Properties

Label 6825.2.a.bg.1.3
Level $6825$
Weight $2$
Character 6825.1
Self dual yes
Analytic conductor $54.498$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6825,2,Mod(1,6825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6825.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6825.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: \(54.4978993795\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.52616\) of defining polynomial
Character \(\chi\) \(=\) 6825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.670843 q^{2} -1.00000 q^{3} -1.54997 q^{4} -0.670843 q^{6} -1.00000 q^{7} -2.38147 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.670843 q^{2} -1.00000 q^{3} -1.54997 q^{4} -0.670843 q^{6} -1.00000 q^{7} -2.38147 q^{8} +1.00000 q^{9} -3.05232 q^{11} +1.54997 q^{12} -1.00000 q^{13} -0.670843 q^{14} +1.50235 q^{16} -1.34169 q^{17} +0.670843 q^{18} +7.60228 q^{19} +1.00000 q^{21} -2.04762 q^{22} -1.84403 q^{23} +2.38147 q^{24} -0.670843 q^{26} -1.00000 q^{27} +1.54997 q^{28} -5.60228 q^{29} +10.2857 q^{31} +5.77078 q^{32} +3.05232 q^{33} -0.900061 q^{34} -1.54997 q^{36} +9.20457 q^{37} +5.09994 q^{38} +1.00000 q^{39} -8.04762 q^{41} +0.670843 q^{42} -1.49765 q^{43} +4.73100 q^{44} -1.23706 q^{46} +3.89166 q^{47} -1.50235 q^{48} +1.00000 q^{49} +1.34169 q^{51} +1.54997 q^{52} -0.502345 q^{53} -0.670843 q^{54} +2.38147 q^{56} -7.60228 q^{57} -3.75825 q^{58} -4.28937 q^{59} -0.683372 q^{61} +6.90006 q^{62} -1.00000 q^{63} +0.866598 q^{64} +2.04762 q^{66} +7.68806 q^{67} +2.07957 q^{68} +1.84403 q^{69} -14.2569 q^{71} -2.38147 q^{72} +12.0189 q^{73} +6.17482 q^{74} -11.7833 q^{76} +3.05232 q^{77} +0.670843 q^{78} +4.91891 q^{79} +1.00000 q^{81} -5.39869 q^{82} +1.20828 q^{83} -1.54997 q^{84} -1.00469 q^{86} +5.60228 q^{87} +7.26900 q^{88} +13.7545 q^{89} +1.00000 q^{91} +2.85819 q^{92} -10.2857 q^{93} +2.61069 q^{94} -5.77078 q^{96} +7.18572 q^{97} +0.670843 q^{98} -3.05232 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9} - 2 q^{11} - 7 q^{12} - 4 q^{13} + q^{14} + 9 q^{16} + 2 q^{17} - q^{18} + 7 q^{19} + 4 q^{21} + 8 q^{22} - 3 q^{23} + 3 q^{24} + q^{26} - 4 q^{27} - 7 q^{28} + q^{29} + 3 q^{31} - 7 q^{32} + 2 q^{33} - 30 q^{34} + 7 q^{36} - 10 q^{37} - 6 q^{38} + 4 q^{39} - 16 q^{41} - q^{42} - 3 q^{43} - 12 q^{44} - 18 q^{46} - 5 q^{47} - 9 q^{48} + 4 q^{49} - 2 q^{51} - 7 q^{52} - 5 q^{53} + q^{54} + 3 q^{56} - 7 q^{57} + 4 q^{58} - 20 q^{59} + 12 q^{61} + 54 q^{62} - 4 q^{63} + 5 q^{64} - 8 q^{66} + 22 q^{67} + 10 q^{68} + 3 q^{69} - 3 q^{72} + 13 q^{73} - 6 q^{74} - 6 q^{76} + 2 q^{77} - q^{78} + 11 q^{79} + 4 q^{81} - 10 q^{82} - q^{83} + 7 q^{84} - 10 q^{86} - q^{87} + 60 q^{88} - 5 q^{89} + 4 q^{91} - 34 q^{92} - 3 q^{93} + 34 q^{94} + 7 q^{96} + 17 q^{97} - q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.670843 0.474358 0.237179 0.971466i \(-0.423777\pi\)
0.237179 + 0.971466i \(0.423777\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.54997 −0.774985
\(5\) 0 0
\(6\) −0.670843 −0.273870
\(7\) −1.00000 −0.377964
\(8\) −2.38147 −0.841978
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.05232 −0.920308 −0.460154 0.887839i \(-0.652206\pi\)
−0.460154 + 0.887839i \(0.652206\pi\)
\(12\) 1.54997 0.447438
\(13\) −1.00000 −0.277350
\(14\) −0.670843 −0.179290
\(15\) 0 0
\(16\) 1.50235 0.375586
\(17\) −1.34169 −0.325407 −0.162703 0.986675i \(-0.552021\pi\)
−0.162703 + 0.986675i \(0.552021\pi\)
\(18\) 0.670843 0.158119
\(19\) 7.60228 1.74408 0.872042 0.489431i \(-0.162796\pi\)
0.872042 + 0.489431i \(0.162796\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −2.04762 −0.436555
\(23\) −1.84403 −0.384507 −0.192254 0.981345i \(-0.561580\pi\)
−0.192254 + 0.981345i \(0.561580\pi\)
\(24\) 2.38147 0.486116
\(25\) 0 0
\(26\) −0.670843 −0.131563
\(27\) −1.00000 −0.192450
\(28\) 1.54997 0.292917
\(29\) −5.60228 −1.04032 −0.520159 0.854069i \(-0.674127\pi\)
−0.520159 + 0.854069i \(0.674127\pi\)
\(30\) 0 0
\(31\) 10.2857 1.84736 0.923679 0.383167i \(-0.125166\pi\)
0.923679 + 0.383167i \(0.125166\pi\)
\(32\) 5.77078 1.02014
\(33\) 3.05232 0.531340
\(34\) −0.900061 −0.154359
\(35\) 0 0
\(36\) −1.54997 −0.258328
\(37\) 9.20457 1.51322 0.756611 0.653865i \(-0.226854\pi\)
0.756611 + 0.653865i \(0.226854\pi\)
\(38\) 5.09994 0.827319
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −8.04762 −1.25683 −0.628414 0.777879i \(-0.716296\pi\)
−0.628414 + 0.777879i \(0.716296\pi\)
\(42\) 0.670843 0.103513
\(43\) −1.49765 −0.228390 −0.114195 0.993458i \(-0.536429\pi\)
−0.114195 + 0.993458i \(0.536429\pi\)
\(44\) 4.73100 0.713224
\(45\) 0 0
\(46\) −1.23706 −0.182394
\(47\) 3.89166 0.567656 0.283828 0.958875i \(-0.408395\pi\)
0.283828 + 0.958875i \(0.408395\pi\)
\(48\) −1.50235 −0.216845
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.34169 0.187874
\(52\) 1.54997 0.214942
\(53\) −0.502345 −0.0690025 −0.0345012 0.999405i \(-0.510984\pi\)
