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Properties

Label 1425.2.a.d.1.1
Level $1425$
Weight $2$
Character 1425.1
Self dual yes
Analytic conductor $11.379$
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] = [1425,2,Mod(1,1425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1425, 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("1425.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.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: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1425.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} +2.00000 q^{21} +2.00000 q^{22} +4.00000 q^{23} +3.00000 q^{24} -4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} +4.00000 q^{29} -5.00000 q^{32} -2.00000 q^{33} +2.00000 q^{34} -1.00000 q^{36} +1.00000 q^{38} +4.00000 q^{39} -2.00000 q^{42} +10.0000 q^{43} +2.00000 q^{44} -4.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -2.00000 q^{51} -4.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} +6.00000 q^{56} -1.00000 q^{57} -4.00000 q^{58} +4.00000 q^{59} +2.00000 q^{61} +2.00000 q^{63} +7.00000 q^{64} +2.00000 q^{66} +16.0000 q^{67} +2.00000 q^{68} +4.00000 q^{69} +3.00000 q^{72} +2.00000 q^{73} +1.00000 q^{76} -4.00000 q^{77} -4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -2.00000 q^{84} -10.0000 q^{86} +4.00000 q^{87} -6.00000 q^{88} +8.00000 q^{91} -4.00000 q^{92} +12.0000 q^{94} -5.00000 q^{96} +16.0000 q^{97} +3.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 2.00000 0.426401
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −5.00000 −0.883883
\(33\) −2.00000 −0.348155
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 1.00000 0.162221
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −2.00000 −0.308607
\(43\) 10.0000 1.52499 0.762493 − 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) −4.00000 −0.554700
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 6.00000 0.801784
\(57\) −1.00000 −0.132453
\(58\) −4.00000 −0.525226
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 16.0000 1.95471 0.977356 − 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) 2.00000 0.242536
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 3.00000 0.353553
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −4.00000 −0.455842
\(78\) −4.00000 −0.452911
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 4.00000 0.428845
\(88\) −6.00000 −0.639602
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) 16.0000 1.62455 0.812277 − 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) 3.00000 0.303046
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 − 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 2.00000 0.198030
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 12.0000 1.17670
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.0000 0.957826 0.478913 − 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) −14.0000 −1.31701 −0.658505 − 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 4.00000 0.369800
\(118\) −4.00000 −0.368230
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) −4.00000 −0.354943 −0.177471 − 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 3.00000 0.265165
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) −14.0000 −1.22319 −0.611593 − 0.791173i \(-0.709471\pi\)
−0.611593 + 0.791173i \(0.709471\pi\)
\(132\) 2.00000 0.174078
\(133\) −2.00000 −0.173422
\(134\) −16.0000 −1.38219
\(135\) 0 0
\(136\) −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
\(139\) 8.00000 0.678551 0.339276 − 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) −3.00000 −0.247436
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 − 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 24.0000 1.95309 0.976546 − 0.215308i \(-0.0690756\pi\)
0.976546 + 0.215308i \(0.0690756\pi\)
\(152\) −3.00000 −0.243332
\(153\) −2.00000 −0.161690
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 18.0000 1.43656 0.718278 − 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 8.00000 0.636446
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) −1.00000 −0.0785674
\(163\) 6.00000 0.469956 0.234978 − 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 8.00000 0.619059 0.309529 − 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 6.00000 0.462910
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) −10.0000 −0.762493
\(173\) −18.0000 −1.36851 −0.684257 − 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) −8.00000 −0.592999
\(183\) 2.00000 0.147844
\(184\) 12.0000 0.884652
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 12.0000 0.875190
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 − 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 7.00000 0.505181
\(193\) −4.00000 −0.287926 −0.143963 − 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) 2.00000 0.142494 0.0712470 − 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 2.00000 0.142134
\(199\) 12.0000 0.850657 0.425329 − 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) −14.0000 −0.985037
\(203\) 8.00000 0.561490
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 4.00000 0.278019
\(208\) −4.00000 −0.277350
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 − 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) −10.0000 −0.668153
