Properties

Label 435.2.a.a.1.1
Level $435$
Weight $2$
Character 435.1
Self dual yes
Analytic conductor $3.473$
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] = [435,2,Mod(1,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, 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("435.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.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: \(3.47349248793\)
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 \(\chi\) \(=\) 435.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +6.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +8.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} -4.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} -6.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} +1.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} -5.00000 q^{32} -2.00000 q^{34} -4.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} -8.00000 q^{38} +6.00000 q^{39} +3.00000 q^{40} +2.00000 q^{41} +4.00000 q^{42} -4.00000 q^{43} +1.00000 q^{45} +4.00000 q^{46} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -12.0000 q^{56} +8.00000 q^{57} -1.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} +6.00000 q^{61} -4.00000 q^{62} -4.00000 q^{63} +7.00000 q^{64} +6.00000 q^{65} -8.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} +4.00000 q^{70} +16.0000 q^{71} +3.00000 q^{72} -6.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -8.00000 q^{76} -6.00000 q^{78} +12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -16.0000 q^{83} +4.00000 q^{84} +2.00000 q^{85} +4.00000 q^{86} +1.00000 q^{87} +2.00000 q^{89} -1.00000 q^{90} -24.0000 q^{91} +4.00000 q^{92} +4.00000 q^{93} +8.00000 q^{95} -5.00000 q^{96} -14.0000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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 \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 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 \(0.187167\pi\)
\(14\) 4.00000 1.06904
\(15\) 1.00000 0.258199
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −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
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 3.00000 0.612372
\(25\) 1.00000 0.200000
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) 1.00000 0.185695
\(30\) −1.00000 −0.182574
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −4.00000 −0.676123
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −8.00000 −1.29777
\(39\) 6.00000 0.960769
\(40\) 3.00000 0.474342
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 4.00000 0.617213
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 4.00000 0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 2.00000 0.280056
\(52\) −6.00000 −0.832050
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −12.0000 −1.60357
\(57\) 8.00000 1.05963
\(58\) −1.00000 −0.131306
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −1.00000 −0.129099
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −4.00000 −0.508001
\(63\) −4.00000 −0.503953
\(64\) 7.00000 0.875000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −2.00000 −0.242536
\(69\) −4.00000 −0.481543
\(70\) 4.00000 0.478091
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 3.00000 0.353553
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 4.00000 0.436436
\(85\) 2.00000 0.216930
\(86\) 4.00000 0.431331
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) −1.00000 −0.105409
\(91\) −24.0000 −2.51588
\(92\) 4.00000 0.417029
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) −5.00000 −0.510310
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −2.00000 −0.198030
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 18.0000 1.76505
\(105\) −4.00000 −0.390360
\(106\) −6.00000 −0.582772
\(107\) −16.0000 −1.54678 −0.773389 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 4.00000 0.377964
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −8.00000 −0.749269
\(115\) −4.00000 −0.373002
\(116\) −1.00000 −0.0928477
\(117\) 6.00000 0.554700
\(118\) 12.0000 1.10469
\(119\) −8.00000 −0.733359
\(120\) 3.00000 0.273861
\(121\) −11.0000 −1.00000
\(122\) −6.00000 −0.543214
\(123\) 2.00000 0.180334
\(124\) −4.00000 −0.359211
\(125\) 1.00000 0.0894427
\(126\) 4.00000 0.356348
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 3.00000 0.265165
\(129\) −4.00000 −0.352180
\(130\) −6.00000 −0.526235
