Properties

Label 345.2.a.d.1.1
Level $345$
Weight $2$
Character 345.1
Self dual yes
Analytic conductor $2.755$
Analytic rank $1$
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] = [345,2,Mod(1,345)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(345, 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("345.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 345.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: \(2.75483886973\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 345.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} -2.00000 q^{12} -1.00000 q^{15} +4.00000 q^{16} -3.00000 q^{17} -8.00000 q^{19} +2.00000 q^{20} -3.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{28} +9.00000 q^{29} -5.00000 q^{31} -4.00000 q^{33} +3.00000 q^{35} -2.00000 q^{36} -9.00000 q^{37} +7.00000 q^{41} +4.00000 q^{43} +8.00000 q^{44} -1.00000 q^{45} -2.00000 q^{47} +4.00000 q^{48} +2.00000 q^{49} -3.00000 q^{51} +13.0000 q^{53} +4.00000 q^{55} -8.00000 q^{57} -3.00000 q^{59} +2.00000 q^{60} -14.0000 q^{61} -3.00000 q^{63} -8.00000 q^{64} +13.0000 q^{67} +6.00000 q^{68} +1.00000 q^{69} -13.0000 q^{71} -4.00000 q^{73} +1.00000 q^{75} +16.0000 q^{76} +12.0000 q^{77} -4.00000 q^{80} +1.00000 q^{81} -1.00000 q^{83} +6.00000 q^{84} +3.00000 q^{85} +9.00000 q^{87} -8.00000 q^{89} -2.00000 q^{92} -5.00000 q^{93} +8.00000 q^{95} +10.0000 q^{97} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.00000 0.577350
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −2.00000 −0.577350
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 4.00000 1.00000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 2.00000 0.447214
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 6.00000 1.13389
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) −2.00000 −0.333333
\(37\) −9.00000 −1.47959 −0.739795 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 8.00000 1.20605
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 4.00000 0.577350
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 13.0000 1.78569 0.892844 0.450367i \(-0.148707\pi\)
0.892844 + 0.450367i \(0.148707\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 2.00000 0.258199
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 13.0000 1.58820 0.794101 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) 6.00000 0.727607
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −13.0000 −1.54282 −0.771408 0.636341i \(-0.780447\pi\)
−0.771408 + 0.636341i \(0.780447\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 16.0000 1.83533
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.00000 −0.109764 −0.0548821 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) 6.00000 0.654654
\(85\) 3.00000 0.325396
\(86\) 0 0
\(87\) 9.00000 0.964901
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) −5.00000 −0.518476
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) −2.00000 −0.200000
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 0 0
\(107\) 11.0000 1.06341 0.531705 0.846930i \(-0.321551\pi\)
0.531705 + 0.846930i \(0.321551\pi\)
\(108\) −2.00000 −0.192450
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) −9.00000 −0.854242
\(112\) −12.0000 −1.13389
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) −18.0000 −1.67126
\(117\) 0 0
\(118\) 0 0
\(119\) 9.00000 0.825029
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 7.00000 0.631169
\(124\) 10.0000 0.898027
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 8.00000 0.696311
\(133\) 24.0000 2.08106
