Properties

Label 9025.2.a.cd.1.1
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9025,2,Mod(1,9025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.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: \(72.0649878242\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 12x^{7} - 4x^{6} + 48x^{5} + 27x^{4} - 72x^{3} - 51x^{2} + 27x + 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.02162\) of defining polynomial
Character \(\chi\) \(=\) 9025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.02162 q^{2} -1.90055 q^{3} +2.08694 q^{4} +3.84218 q^{6} -3.09218 q^{7} -0.175759 q^{8} +0.612075 q^{9} +O(q^{10})\) \(q-2.02162 q^{2} -1.90055 q^{3} +2.08694 q^{4} +3.84218 q^{6} -3.09218 q^{7} -0.175759 q^{8} +0.612075 q^{9} +0.963188 q^{11} -3.96633 q^{12} -2.95920 q^{13} +6.25120 q^{14} -3.81856 q^{16} -0.0389621 q^{17} -1.23738 q^{18} +5.87682 q^{21} -1.94720 q^{22} -3.88160 q^{23} +0.334039 q^{24} +5.98236 q^{26} +4.53836 q^{27} -6.45319 q^{28} +9.32783 q^{29} +9.37788 q^{31} +8.07119 q^{32} -1.83058 q^{33} +0.0787664 q^{34} +1.27736 q^{36} -1.11739 q^{37} +5.62409 q^{39} -12.0747 q^{41} -11.8807 q^{42} +4.95348 q^{43} +2.01012 q^{44} +7.84711 q^{46} -8.01557 q^{47} +7.25735 q^{48} +2.56155 q^{49} +0.0740492 q^{51} -6.17566 q^{52} +6.05981 q^{53} -9.17483 q^{54} +0.543479 q^{56} -18.8573 q^{58} -2.77465 q^{59} -10.0302 q^{61} -18.9585 q^{62} -1.89264 q^{63} -8.67975 q^{64} +3.70074 q^{66} +9.34881 q^{67} -0.0813115 q^{68} +7.37716 q^{69} -12.5894 q^{71} -0.107578 q^{72} -0.897116 q^{73} +2.25894 q^{74} -2.97835 q^{77} -11.3698 q^{78} -8.89241 q^{79} -10.4616 q^{81} +24.4104 q^{82} +9.60447 q^{83} +12.2646 q^{84} -10.0140 q^{86} -17.7280 q^{87} -0.169289 q^{88} +8.74675 q^{89} +9.15035 q^{91} -8.10067 q^{92} -17.8231 q^{93} +16.2044 q^{94} -15.3397 q^{96} +7.78824 q^{97} -5.17848 q^{98} +0.589543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{3} + 6 q^{4} - 12 q^{6} + 12 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 3 q^{3} + 6 q^{4} - 12 q^{6} + 12 q^{8} + 6 q^{9} - 6 q^{12} + 3 q^{13} + 12 q^{14} - 12 q^{16} + 9 q^{17} - 6 q^{18} + 12 q^{21} - 12 q^{22} + 15 q^{24} + 21 q^{26} - 6 q^{27} + 15 q^{28} + 15 q^{29} + 30 q^{31} + 9 q^{32} - 9 q^{33} - 6 q^{36} - 30 q^{37} + 6 q^{39} + 18 q^{41} - 36 q^{42} + 6 q^{43} - 24 q^{44} + 21 q^{46} - 21 q^{47} - 15 q^{48} + 3 q^{49} + 18 q^{51} + 3 q^{52} + 9 q^{53} - 9 q^{54} + 36 q^{56} - 18 q^{58} + 27 q^{59} + 12 q^{61} + 6 q^{62} + 15 q^{63} + 24 q^{64} + 3 q^{66} - 36 q^{67} - 3 q^{68} + 27 q^{69} - 6 q^{71} - 12 q^{72} + 9 q^{73} - 9 q^{74} - 12 q^{77} - 54 q^{78} + 45 q^{79} - 15 q^{81} + 48 q^{82} - 12 q^{84} - 9 q^{86} - 45 q^{87} - 39 q^{88} - 9 q^{89} + 51 q^{91} + 54 q^{92} - 9 q^{93} + 33 q^{94} - 9 q^{96} - 45 q^{97} - 33 q^{98} + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.02162 −1.42950 −0.714750 0.699380i \(-0.753460\pi\)