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 1.00000 0.0662266
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 12.0000 0.787839
\(233\) 18.0000 1.17922 0.589610 − 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) −8.00000 −0.519656
\(238\) 4.00000 0.259281
\(239\) −6.00000 −0.388108 −0.194054 − 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 − 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 22.0000 1.38863 0.694314 − 0.719672i \(-0.255708\pi\)
0.694314 + 0.719672i \(0.255708\pi\)
\(252\) −2.00000 −0.125988
\(253\) −8.00000 −0.502956
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −22.0000 −1.37232 −0.686161 − 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) −10.0000 −0.622573
\(259\) 0 0
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 14.0000 0.864923
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) −16.0000 −0.977356
\(269\) 4.00000 0.243884 0.121942 − 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 − 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 2.00000 0.121268
\(273\) 8.00000 0.484182
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) −22.0000 −1.32185 −0.660926 − 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −8.00000 −0.479808
\(279\) 0 0
\(280\) 0 0
\(281\) 28.0000 1.67034 0.835170 − 0.549992i \(-0.185369\pi\)
0.835170 + 0.549992i \(0.185369\pi\)
\(282\) 12.0000 0.714590
\(283\) −26.0000 −1.54554 −0.772770 − 0.634686i \(-0.781129\pi\)
−0.772770 + 0.634686i \(0.781129\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) −2.00000 −0.117041
\(293\) −26.0000 −1.51894 −0.759468 − 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) 18.0000 1.04271
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) 20.0000 1.15278
\(302\) −24.0000 −1.38104
\(303\) 14.0000 0.804279
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 4.00000 0.227921
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 − 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 12.0000 0.679366
\(313\) 14.0000 0.791327 0.395663 − 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −14.0000 −0.786318 −0.393159 − 0.919470i \(-0.628618\pi\)
−0.393159 + 0.919470i \(0.628618\pi\)
\(318\) −2.00000 −0.112154
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) −8.00000 −0.445823
\(323\) 2.00000 0.111283
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −6.00000 −0.332309
\(327\) 10.0000 0.553001
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 − 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) −16.0000 −0.871576 −0.435788 − 0.900049i \(-0.643530\pi\)
−0.435788 + 0.900049i \(0.643530\pi\)
\(338\) −3.00000 −0.163178
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 0 0
\(342\) 1.00000 0.0540738
\(343\) −20.0000 −1.07990
\(344\) 30.0000 1.61749
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 28.0000 1.50312 0.751559 − 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −4.00000 −0.214423
\(349\) −18.0000 −0.963518 −0.481759 − 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 10.0000 0.533002
\(353\) 2.00000 0.106449 0.0532246 − 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 12.0000 0.634220
\(359\) −2.00000 −0.105556 −0.0527780 − 0.998606i \(-0.516808\pi\)
−0.0527780 + 0.998606i \(0.516808\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −10.0000 −0.525588
\(363\) −7.00000 −0.367405
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) −10.0000 −0.521996 −0.260998 − 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) −36.0000 −1.85656
\(377\) 16.0000 0.824042
\(378\) −2.00000 −0.102869
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 18.0000 0.920960
\(383\) −8.00000 −0.408781 −0.204390 − 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 10.0000 0.508329
\(388\) −16.0000 −0.812277
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) −9.00000 −0.454569
\(393\) −14.0000 −0.706207
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 2.00000 0.100377 0.0501886 − 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −12.0000 −0.601506
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 − 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) −16.0000 −0.798007
\(403\) 0 0
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) 0 0
\(408\) −6.00000 −0.297044
\(409\) −26.0000 −1.28562 −0.642809 − 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) −8.00000 −0.394132
\(413\) 8.00000 0.393654
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) −20.0000 −0.980581
\(417\) 8.00000 0.391762
\(418\) −2.00000 −0.0978232
\(419\) −26.0000 −1.27018 −0.635092 − 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 − 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 12.0000 0.584151
\(423\) −12.0000 −0.583460
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) −12.0000 −0.580042
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) −4.00000 −0.192673 −0.0963366 − 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 36.0000 1.73005 0.865025 − 0.501729i \(-0.167303\pi\)
0.865025 + 0.501729i \(0.167303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) −4.00000 −0.191346
\(438\) −2.00000 −0.0955637
