Properties

Label 170.2.a.c.1.1
Level $170$
Weight $2$
Character 170.1
Self dual yes
Analytic conductor $1.357$
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] = [170,2,Mod(1,170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(170, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("170.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.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: \(1.35745683436\)
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 \(\chi\) \(=\) 170.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +5.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +2.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} +6.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -5.00000 q^{26} -5.00000 q^{27} +2.00000 q^{28} -9.00000 q^{29} -1.00000 q^{30} -1.00000 q^{31} -1.00000 q^{32} +1.00000 q^{34} +2.00000 q^{35} -2.00000 q^{36} -4.00000 q^{37} +1.00000 q^{38} +5.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} -2.00000 q^{42} +2.00000 q^{43} -2.00000 q^{45} -6.00000 q^{46} -9.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} +5.00000 q^{52} -9.00000 q^{53} +5.00000 q^{54} -2.00000 q^{56} -1.00000 q^{57} +9.00000 q^{58} +3.00000 q^{59} +1.00000 q^{60} -7.00000 q^{61} +1.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +5.00000 q^{65} +14.0000 q^{67} -1.00000 q^{68} +6.00000 q^{69} -2.00000 q^{70} +3.00000 q^{71} +2.00000 q^{72} +11.0000 q^{73} +4.00000 q^{74} +1.00000 q^{75} -1.00000 q^{76} -5.00000 q^{78} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +2.00000 q^{84} -1.00000 q^{85} -2.00000 q^{86} -9.00000 q^{87} -9.00000 q^{89} +2.00000 q^{90} +10.0000 q^{91} +6.00000 q^{92} -1.00000 q^{93} +9.00000 q^{94} -1.00000 q^{95} -1.00000 q^{96} -7.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) −1.00000 −0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) −2.00000 −0.534522
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) 2.00000 0.471405
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −5.00000 −0.980581
\(27\) −5.00000 −0.962250
\(28\) 2.00000 0.377964
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) −1.00000 −0.182574
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.00000 0.171499
\(35\) 2.00000 0.338062
\(36\) −2.00000 −0.333333
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 1.00000 0.162221
\(39\) 5.00000 0.800641
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −2.00000 −0.308607
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) −6.00000 −0.884652
\(47\) −9.00000 −1.31278 −0.656392 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) −1.00000 −0.140028
\(52\) 5.00000 0.693375
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) −1.00000 −0.132453
\(58\) 9.00000 1.18176
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 1.00000 0.129099
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 1.00000 0.127000
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 5.00000 0.620174
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) −1.00000 −0.121268
\(69\) 6.00000 0.722315
\(70\) −2.00000 −0.239046
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 2.00000 0.235702
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 4.00000 0.464991
\(75\) 1.00000 0.115470
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) −5.00000 −0.566139
\(79\) 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
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 2.00000 0.218218
\(85\) −1.00000 −0.108465
\(86\) −2.00000 −0.215666
\(87\) −9.00000 −0.964901
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 2.00000 0.210819
\(91\) 10.0000 1.04828
\(92\) 6.00000 0.625543
\(93\) −1.00000 −0.103695
\(94\) 9.00000 0.928279
\(95\) −1.00000 −0.102598
\(96\) −1.00000 −0.102062
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 1.00000 0.0990148
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −5.00000 −0.490290
\(105\) 2.00000 0.195180
\(106\) 9.00000 0.874157