−0.714750 + 0.699380i \(0.753460\pi\)
\(3\) −1.90055 −1.09728 −0.548640 0.836058i \(-0.684854\pi\)
−0.548640 + 0.836058i \(0.684854\pi\)
\(4\) 2.08694 1.04347
\(5\) 0 0
\(6\) 3.84218 1.56856
\(7\) −3.09218 −1.16873 −0.584366 0.811490i \(-0.698657\pi\)
−0.584366 + 0.811490i \(0.698657\pi\)
\(8\) −0.175759 −0.0621403
\(9\) 0.612075 0.204025
\(10\) 0 0
\(11\) 0.963188 0.290412 0.145206 0.989401i \(-0.453615\pi\)
0.145206 + 0.989401i \(0.453615\pi\)
\(12\) −3.96633 −1.14498
\(13\) −2.95920 −0.820733 −0.410367 0.911921i \(-0.634599\pi\)
−0.410367 + 0.911921i \(0.634599\pi\)
\(14\) 6.25120 1.67070
\(15\) 0 0
\(16\) −3.81856 −0.954640
\(17\) −0.0389621 −0.00944969 −0.00472484 0.999989i \(-0.501504\pi\)
−0.00472484 + 0.999989i \(0.501504\pi\)
\(18\) −1.23738 −0.291654
\(19\) 0 0
\(20\) 0 0
\(21\) 5.87682 1.28243
\(22\) −1.94720 −0.415144
\(23\) −3.88160 −0.809370 −0.404685 0.914456i \(-0.632619\pi\)
−0.404685 + 0.914456i \(0.632619\pi\)
\(24\) 0.334039 0.0681853
\(25\) 0 0
\(26\) 5.98236 1.17324
\(27\) 4.53836 0.873408
\(28\) −6.45319 −1.21954
\(29\) 9.32783 1.73213 0.866067 0.499928i \(-0.166640\pi\)
0.866067 + 0.499928i \(0.166640\pi\)
\(30\) 0 0
\(31\) 9.37788 1.68432 0.842158 0.539231i \(-0.181285\pi\)
0.842158 + 0.539231i \(0.181285\pi\)
\(32\) 8.07119 1.42680
\(33\) −1.83058 −0.318664
\(34\) 0.0787664 0.0135083
\(35\) 0 0
\(36\) 1.27736 0.212894
\(37\) −1.11739 −0.183698 −0.0918491 0.995773i \(-0.529278\pi\)
−0.0918491 + 0.995773i \(0.529278\pi\)
\(38\) 0 0
\(39\) 5.62409 0.900575
\(40\) 0 0
\(41\) −12.0747 −1.88575 −0.942873 0.333152i \(-0.891888\pi\)
−0.942873 + 0.333152i \(0.891888\pi\)
\(42\) −11.8807 −1.83323
\(43\) 4.95348 0.755399 0.377699 0.925928i \(-0.376715\pi\)
0.377699 + 0.925928i \(0.376715\pi\)
\(44\) 2.01012 0.303036
\(45\) 0 0
\(46\) 7.84711 1.15699
\(47\) −8.01557 −1.16919 −0.584595 0.811325i \(-0.698747\pi\)
−0.584595 + 0.811325i \(0.698747\pi\)
\(48\) 7.25735 1.04751
\(49\) 2.56155 0.365936
\(50\) 0 0
\(51\) 0.0740492 0.0103690
\(52\) −6.17566 −0.856410
\(53\) 6.05981 0.832379 0.416189 0.909278i \(-0.363365\pi\)
0.416189 + 0.909278i \(0.363365\pi\)
\(54\) −9.17483 −1.24854
\(55\) 0 0
\(56\) 0.543479 0.0726254
\(57\) 0 0
\(58\) −18.8573 −2.47609
\(59\) −2.77465 −0.361229 −0.180615 0.983554i \(-0.557809\pi\)
−0.180615 + 0.983554i \(0.557809\pi\)
\(60\) 0 0
\(61\) −10.0302 −1.28424 −0.642119 0.766605i \(-0.721945\pi\)
−0.642119 + 0.766605i \(0.721945\pi\)
\(62\) −18.9585 −2.40773
\(63\) −1.89264 −0.238451
\(64\) −8.67975 −1.08497
\(65\) 0 0
\(66\) 3.70074 0.455530
\(67\) 9.34881 1.14214 0.571070 0.820902i \(-0.306529\pi\)
0.571070 + 0.820902i \(0.306529\pi\)
\(68\) −0.0813115 −0.00986047
\(69\) 7.37716 0.888106
\(70\) 0 0
\(71\) −12.5894 −1.49409 −0.747043 0.664775i \(-0.768527\pi\)