\(439\) −40.0000 −1.90910 −0.954548 − 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 8.00000 0.380521
\(443\) −16.0000 −0.760183 −0.380091 − 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) −18.0000 −0.851371
\(448\) 14.0000 0.661438
\(449\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 14.0000 0.658505
\(453\) 24.0000 1.12762
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −3.00000 −0.140488
\(457\) 14.0000 0.654892 0.327446 − 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) 14.0000 0.654177
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 − 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 4.00000 0.186097
\(463\) 22.0000 1.02243 0.511213 − 0.859454i \(-0.329196\pi\)
0.511213 + 0.859454i \(0.329196\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) −16.0000 −0.740392 −0.370196 − 0.928954i \(-0.620709\pi\)
−0.370196 + 0.928954i \(0.620709\pi\)
\(468\) −4.00000 −0.184900
\(469\) 32.0000 1.47762
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 12.0000 0.552345
\(473\) −20.0000 −0.919601
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 2.00000 0.0915737
\(478\) 6.00000 0.274434
\(479\) 38.0000 1.73626 0.868132 − 0.496333i \(-0.165321\pi\)
0.868132 + 0.496333i \(0.165321\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −26.0000 −1.18427
\(483\) 8.00000 0.364013
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −8.00000 −0.362515 −0.181257 − 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 6.00000 0.271607
\(489\) 6.00000 0.271329
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 − 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 0 0
\(493\) −8.00000 −0.360302
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) 32.0000 1.43252 0.716258 − 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) −22.0000 −0.981908
\(503\) 8.00000 0.356702 0.178351 − 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) 6.00000 0.267261
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) 3.00000 0.133235
\(508\) 4.00000 0.177471
\(509\) −24.0000 −1.06378 −0.531891 − 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 11.0000 0.486136
\(513\) −1.00000 −0.0441511
\(514\) 22.0000 0.970378
\(515\) 0 0
\(516\) −10.0000 −0.440225
\(517\) 24.0000 1.05552
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 − 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) −4.00000 −0.175075
\(523\) 28.0000 1.22435 0.612177 − 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) 14.0000 0.611593
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 2.00000 0.0867110
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 48.0000 2.07328
\(537\) −12.0000 −0.517838
\(538\) −4.00000 −0.172452
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 − 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) −12.0000 −0.515444
\(543\) 10.0000 0.429141
\(544\) 10.0000 0.428746
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) −16.0000 −0.684111 −0.342055 − 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 6.00000 0.256307
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) 12.0000 0.510754
\(553\) −16.0000 −0.680389
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) 38.0000 1.61011 0.805056 − 0.593199i \(-0.202135\pi\)
0.805056 + 0.593199i \(0.202135\pi\)
\(558\) 0 0
\(559\) 40.0000 1.69182
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) −28.0000 −1.18111
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) 26.0000 1.09286
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −24.0000 −1.00437 −0.502184 − 0.864761i \(-0.667470\pi\)
−0.502184 + 0.864761i \(0.667470\pi\)
\(572\) 8.00000 0.334497
\(573\) −18.0000 −0.751961
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −42.0000 −1.74848 −0.874241 − 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) 13.0000 0.540729
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) −16.0000 −0.663221
\(583\) −4.00000 −0.165663
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 26.0000 1.07405
\(587\) 32.0000 1.32078 0.660391 − 0.750922i \(-0.270391\pi\)
0.660391 + 0.750922i \(0.270391\pi\)
\(588\) 3.00000 0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 − 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 12.0000 0.491127
\(598\) −16.0000 −0.654289
\(599\) 36.0000 1.47092 0.735460 − 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 − 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) −20.0000 −0.815139
\(603\) 16.0000 0.651570
\(604\) −24.0000 −0.976546
\(605\) 0 0
\(606\) −14.0000 −0.568711
\(607\) 16.0000 0.649420 0.324710 − 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 5.00000 0.202777
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 2.00000 0.0808452
\(613\) 18.0000 0.727013 0.363507 − 0.931592i \(-0.381579\pi\)
0.363507 + 0.931592i \(0.381579\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) 6.00000 0.241551 0.120775 − 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) −8.00000 −0.321807
\(619\) 16.0000 0.643094 0.321547 − 0.946894i \(-0.395797\pi\)
0.321547 + 0.946894i \(0.395797\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 18.0000 0.721734
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) −14.0000 −0.559553
\(627\) 2.00000 0.0798723
\(628\) −18.0000 −0.718278
\(629\) 0 0