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −32.0000 −2.77475
\(134\) 8.00000 0.691095
\(135\) 1.00000 0.0860663
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 4.00000 0.340503
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) −16.0000 −1.34269
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 1.00000 0.0830455
\(146\) 6.00000 0.496564
\(147\) 9.00000 0.742307
\(148\) −6.00000 −0.493197
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 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 \(0.311315\pi\)
\(158\) −12.0000 −0.954669
\(159\) 6.00000 0.475831
\(160\) −5.00000 −0.395285
\(161\) 16.0000 1.26098
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) −12.0000 −0.925820
\(169\) 23.0000 1.76923
\(170\) −2.00000 −0.153393
\(171\) 8.00000 0.611775
\(172\) 4.00000 0.304997
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) −1.00000 −0.0758098
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) −2.00000 −0.149906
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 24.0000 1.77900
\(183\) 6.00000 0.443533
\(184\) −12.0000 −0.884652
\(185\) 6.00000 0.441129
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) −8.00000 −0.580381
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 7.00000 0.505181
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 14.0000 1.00514
\(195\) 6.00000 0.429669
\(196\) −9.00000 −0.642857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 3.00000 0.212132
\(201\) −8.00000 −0.564276
\(202\) 10.0000 0.703598
\(203\) −4.00000 −0.280745
\(204\) −2.00000 −0.140028
\(205\) 2.00000 0.139686
\(206\) −4.00000 −0.278693
\(207\) −4.00000 −0.278019
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) 4.00000 0.276026
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −6.00000 −0.412082
\(213\) 16.0000 1.09630
\(214\) 16.0000 1.09374
\(215\) −4.00000 −0.272798
\(216\) 3.00000 0.204124
\(217\) −16.0000 −1.08615
\(218\) 2.00000 0.135457
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) −6.00000 −0.402694
\(223\) 12.0000 0.803579 0.401790 0.915732i \(-0.368388\pi\)
0.401790 + 0.915732i \(0.368388\pi\)
\(224\) 20.0000 1.33631
\(225\) 1.00000 0.0666667
\(226\) −2.00000 −0.133038
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) −8.00000 −0.529813
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 12.0000 0.779484
\(238\) 8.00000 0.518563
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 11.0000 0.707107
\(243\) 1.00000 0.0641500
\(244\) −6.00000 −0.384111
\(245\) 9.00000 0.574989
\(246\) −2.00000 −0.127515
\(247\) 48.0000 3.05417
\(248\) 12.0000 0.762001
\(249\) −16.0000 −1.01396
\(250\) −1.00000 −0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 2.00000 0.125245
\(256\) −17.0000 −1.06250
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 4.00000 0.249029
\(259\) −24.0000 −1.49129
\(260\) −6.00000 −0.372104
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 32.0000 1.96205
\(267\) 2.00000 0.122398
\(268\) 8.00000 0.488678
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −2.00000 −0.121268
\(273\) −24.0000 −1.45255
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −20.0000 −1.19952
\(279\) 4.00000 0.239474
\(280\) −12.0000 −0.717137
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) −16.0000 −0.949425
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) −5.00000 −0.294628
\(289\) −13.0000 −0.764706
\(290\) −1.00000 −0.0587220
\(291\) −14.0000 −0.820695
\(292\) 6.00000 0.351123
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) −9.00000 −0.524891
\(295\) −12.0000 −0.698667
\(296\) 18.0000 1.04623
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) −24.0000 −1.38796
\(300\) −1.00000 −0.0577350
\(301\) 16.0000 0.922225
\(302\) −16.0000 −0.920697
\(303\) −10.0000 −0.574485
\(304\) −8.00000 −0.458831
\(305\) 6.00000 0.343559
\(306\) −2.00000 −0.114332
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) −4.00000 −0.227185
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 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
\(315\) −4.00000 −0.225374
\(316\) −12.0000 −0.675053
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) 7.00000 0.391312
\(321\) −16.0000 −0.893033
\(322\) −16.0000 −0.891645
\(323\) 16.0000 0.890264
\(324\) −1.00000 −0.0555556
\(325\) 6.00000 0.332820
\(326\) −12.0000 −0.664619
\(327\) −2.00000 −0.110600
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 16.0000 0.878114