−0.0345012 + 0.999405i \(0.510984\pi\)
\(54\) −0.670843 −0.0912902
\(55\) 0 0
\(56\) 2.38147 0.318238
\(57\) −7.60228 −1.00695
\(58\) −3.75825 −0.493483
\(59\) −4.28937 −0.558429 −0.279214 0.960229i \(-0.590074\pi\)
−0.279214 + 0.960229i \(0.590074\pi\)
\(60\) 0 0
\(61\) −0.683372 −0.0874968 −0.0437484 0.999043i \(-0.513930\pi\)
−0.0437484 + 0.999043i \(0.513930\pi\)
\(62\) 6.90006 0.876309
\(63\) −1.00000 −0.125988
\(64\) 0.866598 0.108325
\(65\) 0 0
\(66\) 2.04762 0.252045
\(67\) 7.68806 0.939246 0.469623 0.882867i \(-0.344390\pi\)
0.469623 + 0.882867i \(0.344390\pi\)
\(68\) 2.07957 0.252185
\(69\) 1.84403 0.221995
\(70\) 0 0
\(71\) −14.2569 −1.69198 −0.845990 0.533198i \(-0.820990\pi\)
−0.845990 + 0.533198i \(0.820990\pi\)
\(72\) −2.38147 −0.280659
\(73\) 12.0189 1.40670 0.703350 0.710844i \(-0.251687\pi\)
0.703350 + 0.710844i \(0.251687\pi\)
\(74\) 6.17482 0.717808
\(75\) 0 0
\(76\) −11.7833 −1.35164
\(77\) 3.05232 0.347844
\(78\) 0.670843 0.0759580
\(79\) 4.91891 0.553421 0.276710 0.960953i \(-0.410756\pi\)
0.276710 + 0.960953i \(0.410756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −5.39869 −0.596186
\(83\) 1.20828 0.132626 0.0663132 0.997799i \(-0.478876\pi\)
0.0663132 + 0.997799i \(0.478876\pi\)
\(84\) −1.54997 −0.169116
\(85\) 0 0
\(86\) −1.00469 −0.108339
\(87\) 5.60228 0.600628
\(88\) 7.26900 0.774878
\(89\) 13.7545 1.45798 0.728989 0.684525i \(-0.239990\pi\)
0.728989 + 0.684525i \(0.239990\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 2.85819 0.297987
\(93\) −10.2857 −1.06657
\(94\) 2.61069 0.269272
\(95\) 0 0
\(96\) −5.77078 −0.588978
\(97\) 7.18572 0.729599 0.364800 0.931086i \(-0.381138\pi\)
0.364800 + 0.931086i \(0.381138\pi\)
\(98\) 0.670843 0.0677654
\(99\) −3.05232 −0.306769
\(100\) 0 0
\(101\) −18.4416 −1.83501 −0.917505 0.397724i \(-0.869800\pi\)
−0.917505 + 0.397724i \(0.869800\pi\)
\(102\) 0.900061 0.0891193
\(103\) −1.89537 −0.186756 −0.0933782 0.995631i \(-0.529767\pi\)
−0.0933782 + 0.995631i \(0.529767\pi\)
\(104\) 2.38147 0.233523
\(105\) 0 0
\(106\) −0.336995 −0.0327318
\(107\) −18.5463 −1.79293 −0.896467 0.443110i \(-0.853875\pi\)
−0.896467 + 0.443110i \(0.853875\pi\)
\(108\) 1.54997 0.149146
\(109\) −1.42126 −0.136132 −0.0680659 0.997681i \(-0.521683\pi\)
−0.0680659 + 0.997681i \(0.521683\pi\)
\(110\) 0 0
\(111\) −9.20457 −0.873659
\(112\) −1.50235 −0.141958
\(113\) −5.60228 −0.527019 −0.263509 0.964657i \(-0.584880\pi\)
−0.263509 + 0.964657i \(0.584880\pi\)
\(114\) −5.09994 −0.477653
\(115\) 0 0
\(116\) 8.68337 0.806231
\(117\) −1.00000 −0.0924500
\(118\) −2.87749 −0.264895
\(119\) 1.34169 0.122992
\(120\) 0 0
\(121\) −1.68337 −0.153034
\(122\) −0.458435 −0.0415048
\(123\) 8.04762 0.725630
\(124\) −15.9425 −1.43167
\(125\) 0 0
\(126\) −0.670843 −0.0597634
\(127\) −14.1046 −1.25158 −0.625792 0.779990i \(-0.715224\pi\)
−0.625792 + 0.779990i \(0.715224\pi\)
\(128\) −10.9602 −0.968755
\(129\) 1.49765 0.131861
\(130\) 0 0
\(131\) −8.78800 −0.767811 −0.383906 0.923372i \(-0.625421\pi\)
−0.383906 + 0.923372i \(0.625421\pi\)
\(132\) −4.73100 −0.411780
\(133\) −7.60228 −0.659202
\(134\) 5.15748 0.445539
\(135\) 0 0
\(136\) 3.19519 0.273985
\(137\) 1.97743 0.168944 0.0844718 0.996426i \(-0.473080\pi\)
0.0844718 + 0.996426i \(0.473080\pi\)
\(138\) 1.23706 0.105305
\(139\) 13.0999 1.11112 0.555561 0.831476i \(-0.312503\pi\)
0.555561 + 0.831476i \(0.312503\pi\)
\(140\) 0 0
\(141\) −3.89166 −0.327737
\(142\) −9.56413 −0.802604
\(143\) 3.05232 0.255247
\(144\) 1.50235 0.125195
\(145\) 0 0
\(146\) 8.06276 0.667279
\(147\) −1.00000 −0.0824786
\(148\) −14.2668 −1.17272
\(149\) 14.5939 1.19558 0.597789 0.801654i \(-0.296046\pi\)
0.597789 + 0.801654i \(0.296046\pi\)
\(150\) 0 0
\(151\) −5.41188 −0.440412 −0.220206 0.975453i \(-0.570673\pi\)
−0.220206 + 0.975453i \(0.570673\pi\)
\(152\) −18.1046 −1.46848
\(153\) −1.34169 −0.108469
\(154\) 2.04762 0.165002
\(155\) 0 0
\(156\) −1.54997 −0.124097
\(157\) −23.4044 −1.86788 −0.933939 0.357432i \(-0.883652\pi\)
−0.933939 + 0.357432i \(0.883652\pi\)
\(158\) 3.29982 0.262519
\(159\) 0.502345 0.0398386
\(160\) 0 0
\(161\) 1.84403 0.145330
\(162\) 0.670843 0.0527064
\(163\) 10.1046 0.791456 0.395728 0.918368i \(-0.370492\pi\)
0.395728 + 0.918368i \(0.370492\pi\)
\(164\) 12.4736 0.974022
\(165\) 0 0
\(166\) 0.810569 0.0629123
\(167\) 3.09623 0.239593 0.119797 0.992798i \(-0.461776\pi\)
0.119797 + 0.992798i \(0.461776\pi\)
\(168\) −2.38147 −0.183735
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 7.60228 0.581361
\(172\) 2.32132 0.176999
\(173\) −2.75356 −0.209349 −0.104675 0.994507i \(-0.533380\pi\)
−0.104675 + 0.994507i \(0.533380\pi\)
\(174\) 3.75825 0.284912
\(175\) 0 0
\(176\) −4.58563 −0.345655
\(177\) 4.28937 0.322409
\(178\) 9.22714 0.691603
\(179\) −1.57723 −0.117887 −0.0589437 0.998261i \(-0.518773\pi\)
−0.0589437 + 0.998261i \(0.518773\pi\)
