Properties

Label 3234.2.e.c.2155.5
Level $3234$
Weight $2$
Character 3234.2155
Analytic conductor $25.824$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3234,2,Mod(2155,3234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3234, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3234.2155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.e (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2155.5
Character \(\chi\) \(=\) 3234.2155
Dual form 3234.2.e.c.2155.20

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -0.942563i q^{5} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -0.942563i q^{5} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -0.942563 q^{10} +(1.67140 - 2.86468i) q^{11} +1.00000i q^{12} +2.81702 q^{13} -0.942563 q^{15} +1.00000 q^{16} +4.30608 q^{17} +1.00000i q^{18} +2.51637 q^{19} +0.942563i q^{20} +(-2.86468 - 1.67140i) q^{22} +4.33514 q^{23} +1.00000 q^{24} +4.11157 q^{25} -2.81702i q^{26} +1.00000i q^{27} +8.01022i q^{29} +0.942563i q^{30} +6.05783i q^{31} -1.00000i q^{32} +(-2.86468 - 1.67140i) q^{33} -4.30608i q^{34} +1.00000 q^{36} +10.4379 q^{37} -2.51637i q^{38} -2.81702i q^{39} +0.942563 q^{40} -5.20158 q^{41} +11.3608i q^{43} +(-1.67140 + 2.86468i) q^{44} +0.942563i q^{45} -4.33514i q^{46} +8.96951i q^{47} -1.00000i q^{48} -4.11157i q^{50} -4.30608i q^{51} -2.81702 q^{52} +4.55271 q^{53} +1.00000 q^{54} +(-2.70014 - 1.57540i) q^{55} -2.51637i q^{57} +8.01022 q^{58} -13.4801i q^{59} +0.942563 q^{60} -2.41499 q^{61} +6.05783 q^{62} -1.00000 q^{64} -2.65522i q^{65} +(-1.67140 + 2.86468i) q^{66} -2.22408 q^{67} -4.30608 q^{68} -4.33514i q^{69} -4.80437 q^{71} -1.00000i q^{72} +4.63368 q^{73} -10.4379i q^{74} -4.11157i q^{75} -2.51637 q^{76} -2.81702 q^{78} +9.26447i q^{79} -0.942563i q^{80} +1.00000 q^{81} +5.20158i q^{82} +8.84143 q^{83} -4.05875i q^{85} +11.3608 q^{86} +8.01022 q^{87} +(2.86468 + 1.67140i) q^{88} -11.5356i q^{89} +0.942563 q^{90} -4.33514 q^{92} +6.05783 q^{93} +8.96951 q^{94} -2.37184i q^{95} -1.00000 q^{96} -12.5496i q^{97} +(-1.67140 + 2.86468i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 24 q^{4} - 24 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 24 q^{4} - 24 q^{6} - 24 q^{9} + 24 q^{16} + 16 q^{17} - 32 q^{19} - 8 q^{22} + 24 q^{24} - 8 q^{25} - 8 q^{33} + 24 q^{36} + 16 q^{37} - 16 q^{41} + 24 q^{54} + 16 q^{55} - 16 q^{62} - 24 q^{64} - 64 q^{67} - 16 q^{68} + 64 q^{71} + 32 q^{76} + 24 q^{81} - 16 q^{83} + 8 q^{88} - 16 q^{93} + 64 q^{94} - 24 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3234\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(1079\) \(2059\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0.942563i 0.421527i −0.977537 0.210764i \(-0.932405\pi\)
0.977537 0.210764i \(-0.0675950\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) −0.942563 −0.298065
\(11\) 1.67140 2.86468i 0.503948 0.863734i
\(12\) 1.00000i 0.288675i
\(13\) 2.81702 0.781301 0.390651 0.920539i \(-0.372250\pi\)
0.390651 + 0.920539i \(0.372250\pi\)
\(14\) 0 0
\(15\) −0.942563 −0.243369
\(16\) 1.00000 0.250000
\(17\) 4.30608 1.04438 0.522188 0.852830i \(-0.325116\pi\)
0.522188 + 0.852830i \(0.325116\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 2.51637 0.577295 0.288648 0.957435i \(-0.406794\pi\)
0.288648 + 0.957435i \(0.406794\pi\)
\(20\) 0.942563i 0.210764i
\(21\) 0 0
\(22\) −2.86468 1.67140i −0.610752 0.356345i
\(23\) 4.33514 0.903938 0.451969 0.892034i \(-0.350722\pi\)
0.451969 + 0.892034i \(0.350722\pi\)
\(24\) 1.00000 0.204124
\(25\) 4.11157 0.822315
\(26\) 2.81702i 0.552463i
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 8.01022i 1.48746i 0.668480 + 0.743730i \(0.266945\pi\)
−0.668480 + 0.743730i \(0.733055\pi\)
\(30\) 0.942563i 0.172088i
\(31\) 6.05783i 1.08802i 0.839079 + 0.544009i \(0.183094\pi\)
−0.839079 + 0.544009i \(0.816906\pi\)
\(32\) 1.00000i 0.176777i
\(33\) −2.86468 1.67140i −0.498677 0.290954i
\(34\) 4.30608i 0.738486i
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.4379 1.71599 0.857993 0.513662i \(-0.171711\pi\)
0.857993 + 0.513662i \(0.171711\pi\)
\(38\) 2.51637i 0.408209i
\(39\) 2.81702i 0.451084i
\(40\) 0.942563 0.149032
\(41\) −5.20158 −0.812351 −0.406175 0.913795i \(-0.633138\pi\)
−0.406175 + 0.913795i \(0.633138\pi\)
\(42\) 0 0
\(43\) 11.3608i 1.73250i 0.499608 + 0.866251i \(0.333477\pi\)
−0.499608 + 0.866251i \(0.666523\pi\)
\(44\) −1.67140 + 2.86468i −0.251974 + 0.431867i
\(45\) 0.942563i 0.140509i
\(46\) 4.33514i 0.639181i
\(47\) 8.96951i 1.30834i 0.756349 + 0.654169i \(0.226981\pi\)
−0.756349 + 0.654169i \(0.773019\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 0 0
\(50\) 4.11157i 0.581464i
\(51\) 4.30608i 0.602971i
\(52\) −2.81702 −0.390651
\(53\) 4.55271 0.625363 0.312682 0.949858i \(-0.398773\pi\)
0.312682 + 0.949858i \(0.398773\pi\)
\(54\) 1.00000 0.136083
\(55\) −2.70014 1.57540i −0.364087 0.212428i
\(56\) 0 0
\(57\) 2.51637i 0.333302i
\(58\) 8.01022 1.05179
\(59\) 13.4801i 1.75496i −0.479613 0.877480i \(-0.659223\pi\)
0.479613 0.877480i \(-0.340777\pi\)
\(60\) 0.942563 0.121684
\(61\) −2.41499 −0.309208 −0.154604 0.987976i \(-0.549410\pi\)
−0.154604 + 0.987976i \(0.549410\pi\)
\(62\) 6.05783 0.769345