\(630\) 0 0
\(631\) 48.0000 1.91085 0.955425 − 0.295234i \(-0.0953977\pi\)
0.955425 + 0.295234i \(0.0953977\pi\)
\(632\) −24.0000 −0.954669
\(633\) −12.0000 −0.476957
\(634\) 14.0000 0.556011
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) −12.0000 −0.475457
\(638\) 8.00000 0.316723
\(639\) 0 0
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 − 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) −12.0000 −0.473602
\(643\) −26.0000 −1.02534 −0.512670 − 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) −2.00000 −0.0786889
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 3.00000 0.117851
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) −6.00000 −0.234978
\(653\) 18.0000 0.704394 0.352197 − 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −10.0000 −0.391031
\(655\) 0 0
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 24.0000 0.935617
\(659\) 16.0000 0.623272 0.311636 − 0.950202i \(-0.399123\pi\)
0.311636 + 0.950202i \(0.399123\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 − 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 28.0000 1.08825
\(663\) −8.00000 −0.310694
\(664\) 36.0000 1.39707
\(665\) 0 0
\(666\) 0 0
\(667\) 16.0000 0.619522
\(668\) −8.00000 −0.309529
\(669\) −24.0000 −0.927894
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) −10.0000 −0.385758
\(673\) −24.0000 −0.925132 −0.462566 − 0.886585i \(-0.653071\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(674\) 16.0000 0.616297
\(675\) 0 0
\(676\) −3.00000 −0.115385
\(677\) 50.0000 1.92166 0.960828 − 0.277145i \(-0.0893883\pi\)
0.960828 + 0.277145i \(0.0893883\pi\)
\(678\) 14.0000 0.537667
\(679\) 32.0000 1.22805
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) −14.0000 −0.534133
\(688\) −10.0000 −0.381246
\(689\) 8.00000 0.304776
\(690\) 0 0
\(691\) −16.0000 −0.608669 −0.304334 − 0.952565i \(-0.598434\pi\)
−0.304334 + 0.952565i \(0.598434\pi\)
\(692\) 18.0000 0.684257
\(693\) −4.00000 −0.151947
\(694\) −28.0000 −1.06287
\(695\) 0 0
\(696\) 12.0000 0.454859
\(697\) 0 0
\(698\) 18.0000 0.681310
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 − 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) −4.00000 −0.150970
\(703\) 0 0
\(704\) −14.0000 −0.527645
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) 28.0000 1.05305
\(708\) −4.00000 −0.150329
\(709\) 18.0000 0.676004 0.338002 − 0.941145i \(-0.390249\pi\)
0.338002 + 0.941145i \(0.390249\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −6.00000 −0.224074
\(718\) 2.00000 0.0746393
\(719\) 42.0000 1.56634 0.783168 − 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) −1.00000 −0.0372161
\(723\) 26.0000 0.966950
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 7.00000 0.259794
\(727\) 14.0000 0.519231 0.259616 − 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 24.0000 0.889499
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) −2.00000 −0.0739221
\(733\) −34.0000 −1.25582 −0.627909 − 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 10.0000 0.369107
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) −32.0000 −1.17874
\(738\) 0 0
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) −4.00000 −0.146845
\(743\) 8.00000 0.293492 0.146746 − 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) 12.0000 0.439057
\(748\) −4.00000 −0.146254
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 12.0000 0.437595
\(753\) 22.0000 0.801725
\(754\) −16.0000 −0.582686
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 20.0000 0.726433
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 − 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 4.00000 0.144905
\(763\) 20.0000 0.724049
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) 16.0000 0.577727
\(768\) −17.0000 −0.613435
\(769\) −46.0000 −1.65880 −0.829401 − 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) 4.00000 0.143963
\(773\) −14.0000 −0.503545 −0.251773 − 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) −10.0000 −0.359443
\(775\) 0 0
\(776\) 48.0000 1.72310
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) 4.00000 0.142948
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 14.0000 0.499363
\(787\) −28.0000 −0.998092 −0.499046 − 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) −28.0000 −0.995565
\(792\) −6.00000 −0.213201
\(793\) 8.00000 0.284088
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −12.0000 −0.425329
\(797\) 14.0000 0.495905 0.247953 − 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(798\) 2.00000 0.0707992
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) −12.0000 −0.423735
\(803\) −4.00000 −0.141157
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) 0 0
\(807\) 4.00000 0.140807
\(808\) 42.0000 1.47755
\(809\) 2.00000 0.0703163 0.0351581 − 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 − 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) −8.00000 −0.280745
\(813\) 12.0000 0.420858
\(814\) 0 0
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) −10.0000 −0.349856
\(818\) 26.0000 0.909069
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 − 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 6.00000 0.209274