−0.747043 + 0.664775i \(0.768527\pi\)
\(72\) −0.107578 −0.0126782
\(73\) −0.897116 −0.104999 −0.0524997 0.998621i \(-0.516719\pi\)
−0.0524997 + 0.998621i \(0.516719\pi\)
\(74\) 2.25894 0.262597
\(75\) 0 0
\(76\) 0 0
\(77\) −2.97835 −0.339414
\(78\) −11.3698 −1.28737
\(79\) −8.89241 −1.00047 −0.500237 0.865888i \(-0.666754\pi\)
−0.500237 + 0.865888i \(0.666754\pi\)
\(80\) 0 0
\(81\) −10.4616 −1.16240
\(82\) 24.4104 2.69567
\(83\) 9.60447 1.05423 0.527114 0.849795i \(-0.323274\pi\)
0.527114 + 0.849795i \(0.323274\pi\)
\(84\) 12.2646 1.33817
\(85\) 0 0
\(86\) −10.0140 −1.07984
\(87\) −17.7280 −1.90064
\(88\) −0.169289 −0.0180463
\(89\) 8.74675 0.927153 0.463577 0.886057i \(-0.346566\pi\)
0.463577 + 0.886057i \(0.346566\pi\)
\(90\) 0 0
\(91\) 9.15035 0.959218
\(92\) −8.10067 −0.844553
\(93\) −17.8231 −1.84817
\(94\) 16.2044 1.67136
\(95\) 0 0
\(96\) −15.3397 −1.56560
\(97\) 7.78824 0.790776 0.395388 0.918514i \(-0.370610\pi\)
0.395388 + 0.918514i \(0.370610\pi\)
\(98\) −5.17848 −0.523105
\(99\) 0.589543 0.0592513
\(100\) 0 0
\(101\) −3.04363 −0.302852 −0.151426 0.988469i \(-0.548387\pi\)
−0.151426 + 0.988469i \(0.548387\pi\)
\(102\) −0.149699 −0.0148224
\(103\) −0.328573 −0.0323753 −0.0161876 0.999869i \(-0.505153\pi\)
−0.0161876 + 0.999869i \(0.505153\pi\)
\(104\) 0.520106 0.0510006
\(105\) 0 0
\(106\) −12.2506 −1.18989
\(107\) −12.7873 −1.23620 −0.618099 0.786101i \(-0.712097\pi\)
−0.618099 + 0.786101i \(0.712097\pi\)
\(108\) 9.47129 0.911375
\(109\) 15.2412 1.45984 0.729919 0.683534i \(-0.239558\pi\)
0.729919 + 0.683534i \(0.239558\pi\)
\(110\) 0 0
\(111\) 2.12366 0.201569
\(112\) 11.8077 1.11572
\(113\) 2.56380 0.241182 0.120591 0.992702i \(-0.461521\pi\)
0.120591 + 0.992702i \(0.461521\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 19.4666 1.80743
\(117\) −1.81125 −0.167450
\(118\) 5.60929 0.516377
\(119\) 0.120478 0.0110442
\(120\) 0 0
\(121\) −10.0723 −0.915661
\(122\) 20.2773 1.83582
\(123\) 22.9485 2.06919
\(124\) 19.5711 1.75753
\(125\) 0 0
\(126\) 3.82620 0.340865
\(127\) −15.4761 −1.37328 −0.686641 0.726997i \(-0.740916\pi\)
−0.686641 + 0.726997i \(0.740916\pi\)
\(128\) 1.40475 0.124163
\(129\) −9.41432 −0.828884
\(130\) 0 0
\(131\) 0.842204 0.0735837 0.0367919 0.999323i \(-0.488286\pi\)
0.0367919 + 0.999323i \(0.488286\pi\)
\(132\) −3.82032 −0.332516
\(133\) 0 0
\(134\) −18.8997 −1.63269
\(135\) 0 0
\(136\) 0.00684795 0.000587206 0
\(137\) 5.07379 0.433483 0.216742 0.976229i \(-0.430457\pi\)
0.216742 + 0.976229i \(0.430457\pi\)
\(138\) −14.9138 −1.26955
\(139\) 2.73039 0.231589 0.115794 0.993273i \(-0.463059\pi\)
0.115794 + 0.993273i \(0.463059\pi\)
\(140\) 0 0
\(141\) 15.2340 1.28293
\(142\) 25.4510 2.13580
\(143\) −2.85026 −0.238351
\(144\) −2.33725 −0.194770
\(145\) 0 0
\(146\) 1.81363 0.150097
\(147\) −4.86835 −0.401534
\(148\) −2.33193 −0.191684