\(333\) 6.00000 0.328798
\(334\) 20.0000 1.09435
\(335\) −8.00000 −0.437087
\(336\) 4.00000 0.218218
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) −23.0000 −1.25104
\(339\) 2.00000 0.108625
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) −8.00000 −0.432590
\(343\) −8.00000 −0.431959
\(344\) −12.0000 −0.646997
\(345\) −4.00000 −0.215353
\(346\) 2.00000 0.107521
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) −1.00000 −0.0536056
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 4.00000 0.213809
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 12.0000 0.637793
\(355\) 16.0000 0.849192
\(356\) −2.00000 −0.106000
\(357\) −8.00000 −0.423405
\(358\) 12.0000 0.634220
\(359\) −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
\(362\) 10.0000 0.525588
\(363\) −11.0000 −0.577350
\(364\) 24.0000 1.25794
\(365\) −6.00000 −0.314054
\(366\) −6.00000 −0.313625
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 4.00000 0.208514
\(369\) 2.00000 0.104116
\(370\) −6.00000 −0.311925
\(371\) −24.0000 −1.24602
\(372\) −4.00000 −0.207390
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 4.00000 0.205738
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −8.00000 −0.410391
\(381\) −8.00000 −0.409852
\(382\) 4.00000 0.204658
\(383\) 28.0000 1.43073 0.715367 0.698749i \(-0.246260\pi\)
0.715367 + 0.698749i \(0.246260\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −4.00000 −0.203331
\(388\) 14.0000 0.710742
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) −6.00000 −0.303822
\(391\) −8.00000 −0.404577
\(392\) 27.0000 1.36371
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 16.0000 0.802008
\(399\) −32.0000 −1.60200
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 8.00000 0.399004
\(403\) 24.0000 1.19553
\(404\) 10.0000 0.497519
\(405\) 1.00000 0.0496904
\(406\) 4.00000 0.198517
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −2.00000 −0.0987730
\(411\) −6.00000 −0.295958
\(412\) −4.00000 −0.197066
\(413\) 48.0000 2.36193
\(414\) 4.00000 0.196589
\(415\) −16.0000 −0.785409
\(416\) −30.0000 −1.47087
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 4.00000 0.195180
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) 2.00000 0.0970143
\(426\) −16.0000 −0.775203
\(427\) −24.0000 −1.16144
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 16.0000 0.768025
\(435\) 1.00000 0.0479463
\(436\) 2.00000 0.0957826
\(437\) −32.0000 −1.53077
\(438\) 6.00000 0.286691
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −12.0000 −0.570782
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) −6.00000 −0.284747
\(445\) 2.00000 0.0948091
\(446\) −12.0000 −0.568216
\(447\) −10.0000 −0.472984
\(448\) −28.0000 −1.32288
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) −2.00000 −0.0940721
\(453\) 16.0000 0.751746
\(454\) −8.00000 −0.375459
\(455\) −24.0000 −1.12514
\(456\) 24.0000 1.12390
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −14.0000 −0.654177
\(459\) 2.00000 0.0933520
\(460\) 4.00000 0.186501
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 4.00000 0.185496
\(466\) −18.0000 −0.833834
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −6.00000 −0.277350
\(469\) 32.0000 1.47762
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) −36.0000 −1.65703
\(473\) 0 0
\(474\) −12.0000 −0.551178
\(475\) 8.00000 0.367065
\(476\) 8.00000 0.366679
\(477\) 6.00000 0.274721
\(478\) −24.0000 −1.09773
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) −5.00000 −0.228218
\(481\) 36.0000 1.64146
\(482\) −2.00000 −0.0910975
\(483\) 16.0000 0.728025
\(484\) 11.0000 0.500000
\(485\) −14.0000 −0.635707
\(486\) −1.00000 −0.0453609
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 18.0000 0.814822
\(489\) 12.0000 0.542659
\(490\) −9.00000 −0.406579
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 2.00000 0.0900755
\(494\) −48.0000 −2.15962
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −64.0000 −2.87079
\(498\) 16.0000 0.716977
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −20.0000 −0.893534
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −12.0000 −0.534522
\(505\) −10.0000 −0.444994
\(506\) 0 0
\(507\) 23.0000 1.02147
\(508\) 8.00000 0.354943
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 24.0000 1.06170
\(512\) 11.0000 0.486136
\(513\) 8.00000 0.353209
\(514\) −26.0000 −1.14681
\(515\) 4.00000 0.176261