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 2.65522i 0.329340i
\(66\) −1.67140 + 2.86468i −0.205736 + 0.352618i
\(67\) −2.22408 −0.271715 −0.135857 0.990728i \(-0.543379\pi\)
−0.135857 + 0.990728i \(0.543379\pi\)
\(68\) −4.30608 −0.522188
\(69\) 4.33514i 0.521889i
\(70\) 0 0
\(71\) −4.80437 −0.570173 −0.285087 0.958502i \(-0.592022\pi\)
−0.285087 + 0.958502i \(0.592022\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 4.63368 0.542331 0.271165 0.962533i \(-0.412591\pi\)
0.271165 + 0.962533i \(0.412591\pi\)
\(74\) 10.4379i 1.21338i
\(75\) 4.11157i 0.474764i
\(76\) −2.51637 −0.288648
\(77\) 0 0
\(78\) −2.81702 −0.318965
\(79\) 9.26447i 1.04233i 0.853455 + 0.521167i \(0.174503\pi\)
−0.853455 + 0.521167i \(0.825497\pi\)
\(80\) 0.942563i 0.105382i
\(81\) 1.00000 0.111111
\(82\) 5.20158i 0.574419i
\(83\) 8.84143 0.970473 0.485237 0.874383i \(-0.338733\pi\)
0.485237 + 0.874383i \(0.338733\pi\)
\(84\) 0 0
\(85\) 4.05875i 0.440233i
\(86\) 11.3608 1.22506
\(87\) 8.01022 0.858786
\(88\) 2.86468 + 1.67140i 0.305376 + 0.178172i
\(89\) 11.5356i 1.22277i −0.791334 0.611384i \(-0.790613\pi\)
0.791334 0.611384i \(-0.209387\pi\)
\(90\) 0.942563 0.0993549
\(91\) 0 0
\(92\) −4.33514 −0.451969
\(93\) 6.05783 0.628168
\(94\) 8.96951 0.925134
\(95\) 2.37184i 0.243346i
\(96\) −1.00000 −0.102062
\(97\) 12.5496i 1.27422i −0.770772 0.637112i \(-0.780129\pi\)
0.770772 0.637112i \(-0.219871\pi\)
\(98\) 0 0
\(99\) −1.67140 + 2.86468i −0.167983 + 0.287911i
\(100\) −4.11157 −0.411157
\(101\) −8.65999 −0.861701 −0.430851 0.902423i \(-0.641786\pi\)
−0.430851 + 0.902423i \(0.641786\pi\)
\(102\) −4.30608 −0.426365
\(103\) 11.4902i 1.13216i −0.824351 0.566079i \(-0.808460\pi\)
0.824351 0.566079i \(-0.191540\pi\)
\(104\) 2.81702i 0.276232i
\(105\) 0 0
\(106\) 4.55271i 0.442198i
\(107\) 8.51031i 0.822723i −0.911472 0.411361i \(-0.865053\pi\)
0.911472 0.411361i \(-0.134947\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 0.950748i 0.0910651i 0.998963 + 0.0455326i \(0.0144985\pi\)
−0.998963 + 0.0455326i \(0.985502\pi\)
\(110\) −1.57540 + 2.70014i −0.150209 + 0.257449i
\(111\) 10.4379i 0.990725i
\(112\) 0 0
\(113\) −8.55994 −0.805251 −0.402626 0.915365i \(-0.631902\pi\)
−0.402626 + 0.915365i \(0.631902\pi\)
\(114\) −2.51637 −0.235680
\(115\) 4.08614i 0.381035i
\(116\) 8.01022i 0.743730i
\(117\) −2.81702 −0.260434
\(118\) −13.4801 −1.24094
\(119\) 0 0
\(120\) 0.942563i 0.0860438i
\(121\) −5.41281 9.57609i −0.492074 0.870554i
\(122\) 2.41499i 0.218643i
\(123\) 5.20158i 0.469011i
\(124\) 6.05783i 0.544009i
\(125\) 8.58823i 0.768155i
\(126\) 0 0
\(127\) 14.4372i 1.28109i 0.767919 + 0.640547i \(0.221292\pi\)
−0.767919 + 0.640547i \(0.778708\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 11.3608 1.00026
\(130\) −2.65522 −0.232878
\(131\) 9.54305 0.833780 0.416890 0.908957i \(-0.363120\pi\)
0.416890 + 0.908957i \(0.363120\pi\)
\(132\) 2.86468 + 1.67140i 0.249339 + 0.145477i
\(133\) 0 0
\(134\) 2.22408i 0.192131i
\(135\) 0.942563 0.0811229
\(136\) 4.30608i 0.369243i
\(137\) 18.7883 1.60519 0.802595 0.596525i \(-0.203452\pi\)
0.802595 + 0.596525i \(0.203452\pi\)
\(138\) −4.33514 −0.369031
\(139\) −19.9086 −1.68863 −0.844315 0.535848i \(-0.819992\pi\)
−0.844315 + 0.535848i \(0.819992\pi\)
\(140\) 0 0
\(141\) 8.96951 0.755369
\(142\) 4.80437i 0.403173i
\(143\) 4.70838 8.06987i 0.393735 0.674837i
\(144\) −1.00000 −0.0833333
\(145\) 7.55014 0.627005
\(146\) 4.63368i 0.383486i
\(147\) 0 0
\(148\) −10.4379 −0.857993
\(149\) 3.86561i 0.316683i 0.987384 + 0.158341i \(0.0506147\pi\)
−0.987384 + 0.158341i \(0.949385\pi\)
\(150\) −4.11157 −0.335709
\(151\) 1.64273i 0.133684i −0.997764 0.0668418i \(-0.978708\pi\)
0.997764 0.0668418i \(-0.0212923\pi\)
\(152\) 2.51637i 0.204105i
\(153\) −4.30608 −0.348126
\(154\) 0 0
\(155\) 5.70989 0.458629
\(156\) 2.81702i 0.225542i
\(157\) 4.01593i 0.320506i −0.987076 0.160253i \(-0.948769\pi\)
0.987076 0.160253i \(-0.0512311\pi\)
\(158\) 9.26447 0.737042
\(159\) 4.55271i 0.361054i
\(160\) −0.942563 −0.0745162
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) 12.6206 0.988520 0.494260 0.869314i \(-0.335439\pi\)
0.494260 + 0.869314i \(0.335439\pi\)
\(164\) 5.20158 0.406175
\(165\) −1.57540 + 2.70014i −0.122645 + 0.210206i
\(166\) 8.84143i 0.686228i
\(167\) −15.1392 −1.17150 −0.585752 0.810490i \(-0.699201\pi\)
−0.585752 + 0.810490i \(0.699201\pi\)
\(168\) 0 0
\(169\) −5.06439 −0.389569
\(170\) −4.05875 −0.311292
\(171\) −2.51637 −0.192432
\(172\) 11.3608i 0.866251i
\(173\) −7.73438 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(174\) 8.01022i 0.607253i
\(175\) 0 0
\(176\) 1.67140 2.86468i 0.125987 0.215934i
\(177\) −13.4801 −1.01323
\(178\) −11.5356 −0.864627
\(179\) 16.8680 1.26078 0.630388 0.776280i \(-0.282896\pi\)
0.630388 + 0.776280i \(0.282896\pi\)
\(180\) 0.942563i 0.0702545i
\(181\) 1.82684i 0.135788i −0.997693 0.0678938i \(-0.978372\pi\)
0.997693 0.0678938i \(-0.0216279\pi\)
\(182\) 0 0
\(183\) 2.41499i 0.178522i
\(184\) 4.33514i 0.319591i
\(185\) 9.83841i 0.723334i
\(186\) 6.05783i 0.444182i
\(187\) 7.19720 12.3355i 0.526311 0.902064i
\(188\) 8.96951i 0.654169i
\(189\) 0 0
\(190\) −2.37184 −0.172071
\(191\) −14.2909 −1.03406 −0.517028 0.855968i \(-0.672962\pi\)