\(180\) 0 0
\(181\) 9.51651 0.707356 0.353678 0.935367i \(-0.384931\pi\)
0.353678 + 0.935367i \(0.384931\pi\)
\(182\) 0.670843 0.0497262
\(183\) 0.683372 0.0505163
\(184\) 4.39151 0.323746
\(185\) 0 0
\(186\) −6.90006 −0.505937
\(187\) 4.09525 0.299474
\(188\) −6.03195 −0.439925
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −22.8508 −1.65342 −0.826712 0.562626i \(-0.809791\pi\)
−0.826712 + 0.562626i \(0.809791\pi\)
\(192\) −0.866598 −0.0625413
\(193\) −19.5760 −1.40911 −0.704556 0.709649i \(-0.748854\pi\)
−0.704556 + 0.709649i \(0.748854\pi\)
\(194\) 4.82049 0.346091
\(195\) 0 0
\(196\) −1.54997 −0.110712
\(197\) 21.2271 1.51237 0.756185 0.654357i \(-0.227061\pi\)
0.756185 + 0.654357i \(0.227061\pi\)
\(198\) −2.04762 −0.145518
\(199\) −6.68337 −0.473772 −0.236886 0.971537i \(-0.576127\pi\)
−0.236886 + 0.971537i \(0.576127\pi\)
\(200\) 0 0
\(201\) −7.68806 −0.542274
\(202\) −12.3714 −0.870451
\(203\) 5.60228 0.393203
\(204\) −2.07957 −0.145599
\(205\) 0 0
\(206\) −1.27150 −0.0885893
\(207\) −1.84403 −0.128169
\(208\) −1.50235 −0.104169
\(209\) −23.2046 −1.60509
\(210\) 0 0
\(211\) −4.60698 −0.317157 −0.158579 0.987346i \(-0.550691\pi\)
−0.158579 + 0.987346i \(0.550691\pi\)
\(212\) 0.778620 0.0534759
\(213\) 14.2569 0.976866
\(214\) −12.4416 −0.850492
\(215\) 0 0
\(216\) 2.38147 0.162039
\(217\) −10.2857 −0.698236
\(218\) −0.953441 −0.0645752
\(219\) −12.0189 −0.812159
\(220\) 0 0
\(221\) 1.34169 0.0902516
\(222\) −6.17482 −0.414427
\(223\) −8.60698 −0.576366 −0.288183 0.957575i \(-0.593051\pi\)
−0.288183 + 0.957575i \(0.593051\pi\)
\(224\) −5.77078 −0.385577
\(225\) 0 0
\(226\) −3.75825 −0.249995
\(227\) 14.4892 0.961685 0.480843 0.876807i \(-0.340331\pi\)
0.480843 + 0.876807i \(0.340331\pi\)
\(228\) 11.7833 0.780369
\(229\) −20.1999 −1.33485 −0.667423 0.744679i \(-0.732603\pi\)
−0.667423 + 0.744679i \(0.732603\pi\)
\(230\) 0 0
\(231\) −3.05232 −0.200828
\(232\) 13.3417 0.875925
\(233\) −7.28097 −0.476992 −0.238496 0.971143i \(-0.576654\pi\)
−0.238496 + 0.971143i \(0.576654\pi\)
\(234\) −0.670843 −0.0438544
\(235\) 0 0
\(236\) 6.64839 0.432774
\(237\) −4.91891 −0.319518
\(238\) 0.900061 0.0583423
\(239\) 27.6688 1.78974 0.894872 0.446323i \(-0.147267\pi\)
0.894872 + 0.446323i \(0.147267\pi\)
\(240\) 0 0
\(241\) −22.2187 −1.43123 −0.715617 0.698493i \(-0.753854\pi\)
−0.715617 + 0.698493i \(0.753854\pi\)
\(242\) −1.12928 −0.0725928
\(243\) −1.00000 −0.0641500
\(244\) 1.05921 0.0678087
\(245\) 0 0
\(246\) 5.39869 0.344208
\(247\) −7.60228 −0.483722
\(248\) −24.4950 −1.55543
\(249\) −1.20828 −0.0765719
\(250\) 0 0
\(251\) 17.3092 1.09255 0.546274 0.837607i \(-0.316046\pi\)
0.546274 + 0.837607i \(0.316046\pi\)
\(252\) 1.54997 0.0976389
\(253\) 5.62857 0.353865
\(254\) −9.46199 −0.593698
\(255\) 0 0
\(256\) −9.08578 −0.567861
\(257\) 5.55837 0.346722 0.173361 0.984858i \(-0.444537\pi\)
0.173361 + 0.984858i \(0.444537\pi\)
\(258\) 1.00469 0.0625493
\(259\) −9.20457 −0.571944
\(260\) 0 0
\(261\) −5.60228 −0.346773
\(262\) −5.89537 −0.364217
\(263\) −25.0486 −1.54456 −0.772281 0.635281i \(-0.780884\pi\)
−0.772281 + 0.635281i \(0.780884\pi\)
\(264\) −7.26900 −0.447376
\(265\) 0 0
\(266\) −5.09994 −0.312697
\(267\) −13.7545 −0.841764
\(268\) −11.9163 −0.727902
\(269\) −4.55368 −0.277643 −0.138821 0.990317i \(-0.544331\pi\)
−0.138821 + 0.990317i \(0.544331\pi\)
\(270\) 0 0
\(271\) 8.88325 0.539619 0.269810 0.962914i \(-0.413039\pi\)
0.269810 + 0.962914i \(0.413039\pi\)
\(272\) −2.01568 −0.122218
\(273\) −1.00000 −0.0605228
\(274\) 1.32655 0.0801397
\(275\) 0 0
\(276\) −2.85819 −0.172043
\(277\) 5.38560 0.323589 0.161795 0.986824i \(-0.448272\pi\)
0.161795 + 0.986824i \(0.448272\pi\)
\(278\) 8.78800 0.527069
\(279\) 10.2857 0.615786
\(280\) 0 0
\(281\) −12.9727 −0.773889 −0.386944 0.922103i \(-0.626469\pi\)
−0.386944 + 0.922103i \(0.626469\pi\)
\(282\) −2.61069 −0.155464
\(283\) 1.52589 0.0907047 0.0453523 0.998971i \(-0.485559\pi\)
0.0453523 + 0.998971i \(0.485559\pi\)
\(284\) 22.0977 1.31126
\(285\) 0 0
\(286\) 2.04762 0.121079
\(287\) 8.04762 0.475036
\(288\) 5.77078 0.340047
\(289\) −15.1999 −0.894111
\(290\) 0 0
\(291\) −7.18572 −0.421234
\(292\) −18.6289 −1.09017
\(293\) −18.8545 −1.10149 −0.550745 0.834673i \(-0.685656\pi\)
−0.550745 + 0.834673i \(0.685656\pi\)
\(294\) −0.670843 −0.0391244
\(295\) 0 0
\(296\) −21.9204 −1.27410
\(297\) 3.05232 0.177113
\(298\) 9.79020 0.567131
\(299\) 1.84403 0.106643
\(300\) 0 0
\(301\) 1.49765 0.0863234
\(302\) −3.63052 −0.208913
\(303\) 18.4416 1.05944
\(304\) 11.4213 0.655054
\(305\) 0 0
\(306\) −0.900061 −0.0514530
\(307\) −15.3856 −0.878102 −0.439051 0.898462i \(-0.644685\pi\)
−0.439051 + 0.898462i \(0.644685\pi\)
\(308\) −4.73100 −0.269574
\(309\) 1.89537 0.107824
\(310\) 0 0
\(311\) −17.0999 −0.969649 −0.484824 0.874612i \(-0.661116\pi\)
−0.484824 + 0.874612i \(0.661116\pi\)
\(312\) −2.38147 −0.134824