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −5.00000 −0.481125
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 2.00000 0.188982
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 1.00000 0.0936586
\(115\) 6.00000 0.559503
\(116\) −9.00000 −0.835629
\(117\) −10.0000 −0.924500
\(118\) −3.00000 −0.276172
\(119\) −2.00000 −0.183340
\(120\) −1.00000 −0.0912871
\(121\) −11.0000 −1.00000
\(122\) 7.00000 0.633750
\(123\) −6.00000 −0.541002
\(124\) −1.00000 −0.0898027
\(125\) 1.00000 0.0894427
\(126\) 4.00000 0.356348
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.00000 0.176090
\(130\) −5.00000 −0.438529
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) −14.0000 −1.20942
\(135\) −5.00000 −0.430331
\(136\) 1.00000 0.0857493
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −6.00000 −0.510754
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 2.00000 0.169031
\(141\) −9.00000 −0.757937
\(142\) −3.00000 −0.251754
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) −9.00000 −0.747409
\(146\) −11.0000 −0.910366
\(147\) −3.00000 −0.247436
\(148\) −4.00000 −0.328798
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 1.00000 0.0811107
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) −1.00000 −0.0803219
\(156\) 5.00000 0.400320
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −8.00000 −0.636446
\(159\) −9.00000 −0.713746
\(160\) −1.00000 −0.0790569
\(161\) 12.0000 0.945732
\(162\) −1.00000 −0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) −2.00000 −0.154303
\(169\) 12.0000 0.923077
\(170\) 1.00000 0.0766965
\(171\) 2.00000 0.152944
\(172\) 2.00000 0.152499
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 9.00000 0.682288
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) 3.00000 0.225494
\(178\) 9.00000 0.674579
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −2.00000 −0.149071
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −10.0000 −0.741249
\(183\) −7.00000 −0.517455
\(184\) −6.00000 −0.442326
\(185\) −4.00000 −0.294086
\(186\) 1.00000 0.0733236
\(187\) 0 0
\(188\) −9.00000 −0.656392
\(189\) −10.0000 −0.727393
\(190\) 1.00000 0.0725476
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 7.00000 0.502571
\(195\) 5.00000 0.358057
\(196\) −3.00000 −0.214286
\(197\) 24.0000 1.70993 0.854965 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 0 0
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 14.0000 0.987484
\(202\) −18.0000 −1.26648
\(203\) −18.0000 −1.26335
\(204\) −1.00000 −0.0700140
\(205\) −6.00000 −0.419058
\(206\) 16.0000 1.11477
\(207\) −12.0000 −0.834058
\(208\) 5.00000 0.346688
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) −9.00000 −0.618123
\(213\) 3.00000 0.205557
\(214\) −12.0000 −0.820303
\(215\) 2.00000 0.136399
\(216\) 5.00000 0.340207
\(217\) −2.00000 −0.135769
\(218\) 7.00000 0.474100
\(219\) 11.0000 0.743311
\(220\) 0 0
\(221\) −5.00000 −0.336336
\(222\) 4.00000 0.268462
\(223\) −19.0000 −1.27233 −0.636167 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) −2.00000 −0.133631
\(225\) −2.00000 −0.133333
\(226\) −9.00000 −0.598671
\(227\) 27.0000 1.79205 0.896026 0.444001i \(-0.146441\pi\)
0.896026 + 0.444001i \(0.146441\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) 9.00000 0.590879
\(233\) 9.00000 0.589610 0.294805 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) 10.0000 0.653720
\(235\) −9.00000 −0.587095
\(236\) 3.00000 0.195283
\(237\) 8.00000 0.519656
\(238\) 2.00000 0.129641
\(239\) −30.0000 −1.94054 −0.970269 0.242028i \(-0.922188\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) 1.00000 0.0645497
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 11.0000 0.707107
\(243\) 16.0000 1.02640
\(244\) −7.00000 −0.448129
\(245\) −3.00000 −0.191663
\(246\) 6.00000 0.382546
\(247\) −5.00000 −0.318142
\(248\) 1.00000 0.0635001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −4.00000 −0.251976
\(253\) 0 0
\(254\) 7.00000 0.439219
\(255\) −1.00000 −0.0626224