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 24.0000 1.05450
\(519\) −2.00000 −0.0877903
\(520\) 18.0000 0.789352
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −1.00000 −0.0437688
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) −8.00000 −0.348817
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −6.00000 −0.260623
\(531\) −12.0000 −0.520756
\(532\) 32.0000 1.38738
\(533\) 12.0000 0.519778
\(534\) −2.00000 −0.0865485
\(535\) −16.0000 −0.691740
\(536\) −24.0000 −1.03664
\(537\) −12.0000 −0.517838
\(538\) 18.0000 0.776035
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 20.0000 0.859074
\(543\) −10.0000 −0.429141
\(544\) −10.0000 −0.428746
\(545\) −2.00000 −0.0856706
\(546\) 24.0000 1.02711
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) 6.00000 0.256307
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) −12.0000 −0.510754
\(553\) −48.0000 −2.04117
\(554\) 2.00000 0.0849719
\(555\) 6.00000 0.254686
\(556\) −20.0000 −0.848189
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) −4.00000 −0.169334
\(559\) −24.0000 −1.01509
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) −8.00000 −0.336265
\(567\) −4.00000 −0.167984
\(568\) 48.0000 2.01404
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) −8.00000 −0.335083
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) −4.00000 −0.167102
\(574\) 8.00000 0.333914
\(575\) −4.00000 −0.166812
\(576\) 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
\(579\) 2.00000 0.0831172
\(580\) −1.00000 −0.0415227
\(581\) 64.0000 2.65517
\(582\) 14.0000 0.580319
\(583\) 0 0
\(584\) −18.0000 −0.744845
\(585\) 6.00000 0.248069
\(586\) 26.0000 1.07405
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) −9.00000 −0.371154
\(589\) 32.0000 1.31854
\(590\) 12.0000 0.494032
\(591\) 6.00000 0.246807
\(592\) −6.00000 −0.246598
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) −8.00000 −0.327968
\(596\) 10.0000 0.409616
\(597\) −16.0000 −0.654836
\(598\) 24.0000 0.981433
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 3.00000 0.122474
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −16.0000 −0.652111
\(603\) −8.00000 −0.325785
\(604\) −16.0000 −0.651031
\(605\) −11.0000 −0.447214
\(606\) 10.0000 0.406222
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) −40.0000 −1.62221
\(609\) −4.00000 −0.162088
\(610\) −6.00000 −0.242933
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 20.0000 0.807134
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) −4.00000 −0.160904
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) −4.00000 −0.160644
\(621\) −4.00000 −0.160514
\(622\) 28.0000 1.12270
\(623\) −8.00000 −0.320513
\(624\) −6.00000 −0.240192
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 12.0000 0.478471
\(630\) 4.00000 0.159364
\(631\) 48.0000 1.91085 0.955425 0.295234i \(-0.0953977\pi\)
0.955425 + 0.295234i \(0.0953977\pi\)
\(632\) 36.0000 1.43200
\(633\) 8.00000 0.317971
\(634\) −30.0000 −1.19145
\(635\) −8.00000 −0.317470
\(636\) −6.00000 −0.237915
\(637\) 54.0000 2.13956
\(638\) 0 0
\(639\) 16.0000 0.632950
\(640\) 3.00000 0.118585
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 16.0000 0.631470
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) −16.0000 −0.630488
\(645\) −4.00000 −0.157500
\(646\) −16.0000 −0.629512
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 3.00000 0.117851
\(649\) 0 0
\(650\) −6.00000 −0.235339
\(651\) −16.0000 −0.627089
\(652\) −12.0000 −0.469956
\(653\) −50.0000 −1.95665 −0.978326 0.207072i \(-0.933606\pi\)
−0.978326 + 0.207072i \(0.933606\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −48.0000 −1.86981 −0.934907 0.354892i \(-0.884518\pi\)
−0.934907 + 0.354892i \(0.884518\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 16.0000 0.621858
\(663\) 12.0000 0.466041
\(664\) −48.0000 −1.86276
\(665\) −32.0000 −1.24091
\(666\) −6.00000 −0.232495
\(667\) −4.00000 −0.154881
\(668\) 20.0000 0.773823
\(669\) 12.0000 0.463947
\(670\) 8.00000 0.309067
\(671\) 0 0
\(672\) 20.0000 0.771517
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 30.0000 1.15556
\(675\) 1.00000 0.0384900
\(676\) −23.0000 −0.884615
\(677\) −26.0000 −0.999261 −0.499631 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) −2.00000 −0.0768095
\(679\) 56.0000 2.14908
\(680\) 6.00000 0.230089
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) −8.00000 −0.305888
