 7.2 even 3
1575.2.a.h.1.1 1 105.104 even 2
1575.2.d.b.1324.1 2 105.62 odd 4
1575.2.d.b.1324.2 2 105.83 odd 4
1680.2.a.f.1.1 1 28.27 even 2
2205.2.a.b.1.1 1 3.2 odd 2
3675.2.a.f.1.1 1 5.4 even 2
5040.2.a.d.1.1 1 84.83 odd 2
6720.2.a.p.1.1 1 56.13 odd 2
6720.2.a.bk.1.1 1 56.27 even 2
8400.2.a.co.1.1 1 140.139 even 2