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −2.00000 −0.124515
\(259\) −8.00000 −0.497096
\(260\) 5.00000 0.310087
\(261\) 18.0000 1.11417
\(262\) −6.00000 −0.370681
\(263\) 3.00000 0.184988 0.0924940 0.995713i \(-0.470516\pi\)
0.0924940 + 0.995713i \(0.470516\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 2.00000 0.122628
\(267\) −9.00000 −0.550791
\(268\) 14.0000 0.855186
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 5.00000 0.304290
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 10.0000 0.605228
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) −20.0000 −1.19952
\(279\) 2.00000 0.119737
\(280\) −2.00000 −0.119523
\(281\) −21.0000 −1.25275 −0.626377 0.779520i \(-0.715463\pi\)
−0.626377 + 0.779520i \(0.715463\pi\)
\(282\) 9.00000 0.535942
\(283\) 11.0000 0.653882 0.326941 0.945045i \(-0.393982\pi\)
0.326941 + 0.945045i \(0.393982\pi\)
\(284\) 3.00000 0.178017
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 2.00000 0.117851
\(289\) 1.00000 0.0588235
\(290\) 9.00000 0.528498
\(291\) −7.00000 −0.410347
\(292\) 11.0000 0.643726
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 3.00000 0.174964
\(295\) 3.00000 0.174667
\(296\) 4.00000 0.232495
\(297\) 0 0
\(298\) 12.0000 0.695141
\(299\) 30.0000 1.73494
\(300\) 1.00000 0.0577350
\(301\) 4.00000 0.230556
\(302\) −14.0000 −0.805609
\(303\) 18.0000 1.03407
\(304\) −1.00000 −0.0573539
\(305\) −7.00000 −0.400819
\(306\) −2.00000 −0.114332
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 1.00000 0.0567962
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −5.00000 −0.283069
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −2.00000 −0.112867
\(315\) −4.00000 −0.225374
\(316\) 8.00000 0.450035
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 9.00000 0.504695
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 12.0000 0.669775
\(322\) −12.0000 −0.668734
\(323\) 1.00000 0.0556415
\(324\) 1.00000 0.0555556
\(325\) 5.00000 0.277350
\(326\) 16.0000 0.886158
\(327\) −7.00000 −0.387101
\(328\) 6.00000 0.331295
\(329\) −18.0000 −0.992372
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 0 0
\(333\) 8.00000 0.438397
\(334\) −6.00000 −0.328305
\(335\) 14.0000 0.764902
\(336\) 2.00000 0.109109
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) −12.0000 −0.652714
\(339\) 9.00000 0.488813
\(340\) −1.00000 −0.0542326
\(341\) 0 0
\(342\) −2.00000 −0.108148
\(343\) −20.0000 −1.07990
\(344\) −2.00000 −0.107833
\(345\) 6.00000 0.323029
\(346\) −6.00000 −0.322562
\(347\) 33.0000 1.77153 0.885766 0.464131i \(-0.153633\pi\)
0.885766 + 0.464131i \(0.153633\pi\)
\(348\) −9.00000 −0.482451
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) −2.00000 −0.106904
\(351\) −25.0000 −1.33440
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) −3.00000 −0.159448
\(355\) 3.00000 0.159223
\(356\) −9.00000 −0.476999
\(357\) −2.00000 −0.105851
\(358\) −12.0000 −0.634220
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 2.00000 0.105409
\(361\) −18.0000 −0.947368
\(362\) −2.00000 −0.105118
\(363\) −11.0000 −0.577350
\(364\) 10.0000 0.524142
\(365\) 11.0000 0.575766
\(366\) 7.00000 0.365896
\(367\) 26.0000 1.35719 0.678594 0.734513i \(-0.262589\pi\)
0.678594 + 0.734513i \(0.262589\pi\)
\(368\) 6.00000 0.312772
\(369\) 12.0000 0.624695
\(370\) 4.00000 0.207950
\(371\) −18.0000 −0.934513
\(372\) −1.00000 −0.0518476
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 9.00000 0.464140
\(377\) −45.0000 −2.31762
\(378\) 10.0000 0.514344
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −1.00000 −0.0512989
\(381\) −7.00000 −0.358621
\(382\) −24.0000 −1.22795
\(383\) −27.0000 −1.37964 −0.689818 0.723983i \(-0.742309\pi\)
−0.689818 + 0.723983i \(0.742309\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −4.00000 −0.203331
\(388\) −7.00000 −0.355371
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) −5.00000 −0.253185
\(391\) −6.00000 −0.303433
\(392\) 3.00000 0.151523
\(393\) 6.00000 0.302660
\(394\) −24.0000 −1.20910