\(313\) −4.89263 −0.276548 −0.138274 0.990394i \(-0.544155\pi\)
−0.138274 + 0.990394i \(0.544155\pi\)
\(314\) −15.7007 −0.886042
\(315\) 0 0
\(316\) −7.62417 −0.428893
\(317\) −1.44382 −0.0810933 −0.0405466 0.999178i \(-0.512910\pi\)
−0.0405466 + 0.999178i \(0.512910\pi\)
\(318\) 0.336995 0.0188977
\(319\) 17.0999 0.957413
\(320\) 0 0
\(321\) 18.5463 1.03515
\(322\) 1.23706 0.0689384
\(323\) −10.1999 −0.567536
\(324\) −1.54997 −0.0861094
\(325\) 0 0
\(326\) 6.77862 0.375433
\(327\) 1.42126 0.0785958
\(328\) 19.1652 1.05822
\(329\) −3.89166 −0.214554
\(330\) 0 0
\(331\) −14.9953 −0.824217 −0.412108 0.911135i \(-0.635207\pi\)
−0.412108 + 0.911135i \(0.635207\pi\)
\(332\) −1.87280 −0.102783
\(333\) 9.20457 0.504407
\(334\) 2.07708 0.113653
\(335\) 0 0
\(336\) 1.50235 0.0819597
\(337\) 14.9115 0.812280 0.406140 0.913811i \(-0.366874\pi\)
0.406140 + 0.913811i \(0.366874\pi\)
\(338\) 0.670843 0.0364890
\(339\) 5.60228 0.304274
\(340\) 0 0
\(341\) −31.3951 −1.70014
\(342\) 5.09994 0.275773
\(343\) −1.00000 −0.0539949
\(344\) 3.56662 0.192299
\(345\) 0 0
\(346\) −1.84721 −0.0993065
\(347\) −7.07488 −0.379800 −0.189900 0.981803i \(-0.560816\pi\)
−0.189900 + 0.981803i \(0.560816\pi\)
\(348\) −8.68337 −0.465478
\(349\) 21.1188 1.13046 0.565232 0.824932i \(-0.308787\pi\)
0.565232 + 0.824932i \(0.308787\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) −17.6142 −0.938843
\(353\) 17.3643 0.924206 0.462103 0.886826i \(-0.347095\pi\)
0.462103 + 0.886826i \(0.347095\pi\)
\(354\) 2.87749 0.152937
\(355\) 0 0
\(356\) −21.3191 −1.12991
\(357\) −1.34169 −0.0710096
\(358\) −1.05807 −0.0559208
\(359\) 18.3521 0.968589 0.484294 0.874905i \(-0.339076\pi\)
0.484294 + 0.874905i \(0.339076\pi\)
\(360\) 0 0
\(361\) 38.7947 2.04183
\(362\) 6.38408 0.335540
\(363\) 1.68337 0.0883541
\(364\) −1.54997 −0.0812405
\(365\) 0 0
\(366\) 0.458435 0.0239628
\(367\) −8.40719 −0.438852 −0.219426 0.975629i \(-0.570418\pi\)
−0.219426 + 0.975629i \(0.570418\pi\)
\(368\) −2.77037 −0.144416
\(369\) −8.04762 −0.418943
\(370\) 0 0
\(371\) 0.502345 0.0260805
\(372\) 15.9425 0.826578
\(373\) −27.0925 −1.40280 −0.701399 0.712769i \(-0.747441\pi\)
−0.701399 + 0.712769i \(0.747441\pi\)
\(374\) 2.74727 0.142058
\(375\) 0 0
\(376\) −9.26787 −0.477954
\(377\) 5.60228 0.288532
\(378\) 0.670843 0.0345044
\(379\) −6.41657 −0.329597 −0.164798 0.986327i \(-0.552697\pi\)
−0.164798 + 0.986327i \(0.552697\pi\)
\(380\) 0 0
\(381\) 14.1046 0.722602
\(382\) −15.3293 −0.784314
\(383\) −32.5939 −1.66547 −0.832735 0.553672i \(-0.813226\pi\)
−0.832735 + 0.553672i \(0.813226\pi\)
\(384\) 10.9602 0.559311
\(385\) 0 0
\(386\) −13.1324 −0.668423
\(387\) −1.49765 −0.0761301
\(388\) −11.1376 −0.565428
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 2.47411 0.125121
\(392\) −2.38147 −0.120283
\(393\) 8.78800 0.443296
\(394\) 14.2401 0.717405
\(395\) 0 0
\(396\) 4.73100 0.237741
\(397\) −15.7520 −0.790573 −0.395286 0.918558i \(-0.629355\pi\)
−0.395286 + 0.918558i \(0.629355\pi\)
\(398\) −4.48349 −0.224737
\(399\) 7.60228 0.380590
\(400\) 0 0
\(401\) 26.1867 1.30770 0.653851 0.756624i \(-0.273152\pi\)
0.653851 + 0.756624i \(0.273152\pi\)
\(402\) −5.15748 −0.257232
\(403\) −10.2857 −0.512365
\(404\) 28.5840 1.42211
\(405\) 0 0
\(406\) 3.75825 0.186519
\(407\) −28.0952 −1.39263
\(408\) −3.19519 −0.158185
\(409\) −17.3856 −0.859662 −0.429831 0.902909i \(-0.641427\pi\)
−0.429831 + 0.902909i \(0.641427\pi\)
\(410\) 0 0
\(411\) −1.97743 −0.0975396
\(412\) 2.93777 0.144733
\(413\) 4.28937 0.211066
\(414\) −1.23706 −0.0607980
\(415\) 0 0
\(416\) −5.77078 −0.282936
\(417\) −13.0999 −0.641507
\(418\) −15.5666 −0.761388
\(419\) −12.4761 −0.609496 −0.304748 0.952433i \(-0.598572\pi\)
−0.304748 + 0.952433i \(0.598572\pi\)
\(420\) 0 0
\(421\) 9.57405 0.466611 0.233305 0.972404i \(-0.425046\pi\)
0.233305 + 0.972404i \(0.425046\pi\)
\(422\) −3.09056 −0.150446
\(423\) 3.89166 0.189219
\(424\) 1.19632 0.0580985
\(425\) 0 0
\(426\) 9.56413 0.463384
\(427\) 0.683372 0.0330707
\(428\) 28.7461 1.38950
\(429\) −3.05232 −0.147367
\(430\) 0 0
\(431\) 5.20208 0.250575 0.125288 0.992120i \(-0.460015\pi\)
0.125288 + 0.992120i \(0.460015\pi\)
\(432\) −1.50235 −0.0722816
\(433\) 17.2547 0.829207 0.414604 0.910002i \(-0.363920\pi\)
0.414604 + 0.910002i \(0.363920\pi\)
\(434\) −6.90006 −0.331214
\(435\) 0 0
\(436\) 2.20291 0.105500
\(437\) −14.0189 −0.670613
\(438\) −8.06276 −0.385254
\(439\) −22.1925 −1.05919 −0.529594 0.848251i \(-0.677656\pi\)
−0.529594 + 0.848251i \(0.677656\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0.900061 0.0428115
\(443\) 37.9319 1.80220 0.901098 0.433615i \(-0.142762\pi\)
0.901098 + 0.433615i \(0.142762\pi\)
\(444\) 14.2668 0.677073
\(445\) 0 0
\(446\) −5.77393 −0.273403
\(447\) −14.5939 −0.690267
\(448\) −0.866598 −0.0409429
\(449\) −33.3150 −1.57223 −0.786115 0.618080i \(-0.787911\pi\)
−0.786115 + 0.618080i \(0.787911\pi\)