\(685\) −6.00000 −0.229248
\(686\) 8.00000 0.305441
\(687\) 14.0000 0.534133
\(688\) 4.00000 0.152499
\(689\) 36.0000 1.37149
\(690\) 4.00000 0.152277
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) 20.0000 0.758643
\(696\) 3.00000 0.113715
\(697\) 4.00000 0.151511
\(698\) 2.00000 0.0757011
\(699\) 18.0000 0.680823
\(700\) 4.00000 0.151186
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) −6.00000 −0.226455
\(703\) 48.0000 1.81035
\(704\) 0 0
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 40.0000 1.50435
\(708\) 12.0000 0.450988
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −16.0000 −0.600469
\(711\) 12.0000 0.450035
\(712\) 6.00000 0.224860
\(713\) −16.0000 −0.599205
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 24.0000 0.896296
\(718\) 36.0000 1.34351
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −16.0000 −0.595871
\(722\) −45.0000 −1.67473
\(723\) 2.00000 0.0743808
\(724\) 10.0000 0.371647
\(725\) 1.00000 0.0371391
\(726\) 11.0000 0.408248
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) −72.0000 −2.66850
\(729\) 1.00000 0.0370370
\(730\) 6.00000 0.222070
\(731\) −8.00000 −0.295891
\(732\) −6.00000 −0.221766
\(733\) 46.0000 1.69905 0.849524 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) 24.0000 0.885856
\(735\) 9.00000 0.331970
\(736\) 20.0000 0.737210
\(737\) 0 0
\(738\) −2.00000 −0.0736210
\(739\) −24.0000 −0.882854 −0.441427 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) −6.00000 −0.220564
\(741\) 48.0000 1.76332
\(742\) 24.0000 0.881068
\(743\) −32.0000 −1.17397 −0.586983 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 12.0000 0.439941
\(745\) −10.0000 −0.366372
\(746\) 34.0000 1.24483
\(747\) −16.0000 −0.585409
\(748\) 0 0
\(749\) 64.0000 2.33851
\(750\) −1.00000 −0.0365148
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −6.00000 −0.218507
\(755\) 16.0000 0.582300
\(756\) 4.00000 0.145479
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 8.00000 0.289809
\(763\) 8.00000 0.289619
\(764\) 4.00000 0.144715
\(765\) 2.00000 0.0723102
\(766\) −28.0000 −1.01168
\(767\) −72.0000 −2.59977
\(768\) −17.0000 −0.613435
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) 26.0000 0.936367
\(772\) −2.00000 −0.0719816
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 4.00000 0.143777
\(775\) 4.00000 0.143684
\(776\) −42.0000 −1.50771
\(777\) −24.0000 −0.860995
\(778\) 26.0000 0.932145
\(779\) 16.0000 0.573259
\(780\) −6.00000 −0.214834
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) 1.00000 0.0357371
\(784\) −9.00000 −0.321429
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) −24.0000 −0.855508 −0.427754 0.903895i \(-0.640695\pi\)
−0.427754 + 0.903895i \(0.640695\pi\)
\(788\) −6.00000 −0.213741
\(789\) 8.00000 0.284808
\(790\) −12.0000 −0.426941
\(791\) −8.00000 −0.284447
\(792\) 0 0
\(793\) 36.0000 1.27840
\(794\) −22.0000 −0.780751
\(795\) 6.00000 0.212798
\(796\) 16.0000 0.567105
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 32.0000 1.13279
\(799\) 0 0
\(800\) −5.00000 −0.176777
\(801\) 2.00000 0.0706665
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) 16.0000 0.563926
\(806\) −24.0000 −0.845364
\(807\) −18.0000 −0.633630
\(808\) −30.0000 −1.05540
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 4.00000 0.140372
\(813\) −20.0000 −0.701431
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) −2.00000 −0.0700140
\(817\) −32.0000 −1.11954
\(818\) 6.00000 0.209785
\(819\) −24.0000 −0.838628
\(820\) −2.00000 −0.0698430
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 6.00000 0.209274
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 4.00000 0.139010
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 16.0000 0.555368
\(831\) −2.00000 −0.0693792
\(832\) 42.0000 1.45609
\(833\) 18.0000 0.623663
\(834\) −20.0000 −0.692543
\(835\) −20.0000 −0.692129
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) −4.00000 −0.138178
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) −12.0000 −0.414039
\(841\) 1.00000 0.0344828
\(842\) −30.0000 −1.03387
\(843\) −6.00000 −0.206651
\(844\) −8.00000 −0.275371
\(845\) 23.0000 0.791224
\(846\) 0 0
\(847\) 44.0000 1.51186
\(848\) −6.00000 −0.206041
\(849\) 8.00000 0.274559
\(850\) −2.00000 −0.0685994
\(851\) −24.0000 −0.822709
\(852\) −16.0000 −0.548151
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 24.0000 0.821263