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −16.0000 −0.803017 −0.401508 0.915855i \(-0.631514\pi\)
−0.401508 + 0.915855i \(0.631514\pi\)
\(398\) 7.00000 0.350878
\(399\) −2.00000 −0.100125
\(400\) 1.00000 0.0500000
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) −14.0000 −0.698257
\(403\) −5.00000 −0.249068
\(404\) 18.0000 0.895533
\(405\) 1.00000 0.0496904
\(406\) 18.0000 0.893325
\(407\) 0 0
\(408\) 1.00000 0.0495074
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 6.00000 0.296319
\(411\) −6.00000 −0.295958
\(412\) −16.0000 −0.788263
\(413\) 6.00000 0.295241
\(414\) 12.0000 0.589768
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 2.00000 0.0975900
\(421\) −16.0000 −0.779792 −0.389896 0.920859i \(-0.627489\pi\)
−0.389896 + 0.920859i \(0.627489\pi\)
\(422\) 22.0000 1.07094
\(423\) 18.0000 0.875190
\(424\) 9.00000 0.437079
\(425\) −1.00000 −0.0485071
\(426\) −3.00000 −0.145350
\(427\) −14.0000 −0.677507
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −2.00000 −0.0964486
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −5.00000 −0.240563
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 2.00000 0.0960031
\(435\) −9.00000 −0.431517
\(436\) −7.00000 −0.335239
\(437\) −6.00000 −0.287019
\(438\) −11.0000 −0.525600
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 5.00000 0.237826
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) −4.00000 −0.189832
\(445\) −9.00000 −0.426641
\(446\) 19.0000 0.899676
\(447\) −12.0000 −0.567581
\(448\) 2.00000 0.0944911
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 2.00000 0.0942809
\(451\) 0 0
\(452\) 9.00000 0.423324
\(453\) 14.0000 0.657777
\(454\) −27.0000 −1.26717
\(455\) 10.0000 0.468807
\(456\) 1.00000 0.0468293
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −14.0000 −0.654177
\(459\) 5.00000 0.233380
\(460\) 6.00000 0.279751
\(461\) 42.0000 1.95614 0.978068 0.208288i \(-0.0667892\pi\)
0.978068 + 0.208288i \(0.0667892\pi\)
\(462\) 0 0
\(463\) −25.0000 −1.16185 −0.580924 0.813958i \(-0.697309\pi\)
−0.580924 + 0.813958i \(0.697309\pi\)
\(464\) −9.00000 −0.417815
\(465\) −1.00000 −0.0463739
\(466\) −9.00000 −0.416917
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) −10.0000 −0.462250
\(469\) 28.0000 1.29292
\(470\) 9.00000 0.415139
\(471\) 2.00000 0.0921551
\(472\) −3.00000 −0.138086
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) −1.00000 −0.0458831
\(476\) −2.00000 −0.0916698
\(477\) 18.0000 0.824163
\(478\) 30.0000 1.37217
\(479\) 9.00000 0.411220 0.205610 0.978634i \(-0.434082\pi\)
0.205610 + 0.978634i \(0.434082\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −20.0000 −0.911922
\(482\) −26.0000 −1.18427
\(483\) 12.0000 0.546019
\(484\) −11.0000 −0.500000
\(485\) −7.00000 −0.317854
\(486\) −16.0000 −0.725775
\(487\) −34.0000 −1.54069 −0.770344 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) 7.00000 0.316875
\(489\) −16.0000 −0.723545
\(490\) 3.00000 0.135526
\(491\) 33.0000 1.48927 0.744635 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) −6.00000 −0.270501
\(493\) 9.00000 0.405340
\(494\) 5.00000 0.224961
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) 38.0000 1.70111 0.850557 0.525883i \(-0.176265\pi\)
0.850557 + 0.525883i \(0.176265\pi\)
\(500\) 1.00000 0.0447214
\(501\) 6.00000 0.268060
\(502\) 12.0000 0.535586
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 4.00000 0.178174
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) −7.00000 −0.310575
\(509\) −36.0000 −1.59567 −0.797836 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 1.00000 0.0442807
\(511\) 22.0000 0.973223
\(512\) −1.00000 −0.0441942
\(513\) 5.00000 0.220755
\(514\) −18.0000 −0.793946
\(515\) −16.0000 −0.705044
\(516\) 2.00000 0.0880451
\(517\) 0 0
\(518\) 8.00000 0.351500
\(519\) 6.00000 0.263371
\(520\) −5.00000 −0.219265
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) −18.0000 −0.787839
\(523\) 38.0000 1.66162 0.830812 0.556553i \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) 6.00000 0.262111