\(450\) 0 0
\(451\) 24.5639 1.15667
\(452\) 8.68337 0.408431
\(453\) 5.41188 0.254272
\(454\) 9.72001 0.456183
\(455\) 0 0
\(456\) 18.1046 0.847827
\(457\) 7.79269 0.364527 0.182263 0.983250i \(-0.441658\pi\)
0.182263 + 0.983250i \(0.441658\pi\)
\(458\) −13.5509 −0.633194
\(459\) 1.34169 0.0626245
\(460\) 0 0
\(461\) 30.4568 1.41851 0.709256 0.704951i \(-0.249031\pi\)
0.709256 + 0.704951i \(0.249031\pi\)
\(462\) −2.04762 −0.0952641
\(463\) −39.4639 −1.83405 −0.917023 0.398835i \(-0.869415\pi\)
−0.917023 + 0.398835i \(0.869415\pi\)
\(464\) −8.41657 −0.390729
\(465\) 0 0
\(466\) −4.88438 −0.226265
\(467\) 9.93307 0.459648 0.229824 0.973232i \(-0.426185\pi\)
0.229824 + 0.973232i \(0.426185\pi\)
\(468\) 1.54997 0.0716474
\(469\) −7.68806 −0.355002
\(470\) 0 0
\(471\) 23.4044 1.07842
\(472\) 10.2150 0.470184
\(473\) 4.57131 0.210189
\(474\) −3.29982 −0.151566
\(475\) 0 0
\(476\) −2.07957 −0.0953171
\(477\) −0.502345 −0.0230008
\(478\) 18.5614 0.848978
\(479\) −0.941479 −0.0430173 −0.0215086 0.999769i \(-0.506847\pi\)
−0.0215086 + 0.999769i \(0.506847\pi\)
\(480\) 0 0
\(481\) −9.20457 −0.419692
\(482\) −14.9053 −0.678917
\(483\) −1.84403 −0.0839063
\(484\) 2.60918 0.118599
\(485\) 0 0
\(486\) −0.670843 −0.0304301
\(487\) 39.3499 1.78312 0.891558 0.452907i \(-0.149613\pi\)
0.891558 + 0.452907i \(0.149613\pi\)
\(488\) 1.62743 0.0736703
\(489\) −10.1046 −0.456947
\(490\) 0 0
\(491\) 5.45374 0.246124 0.123062 0.992399i \(-0.460729\pi\)
0.123062 + 0.992399i \(0.460729\pi\)
\(492\) −12.4736 −0.562352
\(493\) 7.51651 0.338526
\(494\) −5.09994 −0.229457
\(495\) 0 0
\(496\) 15.4526 0.693843
\(497\) 14.2569 0.639509
\(498\) −0.810569 −0.0363225
\(499\) −31.3574 −1.40375 −0.701874 0.712301i \(-0.747653\pi\)
−0.701874 + 0.712301i \(0.747653\pi\)
\(500\) 0 0
\(501\) −3.09623 −0.138329
\(502\) 11.6118 0.518258
\(503\) −43.9926 −1.96153 −0.980766 0.195188i \(-0.937468\pi\)
−0.980766 + 0.195188i \(0.937468\pi\)
\(504\) 2.38147 0.106079
\(505\) 0 0
\(506\) 3.77588 0.167858
\(507\) −1.00000 −0.0444116
\(508\) 21.8617 0.969958
\(509\) −41.8498 −1.85496 −0.927480 0.373874i \(-0.878029\pi\)
−0.927480 + 0.373874i \(0.878029\pi\)
\(510\) 0 0
\(511\) −12.0189 −0.531683
\(512\) 15.8253 0.699386
\(513\) −7.60228 −0.335649
\(514\) 3.72880 0.164470
\(515\) 0 0
\(516\) −2.32132 −0.102190
\(517\) −11.8786 −0.522418
\(518\) −6.17482 −0.271306
\(519\) 2.75356 0.120868
\(520\) 0 0
\(521\) 8.32957 0.364925 0.182462 0.983213i \(-0.441593\pi\)
0.182462 + 0.983213i \(0.441593\pi\)
\(522\) −3.75825 −0.164494
\(523\) −25.3092 −1.10669 −0.553347 0.832951i \(-0.686650\pi\)
−0.553347 + 0.832951i \(0.686650\pi\)
\(524\) 13.6211 0.595042
\(525\) 0 0
\(526\) −16.8037 −0.732675
\(527\) −13.8001 −0.601143
\(528\) 4.58563 0.199564
\(529\) −19.5995 −0.852154
\(530\) 0 0
\(531\) −4.28937 −0.186143
\(532\) 11.7833 0.510871
\(533\) 8.04762 0.348581
\(534\) −9.22714 −0.399297
\(535\) 0 0
\(536\) −18.3089 −0.790824
\(537\) 1.57723 0.0680624
\(538\) −3.05481 −0.131702
\(539\) −3.05232 −0.131473
\(540\) 0 0
\(541\) 27.1501 1.16727 0.583636 0.812015i \(-0.301630\pi\)
0.583636 + 0.812015i \(0.301630\pi\)
\(542\) 5.95927 0.255972
\(543\) −9.51651 −0.408392
\(544\) −7.74258 −0.331960
\(545\) 0 0
\(546\) −0.670843 −0.0287094
\(547\) −27.5949 −1.17987 −0.589935 0.807450i \(-0.700847\pi\)
−0.589935 + 0.807450i \(0.700847\pi\)
\(548\) −3.06496 −0.130929
\(549\) −0.683372 −0.0291656
\(550\) 0 0
\(551\) −42.5902 −1.81440
\(552\) −4.39151 −0.186915
\(553\) −4.91891 −0.209173
\(554\) 3.61289 0.153497
\(555\) 0 0
\(556\) −20.3045 −0.861103
\(557\) −7.19881 −0.305024 −0.152512 0.988302i \(-0.548736\pi\)
−0.152512 + 0.988302i \(0.548736\pi\)
\(558\) 6.90006 0.292103
\(559\) 1.49765 0.0633440
\(560\) 0 0
\(561\) −4.09525 −0.172902
\(562\) −8.70267 −0.367100
\(563\) 20.2594 0.853831 0.426915 0.904292i \(-0.359600\pi\)
0.426915 + 0.904292i \(0.359600\pi\)
\(564\) 6.03195 0.253991
\(565\) 0 0
\(566\) 1.02363 0.0430265
\(567\) −1.00000 −0.0419961
\(568\) 33.9524 1.42461
\(569\) 7.28097 0.305234 0.152617 0.988285i \(-0.451230\pi\)
0.152617 + 0.988285i \(0.451230\pi\)
\(570\) 0 0
\(571\) −22.1141 −0.925446 −0.462723 0.886503i \(-0.653128\pi\)
−0.462723 + 0.886503i \(0.653128\pi\)
\(572\) −4.73100 −0.197813
\(573\) 22.8508 0.954604
\(574\) 5.39869 0.225337
\(575\) 0 0
\(576\) 0.866598 0.0361082
\(577\) −20.3139 −0.845678 −0.422839 0.906205i \(-0.638966\pi\)
−0.422839 + 0.906205i \(0.638966\pi\)
\(578\) −10.1967 −0.424128
\(579\) 19.5760 0.813551
\(580\) 0 0
\(581\) −1.20828 −0.0501281
\(582\) −4.82049 −0.199816
\(583\) 1.53332 0.0635035
\(584\) −28.6226 −1.18441
\(585\) 0 0
\(586\) −12.6484 −0.522500
\(587\) −35.8842 −1.48110 −0.740550 0.672001i \(-0.765435\pi\)
−0.740550 + 0.672001i \(0.765435\pi\)
\(588\) 1.54997 0.0639197
\(589\) 78.1945 3.22195
\(590\) 0 0
\(591\) −21.2271 −0.873168
\(592\) 13.8284 0.568346