−0.517028 + 0.855968i \(0.672962\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 3.68856i 0.265509i 0.991149 + 0.132754i \(0.0423821\pi\)
−0.991149 + 0.132754i \(0.957618\pi\)
\(194\) −12.5496 −0.901012
\(195\) −2.65522 −0.190144
\(196\) 0 0
\(197\) 1.22462i 0.0872507i 0.999048 + 0.0436253i \(0.0138908\pi\)
−0.999048 + 0.0436253i \(0.986109\pi\)
\(198\) 2.86468 + 1.67140i 0.203584 + 0.118782i
\(199\) 12.8195i 0.908752i 0.890810 + 0.454376i \(0.150138\pi\)
−0.890810 + 0.454376i \(0.849862\pi\)
\(200\) 4.11157i 0.290732i
\(201\) 2.22408i 0.156874i
\(202\) 8.65999i 0.609315i
\(203\) 0 0
\(204\) 4.30608i 0.301486i
\(205\) 4.90282i 0.342428i
\(206\) −11.4902 −0.800557
\(207\) −4.33514 −0.301313
\(208\) 2.81702 0.195325
\(209\) 4.20588 7.20861i 0.290927 0.498630i
\(210\) 0 0
\(211\) 17.0672i 1.17495i 0.809241 + 0.587476i \(0.199878\pi\)
−0.809241 + 0.587476i \(0.800122\pi\)
\(212\) −4.55271 −0.312682
\(213\) 4.80437i 0.329190i
\(214\) −8.51031 −0.581753
\(215\) 10.7083 0.730297
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 0.950748 0.0643928
\(219\) 4.63368i 0.313115i
\(220\) 2.70014 + 1.57540i 0.182044 + 0.106214i
\(221\) 12.1303 0.815973
\(222\) −10.4379 −0.700548
\(223\) 8.56595i 0.573619i −0.957988 0.286809i \(-0.907405\pi\)
0.957988 0.286809i \(-0.0925946\pi\)
\(224\) 0 0
\(225\) −4.11157 −0.274105
\(226\) 8.55994i 0.569398i
\(227\) −16.9377 −1.12419 −0.562097 0.827071i \(-0.690005\pi\)
−0.562097 + 0.827071i \(0.690005\pi\)
\(228\) 2.51637i 0.166651i
\(229\) 0.723697i 0.0478233i 0.999714 + 0.0239116i \(0.00761203\pi\)
−0.999714 + 0.0239116i \(0.992388\pi\)
\(230\) −4.08614 −0.269432
\(231\) 0 0
\(232\) −8.01022 −0.525897
\(233\) 13.7409i 0.900199i −0.892978 0.450100i \(-0.851388\pi\)
0.892978 0.450100i \(-0.148612\pi\)
\(234\) 2.81702i 0.184154i
\(235\) 8.45433 0.551499
\(236\) 13.4801i 0.877480i
\(237\) 9.26447 0.601792
\(238\) 0 0
\(239\) 0.273841i 0.0177133i 0.999961 + 0.00885667i \(0.00281920\pi\)
−0.999961 + 0.00885667i \(0.997181\pi\)
\(240\) −0.942563 −0.0608422
\(241\) −0.693253 −0.0446563 −0.0223282 0.999751i \(-0.507108\pi\)
−0.0223282 + 0.999751i \(0.507108\pi\)
\(242\) −9.57609 + 5.41281i −0.615574 + 0.347949i
\(243\) 1.00000i 0.0641500i
\(244\) 2.41499 0.154604
\(245\) 0 0
\(246\) 5.20158 0.331641
\(247\) 7.08867 0.451041
\(248\) −6.05783 −0.384673
\(249\) 8.84143i 0.560303i
\(250\) −8.58823 −0.543168
\(251\) 4.79326i 0.302548i 0.988492 + 0.151274i \(0.0483376\pi\)
−0.988492 + 0.151274i \(0.951662\pi\)
\(252\) 0 0
\(253\) 7.24577 12.4188i 0.455538 0.780763i
\(254\) 14.4372 0.905870
\(255\) −4.05875 −0.254169
\(256\) 1.00000 0.0625000
\(257\) 15.9768i 0.996605i 0.867003 + 0.498302i \(0.166043\pi\)
−0.867003 + 0.498302i \(0.833957\pi\)
\(258\) 11.3608i 0.707291i
\(259\) 0 0
\(260\) 2.65522i 0.164670i
\(261\) 8.01022i 0.495820i
\(262\) 9.54305i 0.589572i
\(263\) 17.1219i 1.05578i −0.849312 0.527891i \(-0.822983\pi\)
0.849312 0.527891i \(-0.177017\pi\)
\(264\) 1.67140 2.86468i 0.102868 0.176309i
\(265\) 4.29122i 0.263607i
\(266\) 0 0
\(267\) −11.5356 −0.705965
\(268\) 2.22408 0.135857
\(269\) 11.8794i 0.724300i −0.932120 0.362150i \(-0.882043\pi\)
0.932120 0.362150i \(-0.117957\pi\)
\(270\) 0.942563i 0.0573626i
\(271\) −12.9150 −0.784531 −0.392266 0.919852i \(-0.628309\pi\)
−0.392266 + 0.919852i \(0.628309\pi\)
\(272\) 4.30608 0.261094
\(273\) 0 0
\(274\) 18.7883i 1.13504i
\(275\) 6.87211 11.7784i 0.414404 0.710262i
\(276\) 4.33514i 0.260945i
\(277\) 21.4574i 1.28925i 0.764499 + 0.644625i \(0.222987\pi\)
−0.764499 + 0.644625i \(0.777013\pi\)
\(278\) 19.9086i 1.19404i
\(279\) 6.05783i 0.362673i
\(280\) 0 0
\(281\) 0.770084i 0.0459393i −0.999736 0.0229697i \(-0.992688\pi\)
0.999736 0.0229697i \(-0.00731212\pi\)
\(282\) 8.96951i 0.534126i
\(283\) −16.7151 −0.993609 −0.496804 0.867863i \(-0.665493\pi\)
−0.496804 + 0.867863i \(0.665493\pi\)
\(284\) 4.80437 0.285087
\(285\) −2.37184 −0.140496
\(286\) −8.06987 4.70838i −0.477181 0.278413i
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) 1.54228 0.0907225
\(290\) 7.55014i 0.443359i
\(291\) −12.5496 −0.735673
\(292\) −4.63368 −0.271165
\(293\) 24.6448 1.43976 0.719881 0.694097i \(-0.244196\pi\)
0.719881 + 0.694097i \(0.244196\pi\)
\(294\) 0 0
\(295\) −12.7058 −0.739763
\(296\) 10.4379i 0.606692i
\(297\) 2.86468 + 1.67140i 0.166226 + 0.0969847i
\(298\) 3.86561 0.223929
\(299\) 12.2122 0.706248
\(300\) 4.11157i 0.237382i
\(301\) 0 0
\(302\) −1.64273 −0.0945285
\(303\) 8.65999i 0.497504i
\(304\) 2.51637 0.144324
\(305\) 2.27628i 0.130340i
\(306\) 4.30608i 0.246162i
\(307\) −15.2768 −0.871893 −0.435947 0.899973i \(-0.643586\pi\)
−0.435947 + 0.899973i \(0.643586\pi\)
\(308\) 0 0
\(309\) −11.4902 −0.653652
\(310\) 5.70989i 0.324300i
\(311\) 24.0083i 1.36139i 0.732569 + 0.680693i \(0.238321\pi\)
−0.732569 + 0.680693i \(0.761679\pi\)
\(312\) 2.81702 0.159482
\(313\) 1.80468i 0.102007i −0.998698 0.0510034i \(-0.983758\pi\)
0.998698 0.0510034i \(-0.0162419\pi\)
\(314\) −4.01593 −0.226632
\(315\) 0 0
\(316\) 9.26447i 0.521167i
\(317\) −21.4013 −1.20202 −0.601009 0.799242i \(-0.705235\pi\)
−0.601009 + 0.799242i \(0.705235\pi\)
\(318\) −4.55271 −0.255303