\(525\) 2.00000 0.0872872
\(526\) −3.00000 −0.130806
\(527\) 1.00000 0.0435607
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 9.00000 0.390935
\(531\) −6.00000 −0.260378
\(532\) −2.00000 −0.0867110
\(533\) −30.0000 −1.29944
\(534\) 9.00000 0.389468
\(535\) 12.0000 0.518805
\(536\) −14.0000 −0.604708
\(537\) 12.0000 0.517838
\(538\) 3.00000 0.129339
\(539\) 0 0
\(540\) −5.00000 −0.215166
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 28.0000 1.20270
\(543\) 2.00000 0.0858282
\(544\) 1.00000 0.0428746
\(545\) −7.00000 −0.299847
\(546\) −10.0000 −0.427960
\(547\) −7.00000 −0.299298 −0.149649 0.988739i \(-0.547814\pi\)
−0.149649 + 0.988739i \(0.547814\pi\)
\(548\) −6.00000 −0.256307
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) 9.00000 0.383413
\(552\) −6.00000 −0.255377
\(553\) 16.0000 0.680389
\(554\) −14.0000 −0.594803
\(555\) −4.00000 −0.169791
\(556\) 20.0000 0.848189
\(557\) 21.0000 0.889799 0.444899 0.895581i \(-0.353239\pi\)
0.444899 + 0.895581i \(0.353239\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 10.0000 0.422955
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 21.0000 0.885832
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) −9.00000 −0.378968
\(565\) 9.00000 0.378633
\(566\) −11.0000 −0.462364
\(567\) 2.00000 0.0839921
\(568\) −3.00000 −0.125877
\(569\) −21.0000 −0.880366 −0.440183 0.897908i \(-0.645086\pi\)
−0.440183 + 0.897908i \(0.645086\pi\)
\(570\) 1.00000 0.0418854
\(571\) −34.0000 −1.42286 −0.711428 0.702759i \(-0.751951\pi\)
−0.711428 + 0.702759i \(0.751951\pi\)
\(572\) 0 0
\(573\) 24.0000 1.00261
\(574\) 12.0000 0.500870
\(575\) 6.00000 0.250217
\(576\) −2.00000 −0.0833333
\(577\) 44.0000 1.83174 0.915872 0.401470i \(-0.131501\pi\)
0.915872 + 0.401470i \(0.131501\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 2.00000 0.0831172
\(580\) −9.00000 −0.373705
\(581\) 0 0
\(582\) 7.00000 0.290159
\(583\) 0 0
\(584\) −11.0000 −0.455183
\(585\) −10.0000 −0.413449
\(586\) 9.00000 0.371787
\(587\) 30.0000 1.23823 0.619116 0.785299i \(-0.287491\pi\)
0.619116 + 0.785299i \(0.287491\pi\)
\(588\) −3.00000 −0.123718
\(589\) 1.00000 0.0412043
\(590\) −3.00000 −0.123508
\(591\) 24.0000 0.987228
\(592\) −4.00000 −0.164399
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) −12.0000 −0.491539
\(597\) −7.00000 −0.286491
\(598\) −30.0000 −1.22679
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 20.0000 0.815817 0.407909 0.913023i \(-0.366258\pi\)
0.407909 + 0.913023i \(0.366258\pi\)
\(602\) −4.00000 −0.163028
\(603\) −28.0000 −1.14025
\(604\) 14.0000 0.569652
\(605\) −11.0000 −0.447214
\(606\) −18.0000 −0.731200
\(607\) 2.00000 0.0811775 0.0405887 0.999176i \(-0.487077\pi\)
0.0405887 + 0.999176i \(0.487077\pi\)
\(608\) 1.00000 0.0405554
\(609\) −18.0000 −0.729397
\(610\) 7.00000 0.283422
\(611\) −45.0000 −1.82051
\(612\) 2.00000 0.0808452
\(613\) 5.00000 0.201948 0.100974 0.994889i \(-0.467804\pi\)
0.100974 + 0.994889i \(0.467804\pi\)
\(614\) 16.0000 0.645707
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) −15.0000 −0.603877 −0.301939 0.953327i \(-0.597634\pi\)
−0.301939 + 0.953327i \(0.597634\pi\)
\(618\) 16.0000 0.643614
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) −1.00000 −0.0401610
\(621\) −30.0000 −1.20386
\(622\) 24.0000 0.962312
\(623\) −18.0000 −0.721155
\(624\) 5.00000 0.200160
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 4.00000 0.159490
\(630\) 4.00000 0.159364
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) −8.00000 −0.318223
\(633\) −22.0000 −0.874421
\(634\) −24.0000 −0.953162
\(635\) −7.00000 −0.277787
\(636\) −9.00000 −0.356873
\(637\) −15.0000 −0.594322
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) −1.00000 −0.0395285
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) −12.0000 −0.473602
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 12.0000 0.472866
\(645\) 2.00000 0.0787499
\(646\) −1.00000 −0.0393445
\(647\) −27.0000 −1.06148 −0.530740 0.847535i \(-0.678086\pi\)