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) −3.00000 −0.254457 −0.127228 0.991873i \(-0.540608\pi\)
−0.127228 + 0.991873i \(0.540608\pi\)
\(140\) −6.00000 −0.507093
\(141\) −2.00000 −0.168430
\(142\) 0 0
\(143\) 0 0
\(144\) 4.00000 0.333333
\(145\) −9.00000 −0.747409
\(146\) 0 0
\(147\) 2.00000 0.164957
\(148\) 18.0000 1.47959
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 5.00000 0.401610
\(156\) 0 0
\(157\) −3.00000 −0.239426 −0.119713 0.992809i \(-0.538197\pi\)
−0.119713 + 0.992809i \(0.538197\pi\)
\(158\) 0 0
\(159\) 13.0000 1.03097
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) −14.0000 −1.09322
\(165\) 4.00000 0.311400
\(166\) 0 0
\(167\) −22.0000 −1.70241 −0.851206 0.524832i \(-0.824128\pi\)
−0.851206 + 0.524832i \(0.824128\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) −8.00000 −0.609994
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) −16.0000 −1.20605
\(177\) −3.00000 −0.225494
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 2.00000 0.149071
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) 0 0
\(185\) 9.00000 0.661693
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 4.00000 0.291730
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) −2.00000 −0.144715 −0.0723575 0.997379i \(-0.523052\pi\)
−0.0723575 + 0.997379i \(0.523052\pi\)
\(192\) −8.00000 −0.577350
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −4.00000 −0.285714
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 0 0
\(201\) 13.0000 0.916949
\(202\) 0 0
\(203\) −27.0000 −1.89503
\(204\) 6.00000 0.420084
\(205\) −7.00000 −0.488901
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 32.0000 2.21349
\(210\) 0 0
\(211\) 7.00000 0.481900 0.240950 0.970538i \(-0.422541\pi\)
0.240950 + 0.970538i \(0.422541\pi\)
\(212\) −26.0000 −1.78569
\(213\) −13.0000 −0.890745
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 15.0000 1.01827
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) −8.00000 −0.539360
\(221\) 0 0
\(222\) 0 0
\(223\) 10.0000 0.669650 0.334825 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 16.0000 1.05963
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) 0 0
\(233\) −8.00000 −0.524097 −0.262049 0.965055i \(-0.584398\pi\)
−0.262049 + 0.965055i \(0.584398\pi\)
\(234\) 0 0
\(235\) 2.00000 0.130466
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) 0 0
\(239\) 13.0000 0.840900 0.420450 0.907316i \(-0.361872\pi\)
0.420450 + 0.907316i \(0.361872\pi\)
\(240\) −4.00000 −0.258199
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 28.0000 1.79252
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −1.00000 −0.0633724
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 6.00000 0.377964
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) 3.00000 0.187867
\(256\) 16.0000 1.00000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 27.0000 1.67770
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) 0 0
\(263\) −23.0000 −1.41824 −0.709120 0.705087i \(-0.750908\pi\)
−0.709120 + 0.705087i \(0.750908\pi\)
\(264\) 0 0
\(265\) −13.0000 −0.798584
\(266\) 0 0
\(267\) −8.00000 −0.489592
\(268\) −26.0000 −1.58820
\(269\) −17.0000 −1.03651 −0.518254 0.855227i \(-0.673418\pi\)
−0.518254 + 0.855227i \(0.673418\pi\)
\(270\) 0 0
\(271\) −23.0000 −1.39715 −0.698575 0.715537i \(-0.746182\pi\)
−0.698575 + 0.715537i \(0.746182\pi\)
\(272\) −12.0000 −0.727607
\(273\) 0 0
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) −2.00000 −0.120386
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 0 0