\(319\) 22.9467 + 13.3883i 1.28477 + 0.749602i
\(320\) 0.942563i 0.0526909i
\(321\) −8.51031 −0.474999
\(322\) 0 0
\(323\) 10.8357 0.602914
\(324\) −1.00000 −0.0555556
\(325\) 11.5824 0.642476
\(326\) 12.6206i 0.698989i
\(327\) 0.950748 0.0525765
\(328\) 5.20158i 0.287209i
\(329\) 0 0
\(330\) 2.70014 + 1.57540i 0.148638 + 0.0867232i
\(331\) −24.5734 −1.35068 −0.675338 0.737508i \(-0.736002\pi\)
−0.675338 + 0.737508i \(0.736002\pi\)
\(332\) −8.84143 −0.485237
\(333\) −10.4379 −0.571995
\(334\) 15.1392i 0.828379i
\(335\) 2.09634i 0.114535i
\(336\) 0 0
\(337\) 33.4718i 1.82332i 0.410940 + 0.911662i \(0.365201\pi\)
−0.410940 + 0.911662i \(0.634799\pi\)
\(338\) 5.06439i 0.275467i
\(339\) 8.55994i 0.464912i
\(340\) 4.05875i 0.220116i
\(341\) 17.3538 + 10.1251i 0.939759 + 0.548304i
\(342\) 2.51637i 0.136070i
\(343\) 0 0
\(344\) −11.3608 −0.612532
\(345\) −4.08614 −0.219990
\(346\) 7.73438i 0.415803i
\(347\) 19.4917i 1.04637i −0.852220 0.523184i \(-0.824744\pi\)
0.852220 0.523184i \(-0.175256\pi\)
\(348\) −8.01022 −0.429393
\(349\) −7.90787 −0.423299 −0.211649 0.977346i \(-0.567883\pi\)
−0.211649 + 0.977346i \(0.567883\pi\)
\(350\) 0 0
\(351\) 2.81702i 0.150361i
\(352\) −2.86468 1.67140i −0.152688 0.0890862i
\(353\) 24.9608i 1.32853i −0.747498 0.664265i \(-0.768745\pi\)
0.747498 0.664265i \(-0.231255\pi\)
\(354\) 13.4801i 0.716459i
\(355\) 4.52842i 0.240343i
\(356\) 11.5356i 0.611384i
\(357\) 0 0
\(358\) 16.8680i 0.891504i
\(359\) 36.2082i 1.91099i −0.295001 0.955497i \(-0.595320\pi\)
0.295001 0.955497i \(-0.404680\pi\)
\(360\) −0.942563 −0.0496774
\(361\) −12.6679 −0.666730
\(362\) −1.82684 −0.0960164
\(363\) −9.57609 + 5.41281i −0.502614 + 0.284099i
\(364\) 0 0
\(365\) 4.36753i 0.228607i
\(366\) 2.41499 0.126234
\(367\) 3.99407i 0.208489i 0.994552 + 0.104244i \(0.0332424\pi\)
−0.994552 + 0.104244i \(0.966758\pi\)
\(368\) 4.33514 0.225985
\(369\) 5.20158 0.270784
\(370\) −9.83841 −0.511474
\(371\) 0 0
\(372\) −6.05783 −0.314084
\(373\) 13.7307i 0.710951i 0.934686 + 0.355476i \(0.115681\pi\)
−0.934686 + 0.355476i \(0.884319\pi\)
\(374\) −12.3355 7.19720i −0.637855 0.372158i
\(375\) −8.58823 −0.443495
\(376\) −8.96951 −0.462567
\(377\) 22.5650i 1.16215i
\(378\) 0 0
\(379\) 9.64244 0.495299 0.247649 0.968850i \(-0.420342\pi\)
0.247649 + 0.968850i \(0.420342\pi\)
\(380\) 2.37184i 0.121673i
\(381\) 14.4372 0.739639
\(382\) 14.2909i 0.731188i
\(383\) 7.81472i 0.399313i −0.979866 0.199657i \(-0.936017\pi\)
0.979866 0.199657i \(-0.0639827\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 3.68856 0.187743
\(387\) 11.3608i 0.577501i
\(388\) 12.5496i 0.637112i
\(389\) 12.9955 0.658896 0.329448 0.944174i \(-0.393137\pi\)
0.329448 + 0.944174i \(0.393137\pi\)
\(390\) 2.65522i 0.134452i
\(391\) 18.6674 0.944052
\(392\) 0 0
\(393\) 9.54305i 0.481383i
\(394\) 1.22462 0.0616956
\(395\) 8.73235 0.439372
\(396\) 1.67140 2.86468i 0.0839913 0.143956i
\(397\) 32.8955i 1.65098i −0.564417 0.825490i \(-0.690899\pi\)
0.564417 0.825490i \(-0.309101\pi\)
\(398\) 12.8195 0.642585
\(399\) 0 0
\(400\) 4.11157 0.205579
\(401\) 28.0768 1.40209 0.701045 0.713117i \(-0.252717\pi\)
0.701045 + 0.713117i \(0.252717\pi\)
\(402\) 2.22408 0.110927
\(403\) 17.0650i 0.850070i
\(404\) 8.65999 0.430851
\(405\) 0.942563i 0.0468363i
\(406\) 0 0
\(407\) 17.4460 29.9014i 0.864767 1.48216i
\(408\) 4.30608 0.213182
\(409\) −39.1324 −1.93497 −0.967485 0.252928i \(-0.918607\pi\)
−0.967485 + 0.252928i \(0.918607\pi\)
\(410\) 4.90282 0.242133
\(411\) 18.7883i 0.926757i
\(412\) 11.4902i 0.566079i
\(413\) 0 0
\(414\) 4.33514i 0.213060i
\(415\) 8.33361i 0.409081i
\(416\) 2.81702i 0.138116i
\(417\) 19.9086i 0.974931i
\(418\) −7.20861 4.20588i −0.352584 0.205716i
\(419\) 27.5652i 1.34665i 0.739347 + 0.673324i \(0.235134\pi\)
−0.739347 + 0.673324i \(0.764866\pi\)
\(420\) 0 0
\(421\) −6.46286 −0.314980 −0.157490 0.987521i \(-0.550340\pi\)
−0.157490 + 0.987521i \(0.550340\pi\)
\(422\) 17.0672 0.830817
\(423\) 8.96951i 0.436112i
\(424\) 4.55271i 0.221099i
\(425\) 17.7047 0.858807
\(426\) 4.80437 0.232772
\(427\) 0 0
\(428\) 8.51031i 0.411361i
\(429\) −8.06987 4.70838i −0.389617 0.227323i
\(430\) 10.7083i 0.516398i
\(431\) 5.81850i 0.280267i 0.990133 + 0.140134i \(0.0447532\pi\)
−0.990133 + 0.140134i \(0.955247\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 27.1792i 1.30615i −0.757293 0.653075i \(-0.773478\pi\)
0.757293 0.653075i \(-0.226522\pi\)
\(434\) 0 0
\(435\) 7.55014i 0.362001i
\(436\) 0.950748i 0.0455326i
\(437\) 10.9088 0.521839
\(438\) −4.63368 −0.221406
\(439\) 24.3998 1.16454 0.582270 0.812996i \(-0.302165\pi\)
0.582270 + 0.812996i \(0.302165\pi\)
\(440\) 1.57540 2.70014i 0.0751045 0.128724i
\(441\) 0 0
\(442\) 12.1303i 0.576980i
\(443\) 7.40742 0.351937 0.175969 0.984396i \(-0.443694\pi\)
0.175969 + 0.984396i \(0.443694\pi\)
\(444\) 10.4379i 0.495362i
\(445\) −10.8730 −0.515429
\(446\) −8.56595 −0.405610
\(447\) 3.86561 0.182837
\(448\) 0 0
\(449\) −8.04219 −0.379535 −0.189767 0.981829i \(-0.560773\pi\)
−0.189767 + 0.981829i \(0.560773\pi\)
\(450\) 4.11157i 0.193821i
\(451\) −8.69395 + 14.9009i −0.409382 + 0.701655i
\(452\) 8.55994 0.402626
\(453\) −1.64273 −0.0771822
\(454\) 16.9377i 0.794926i
\(455\) 0 0
\(456\) 2.51637 0.117840