\(593\) −20.2664 −0.832239 −0.416120 0.909310i \(-0.636610\pi\)
−0.416120 + 0.909310i \(0.636610\pi\)
\(594\) 2.04762 0.0840150
\(595\) 0 0
\(596\) −22.6201 −0.926554
\(597\) 6.68337 0.273532
\(598\) 1.23706 0.0505870
\(599\) 14.9365 0.610291 0.305145 0.952306i \(-0.401295\pi\)
0.305145 + 0.952306i \(0.401295\pi\)
\(600\) 0 0
\(601\) 4.89263 0.199575 0.0997873 0.995009i \(-0.468184\pi\)
0.0997873 + 0.995009i \(0.468184\pi\)
\(602\) 1.00469 0.0409481
\(603\) 7.68806 0.313082
\(604\) 8.38824 0.341313
\(605\) 0 0
\(606\) 12.3714 0.502555
\(607\) −20.6164 −0.836796 −0.418398 0.908264i \(-0.637408\pi\)
−0.418398 + 0.908264i \(0.637408\pi\)
\(608\) 43.8711 1.77921
\(609\) −5.60228 −0.227016
\(610\) 0 0
\(611\) −3.89166 −0.157440
\(612\) 2.07957 0.0840617
\(613\) 9.73818 0.393321 0.196661 0.980472i \(-0.436990\pi\)
0.196661 + 0.980472i \(0.436990\pi\)
\(614\) −10.3213 −0.416535
\(615\) 0 0
\(616\) −7.26900 −0.292877
\(617\) 0.763482 0.0307366 0.0153683 0.999882i \(-0.495108\pi\)
0.0153683 + 0.999882i \(0.495108\pi\)
\(618\) 1.27150 0.0511470
\(619\) −41.0925 −1.65165 −0.825824 0.563928i \(-0.809289\pi\)
−0.825824 + 0.563928i \(0.809289\pi\)
\(620\) 0 0
\(621\) 1.84403 0.0739984
\(622\) −11.4714 −0.459960
\(623\) −13.7545 −0.551064
\(624\) 1.50235 0.0601420
\(625\) 0 0
\(626\) −3.28219 −0.131183
\(627\) 23.2046 0.926701
\(628\) 36.2762 1.44758
\(629\) −12.3496 −0.492412
\(630\) 0 0
\(631\) −14.4667 −0.575910 −0.287955 0.957644i \(-0.592975\pi\)
−0.287955 + 0.957644i \(0.592975\pi\)
\(632\) −11.7143 −0.465968
\(633\) 4.60698 0.183111
\(634\) −0.968580 −0.0384672
\(635\) 0 0
\(636\) −0.778620 −0.0308743
\(637\) −1.00000 −0.0396214
\(638\) 11.4714 0.454156
\(639\) −14.2569 −0.563994
\(640\) 0 0
\(641\) 35.7069 1.41034 0.705169 0.709039i \(-0.250871\pi\)
0.705169 + 0.709039i \(0.250871\pi\)
\(642\) 12.4416 0.491032
\(643\) 26.4091 1.04147 0.520737 0.853717i \(-0.325657\pi\)
0.520737 + 0.853717i \(0.325657\pi\)
\(644\) −2.85819 −0.112629
\(645\) 0 0
\(646\) −6.84252 −0.269215
\(647\) −37.3543 −1.46855 −0.734275 0.678852i \(-0.762478\pi\)
−0.734275 + 0.678852i \(0.762478\pi\)
\(648\) −2.38147 −0.0935531
\(649\) 13.0925 0.513926
\(650\) 0 0
\(651\) 10.2857 0.403127
\(652\) −15.6619 −0.613366
\(653\) 0.987881 0.0386588 0.0193294 0.999813i \(-0.493847\pi\)
0.0193294 + 0.999813i \(0.493847\pi\)
\(654\) 0.953441 0.0372825
\(655\) 0 0
\(656\) −12.0903 −0.472047
\(657\) 12.0189 0.468900
\(658\) −2.61069 −0.101775
\(659\) −1.24653 −0.0485578 −0.0242789 0.999705i \(-0.507729\pi\)
−0.0242789 + 0.999705i \(0.507729\pi\)
\(660\) 0 0
\(661\) −33.8994 −1.31853 −0.659266 0.751910i \(-0.729133\pi\)
−0.659266 + 0.751910i \(0.729133\pi\)
\(662\) −10.0595 −0.390973
\(663\) −1.34169 −0.0521068
\(664\) −2.87749 −0.111668
\(665\) 0 0
\(666\) 6.17482 0.239269
\(667\) 10.3308 0.400010
\(668\) −4.79906 −0.185681
\(669\) 8.60698 0.332765
\(670\) 0 0
\(671\) 2.08587 0.0805240
\(672\) 5.77078 0.222613
\(673\) −42.5808 −1.64137 −0.820684 0.571382i \(-0.806408\pi\)
−0.820684 + 0.571382i \(0.806408\pi\)
\(674\) 10.0033 0.385311
\(675\) 0 0
\(676\) −1.54997 −0.0596142
\(677\) 33.8629 1.30146 0.650728 0.759311i \(-0.274464\pi\)
0.650728 + 0.759311i \(0.274464\pi\)
\(678\) 3.75825 0.144335
\(679\) −7.18572 −0.275763
\(680\) 0 0
\(681\) −14.4892 −0.555229
\(682\) −21.0612 −0.806473
\(683\) 34.5163 1.32073 0.660364 0.750946i \(-0.270402\pi\)
0.660364 + 0.750946i \(0.270402\pi\)
\(684\) −11.7833 −0.450546
\(685\) 0 0
\(686\) −0.670843 −0.0256129
\(687\) 20.1999 0.770673
\(688\) −2.24999 −0.0857802
\(689\) 0.502345 0.0191378
\(690\) 0 0
\(691\) −7.90679 −0.300789 −0.150394 0.988626i \(-0.548054\pi\)
−0.150394 + 0.988626i \(0.548054\pi\)
\(692\) 4.26794 0.162243
\(693\) 3.05232 0.115948
\(694\) −4.74613 −0.180161
\(695\) 0 0
\(696\) −13.3417 −0.505715
\(697\) 10.7974 0.408980
\(698\) 14.1674 0.536244
\(699\) 7.28097 0.275391
\(700\) 0 0
\(701\) −5.19510 −0.196216 −0.0981081 0.995176i \(-0.531279\pi\)
−0.0981081 + 0.995176i \(0.531279\pi\)
\(702\) 0.670843 0.0253193
\(703\) 69.9758 2.63919
\(704\) −2.64513 −0.0996921
\(705\) 0 0
\(706\) 11.6487 0.438404
\(707\) 18.4416 0.693569
\(708\) −6.64839 −0.249862
\(709\) 33.1971 1.24674 0.623372 0.781925i \(-0.285762\pi\)
0.623372 + 0.781925i \(0.285762\pi\)
\(710\) 0 0
\(711\) 4.91891 0.184474
\(712\) −32.7561 −1.22759
\(713\) −18.9671 −0.710323
\(714\) −0.900061 −0.0336839
\(715\) 0 0
\(716\) 2.44465 0.0913610
\(717\) −27.6688 −1.03331
\(718\) 12.3114 0.459457
\(719\) 30.7261 1.14589 0.572944 0.819594i \(-0.305801\pi\)
0.572944 + 0.819594i \(0.305801\pi\)
\(720\) 0 0
\(721\) 1.89537 0.0705873
\(722\) 26.0252 0.968557
\(723\) 22.2187 0.826324
\(724\) −14.7503 −0.548190
\(725\) 0 0
\(726\) 1.12928 0.0419114
\(727\) −1.83783 −0.0681612 −0.0340806 0.999419i \(-0.510850\pi\)
−0.0340806 + 0.999419i \(0.510850\pi\)
\(728\) −2.38147 −0.0882632