\(855\) 8.00000 0.273594
\(856\) −48.0000 −1.64061
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 4.00000 0.136399
\(861\) −8.00000 −0.272639
\(862\) −32.0000 −1.08992
\(863\) −52.0000 −1.77010 −0.885050 0.465495i \(-0.845876\pi\)
−0.885050 + 0.465495i \(0.845876\pi\)
\(864\) −5.00000 −0.170103
\(865\) −2.00000 −0.0680020
\(866\) 14.0000 0.475739
\(867\) −13.0000 −0.441503
\(868\) 16.0000 0.543075
\(869\) 0 0
\(870\) −1.00000 −0.0339032
\(871\) −48.0000 −1.62642
\(872\) −6.00000 −0.203186
\(873\) −14.0000 −0.473828
\(874\) 32.0000 1.08242
\(875\) −4.00000 −0.135225
\(876\) 6.00000 0.202721
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 24.0000 0.809961
\(879\) −26.0000 −0.876958
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) −9.00000 −0.303046
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) −12.0000 −0.403604
\(885\) −12.0000 −0.403376
\(886\) −20.0000 −0.671913
\(887\) −32.0000 −1.07445 −0.537227 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 18.0000 0.604040
\(889\) 32.0000 1.07325
\(890\) −2.00000 −0.0670402
\(891\) 0 0
\(892\) −12.0000 −0.401790
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) −12.0000 −0.401116
\(896\) −12.0000 −0.400892
\(897\) −24.0000 −0.801337
\(898\) 6.00000 0.200223
\(899\) 4.00000 0.133407
\(900\) −1.00000 −0.0333333
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 16.0000 0.532447
\(904\) 6.00000 0.199557
\(905\) −10.0000 −0.332411
\(906\) −16.0000 −0.531564
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −8.00000 −0.265489
\(909\) −10.0000 −0.331679
\(910\) 24.0000 0.795592
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) −8.00000 −0.264906
\(913\) 0 0
\(914\) 22.0000 0.727695
\(915\) 6.00000 0.198354
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) −12.0000 −0.395628
\(921\) −20.0000 −0.659022
\(922\) 18.0000 0.592798
\(923\) 96.0000 3.15988
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) −36.0000 −1.18303
\(927\) 4.00000 0.131377
\(928\) −5.00000 −0.164133
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) −4.00000 −0.131165
\(931\) 72.0000 2.35970
\(932\) −18.0000 −0.589610
\(933\) −28.0000 −0.916679
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 18.0000 0.588348
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) −32.0000 −1.04484
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) −14.0000 −0.456145
\(943\) −8.00000 −0.260516
\(944\) 12.0000 0.390567
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) −12.0000 −0.389742
\(949\) −36.0000 −1.16861
\(950\) −8.00000 −0.259554
\(951\) 30.0000 0.972817
\(952\) −24.0000 −0.777844
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) −6.00000 −0.194257
\(955\) −4.00000 −0.129437
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) −12.0000 −0.387702
\(959\) 24.0000 0.775000
\(960\) 7.00000 0.225924
\(961\) −15.0000 −0.483871
\(962\) −36.0000 −1.16069
\(963\) −16.0000 −0.515593
\(964\) −2.00000 −0.0644157
\(965\) 2.00000 0.0643823
\(966\) −16.0000 −0.514792
\(967\) 56.0000 1.80084 0.900419 0.435023i \(-0.143260\pi\)
0.900419 + 0.435023i \(0.143260\pi\)
\(968\) −33.0000 −1.06066
\(969\) 16.0000 0.513994
\(970\) 14.0000 0.449513
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −80.0000 −2.56468
\(974\) 20.0000 0.640841
\(975\) 6.00000 0.192154
\(976\) −6.00000 −0.192055
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) −12.0000 −0.383718
\(979\) 0 0
\(980\) −9.00000 −0.287494
\(981\) −2.00000 −0.0638551
\(982\) 24.0000 0.765871
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 6.00000 0.191273
\(985\) 6.00000 0.191176
\(986\) −2.00000 −0.0636930
\(987\) 0 0
\(988\) −48.0000 −1.52708
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −20.0000 −0.635001
\(993\) −16.0000 −0.507745
\(994\) 64.0000 2.02996
\(995\) −16.0000 −0.507234
\(996\) 16.0000 0.506979
\(997\) 38.0000 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) −36.0000 −1.13956
\(999\) 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 435.2.a.a.1.1 1
3.2 odd 2 1305.2.a.e.1.1 1
4.3 odd 2 6960.2.a.w.1.1 1
5.2 odd 4 2175.2.c.a.349.1 2
5.3 odd 4 2175.2.c.a.349.2 2
5.4 even 2 2175.2.a.h.1.1 1
15.14 odd 2 6525.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.a.1.1 1 1.1 even 1 trivial
1305.2.a.e.1.1 1 3.2 odd 2
2175.2.a.h.1.1 1 5.4 even 2
2175.2.c.a.349.1 2 5.2 odd 4
2175.2.c.a.349.2 2 5.3 odd 4
6525.2.a.e.1.1 1 15.14 odd 2
6960.2.a.w.1.1 1 4.3 odd 2