\(457\) 18.1058i 0.846953i 0.905907 + 0.423476i \(0.139190\pi\)
−0.905907 + 0.423476i \(0.860810\pi\)
\(458\) 0.723697 0.0338162
\(459\) 4.30608i 0.200990i
\(460\) 4.08614i 0.190517i
\(461\) 17.1197 0.797343 0.398672 0.917094i \(-0.369471\pi\)
0.398672 + 0.917094i \(0.369471\pi\)
\(462\) 0 0
\(463\) 28.4136 1.32049 0.660245 0.751050i \(-0.270452\pi\)
0.660245 + 0.751050i \(0.270452\pi\)
\(464\) 8.01022i 0.371865i
\(465\) 5.70989i 0.264790i
\(466\) −13.7409 −0.636537
\(467\) 20.6667i 0.956341i 0.878267 + 0.478171i \(0.158700\pi\)
−0.878267 + 0.478171i \(0.841300\pi\)
\(468\) 2.81702 0.130217
\(469\) 0 0
\(470\) 8.45433i 0.389969i
\(471\) −4.01593 −0.185044
\(472\) 13.4801 0.620472
\(473\) 32.5450 + 18.9885i 1.49642 + 0.873091i
\(474\) 9.26447i 0.425531i
\(475\) 10.3463 0.474719
\(476\) 0 0
\(477\) −4.55271 −0.208454
\(478\) 0.273841 0.0125252
\(479\) 20.7439 0.947811 0.473906 0.880576i \(-0.342844\pi\)
0.473906 + 0.880576i \(0.342844\pi\)
\(480\) 0.942563i 0.0430219i
\(481\) 29.4039 1.34070
\(482\) 0.693253i 0.0315768i
\(483\) 0 0
\(484\) 5.41281 + 9.57609i 0.246037 + 0.435277i
\(485\) −11.8288 −0.537120
\(486\) −1.00000 −0.0453609
\(487\) −36.2719 −1.64364 −0.821818 0.569750i \(-0.807040\pi\)
−0.821818 + 0.569750i \(0.807040\pi\)
\(488\) 2.41499i 0.109322i
\(489\) 12.6206i 0.570723i
\(490\) 0 0
\(491\) 0.584864i 0.0263945i −0.999913 0.0131973i \(-0.995799\pi\)
0.999913 0.0131973i \(-0.00420094\pi\)
\(492\) 5.20158i 0.234505i
\(493\) 34.4926i 1.55347i
\(494\) 7.08867i 0.318934i
\(495\) 2.70014 + 1.57540i 0.121362 + 0.0708092i
\(496\) 6.05783i 0.272005i
\(497\) 0 0
\(498\) −8.84143 −0.396194
\(499\) −21.6453 −0.968979 −0.484489 0.874797i \(-0.660995\pi\)
−0.484489 + 0.874797i \(0.660995\pi\)
\(500\) 8.58823i 0.384078i
\(501\) 15.1392i 0.676368i
\(502\) 4.79326 0.213934
\(503\) −31.5510 −1.40679 −0.703395 0.710800i \(-0.748333\pi\)
−0.703395 + 0.710800i \(0.748333\pi\)
\(504\) 0 0
\(505\) 8.16259i 0.363230i
\(506\) −12.4188 7.24577i −0.552083 0.322114i
\(507\) 5.06439i 0.224918i
\(508\) 14.4372i 0.640547i
\(509\) 21.2731i 0.942914i 0.881889 + 0.471457i \(0.156272\pi\)
−0.881889 + 0.471457i \(0.843728\pi\)
\(510\) 4.05875i 0.179724i
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 2.51637i 0.111101i
\(514\) 15.9768 0.704706
\(515\) −10.8302 −0.477235
\(516\) −11.3608 −0.500131
\(517\) 25.6948 + 14.9917i 1.13006 + 0.659333i
\(518\) 0 0
\(519\) 7.73438i 0.339501i
\(520\) 2.65522 0.116439
\(521\) 9.14346i 0.400582i −0.979736 0.200291i \(-0.935811\pi\)
0.979736 0.200291i \(-0.0641888\pi\)
\(522\) −8.01022 −0.350598
\(523\) −27.9801 −1.22349 −0.611743 0.791057i \(-0.709531\pi\)
−0.611743 + 0.791057i \(0.709531\pi\)
\(524\) −9.54305 −0.416890
\(525\) 0 0
\(526\) −17.1219 −0.746550
\(527\) 26.0855i 1.13630i
\(528\) −2.86468 1.67140i −0.124669 0.0727386i
\(529\) −4.20659 −0.182895
\(530\) −4.29122 −0.186399
\(531\) 13.4801i 0.584987i
\(532\) 0 0
\(533\) −14.6530 −0.634690
\(534\) 11.5356i 0.499193i
\(535\) −8.02151 −0.346800
\(536\) 2.22408i 0.0960656i
\(537\) 16.8680i 0.727910i
\(538\) −11.8794 −0.512157
\(539\) 0 0
\(540\) −0.942563 −0.0405615
\(541\) 27.1482i 1.16719i −0.812044 0.583597i \(-0.801645\pi\)
0.812044 0.583597i \(-0.198355\pi\)
\(542\) 12.9150i 0.554748i
\(543\) −1.82684 −0.0783970
\(544\) 4.30608i 0.184621i
\(545\) 0.896140 0.0383864
\(546\) 0 0
\(547\) 8.17032i 0.349337i 0.984627 + 0.174669i \(0.0558854\pi\)
−0.984627 + 0.174669i \(0.944115\pi\)
\(548\) −18.7883 −0.802595
\(549\) 2.41499 0.103069
\(550\) −11.7784 6.87211i −0.502231 0.293028i
\(551\) 20.1567i 0.858704i
\(552\) 4.33514 0.184516
\(553\) 0 0
\(554\) 21.4574 0.911638
\(555\) −9.83841 −0.417617
\(556\) 19.9086 0.844315
\(557\) 15.2398i 0.645732i −0.946445 0.322866i \(-0.895354\pi\)
0.946445 0.322866i \(-0.104646\pi\)
\(558\) −6.05783 −0.256448
\(559\) 32.0036i 1.35361i
\(560\) 0 0
\(561\) −12.3355 7.19720i −0.520807 0.303866i
\(562\) −0.770084 −0.0324840
\(563\) −28.1605 −1.18683 −0.593413 0.804898i \(-0.702220\pi\)
−0.593413 + 0.804898i \(0.702220\pi\)
\(564\) −8.96951 −0.377684
\(565\) 8.06828i 0.339435i
\(566\) 16.7151i 0.702587i
\(567\) 0 0
\(568\) 4.80437i 0.201587i
\(569\) 45.8500i 1.92213i −0.276317 0.961067i \(-0.589114\pi\)
0.276317 0.961067i \(-0.410886\pi\)
\(570\) 2.37184i 0.0993454i
\(571\) 36.4708i 1.52626i −0.646247 0.763128i \(-0.723663\pi\)
0.646247 0.763128i \(-0.276337\pi\)
\(572\) −4.70838 + 8.06987i −0.196867 + 0.337418i
\(573\) 14.2909i 0.597013i
\(574\) 0 0
\(575\) 17.8242 0.743322
\(576\) 1.00000 0.0416667
\(577\) 38.2106i 1.59073i 0.606132 + 0.795364i \(0.292720\pi\)
−0.606132 + 0.795364i \(0.707280\pi\)
\(578\) 1.54228i 0.0641505i
\(579\) 3.68856 0.153291
\(580\) −7.55014 −0.313502
\(581\) 0 0
\(582\) 12.5496i 0.520199i
\(583\) 7.60942 13.0421i 0.315150 0.540148i
\(584\) 4.63368i 0.191743i
\(585\) 2.65522i 0.109780i
\(586\) 24.6448i 1.01807i
\(587\) 21.0138i 0.867333i 0.901074 + 0.433666i \(0.142780\pi\)
−0.901074 + 0.433666i \(0.857220\pi\)
\(588\) 0 0
\(589\) 15.2438i 0.628108i
\(590\) 12.7058i 0.523092i
\(591\) 1.22462 0.0503742
\(592\) 10.4379 0.428996
\(593\) −2.35790 −0.0968272 −0.0484136 0.998827i \(-0.515417\pi\)
−0.0484136 + 0.998827i \(0.515417\pi\)