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) −5.00000 −0.297219 −0.148610 0.988896i \(-0.547480\pi\)
−0.148610 + 0.988896i \(0.547480\pi\)
\(284\) 26.0000 1.54282
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) −21.0000 −1.23959
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 8.00000 0.468165
\(293\) 19.0000 1.10999 0.554996 0.831853i \(-0.312720\pi\)
0.554996 + 0.831853i \(0.312720\pi\)
\(294\) 0 0
\(295\) 3.00000 0.174667
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 0 0
\(300\) −2.00000 −0.115470
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) 9.00000 0.517036
\(304\) −32.0000 −1.83533
\(305\) 14.0000 0.801638
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −24.0000 −1.36753
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 25.0000 1.41308 0.706542 0.707671i \(-0.250254\pi\)
0.706542 + 0.707671i \(0.250254\pi\)
\(314\) 0 0
\(315\) 3.00000 0.169031
\(316\) 0 0
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) −36.0000 −2.01561
\(320\) 8.00000 0.447214
\(321\) 11.0000 0.613960
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) −2.00000 −0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) −16.0000 −0.884802
\(328\) 0 0
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 7.00000 0.384755 0.192377 0.981321i \(-0.438380\pi\)
0.192377 + 0.981321i \(0.438380\pi\)
\(332\) 2.00000 0.109764
\(333\) −9.00000 −0.493197
\(334\) 0 0
\(335\) −13.0000 −0.710266
\(336\) −12.0000 −0.654654
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 0 0
\(339\) 1.00000 0.0543125
\(340\) −6.00000 −0.325396
\(341\) 20.0000 1.08306
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) −18.0000 −0.964901
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 13.0000 0.689968
\(356\) 16.0000 0.847998
\(357\) 9.00000 0.476331
\(358\) 0 0
\(359\) −2.00000 −0.105556 −0.0527780 0.998606i \(-0.516808\pi\)
−0.0527780 + 0.998606i \(0.516808\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) 31.0000 1.61819 0.809093 0.587680i \(-0.199959\pi\)
0.809093 + 0.587680i \(0.199959\pi\)
\(368\) 4.00000 0.208514
\(369\) 7.00000 0.364405
\(370\) 0 0
\(371\) −39.0000 −2.02478
\(372\) 10.0000 0.518476
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) −16.0000 −0.820783
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) −23.0000 −1.17525 −0.587623 0.809135i \(-0.699936\pi\)
−0.587623 + 0.809135i \(0.699936\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) −20.0000 −1.01535
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) 0 0
\(396\) 8.00000 0.402015
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 0 0
\(399\) 24.0000 1.20150
\(400\) 4.00000 0.200000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −18.0000 −0.895533
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 36.0000 1.78445
\(408\) 0 0
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) −16.0000 −0.788263
\(413\) 9.00000 0.442861
\(414\) 0 0
\(415\) 1.00000 0.0490881
\(416\) 0 0
\(417\) −3.00000 −0.146911
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −6.00000 −0.292770
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) 0 0
\(423\) −2.00000 −0.0972433
\(424\) 0 0
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) 42.0000 2.03252
\(428\) −22.0000 −1.06341
\(429\) 0 0
\(430\) 0 0
\(431\) 20.0000 0.963366 0.481683 0.876346i \(-0.340026\pi\)
0.481683 + 0.876346i \(0.340026\pi\)
\(432\) 4.00000 0.192450
\(433\) −37.0000 −1.77811 −0.889053 0.457804i \(-0.848636\pi\)