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.00938 0.0743197
\(732\) −1.05921 −0.0391494
\(733\) 19.1857 0.708641 0.354320 0.935124i \(-0.384712\pi\)
0.354320 + 0.935124i \(0.384712\pi\)
\(734\) −5.63990 −0.208173
\(735\) 0 0
\(736\) −10.6415 −0.392251
\(737\) −23.4664 −0.864396
\(738\) −5.39869 −0.198729
\(739\) −21.4882 −0.790456 −0.395228 0.918583i \(-0.629334\pi\)
−0.395228 + 0.918583i \(0.629334\pi\)
\(740\) 0 0
\(741\) 7.60228 0.279277
\(742\) 0.336995 0.0123715
\(743\) −19.9430 −0.731637 −0.365819 0.930686i \(-0.619211\pi\)
−0.365819 + 0.930686i \(0.619211\pi\)
\(744\) 24.4950 0.898030
\(745\) 0 0
\(746\) −18.1748 −0.665427
\(747\) 1.20828 0.0442088
\(748\) −6.34751 −0.232088
\(749\) 18.5463 0.677665
\(750\) 0 0
\(751\) 38.4498 1.40305 0.701526 0.712644i \(-0.252502\pi\)
0.701526 + 0.712644i \(0.252502\pi\)
\(752\) 5.84661 0.213204
\(753\) −17.3092 −0.630782
\(754\) 3.75825 0.136868
\(755\) 0 0
\(756\) −1.54997 −0.0563719
\(757\) 16.2931 0.592182 0.296091 0.955160i \(-0.404317\pi\)
0.296091 + 0.955160i \(0.404317\pi\)
\(758\) −4.30451 −0.156347
\(759\) −5.62857 −0.204304
\(760\) 0 0
\(761\) 35.3380 1.28100 0.640500 0.767958i \(-0.278727\pi\)
0.640500 + 0.767958i \(0.278727\pi\)
\(762\) 9.46199 0.342772
\(763\) 1.42126 0.0514530
\(764\) 35.4180 1.28138
\(765\) 0 0
\(766\) −21.8654 −0.790028
\(767\) 4.28937 0.154880
\(768\) 9.08578 0.327855
\(769\) −11.5428 −0.416244 −0.208122 0.978103i \(-0.566735\pi\)
−0.208122 + 0.978103i \(0.566735\pi\)
\(770\) 0 0
\(771\) −5.55837 −0.200180
\(772\) 30.3422 1.09204
\(773\) 5.89786 0.212131 0.106066 0.994359i \(-0.466175\pi\)
0.106066 + 0.994359i \(0.466175\pi\)
\(774\) −1.00469 −0.0361129
\(775\) 0 0
\(776\) −17.1126 −0.614306
\(777\) 9.20457 0.330212
\(778\) 4.02506 0.144305
\(779\) −61.1803 −2.19201
\(780\) 0 0
\(781\) 43.5165 1.55714
\(782\) 1.65974 0.0593522
\(783\) 5.60228 0.200209
\(784\) 1.50235 0.0536552
\(785\) 0 0
\(786\) 5.89537 0.210281
\(787\) −6.33079 −0.225668 −0.112834 0.993614i \(-0.535993\pi\)
−0.112834 + 0.993614i \(0.535993\pi\)
\(788\) −32.9014 −1.17206
\(789\) 25.0486 0.891754
\(790\) 0 0
\(791\) 5.60228 0.199194
\(792\) 7.26900 0.258293
\(793\) 0.683372 0.0242672
\(794\) −10.5672 −0.375014
\(795\) 0 0
\(796\) 10.3590 0.367166
\(797\) 22.0721 0.781835 0.390918 0.920426i \(-0.372158\pi\)
0.390918 + 0.920426i \(0.372158\pi\)
\(798\) 5.09994 0.180536
\(799\) −5.22138 −0.184719
\(800\) 0 0
\(801\) 13.7545 0.485993
\(802\) 17.5672 0.620318
\(803\) −36.6853 −1.29460
\(804\) 11.9163 0.420254
\(805\) 0 0
\(806\) −6.90006 −0.243044
\(807\) 4.55368 0.160297
\(808\) 43.9182 1.54504
\(809\) 36.2019 1.27279 0.636396 0.771363i \(-0.280424\pi\)
0.636396 + 0.771363i \(0.280424\pi\)
\(810\) 0 0
\(811\) −52.4950 −1.84335 −0.921674 0.387964i \(-0.873178\pi\)
−0.921674 + 0.387964i \(0.873178\pi\)
\(812\) −8.68337 −0.304727
\(813\) −8.88325 −0.311549
\(814\) −18.8475 −0.660605
\(815\) 0 0
\(816\) 2.01568 0.0705628
\(817\) −11.3856 −0.398332
\(818\) −11.6630 −0.407787
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) −0.972743 −0.0339490 −0.0169745 0.999856i \(-0.505403\pi\)
−0.0169745 + 0.999856i \(0.505403\pi\)
\(822\) −1.32655 −0.0462687
\(823\) 29.5572 1.03030 0.515150 0.857100i \(-0.327736\pi\)
0.515150 + 0.857100i \(0.327736\pi\)
\(824\) 4.51377 0.157245
\(825\) 0 0
\(826\) 2.87749 0.100121
\(827\) −15.3191 −0.532698 −0.266349 0.963877i \(-0.585817\pi\)
−0.266349 + 0.963877i \(0.585817\pi\)
\(828\) 2.85819 0.0993291
\(829\) 27.0236 0.938570 0.469285 0.883047i \(-0.344512\pi\)
0.469285 + 0.883047i \(0.344512\pi\)
\(830\) 0 0
\(831\) −5.38560 −0.186824
\(832\) −0.866598 −0.0300439
\(833\) −1.34169 −0.0464867
\(834\) −8.78800 −0.304304
\(835\) 0 0
\(836\) 35.9664 1.24392
\(837\) −10.2857 −0.355524
\(838\) −8.36948 −0.289119
\(839\) 46.2200 1.59569 0.797846 0.602862i \(-0.205973\pi\)
0.797846 + 0.602862i \(0.205973\pi\)
\(840\) 0 0
\(841\) 2.38560 0.0822619
\(842\) 6.42268 0.221340
\(843\) 12.9727 0.446805
\(844\) 7.14067 0.245792
\(845\) 0 0
\(846\) 2.61069 0.0897574
\(847\) 1.68337 0.0578413
\(848\) −0.754696 −0.0259164
\(849\) −1.52589 −0.0523684
\(850\) 0 0
\(851\) −16.9735 −0.581845
\(852\) −22.0977 −0.757056
\(853\) 15.1094 0.517336 0.258668 0.965966i \(-0.416716\pi\)
0.258668 + 0.965966i \(0.416716\pi\)
\(854\) 0.458435 0.0156873
\(855\) 0 0
\(856\) 44.1674 1.50961
\(857\) 0.191631 0.00654599 0.00327299 0.999995i \(-0.498958\pi\)
0.00327299 + 0.999995i \(0.498958\pi\)
\(858\) −2.04762 −0.0699047
\(859\) 7.06419 0.241027 0.120514 0.992712i \(-0.461546\pi\)
0.120514 + 0.992712i \(0.461546\pi\)
\(860\) 0 0
\(861\) −8.04762 −0.274262
\(862\) 3.48978 0.118862
\(863\) −41.6142 −1.41657 −0.708283 0.705929i \(-0.750530\pi\)
−0.708283 + 0.705929i \(0.750530\pi\)
\(864\) −5.77078 −0.196326
\(865\) 0 0
\(866\) 11.5752 0.393341
\(867\) 15.1999 0.516215
\(868\) 15.9425 0.541122
\(869\) −15.0141 −0.509318
\(870\) 0 0