\(594\) 1.67140 2.86468i 0.0685786 0.117539i
\(595\) 0 0
\(596\) 3.86561i 0.158341i
\(597\) 12.8195 0.524668
\(598\) 12.2122i 0.499393i
\(599\) −18.4896 −0.755464 −0.377732 0.925915i \(-0.623296\pi\)
−0.377732 + 0.925915i \(0.623296\pi\)
\(600\) 4.11157 0.167854
\(601\) 21.1149 0.861293 0.430647 0.902521i \(-0.358286\pi\)
0.430647 + 0.902521i \(0.358286\pi\)
\(602\) 0 0
\(603\) 2.22408 0.0905715
\(604\) 1.64273i 0.0668418i
\(605\) −9.02607 + 5.10192i −0.366962 + 0.207422i
\(606\) 8.65999 0.351788
\(607\) −33.2254 −1.34858 −0.674290 0.738467i \(-0.735550\pi\)
−0.674290 + 0.738467i \(0.735550\pi\)
\(608\) 2.51637i 0.102052i
\(609\) 0 0
\(610\) 2.27628 0.0921641
\(611\) 25.2673i 1.02221i
\(612\) 4.30608 0.174063
\(613\) 10.4600i 0.422473i 0.977435 + 0.211237i \(0.0677491\pi\)
−0.977435 + 0.211237i \(0.932251\pi\)
\(614\) 15.2768i 0.616522i
\(615\) 4.90282 0.197701
\(616\) 0 0
\(617\) 39.7531 1.60040 0.800200 0.599733i \(-0.204727\pi\)
0.800200 + 0.599733i \(0.204727\pi\)
\(618\) 11.4902i 0.462202i
\(619\) 17.0154i 0.683907i −0.939717 0.341953i \(-0.888911\pi\)
0.939717 0.341953i \(-0.111089\pi\)
\(620\) −5.70989 −0.229315
\(621\) 4.33514i 0.173963i
\(622\) 24.0083 0.962646
\(623\) 0 0
\(624\) 2.81702i 0.112771i
\(625\) 12.4629 0.498517
\(626\) −1.80468 −0.0721296
\(627\) −7.20861 4.20588i −0.287884 0.167967i
\(628\) 4.01593i 0.160253i
\(629\) 44.9465 1.79213
\(630\) 0 0
\(631\) 37.4904 1.49247 0.746234 0.665684i \(-0.231860\pi\)
0.746234 + 0.665684i \(0.231860\pi\)
\(632\) −9.26447 −0.368521
\(633\) 17.0672 0.678359
\(634\) 21.4013i 0.849955i
\(635\) 13.6080 0.540015
\(636\) 4.55271i 0.180527i
\(637\) 0 0
\(638\) 13.3883 22.9467i 0.530049 0.908470i
\(639\) 4.80437 0.190058
\(640\) 0.942563 0.0372581
\(641\) −23.9255 −0.945000 −0.472500 0.881331i \(-0.656648\pi\)
−0.472500 + 0.881331i \(0.656648\pi\)
\(642\) 8.51031i 0.335875i
\(643\) 0.155057i 0.00611485i −0.999995 0.00305742i \(-0.999027\pi\)
0.999995 0.00305742i \(-0.000973210\pi\)
\(644\) 0 0
\(645\) 10.7083i 0.421637i
\(646\) 10.8357i 0.426324i
\(647\) 37.0329i 1.45591i −0.685624 0.727956i \(-0.740471\pi\)
0.685624 0.727956i \(-0.259529\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −38.6162 22.5307i −1.51582 0.884408i
\(650\) 11.5824i 0.454299i
\(651\) 0 0
\(652\) −12.6206 −0.494260
\(653\) 2.65522 0.103907 0.0519534 0.998650i \(-0.483455\pi\)
0.0519534 + 0.998650i \(0.483455\pi\)
\(654\) 0.950748i 0.0371772i
\(655\) 8.99493i 0.351461i
\(656\) −5.20158 −0.203088
\(657\) −4.63368 −0.180777
\(658\) 0 0
\(659\) 13.6694i 0.532485i −0.963906 0.266243i \(-0.914218\pi\)
0.963906 0.266243i \(-0.0857823\pi\)
\(660\) 1.57540 2.70014i 0.0613225 0.105103i
\(661\) 16.4336i 0.639195i −0.947553 0.319597i \(-0.896452\pi\)
0.947553 0.319597i \(-0.103548\pi\)
\(662\) 24.5734i 0.955072i
\(663\) 12.1303i 0.471102i
\(664\) 8.84143i 0.343114i
\(665\) 0 0
\(666\) 10.4379i 0.404462i
\(667\) 34.7254i 1.34457i
\(668\) 15.1392 0.585752
\(669\) −8.56595 −0.331179
\(670\) 2.09634 0.0809885
\(671\) −4.03643 + 6.91819i −0.155825 + 0.267074i
\(672\) 0 0
\(673\) 48.3352i 1.86318i −0.363506 0.931592i \(-0.618420\pi\)
0.363506 0.931592i \(-0.381580\pi\)
\(674\) 33.4718 1.28929
\(675\) 4.11157i 0.158255i
\(676\) 5.06439 0.194784
\(677\) 39.1427 1.50437 0.752187 0.658950i \(-0.228999\pi\)
0.752187 + 0.658950i \(0.228999\pi\)
\(678\) 8.55994 0.328742
\(679\) 0 0
\(680\) 4.05875 0.155646
\(681\) 16.9377i 0.649054i
\(682\) 10.1251 17.3538i 0.387710 0.664510i
\(683\) −4.64928 −0.177900 −0.0889500 0.996036i \(-0.528351\pi\)
−0.0889500 + 0.996036i \(0.528351\pi\)
\(684\) 2.51637 0.0962159
\(685\) 17.7091i 0.676631i
\(686\) 0 0
\(687\) 0.723697 0.0276108
\(688\) 11.3608i 0.433126i
\(689\) 12.8251 0.488597
\(690\) 4.08614i 0.155557i
\(691\) 7.45733i 0.283690i 0.989889 + 0.141845i \(0.0453035\pi\)
−0.989889 + 0.141845i \(0.954696\pi\)
\(692\) 7.73438 0.294017
\(693\) 0 0
\(694\) −19.4917 −0.739894
\(695\) 18.7652i 0.711803i
\(696\) 8.01022i 0.303627i
\(697\) −22.3984 −0.848400
\(698\) 7.90787i 0.299317i
\(699\) −13.7409 −0.519730
\(700\) 0 0
\(701\) 34.1129i 1.28843i −0.764845 0.644214i \(-0.777185\pi\)
0.764845 0.644214i \(-0.222815\pi\)
\(702\) 2.81702 0.106322
\(703\) 26.2657 0.990630
\(704\) −1.67140 + 2.86468i −0.0629934 + 0.107967i
\(705\) 8.45433i 0.318408i
\(706\) −24.9608 −0.939412
\(707\) 0 0
\(708\) 13.4801 0.506613
\(709\) −14.9868 −0.562840 −0.281420 0.959585i \(-0.590805\pi\)
−0.281420 + 0.959585i \(0.590805\pi\)
\(710\) 4.52842 0.169948
\(711\) 9.26447i 0.347445i
\(712\) 11.5356 0.432313
\(713\) 26.2615i 0.983502i
\(714\) 0 0
\(715\) −7.60636 4.43795i −0.284462 0.165970i
\(716\) −16.8680 −0.630388
\(717\) 0.273841 0.0102268
\(718\) −36.2082 −1.35128
\(719\) 28.1244i 1.04886i −0.851452 0.524432i \(-0.824278\pi\)
0.851452 0.524432i \(-0.175722\pi\)
\(720\) 0.942563i 0.0351273i
\(721\) 0 0
\(722\) 12.6679i 0.471449i
\(723\) 0.693253i 0.0257823i
\(724\) 1.82684i 0.0678938i
\(725\) 32.9346i 1.22316i
\(726\) 5.41281 + 9.57609i 0.200888 + 0.355402i
\(727\) 27.0175i 1.00202i 0.865441 + 0.501012i \(0.167039\pi\)
−0.865441 + 0.501012i \(0.832961\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) −4.36753 −0.161650