−0.530740 + 0.847535i \(0.678086\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −5.00000 −0.196116
\(651\) −2.00000 −0.0783862
\(652\) −16.0000 −0.626608
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 7.00000 0.273722
\(655\) 6.00000 0.234439
\(656\) −6.00000 −0.234261
\(657\) −22.0000 −0.858302
\(658\) 18.0000 0.701713
\(659\) −15.0000 −0.584317 −0.292159 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) 0 0
\(661\) 44.0000 1.71140 0.855701 0.517471i \(-0.173126\pi\)
0.855701 + 0.517471i \(0.173126\pi\)
\(662\) 7.00000 0.272063
\(663\) −5.00000 −0.194184
\(664\) 0 0
\(665\) −2.00000 −0.0775567
\(666\) −8.00000 −0.309994
\(667\) −54.0000 −2.09089
\(668\) 6.00000 0.232147
\(669\) −19.0000 −0.734582
\(670\) −14.0000 −0.540867
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) −25.0000 −0.963679 −0.481840 0.876259i \(-0.660031\pi\)
−0.481840 + 0.876259i \(0.660031\pi\)
\(674\) 13.0000 0.500741
\(675\) −5.00000 −0.192450
\(676\) 12.0000 0.461538
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −9.00000 −0.345643
\(679\) −14.0000 −0.537271
\(680\) 1.00000 0.0383482
\(681\) 27.0000 1.03464
\(682\) 0 0
\(683\) −3.00000 −0.114792 −0.0573959 0.998351i \(-0.518280\pi\)
−0.0573959 + 0.998351i \(0.518280\pi\)
\(684\) 2.00000 0.0764719
\(685\) −6.00000 −0.229248
\(686\) 20.0000 0.763604
\(687\) 14.0000 0.534133
\(688\) 2.00000 0.0762493
\(689\) −45.0000 −1.71436
\(690\) −6.00000 −0.228416
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −33.0000 −1.25266
\(695\) 20.0000 0.758643
\(696\) 9.00000 0.341144
\(697\) 6.00000 0.227266
\(698\) −8.00000 −0.302804
\(699\) 9.00000 0.340411
\(700\) 2.00000 0.0755929
\(701\) −12.0000 −0.453234 −0.226617 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 25.0000 0.943564
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) −9.00000 −0.338960
\(706\) 6.00000 0.225813
\(707\) 36.0000 1.35392
\(708\) 3.00000 0.112747
\(709\) −13.0000 −0.488225 −0.244113 0.969747i \(-0.578497\pi\)
−0.244113 + 0.969747i \(0.578497\pi\)
\(710\) −3.00000 −0.112588
\(711\) −16.0000 −0.600047
\(712\) 9.00000 0.337289
\(713\) −6.00000 −0.224702
\(714\) 2.00000 0.0748481
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −30.0000 −1.12037
\(718\) −6.00000 −0.223918
\(719\) 27.0000 1.00693 0.503465 0.864016i \(-0.332058\pi\)
0.503465 + 0.864016i \(0.332058\pi\)
\(720\) −2.00000 −0.0745356
\(721\) −32.0000 −1.19174
\(722\) 18.0000 0.669891
\(723\) 26.0000 0.966950
\(724\) 2.00000 0.0743294
\(725\) −9.00000 −0.334252
\(726\) 11.0000 0.408248
\(727\) 23.0000 0.853023 0.426511 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) −10.0000 −0.370625
\(729\) 13.0000 0.481481
\(730\) −11.0000 −0.407128
\(731\) −2.00000 −0.0739727
\(732\) −7.00000 −0.258727
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −26.0000 −0.959678
\(735\) −3.00000 −0.110657
\(736\) −6.00000 −0.221163
\(737\) 0 0
\(738\) −12.0000 −0.441726
\(739\) −43.0000 −1.58178 −0.790890 0.611958i \(-0.790382\pi\)
−0.790890 + 0.611958i \(0.790382\pi\)
\(740\) −4.00000 −0.147043
\(741\) −5.00000 −0.183680
\(742\) 18.0000 0.660801
\(743\) −18.0000 −0.660356 −0.330178 0.943919i \(-0.607109\pi\)
−0.330178 + 0.943919i \(0.607109\pi\)
\(744\) 1.00000 0.0366618
\(745\) −12.0000 −0.439646
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) 0 0
\(749\) 24.0000 0.876941
\(750\) −1.00000 −0.0365148
\(751\) 53.0000 1.93400 0.966999 0.254781i \(-0.0820034\pi\)
0.966999 + 0.254781i \(0.0820034\pi\)
\(752\) −9.00000 −0.328196
\(753\) −12.0000 −0.437304
\(754\) 45.0000 1.63880
\(755\) 14.0000 0.509512
\(756\) −10.0000 −0.363696
\(757\) 41.0000 1.49017 0.745085 0.666969i \(-0.232409\pi\)
0.745085 + 0.666969i \(0.232409\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) 1.00000 0.0362738
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 7.00000 0.253583
\(763\) −14.0000 −0.506834
\(764\) 24.0000 0.868290
\(765\) 2.00000 0.0723102
\(766\) 27.0000 0.975550
\(767\) 15.0000 0.541619