\(871\) −7.68806 −0.260500
\(872\) 3.38469 0.114620
\(873\) 7.18572 0.243200
\(874\) −9.40445 −0.318110
\(875\) 0 0
\(876\) 18.6289 0.629411
\(877\) 52.5471 1.77439 0.887194 0.461396i \(-0.152651\pi\)
0.887194 + 0.461396i \(0.152651\pi\)
\(878\) −14.8876 −0.502434
\(879\) 18.8545 0.635946
\(880\) 0 0
\(881\) 0.934500 0.0314841 0.0157421 0.999876i \(-0.494989\pi\)
0.0157421 + 0.999876i \(0.494989\pi\)
\(882\) 0.670843 0.0225885
\(883\) 42.6448 1.43511 0.717555 0.696501i \(-0.245261\pi\)
0.717555 + 0.696501i \(0.245261\pi\)
\(884\) −2.07957 −0.0699436
\(885\) 0 0
\(886\) 25.4463 0.854886
\(887\) −24.8833 −0.835498 −0.417749 0.908563i \(-0.637181\pi\)
−0.417749 + 0.908563i \(0.637181\pi\)
\(888\) 21.9204 0.735601
\(889\) 14.1046 0.473054
\(890\) 0 0
\(891\) −3.05232 −0.102256
\(892\) 13.3406 0.446675
\(893\) 29.5855 0.990040
\(894\) −9.79020 −0.327433
\(895\) 0 0
\(896\) 10.9602 0.366155
\(897\) −1.84403 −0.0615704
\(898\) −22.3491 −0.745799
\(899\) −57.6232 −1.92184
\(900\) 0 0
\(901\) 0.673990 0.0224539
\(902\) 16.4785 0.548674
\(903\) −1.49765 −0.0498388
\(904\) 13.3417 0.443738
\(905\) 0 0
\(906\) 3.63052 0.120616
\(907\) −42.2207 −1.40191 −0.700957 0.713203i \(-0.747244\pi\)
−0.700957 + 0.713203i \(0.747244\pi\)
\(908\) −22.4579 −0.745292
\(909\) −18.4416 −0.611670
\(910\) 0 0
\(911\) 14.1108 0.467513 0.233756 0.972295i \(-0.424898\pi\)
0.233756 + 0.972295i \(0.424898\pi\)
\(912\) −11.4213 −0.378196
\(913\) −3.68806 −0.122057
\(914\) 5.22767 0.172916
\(915\) 0 0
\(916\) 31.3092 1.03449
\(917\) 8.78800 0.290205
\(918\) 0.900061 0.0297064
\(919\) −18.1547 −0.598870 −0.299435 0.954117i \(-0.596798\pi\)
−0.299435 + 0.954117i \(0.596798\pi\)
\(920\) 0 0
\(921\) 15.3856 0.506973
\(922\) 20.4317 0.672882
\(923\) 14.2569 0.469271
\(924\) 4.73100 0.155638
\(925\) 0 0
\(926\) −26.4741 −0.869993
\(927\) −1.89537 −0.0622521
\(928\) −32.3296 −1.06127
\(929\) −8.24741 −0.270589 −0.135294 0.990805i \(-0.543198\pi\)
−0.135294 + 0.990805i \(0.543198\pi\)
\(930\) 0 0
\(931\) 7.60228 0.249155
\(932\) 11.2853 0.369662
\(933\) 17.0999 0.559827
\(934\) 6.66353 0.218037
\(935\) 0 0
\(936\) 2.38147 0.0778409
\(937\) 21.6693 0.707905 0.353953 0.935263i \(-0.384837\pi\)
0.353953 + 0.935263i \(0.384837\pi\)
\(938\) −5.15748 −0.168398
\(939\) 4.89263 0.159665
\(940\) 0 0
\(941\) 42.5238 1.38624 0.693118 0.720824i \(-0.256237\pi\)
0.693118 + 0.720824i \(0.256237\pi\)
\(942\) 15.7007 0.511557
\(943\) 14.8401 0.483259
\(944\) −6.44412 −0.209738
\(945\) 0 0
\(946\) 3.06663 0.0997049
\(947\) 6.79792 0.220903 0.110451 0.993882i \(-0.464770\pi\)
0.110451 + 0.993882i \(0.464770\pi\)
\(948\) 7.62417 0.247621
\(949\) −12.0189 −0.390148
\(950\) 0 0
\(951\) 1.44382 0.0468192
\(952\) −3.19519 −0.103557
\(953\) −57.3782 −1.85866 −0.929331 0.369249i \(-0.879615\pi\)
−0.929331 + 0.369249i \(0.879615\pi\)
\(954\) −0.336995 −0.0109106
\(955\) 0 0
\(956\) −42.8857 −1.38702
\(957\) −17.0999 −0.552763
\(958\) −0.631585 −0.0204056
\(959\) −1.97743 −0.0638547
\(960\) 0 0
\(961\) 74.7947 2.41273
\(962\) −6.17482 −0.199084
\(963\) −18.5463 −0.597645
\(964\) 34.4384 1.10918
\(965\) 0 0
\(966\) −1.23706 −0.0398016
\(967\) −7.52393 −0.241953 −0.120977 0.992655i \(-0.538603\pi\)
−0.120977 + 0.992655i \(0.538603\pi\)
\(968\) 4.00890 0.128851
\(969\) 10.1999 0.327667
\(970\) 0 0
\(971\) −18.5807 −0.596283 −0.298141 0.954522i \(-0.596367\pi\)
−0.298141 + 0.954522i \(0.596367\pi\)
\(972\) 1.54997 0.0497153
\(973\) −13.0999 −0.419965
\(974\) 26.3976 0.845835
\(975\) 0 0
\(976\) −1.02666 −0.0328626
\(977\) 38.5939 1.23473 0.617364 0.786678i \(-0.288201\pi\)
0.617364 + 0.786678i \(0.288201\pi\)
\(978\) −6.77862 −0.216756
\(979\) −41.9832 −1.34179
\(980\) 0 0
\(981\) −1.42126 −0.0453773
\(982\) 3.65861 0.116751
\(983\) 0.591837 0.0188767 0.00943834 0.999955i \(-0.496996\pi\)
0.00943834 + 0.999955i \(0.496996\pi\)
\(984\) −19.1652 −0.610964
\(985\) 0 0
\(986\) 5.04240 0.160583
\(987\) 3.89166 0.123873
\(988\) 11.7833 0.374877
\(989\) 2.76172 0.0878177
\(990\) 0 0
\(991\) 9.41686 0.299136 0.149568 0.988751i \(-0.452212\pi\)
0.149568 + 0.988751i \(0.452212\pi\)
\(992\) 59.3563 1.88456
\(993\) 14.9953 0.475862
\(994\) 9.56413 0.303356
\(995\) 0 0
\(996\) 1.87280 0.0593420
\(997\) −22.1716 −0.702180 −0.351090 0.936342i \(-0.614189\pi\)
−0.351090 + 0.936342i \(0.614189\pi\)
\(998\) −21.0359 −0.665879
\(999\) −9.20457 −0.291220
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 6825.2.a.bg.1.3 4
5.4 even 2 273.2.a.e.1.2 4
15.14 odd 2 819.2.a.k.1.3 4
20.19 odd 2 4368.2.a.br.1.4 4
35.34 odd 2 1911.2.a.s.1.2 4
65.64 even 2 3549.2.a.w.1.3 4
105.104 even 2 5733.2.a.bf.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.a.e.1.2 4 5.4 even 2
819.2.a.k.1.3 4 15.14 odd 2
1911.2.a.s.1.2 4 35.34 odd 2
3549.2.a.w.1.3 4 65.64 even 2
4368.2.a.br.1.4 4 20.19 odd 2
5733.2.a.bf.1.3 4 105.104 even 2
6825.2.a.bg.1.3 4 1.1 even 1 trivial