\(731\) 48.9204i 1.80939i
\(732\) 2.41499i 0.0892608i
\(733\) −7.00127 −0.258598 −0.129299 0.991606i \(-0.541273\pi\)
−0.129299 + 0.991606i \(0.541273\pi\)
\(734\) 3.99407 0.147424
\(735\) 0 0
\(736\) 4.33514i 0.159795i
\(737\) −3.71734 + 6.37128i −0.136930 + 0.234689i
\(738\) 5.20158i 0.191473i
\(739\) 41.7535i 1.53593i 0.640494 + 0.767963i \(0.278730\pi\)
−0.640494 + 0.767963i \(0.721270\pi\)
\(740\) 9.83841i 0.361667i
\(741\) 7.08867i 0.260409i
\(742\) 0 0
\(743\) 44.1614i 1.62013i 0.586343 + 0.810063i \(0.300567\pi\)
−0.586343 + 0.810063i \(0.699433\pi\)
\(744\) 6.05783i 0.222091i
\(745\) 3.64358 0.133490
\(746\) 13.7307 0.502718
\(747\) −8.84143 −0.323491
\(748\) −7.19720 + 12.3355i −0.263156 + 0.451032i
\(749\) 0 0
\(750\) 8.58823i 0.313598i
\(751\) −11.8651 −0.432965 −0.216482 0.976287i \(-0.569458\pi\)
−0.216482 + 0.976287i \(0.569458\pi\)
\(752\) 8.96951i 0.327084i
\(753\) 4.79326 0.174676
\(754\) 22.5650 0.821767
\(755\) −1.54838 −0.0563512
\(756\) 0 0
\(757\) 42.4503 1.54288 0.771441 0.636301i \(-0.219536\pi\)
0.771441 + 0.636301i \(0.219536\pi\)
\(758\) 9.64244i 0.350229i
\(759\) −12.4188 7.24577i −0.450774 0.263005i
\(760\) 2.37184 0.0860357
\(761\) −8.21277 −0.297713 −0.148856 0.988859i \(-0.547559\pi\)
−0.148856 + 0.988859i \(0.547559\pi\)
\(762\) 14.4372i 0.523004i
\(763\) 0 0
\(764\) 14.2909 0.517028
\(765\) 4.05875i 0.146744i
\(766\) −7.81472 −0.282357
\(767\) 37.9737i 1.37115i
\(768\) 1.00000i 0.0360844i
\(769\) 35.8809 1.29390 0.646950 0.762533i \(-0.276044\pi\)
0.646950 + 0.762533i \(0.276044\pi\)
\(770\) 0 0
\(771\) 15.9768 0.575390
\(772\) 3.68856i 0.132754i
\(773\) 16.9090i 0.608175i −0.952644 0.304088i \(-0.901648\pi\)
0.952644 0.304088i \(-0.0983516\pi\)
\(774\) −11.3608 −0.408355
\(775\) 24.9072i 0.894694i
\(776\) 12.5496 0.450506
\(777\) 0 0
\(778\) 12.9955i 0.465910i
\(779\) −13.0891 −0.468966
\(780\) 2.65522 0.0950721
\(781\) −8.03004 + 13.7630i −0.287337 + 0.492478i
\(782\) 18.6674i 0.667546i
\(783\) −8.01022 −0.286262
\(784\) 0 0
\(785\) −3.78527 −0.135102
\(786\) −9.54305 −0.340389
\(787\) 35.6858 1.27206 0.636030 0.771664i \(-0.280575\pi\)
0.636030 + 0.771664i \(0.280575\pi\)
\(788\) 1.22462i 0.0436253i
\(789\) −17.1219 −0.609556
\(790\) 8.73235i 0.310683i
\(791\) 0 0
\(792\) −2.86468 1.67140i −0.101792 0.0593908i
\(793\) −6.80309 −0.241585
\(794\) −32.8955 −1.16742
\(795\) −4.29122 −0.152194
\(796\) 12.8195i 0.454376i
\(797\) 25.7624i 0.912550i 0.889839 + 0.456275i \(0.150817\pi\)
−0.889839 + 0.456275i \(0.849183\pi\)
\(798\) 0 0
\(799\) 38.6234i 1.36640i
\(800\) 4.11157i 0.145366i
\(801\) 11.5356i 0.407589i
\(802\) 28.0768i 0.991427i
\(803\) 7.74475 13.2740i 0.273306 0.468430i
\(804\) 2.22408i 0.0784372i
\(805\) 0 0
\(806\) 17.0650 0.601090
\(807\) −11.8794 −0.418175
\(808\) 8.65999i 0.304657i
\(809\) 23.2691i 0.818099i −0.912512 0.409050i \(-0.865860\pi\)
0.912512 0.409050i \(-0.134140\pi\)
\(810\) −0.942563 −0.0331183
\(811\) 1.49575 0.0525229 0.0262614 0.999655i \(-0.491640\pi\)
0.0262614 + 0.999655i \(0.491640\pi\)
\(812\) 0 0
\(813\) 12.9150i 0.452949i
\(814\) −29.9014 17.4460i −1.04804 0.611482i
\(815\) 11.8957i 0.416688i
\(816\) 4.30608i 0.150743i
\(817\) 28.5880i 1.00017i
\(818\) 39.1324i 1.36823i
\(819\) 0 0
\(820\) 4.90282i 0.171214i
\(821\) 34.5952i 1.20738i 0.797219 + 0.603690i \(0.206303\pi\)
−0.797219 + 0.603690i \(0.793697\pi\)
\(822\) −18.7883 −0.655316
\(823\) −33.5957 −1.17107 −0.585537 0.810646i \(-0.699116\pi\)
−0.585537 + 0.810646i \(0.699116\pi\)
\(824\) 11.4902 0.400278
\(825\) −11.7784 6.87211i −0.410070 0.239256i
\(826\) 0 0
\(827\) 5.10053i 0.177363i −0.996060 0.0886814i \(-0.971735\pi\)
0.996060 0.0886814i \(-0.0282653\pi\)
\(828\) 4.33514 0.150656
\(829\) 12.6217i 0.438370i −0.975683 0.219185i \(-0.929660\pi\)
0.975683 0.219185i \(-0.0703398\pi\)
\(830\) −8.33361 −0.289264
\(831\) 21.4574 0.744349
\(832\) −2.81702 −0.0976626
\(833\) 0 0
\(834\) 19.9086 0.689380
\(835\) 14.2696i 0.493821i
\(836\) −4.20588 + 7.20861i −0.145463 + 0.249315i
\(837\) −6.05783 −0.209389
\(838\) 27.5652 0.952224
\(839\) 18.2805i 0.631114i −0.948907 0.315557i \(-0.897809\pi\)
0.948907 0.315557i \(-0.102191\pi\)
\(840\) 0 0
\(841\) −35.1636 −1.21254
\(842\) 6.46286i 0.222725i
\(843\) −0.770084 −0.0265231
\(844\) 17.0672i 0.587476i
\(845\) 4.77351i 0.164214i
\(846\) −8.96951 −0.308378
\(847\) 0 0
\(848\) 4.55271 0.156341
\(849\) 16.7151i 0.573660i
\(850\) 17.7047i 0.607268i
\(851\) 45.2499 1.55115
\(852\) 4.80437i 0.164595i
\(853\) 48.9693 1.67668 0.838338 0.545150i \(-0.183527\pi\)
0.838338 + 0.545150i \(0.183527\pi\)
\(854\) 0 0
\(855\) 2.37184i 0.0811152i
\(856\) 8.51031 0.290876
\(857\) 34.7766 1.18794 0.593972 0.804485i \(-0.297559\pi\)
0.593972 + 0.804485i \(0.297559\pi\)
\(858\) −4.70838 + 8.06987i −0.160742 + 0.275501i
\(859\) 7.70241i 0.262803i −0.991329 0.131401i \(-0.958052\pi\)
0.991329 0.131401i \(-0.0419477\pi\)
\(860\) −10.7083 −0.365148
\(861\) 0 0
\(862\) 5.81850 0.198179
\(863\) 21.8084 0.742365 0.371183 0.928560i \(-0.378952\pi\)
0.371183 + 0.928560i \(0.378952\pi\)
\(864\) 1.00000 0.0340207
\(865\) 7.29014i 0.247872i
\(866\) −27.1792 −0.923588
\(867\) 1.54228i 0.0523787i
\(868\) 0 0
\(869\) 26.5398 + 15.4847i 0.900300 + 0.525282i