\(768\) 1.00000 0.0360844
\(769\) 41.0000 1.47850 0.739249 0.673432i \(-0.235181\pi\)
0.739249 + 0.673432i \(0.235181\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 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
\(775\) −1.00000 −0.0359211
\(776\) 7.00000 0.251285
\(777\) −8.00000 −0.286998
\(778\) 24.0000 0.860442
\(779\) 6.00000 0.214972
\(780\) 5.00000 0.179029
\(781\) 0 0
\(782\) 6.00000 0.214560
\(783\) 45.0000 1.60817
\(784\) −3.00000 −0.107143
\(785\) 2.00000 0.0713831
\(786\) −6.00000 −0.214013
\(787\) −7.00000 −0.249523 −0.124762 0.992187i \(-0.539817\pi\)
−0.124762 + 0.992187i \(0.539817\pi\)
\(788\) 24.0000 0.854965
\(789\) 3.00000 0.106803
\(790\) −8.00000 −0.284627
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) −35.0000 −1.24289
\(794\) 16.0000 0.567819
\(795\) −9.00000 −0.319197
\(796\) −7.00000 −0.248108
\(797\) −54.0000 −1.91278 −0.956389 0.292096i \(-0.905647\pi\)
−0.956389 + 0.292096i \(0.905647\pi\)
\(798\) 2.00000 0.0707992
\(799\) 9.00000 0.318397
\(800\) −1.00000 −0.0353553
\(801\) 18.0000 0.635999
\(802\) 24.0000 0.847469
\(803\) 0 0
\(804\) 14.0000 0.493742
\(805\) 12.0000 0.422944
\(806\) 5.00000 0.176117
\(807\) −3.00000 −0.105605
\(808\) −18.0000 −0.633238
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 14.0000 0.491606 0.245803 0.969320i \(-0.420948\pi\)
0.245803 + 0.969320i \(0.420948\pi\)
\(812\) −18.0000 −0.631676
\(813\) −28.0000 −0.982003
\(814\) 0 0
\(815\) −16.0000 −0.560456
\(816\) −1.00000 −0.0350070
\(817\) −2.00000 −0.0699711
\(818\) −5.00000 −0.174821
\(819\) −20.0000 −0.698857
\(820\) −6.00000 −0.209529
\(821\) −3.00000 −0.104701 −0.0523504 0.998629i \(-0.516671\pi\)
−0.0523504 + 0.998629i \(0.516671\pi\)
\(822\) 6.00000 0.209274
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) −12.0000 −0.417029
\(829\) −28.0000 −0.972480 −0.486240 0.873825i \(-0.661632\pi\)
−0.486240 + 0.873825i \(0.661632\pi\)
\(830\) 0 0
\(831\) 14.0000 0.485655
\(832\) 5.00000 0.173344
\(833\) 3.00000 0.103944
\(834\) −20.0000 −0.692543
\(835\) 6.00000 0.207639
\(836\) 0 0
\(837\) 5.00000 0.172825
\(838\) −6.00000 −0.207267
\(839\) −39.0000 −1.34643 −0.673215 0.739447i \(-0.735087\pi\)
−0.673215 + 0.739447i \(0.735087\pi\)
\(840\) −2.00000 −0.0690066
\(841\) 52.0000 1.79310
\(842\) 16.0000 0.551396
\(843\) −21.0000 −0.723278
\(844\) −22.0000 −0.757271
\(845\) 12.0000 0.412813
\(846\) −18.0000 −0.618853
\(847\) −22.0000 −0.755929
\(848\) −9.00000 −0.309061
\(849\) 11.0000 0.377519
\(850\) 1.00000 0.0342997
\(851\) −24.0000 −0.822709
\(852\) 3.00000 0.102778
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 14.0000 0.479070
\(855\) 2.00000 0.0683986
\(856\) −12.0000 −0.410152
\(857\) −15.0000 −0.512390 −0.256195 0.966625i \(-0.582469\pi\)
−0.256195 + 0.966625i \(0.582469\pi\)
\(858\) 0 0
\(859\) 47.0000 1.60362 0.801810 0.597580i \(-0.203871\pi\)
0.801810 + 0.597580i \(0.203871\pi\)
\(860\) 2.00000 0.0681994
\(861\) −12.0000 −0.408959
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 5.00000 0.170103
\(865\) 6.00000 0.204006
\(866\) −26.0000 −0.883516
\(867\) 1.00000 0.0339618
\(868\) −2.00000 −0.0678844
\(869\) 0 0
\(870\) 9.00000 0.305129
\(871\) 70.0000 2.37186
\(872\) 7.00000 0.237050
\(873\) 14.0000 0.473828
\(874\) 6.00000 0.202953
\(875\) 2.00000 0.0676123
\(876\) 11.0000 0.371656
\(877\) −40.0000 −1.35070 −0.675352 0.737496i \(-0.736008\pi\)
−0.675352 + 0.737496i \(0.736008\pi\)
\(878\) −32.0000 −1.07995
\(879\) −9.00000 −0.303562
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) −6.00000 −0.202031
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) −5.00000 −0.168168
\(885\) 3.00000 0.100844
\(886\) −24.0000 −0.806296
\(887\) −42.0000 −1.41022 −0.705111 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) 4.00000 0.134231
\(889\) −14.0000 −0.469545
\(890\) 9.00000 0.301681
\(891\) 0 0
\(892\) −19.0000 −0.636167
\(893\) 9.00000 0.301174
\(894\) 12.0000 0.401340
\(895\) 12.0000 0.401116
\(896\) −2.00000 −0.0668153