\(823\) 14.0000 0.488009 0.244005 − 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) −36.0000 −1.25184 −0.625921 − 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) −4.00000 −0.139010
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 28.0000 0.970725
\(833\) 6.00000 0.207888
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) 0 0
\(838\) 26.0000 0.898155
\(839\) 24.0000 0.828572 0.414286 − 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 34.0000 1.17172
\(843\) 28.0000 0.964371
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) −14.0000 −0.481046
\(848\) −2.00000 −0.0686803
\(849\) −26.0000 −0.892318
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 − 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) −10.0000 −0.341593 −0.170797 − 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 8.00000 0.273115
\(859\) −12.0000 −0.409435 −0.204717 − 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 4.00000 0.136241
\(863\) −40.0000 −1.36162 −0.680808 − 0.732462i \(-0.738371\pi\)
−0.680808 + 0.732462i \(0.738371\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) −36.0000 −1.22333
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 64.0000 2.16856
\(872\) 30.0000 1.01593
\(873\) 16.0000 0.541518
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) −12.0000 −0.405211 −0.202606 − 0.979260i \(-0.564941\pi\)
−0.202606 + 0.979260i \(0.564941\pi\)
\(878\) 40.0000 1.34993
\(879\) −26.0000 −0.876958
\(880\) 0 0
\(881\) −10.0000 −0.336909 −0.168454 − 0.985709i \(-0.553878\pi\)
−0.168454 + 0.985709i \(0.553878\pi\)
\(882\) 3.00000 0.101015
\(883\) 46.0000 1.54802 0.774012 − 0.633171i \(-0.218247\pi\)
0.774012 + 0.633171i \(0.218247\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) 24.0000 0.805841 0.402921 − 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 24.0000 0.803579
\(893\) 12.0000 0.401565
\(894\) 18.0000 0.602010
\(895\) 0 0
\(896\) 6.00000 0.200446
\(897\) 16.0000 0.534224
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) 20.0000 0.665558
\(904\) −42.0000 −1.39690
\(905\) 0 0
\(906\) −24.0000 −0.797347
\(907\) −52.0000 −1.72663 −0.863316 − 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 12.0000 0.398234
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 1.00000 0.0331133
\(913\) −24.0000 −0.794284
\(914\) −14.0000 −0.463079
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) −28.0000 −0.924641
\(918\) 2.00000 0.0660098
\(919\) 32.0000 1.05558 0.527791 − 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −6.00000 −0.197599
\(923\) 0 0
\(924\) 4.00000 0.131590
\(925\) 0 0
\(926\) −22.0000 −0.722965
\(927\) 8.00000 0.262754
\(928\) −20.0000 −0.656532
\(929\) 10.0000 0.328089 0.164045 − 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) −18.0000 −0.589610
\(933\) −18.0000 −0.589294
\(934\) 16.0000 0.523536
\(935\) 0 0
\(936\) 12.0000 0.392232
\(937\) 42.0000 1.37208 0.686040 − 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) −32.0000 −1.04484
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) −60.0000 −1.95594 −0.977972 − 0.208736i \(-0.933065\pi\)
−0.977972 + 0.208736i \(0.933065\pi\)
\(942\) −18.0000 −0.586472
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 20.0000 0.650256
\(947\) 52.0000 1.68977 0.844886 − 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 8.00000 0.259828
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) −14.0000 −0.453981
\(952\) −12.0000 −0.388922
\(953\) −46.0000 −1.49009 −0.745043 − 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) −8.00000 −0.258603
\(958\) −38.0000 −1.22772
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) −26.0000 −0.837404
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) −34.0000 −1.09337 −0.546683 − 0.837340i \(-0.684110\pi\)
−0.546683 + 0.837340i \(0.684110\pi\)
\(968\) −21.0000 −0.674966
\(969\) 2.00000 0.0642493
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 − 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 16.0000 0.512936
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −18.0000 −0.575871 −0.287936 − 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) −6.00000 −0.191859
\(979\) 0 0
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 6.00000 0.191468
\(983\) −16.0000 −0.510321 −0.255160 − 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 8.00000 0.254772
\(987\) −24.0000 −0.763928
\(988\) 4.00000 0.127257
\(989\) 40.0000 1.27193
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 − 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) −28.0000 −0.888553
\(994\) 0 0
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 2.00000 0.0633406 0.0316703 − 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) −32.0000 −1.01294
\(999\) 0 0
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 1425.2.a.d.1.1 1
3.2 odd 2 4275.2.a.o.1.1 1
5.2 odd 4 1425.2.c.d.799.1 2
5.3 odd 4 1425.2.c.d.799.2 2
5.4 even 2 285.2.a.b.1.1 ✓ 1
15.14 odd 2 855.2.a.b.1.1 1
20.19 odd 2 4560.2.a.v.1.1 1
95.94 odd 2 5415.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.b.1.1 ✓ 1 5.4 even 2
855.2.a.b.1.1 1 15.14 odd 2
1425.2.a.d.1.1 1 1.1 even 1 trivial
1425.2.c.d.799.1 2 5.2 odd 4
1425.2.c.d.799.2 2 5.3 odd 4
4275.2.a.o.1.1 1 3.2 odd 2
4560.2.a.v.1.1 1 20.19 odd 2
5415.2.a.c.1.1 1 95.94 odd 2