\(870\) −7.55014 −0.255974
\(871\) −6.26528 −0.212291
\(872\) −0.950748 −0.0321964
\(873\) 12.5496i 0.424741i
\(874\) 10.9088i 0.368996i
\(875\) 0 0
\(876\) 4.63368i 0.156557i
\(877\) 51.1146i 1.72602i 0.505189 + 0.863009i \(0.331423\pi\)
−0.505189 + 0.863009i \(0.668577\pi\)
\(878\) 24.3998i 0.823454i
\(879\) 24.6448i 0.831247i
\(880\) −2.70014 1.57540i −0.0910218 0.0531069i
\(881\) 14.3532i 0.483573i −0.970329 0.241787i \(-0.922267\pi\)
0.970329 0.241787i \(-0.0777334\pi\)
\(882\) 0 0
\(883\) −13.4009 −0.450977 −0.225488 0.974246i \(-0.572398\pi\)
−0.225488 + 0.974246i \(0.572398\pi\)
\(884\) −12.1303 −0.407986
\(885\) 12.7058i 0.427102i
\(886\) 7.40742i 0.248857i
\(887\) −17.5783 −0.590223 −0.295112 0.955463i \(-0.595357\pi\)
−0.295112 + 0.955463i \(0.595357\pi\)
\(888\) 10.4379 0.350274
\(889\) 0 0
\(890\) 10.8730i 0.364464i
\(891\) 1.67140 2.86468i 0.0559942 0.0959705i
\(892\) 8.56595i 0.286809i
\(893\) 22.5706i 0.755297i
\(894\) 3.86561i 0.129285i
\(895\) 15.8992i 0.531451i
\(896\) 0 0
\(897\) 12.2122i 0.407753i
\(898\) 8.04219i 0.268372i
\(899\) −48.5246 −1.61838
\(900\) 4.11157 0.137052
\(901\) 19.6043 0.653115
\(902\) 14.9009 + 8.69395i 0.496145 + 0.289477i
\(903\) 0 0
\(904\) 8.55994i 0.284699i
\(905\) −1.72191 −0.0572382
\(906\) 1.64273i 0.0545761i
\(907\) −15.3916 −0.511071 −0.255535 0.966800i \(-0.582252\pi\)
−0.255535 + 0.966800i \(0.582252\pi\)
\(908\) 16.9377 0.562097
\(909\) 8.65999 0.287234
\(910\) 0 0
\(911\) −18.3460 −0.607828 −0.303914 0.952699i \(-0.598294\pi\)
−0.303914 + 0.952699i \(0.598294\pi\)
\(912\) 2.51637i 0.0833254i
\(913\) 14.7776 25.3279i 0.489068 0.838231i
\(914\) 18.1058 0.598886
\(915\) 2.27628 0.0752517
\(916\) 0.723697i 0.0239116i
\(917\) 0 0
\(918\) 4.30608 0.142122
\(919\) 19.3005i 0.636664i −0.947979 0.318332i \(-0.896877\pi\)
0.947979 0.318332i \(-0.103123\pi\)
\(920\) 4.08614 0.134716
\(921\) 15.2768i 0.503388i
\(922\) 17.1197i 0.563807i
\(923\) −13.5340 −0.445477
\(924\) 0 0
\(925\) 42.9163 1.41108
\(926\) 28.4136i 0.933728i
\(927\) 11.4902i 0.377386i
\(928\) 8.01022 0.262948
\(929\) 26.6207i 0.873397i −0.899608 0.436698i \(-0.856148\pi\)
0.899608 0.436698i \(-0.143852\pi\)
\(930\) −5.70989 −0.187235
\(931\) 0 0
\(932\) 13.7409i 0.450100i
\(933\) 24.0083 0.785997
\(934\) 20.6667 0.676235
\(935\) −11.6270 6.78381i −0.380244 0.221854i
\(936\) 2.81702i 0.0920772i
\(937\) 45.3628 1.48194 0.740969 0.671539i \(-0.234366\pi\)
0.740969 + 0.671539i \(0.234366\pi\)
\(938\) 0 0
\(939\) −1.80468 −0.0588936
\(940\) −8.45433 −0.275750
\(941\) 47.8871 1.56107 0.780537 0.625110i \(-0.214946\pi\)
0.780537 + 0.625110i \(0.214946\pi\)
\(942\) 4.01593i 0.130846i
\(943\) −22.5496 −0.734315
\(944\) 13.4801i 0.438740i
\(945\) 0 0
\(946\) 18.9885 32.5450i 0.617368 1.05813i
\(947\) −22.5730 −0.733523 −0.366762 0.930315i \(-0.619534\pi\)
−0.366762 + 0.930315i \(0.619534\pi\)
\(948\) −9.26447 −0.300896
\(949\) 13.0532 0.423724
\(950\) 10.3463i 0.335677i
\(951\) 21.4013i 0.693986i
\(952\) 0 0
\(953\) 44.2701i 1.43405i −0.697048 0.717025i \(-0.745503\pi\)
0.697048 0.717025i \(-0.254497\pi\)
\(954\) 4.55271i 0.147399i
\(955\) 13.4701i 0.435883i
\(956\) 0.273841i 0.00885667i
\(957\) 13.3883 22.9467i 0.432783 0.741763i
\(958\) 20.7439i 0.670204i
\(959\) 0 0
\(960\) 0.942563 0.0304211
\(961\) −5.69731 −0.183784
\(962\) 29.4039i 0.948019i
\(963\) 8.51031i 0.274241i
\(964\) 0.693253 0.0223282
\(965\) 3.47670 0.111919
\(966\) 0 0
\(967\) 9.65227i 0.310396i 0.987883 + 0.155198i \(0.0496015\pi\)
−0.987883 + 0.155198i \(0.950398\pi\)
\(968\) 9.57609 5.41281i 0.307787 0.173974i
\(969\) 10.8357i 0.348092i
\(970\) 11.8288i 0.379801i
\(971\) 60.4441i 1.93974i 0.243616 + 0.969872i \(0.421666\pi\)
−0.243616 + 0.969872i \(0.578334\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 0 0
\(974\) 36.2719i 1.16223i
\(975\) 11.5824i 0.370933i
\(976\) −2.41499 −0.0773021
\(977\) −5.39548 −0.172617 −0.0863083 0.996268i \(-0.527507\pi\)
−0.0863083 + 0.996268i \(0.527507\pi\)
\(978\) −12.6206 −0.403562
\(979\) −33.0457 19.2806i −1.05615 0.616211i
\(980\) 0 0
\(981\) 0.950748i 0.0303550i
\(982\) −0.584864 −0.0186638
\(983\) 21.8043i 0.695449i −0.937597 0.347724i \(-0.886954\pi\)
0.937597 0.347724i \(-0.113046\pi\)
\(984\) −5.20158 −0.165820
\(985\) 1.15428 0.0367785
\(986\) 34.4926 1.09847
\(987\) 0 0
\(988\) −7.08867 −0.225521
\(989\) 49.2505i 1.56608i
\(990\) 1.57540 2.70014i 0.0500696 0.0858162i
\(991\) 9.04292 0.287258 0.143629 0.989632i \(-0.454123\pi\)
0.143629 + 0.989632i \(0.454123\pi\)
\(992\) 6.05783 0.192336
\(993\) 24.5734i 0.779813i
\(994\) 0 0
\(995\) 12.0832 0.383063
\(996\) 8.84143i 0.280152i
\(997\) −39.4853 −1.25051 −0.625256 0.780420i \(-0.715005\pi\)
−0.625256 + 0.780420i \(0.715005\pi\)
\(998\) 21.6453i 0.685171i
\(999\) 10.4379i 0.330242i
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 3234.2.e.c.2155.5 24
7.6 odd 2 3234.2.e.d.2155.8 yes 24
11.10 odd 2 3234.2.e.d.2155.17 yes 24
77.76 even 2 inner 3234.2.e.c.2155.20 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.e.c.2155.5 24 1.1 even 1 trivial
3234.2.e.c.2155.20 yes 24 77.76 even 2 inner
3234.2.e.d.2155.8 yes 24 7.6 odd 2
3234.2.e.d.2155.17 yes 24 11.10 odd 2