−0.889053 + 0.457804i \(0.848636\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) 32.0000 1.53252
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 18.0000 0.854242
\(445\) 8.00000 0.379236
\(446\) 0 0
\(447\) −14.0000 −0.662177
\(448\) 24.0000 1.13389
\(449\) −11.0000 −0.519122 −0.259561 0.965727i \(-0.583578\pi\)
−0.259561 + 0.965727i \(0.583578\pi\)
\(450\) 0 0
\(451\) −28.0000 −1.31847
\(452\) −2.00000 −0.0940721
\(453\) −8.00000 −0.375873
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −35.0000 −1.63723 −0.818615 0.574342i \(-0.805258\pi\)
−0.818615 + 0.574342i \(0.805258\pi\)
\(458\) 0 0
\(459\) −3.00000 −0.140028
\(460\) 2.00000 0.0932505
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 36.0000 1.67126
\(465\) 5.00000 0.231869
\(466\) 0 0
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) −39.0000 −1.80085
\(470\) 0 0
\(471\) −3.00000 −0.138233
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) −18.0000 −0.825029
\(477\) 13.0000 0.595229
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −3.00000 −0.136505
\(484\) −10.0000 −0.454545
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 0 0
\(489\) −2.00000 −0.0904431
\(490\) 0 0
\(491\) 3.00000 0.135388 0.0676941 0.997706i \(-0.478436\pi\)
0.0676941 + 0.997706i \(0.478436\pi\)
\(492\) −14.0000 −0.631169
\(493\) −27.0000 −1.21602
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) −20.0000 −0.898027
\(497\) 39.0000 1.74939
\(498\) 0 0
\(499\) 21.0000 0.940089 0.470045 0.882643i \(-0.344238\pi\)
0.470045 + 0.882643i \(0.344238\pi\)
\(500\) 2.00000 0.0894427
\(501\) −22.0000 −0.982888
\(502\) 0 0
\(503\) 21.0000 0.936344 0.468172 0.883637i \(-0.344913\pi\)
0.468172 + 0.883637i \(0.344913\pi\)
\(504\) 0 0
\(505\) −9.00000 −0.400495
\(506\) 0 0
\(507\) −13.0000 −0.577350
\(508\) 36.0000 1.59724
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) 0 0
\(513\) −8.00000 −0.353209
\(514\) 0 0
\(515\) −8.00000 −0.352522
\(516\) −8.00000 −0.352180
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 8.00000 0.349482
\(525\) −3.00000 −0.130931
\(526\) 0 0
\(527\) 15.0000 0.653410
\(528\) −16.0000 −0.696311
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −3.00000 −0.130189
\(532\) −48.0000 −2.08106
\(533\) 0 0
\(534\) 0 0
\(535\) −11.0000 −0.475571
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −8.00000 −0.344584
\(540\) 2.00000 0.0860663
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 16.0000 0.685365
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −4.00000 −0.170872
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) −72.0000 −3.06730
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 9.00000 0.382029
\(556\) 6.00000 0.254457
\(557\) 21.0000 0.889799 0.444899 0.895581i \(-0.353239\pi\)
0.444899 + 0.895581i \(0.353239\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 12.0000 0.507093
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) −11.0000 −0.463595 −0.231797 0.972764i \(-0.574461\pi\)
−0.231797 + 0.972764i \(0.574461\pi\)
\(564\) 4.00000 0.168430
\(565\) −1.00000 −0.0420703
\(566\) 0 0
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) −2.00000 −0.0835512
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) −8.00000 −0.333333
\(577\) −20.0000 −0.832611 −0.416305 0.909225i \(-0.636675\pi\)
−0.416305 + 0.909225i \(0.636675\pi\)
\(578\) 0 0
\(579\) 4.00000 0.166234
\(580\) 18.0000 0.747409
\(581\) 3.00000 0.124461
\(582\) 0 0
\(583\) −52.0000 −2.15362
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.0000 0.577842 0.288921 0.957353i \(-0.406704\pi\)