\(897\) 30.0000 1.00167
\(898\) 0 0
\(899\) 9.00000 0.300167
\(900\) −2.00000 −0.0666667
\(901\) 9.00000 0.299833
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) −9.00000 −0.299336
\(905\) 2.00000 0.0664822
\(906\) −14.0000 −0.465119
\(907\) 23.0000 0.763702 0.381851 0.924224i \(-0.375287\pi\)
0.381851 + 0.924224i \(0.375287\pi\)
\(908\) 27.0000 0.896026
\(909\) −36.0000 −1.19404
\(910\) −10.0000 −0.331497
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) −7.00000 −0.231413
\(916\) 14.0000 0.462573
\(917\) 12.0000 0.396275
\(918\) −5.00000 −0.165025
\(919\) 38.0000 1.25350 0.626752 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) −6.00000 −0.197814
\(921\) −16.0000 −0.527218
\(922\) −42.0000 −1.38320
\(923\) 15.0000 0.493731
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 25.0000 0.821551
\(927\) 32.0000 1.05102
\(928\) 9.00000 0.295439
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 1.00000 0.0327913
\(931\) 3.00000 0.0983210
\(932\) 9.00000 0.294805
\(933\) −24.0000 −0.785725
\(934\) 0 0
\(935\) 0 0
\(936\) 10.0000 0.326860
\(937\) 8.00000 0.261349 0.130674 0.991425i \(-0.458286\pi\)
0.130674 + 0.991425i \(0.458286\pi\)
\(938\) −28.0000 −0.914232
\(939\) −10.0000 −0.326338
\(940\) −9.00000 −0.293548
\(941\) −45.0000 −1.46696 −0.733479 0.679712i \(-0.762105\pi\)
−0.733479 + 0.679712i \(0.762105\pi\)
\(942\) −2.00000 −0.0651635
\(943\) −36.0000 −1.17232
\(944\) 3.00000 0.0976417
\(945\) −10.0000 −0.325300
\(946\) 0 0
\(947\) 3.00000 0.0974869 0.0487435 0.998811i \(-0.484478\pi\)
0.0487435 + 0.998811i \(0.484478\pi\)
\(948\) 8.00000 0.259828
\(949\) 55.0000 1.78538
\(950\) 1.00000 0.0324443
\(951\) 24.0000 0.778253
\(952\) 2.00000 0.0648204
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) −18.0000 −0.582772
\(955\) 24.0000 0.776622
\(956\) −30.0000 −0.970269
\(957\) 0 0
\(958\) −9.00000 −0.290777
\(959\) −12.0000 −0.387500
\(960\) 1.00000 0.0322749
\(961\) −30.0000 −0.967742
\(962\) 20.0000 0.644826
\(963\) −24.0000 −0.773389
\(964\) 26.0000 0.837404
\(965\) 2.00000 0.0643823
\(966\) −12.0000 −0.386094
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 11.0000 0.353553
\(969\) 1.00000 0.0321246
\(970\) 7.00000 0.224756
\(971\) 3.00000 0.0962746 0.0481373 0.998841i \(-0.484672\pi\)
0.0481373 + 0.998841i \(0.484672\pi\)
\(972\) 16.0000 0.513200
\(973\) 40.0000 1.28234
\(974\) 34.0000 1.08943
\(975\) 5.00000 0.160128
\(976\) −7.00000 −0.224065
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 16.0000 0.511624
\(979\) 0 0
\(980\) −3.00000 −0.0958315
\(981\) 14.0000 0.446986
\(982\) −33.0000 −1.05307
\(983\) 18.0000 0.574111 0.287055 0.957914i \(-0.407324\pi\)
0.287055 + 0.957914i \(0.407324\pi\)
\(984\) 6.00000 0.191273
\(985\) 24.0000 0.764704
\(986\) −9.00000 −0.286618
\(987\) −18.0000 −0.572946
\(988\) −5.00000 −0.159071
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) 1.00000 0.0317500
\(993\) −7.00000 −0.222138
\(994\) −6.00000 −0.190308
\(995\) −7.00000 −0.221915
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −38.0000 −1.20287
\(999\) 20.0000 0.632772
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 170.2.a.c.1.1 1
3.2 odd 2 1530.2.a.l.1.1 1
4.3 odd 2 1360.2.a.e.1.1 1
5.2 odd 4 850.2.c.c.749.1 2
5.3 odd 4 850.2.c.c.749.2 2
5.4 even 2 850.2.a.g.1.1 1
7.6 odd 2 8330.2.a.d.1.1 1
8.3 odd 2 5440.2.a.p.1.1 1
8.5 even 2 5440.2.a.i.1.1 1
15.14 odd 2 7650.2.a.i.1.1 1
17.4 even 4 2890.2.b.h.2311.1 2
17.13 even 4 2890.2.b.h.2311.2 2
17.16 even 2 2890.2.a.e.1.1 1
20.19 odd 2 6800.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
170.2.a.c.1.1 1 1.1 even 1 trivial
850.2.a.g.1.1 1 5.4 even 2
850.2.c.c.749.1 2 5.2 odd 4
850.2.c.c.749.2 2 5.3 odd 4
1360.2.a.e.1.1 1 4.3 odd 2
1530.2.a.l.1.1 1 3.2 odd 2
2890.2.a.e.1.1 1 17.16 even 2
2890.2.b.h.2311.1 2 17.4 even 4
2890.2.b.h.2311.2 2 17.13 even 4
5440.2.a.i.1.1 1 8.5 even 2
5440.2.a.p.1.1 1 8.3 odd 2
6800.2.a.s.1.1 1 20.19 odd 2
7650.2.a.i.1.1 1 15.14 odd 2
8330.2.a.d.1.1 1 7.6 odd 2