0.288921 + 0.957353i \(0.406704\pi\)
\(588\) −4.00000 −0.164957
\(589\) 40.0000 1.64817
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) −36.0000 −1.47959
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) −9.00000 −0.368964
\(596\) 28.0000 1.14692
\(597\) −18.0000 −0.736691
\(598\) 0 0
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) −41.0000 −1.67242 −0.836212 0.548406i \(-0.815235\pi\)
−0.836212 + 0.548406i \(0.815235\pi\)
\(602\) 0 0
\(603\) 13.0000 0.529401
\(604\) 16.0000 0.651031
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) 0 0
\(609\) −27.0000 −1.09410
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 0 0
\(615\) −7.00000 −0.282267
\(616\) 0 0
\(617\) −33.0000 −1.32853 −0.664265 0.747497i \(-0.731255\pi\)
−0.664265 + 0.747497i \(0.731255\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) −10.0000 −0.401610
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 32.0000 1.27796
\(628\) 6.00000 0.239426
\(629\) 27.0000 1.07656
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) 0 0
\(633\) 7.00000 0.278225
\(634\) 0 0
\(635\) 18.0000 0.714308
\(636\) −26.0000 −1.03097
\(637\) 0 0
\(638\) 0 0
\(639\) −13.0000 −0.514272
\(640\) 0 0
\(641\) −20.0000 −0.789953 −0.394976 0.918691i \(-0.629247\pi\)
−0.394976 + 0.918691i \(0.629247\pi\)
\(642\) 0 0
\(643\) 13.0000 0.512670 0.256335 0.966588i \(-0.417485\pi\)
0.256335 + 0.966588i \(0.417485\pi\)
\(644\) 6.00000 0.236433
\(645\) −4.00000 −0.157500
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 15.0000 0.587896
\(652\) 4.00000 0.156652
\(653\) 16.0000 0.626128 0.313064 0.949732i \(-0.398644\pi\)
0.313064 + 0.949732i \(0.398644\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) 28.0000 1.09322
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) −8.00000 −0.311400
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −24.0000 −0.930680
\(666\) 0 0
\(667\) 9.00000 0.348481
\(668\) 44.0000 1.70241
\(669\) 10.0000 0.386622
\(670\) 0 0
\(671\) 56.0000 2.16186
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 26.0000 1.00000
\(677\) −45.0000 −1.72949 −0.864745 0.502211i \(-0.832520\pi\)
−0.864745 + 0.502211i \(0.832520\pi\)
\(678\) 0 0
\(679\) −30.0000 −1.15129
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) 26.0000 0.994862 0.497431 0.867503i \(-0.334277\pi\)
0.497431 + 0.867503i \(0.334277\pi\)
\(684\) 16.0000 0.611775
\(685\) −2.00000 −0.0764161
\(686\) 0 0
\(687\) 4.00000 0.152610
\(688\) 16.0000 0.609994
\(689\) 0 0
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 48.0000 1.82469
\(693\) 12.0000 0.455842
\(694\) 0 0
\(695\) 3.00000 0.113796
\(696\) 0 0
\(697\) −21.0000 −0.795432
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 6.00000 0.226779
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 72.0000 2.71553
\(704\) 32.0000 1.20605
\(705\) 2.00000 0.0753244
\(706\) 0 0
\(707\) −27.0000 −1.01544
\(708\) 6.00000 0.225494
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −5.00000 −0.187251
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 13.0000 0.485494
\(718\) 0 0
\(719\) −7.00000 −0.261056 −0.130528 0.991445i \(-0.541667\pi\)
−0.130528 + 0.991445i \(0.541667\pi\)
\(720\) −4.00000 −0.149071
\(721\) −24.0000 −0.893807
\(722\) 0 0
\(723\) 20.0000 0.743808
\(724\) 0 0
\(725\) 9.00000 0.334252
\(726\) 0 0
\(727\) −37.0000 −1.37225 −0.686127 0.727482i \(-0.740691\pi\)
−0.686127 + 0.727482i \(0.740691\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 28.0000 1.03491
\(733\) 51.0000 1.88373 0.941864 0.335994i \(-0.109072\pi\)
0.941864 + 0.335994i \(0.109072\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) −52.0000 −1.91544
\(738\) 0 0
\(739\) 27.0000 0.993211 0.496606 0.867976i \(-0.334580\pi\)
0.496606 + 0.867976i \(0.334580\pi\)
\(740\) −18.0000 −0.661693
\(741\) 0 0
\(742\) 0 0
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) 0 0
\(747\) −1.00000 −0.0365881
\(748\) −24.0000 −0.877527
\(749\) −33.0000 −1.20579
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −8.00000 −0.291730
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 6.00000 0.218218
\(757\) 45.0000 1.63555 0.817776 0.575536i \(-0.195207\pi\)
0.817776 + 0.575536i \(0.195207\pi\)
\(758\) 0 0
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) −41.0000 −1.48625 −0.743124 0.669153i \(-0.766657\pi\)
−0.743124 + 0.669153i \(0.766657\pi\)
\(762\) 0 0
\(763\) 48.0000 1.73772
\(764\) 4.00000 0.144715
\(765\) 3.00000 0.108465
\(766\) 0 0
\(767\) 0 0
\(768\) 16.0000 0.577350
\(769\) 6.00000 0.216366 0.108183 0.994131i \(-0.465497\pi\)
0.108183 + 0.994131i \(0.465497\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) −8.00000 −0.287926
\(773\) −46.0000 −1.65451 −0.827253 0.561830i \(-0.810097\pi\)
−0.827253 + 0.561830i \(0.810097\pi\)
\(774\) 0 0
\(775\) −5.00000 −0.179605
\(776\) 0 0
\(777\) 27.0000 0.968620
\(778\) 0 0
\(779\) −56.0000 −2.00641
\(780\) 0 0
\(781\) 52.0000 1.86071
\(782\) 0 0
\(783\) 9.00000 0.321634
\(784\) 8.00000 0.285714
\(785\) 3.00000 0.107075
\(786\) 0 0
\(787\) −27.0000 −0.962446 −0.481223 0.876598i \(-0.659807\pi\)
−0.481223 + 0.876598i \(0.659807\pi\)
\(788\) −4.00000 −0.142494
\(789\) −23.0000 −0.818822
\(790\) 0 0
\(791\) −3.00000 −0.106668
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −13.0000 −0.461062
\(796\) 36.0000 1.27599
\(797\) 27.0000 0.956389 0.478195 0.878254i \(-0.341291\pi\)
0.478195 + 0.878254i \(0.341291\pi\)
\(798\) 0 0
\(799\) 6.00000 0.212265
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) 0 0
\(803\) 16.0000 0.564628
\(804\) −26.0000 −0.916949
\(805\) 3.00000 0.105736
\(806\) 0 0
\(807\) −17.0000 −0.598428
\(808\) 0 0
\(809\) −39.0000 −1.37117 −0.685583 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) −25.0000 −0.877869 −0.438934 0.898519i \(-0.644644\pi\)
−0.438934 + 0.898519i \(0.644644\pi\)
\(812\) 54.0000 1.89503
\(813\) −23.0000 −0.806645
\(814\) 0 0
\(815\) 2.00000 0.0700569
\(816\) −12.0000 −0.420084
\(817\) −32.0000 −1.11954
\(818\) 0 0
\(819\) 0 0
\(820\) 14.0000 0.488901
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 37.0000 1.28662 0.643308 0.765607i \(-0.277561\pi\)
0.643308 + 0.765607i \(0.277561\pi\)
\(828\) −2.00000 −0.0695048
\(829\) −27.0000 −0.937749 −0.468874 0.883265i \(-0.655340\pi\)
−0.468874 + 0.883265i \(0.655340\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 22.0000 0.761341
\(836\) −64.0000 −2.21349
\(837\) −5.00000 −0.172825
\(838\) 0 0
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 24.0000 0.826604
\(844\) −14.0000 −0.481900
\(845\) 13.0000 0.447214
\(846\) 0 0
\(847\) −15.0000 −0.515406
\(848\) 52.0000 1.78569
\(849\) −5.00000 −0.171600
\(850\) 0 0
\(851\) −9.00000 −0.308516
\(852\) 26.0000 0.890745
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 43.0000 1.46714 0.733571 0.679613i \(-0.237852\pi\)
0.733571 + 0.679613i \(0.237852\pi\)
\(860\) 8.00000 0.272798
\(861\) −21.0000 −0.715678
\(862\) 0 0
\(863\) 54.0000 1.83818 0.919091 0.394046i \(-0.128925\pi\)
0.919091 + 0.394046i \(0.128925\pi\)
\(864\) 0 0
\(865\) 24.0000 0.816024
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) −30.0000 −1.01827
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 8.00000 0.270295
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) 19.0000 0.640854
\(880\) 16.0000 0.539360
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 0 0
\(885\) 3.00000 0.100844
\(886\) 0 0
\(887\) −46.0000 −1.54453 −0.772264 0.635301i \(-0.780876\pi\)
−0.772264 + 0.635301i \(0.780876\pi\)
\(888\) 0 0
\(889\) 54.0000 1.81110
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) −20.0000 −0.669650
\(893\) 16.0000 0.535420
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −45.0000 −1.50083
\(900\) −2.00000 −0.0666667
\(901\) −39.0000 −1.29928
\(902\) 0 0
\(903\) −12.0000 −0.399335
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 53.0000 1.75984 0.879918 0.475125i \(-0.157597\pi\)
0.879918 + 0.475125i \(0.157597\pi\)
\(908\) 8.00000 0.265489
\(909\) 9.00000 0.298511
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −32.0000 −1.05963
\(913\) 4.00000 0.132381
\(914\) 0 0
\(915\) 14.0000 0.462826
\(916\) −8.00000 −0.264327
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) −24.0000 −0.789542
\(925\) −9.00000 −0.295918
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 57.0000 1.87011 0.935055 0.354504i \(-0.115350\pi\)
0.935055 + 0.354504i \(0.115350\pi\)
\(930\) 0 0
\(931\) −16.0000 −0.524379
\(932\) 16.0000 0.524097
\(933\) −8.00000 −0.261908
\(934\) 0 0
\(935\) −12.0000 −0.392442
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 25.0000 0.815844
\(940\) −4.00000 −0.130466
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 0 0
\(943\) 7.00000 0.227951
\(944\) −12.0000 −0.390567
\(945\) 3.00000 0.0975900
\(946\) 0 0
\(947\) −8.00000 −0.259965 −0.129983 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 0 0
\(955\) 2.00000 0.0647185
\(956\) −26.0000 −0.840900
\(957\) −36.0000 −1.16371
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 8.00000 0.258199
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 11.0000 0.354470
\(964\) −40.0000 −1.28831
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) 10.0000 0.321578 0.160789 0.986989i \(-0.448596\pi\)
0.160789 + 0.986989i \(0.448596\pi\)
\(968\) 0 0
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) −2.00000 −0.0641500
\(973\) 9.00000 0.288527
\(974\) 0 0
\(975\) 0 0
\(976\) −56.0000 −1.79252
\(977\) −25.0000 −0.799821 −0.399910 0.916554i \(-0.630959\pi\)
−0.399910 + 0.916554i \(0.630959\pi\)
\(978\) 0 0
\(979\) 32.0000 1.02272
\(980\) 4.00000 0.127775
\(981\) −16.0000 −0.510841
\(982\) 0 0
\(983\) 15.0000 0.478426 0.239213 0.970967i \(-0.423111\pi\)
0.239213 + 0.970967i \(0.423111\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −33.0000 −1.04828 −0.524140 0.851632i \(-0.675613\pi\)
−0.524140 + 0.851632i \(0.675613\pi\)
\(992\) 0 0
\(993\) 7.00000 0.222138
\(994\) 0 0
\(995\) 18.0000 0.570638
\(996\) 2.00000 0.0633724
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) 0 0
\(999\) −9.00000 −0.284747
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 345.2.a.d.1.1 1
3.2 odd 2 1035.2.a.d.1.1 1
4.3 odd 2 5520.2.a.h.1.1 1
5.2 odd 4 1725.2.b.j.1174.1 2
5.3 odd 4 1725.2.b.j.1174.2 2
5.4 even 2 1725.2.a.k.1.1 1
15.14 odd 2 5175.2.a.o.1.1 1
23.22 odd 2 7935.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.a.d.1.1 1 1.1 even 1 trivial
1035.2.a.d.1.1 1 3.2 odd 2
1725.2.a.k.1.1 1 5.4 even 2
1725.2.b.j.1174.1 2 5.2 odd 4
1725.2.b.j.1174.2 2 5.3 odd 4
5175.2.a.o.1.1 1 15.14 odd 2
5520.2.a.h.1.1 1 4.3 odd 2
7935.2.a.i.1.1 1 23.22 odd 2