\(149\) −6.64567 −0.544435 −0.272217 0.962236i \(-0.587757\pi\)
−0.272217 + 0.962236i \(0.587757\pi\)
\(150\) 0 0
\(151\) 5.19059 0.422404 0.211202 0.977442i \(-0.432262\pi\)
0.211202 + 0.977442i \(0.432262\pi\)
\(152\) 0 0
\(153\) −0.0238477 −0.00192797
\(154\) 6.02108 0.485192
\(155\) 0 0
\(156\) 11.7371 0.939723
\(157\) −11.6478 −0.929598 −0.464799 0.885416i \(-0.653873\pi\)
−0.464799 + 0.885416i \(0.653873\pi\)
\(158\) 17.9771 1.43018
\(159\) −11.5169 −0.913353
\(160\) 0 0
\(161\) 12.0026 0.945937
\(162\) 21.1493 1.66165
\(163\) 0.556796 0.0436116 0.0218058 0.999762i \(-0.493058\pi\)
0.0218058 + 0.999762i \(0.493058\pi\)
\(164\) −25.1991 −1.96772
\(165\) 0 0
\(166\) −19.4166 −1.50702
\(167\) 22.1563 1.71451 0.857254 0.514894i \(-0.172169\pi\)
0.857254 + 0.514894i \(0.172169\pi\)
\(168\) −1.03291 −0.0796904
\(169\) −4.24316 −0.326397
\(170\) 0 0
\(171\) 0 0
\(172\) 10.3376 0.788236
\(173\) −11.4561 −0.870994 −0.435497 0.900190i \(-0.643427\pi\)
−0.435497 + 0.900190i \(0.643427\pi\)
\(174\) 35.8392 2.71696
\(175\) 0 0
\(176\) −3.67799 −0.277239
\(177\) 5.27336 0.396370
\(178\) −17.6826 −1.32537
\(179\) −5.96612 −0.445928 −0.222964 0.974827i \(-0.571573\pi\)
−0.222964 + 0.974827i \(0.571573\pi\)
\(180\) 0 0
\(181\) 6.17784 0.459195 0.229597 0.973286i \(-0.426259\pi\)
0.229597 + 0.973286i \(0.426259\pi\)
\(182\) −18.4985 −1.37120
\(183\) 19.0629 1.40917
\(184\) 0.682227 0.0502945
\(185\) 0 0
\(186\) 36.0315 2.64196
\(187\) −0.0375278 −0.00274430
\(188\) −16.7280 −1.22002
\(189\) −14.0334 −1.02078
\(190\) 0 0
\(191\) −18.6639 −1.35047 −0.675235 0.737602i \(-0.735958\pi\)
−0.675235 + 0.737602i \(0.735958\pi\)
\(192\) 16.4963 1.19051
\(193\) −23.9067 −1.72084 −0.860421 0.509583i \(-0.829800\pi\)
−0.860421 + 0.509583i \(0.829800\pi\)
\(194\) −15.7449 −1.13041
\(195\) 0 0
\(196\) 5.34580 0.381843
\(197\) 6.70152 0.477463 0.238732 0.971086i \(-0.423268\pi\)
0.238732 + 0.971086i \(0.423268\pi\)
\(198\) −1.19183 −0.0846997
\(199\) 3.15886 0.223926 0.111963 0.993712i \(-0.464286\pi\)
0.111963 + 0.993712i \(0.464286\pi\)
\(200\) 0 0
\(201\) −17.7678 −1.25325
\(202\) 6.15306 0.432928
\(203\) −28.8433 −2.02440
\(204\) 0.154536 0.0108197
\(205\) 0 0
\(206\) 0.664250 0.0462805
\(207\) −2.37583 −0.165132
\(208\) 11.2999 0.783505
\(209\) 0 0
\(210\) 0 0
\(211\) 11.4580 0.788801 0.394401 0.918939i \(-0.370952\pi\)
0.394401 + 0.918939i \(0.370952\pi\)
\(212\) 12.6465 0.868562
\(213\) 23.9267 1.63943
\(214\) 25.8511 1.76714
\(215\) 0 0
\(216\) −0.797659 −0.0542738
\(217\) −28.9980 −1.96852
\(218\) −30.8118 −2.08684
\(219\) 1.70501 0.115214
\(220\) 0 0
\(221\) 0.115296 0.00775567
\(222\) −4.29322 −0.288142
\(223\) −7.51422 −0.503189 −0.251595 0.967833i \(-0.580955\pi\)
−0.251595 + 0.967833i \(0.580955\pi\)
\(224\) −24.9575 −1.66755
\(225\) 0 0
\(226\) −5.18302 −0.344769
\(227\) −4.20184 −0.278886 −0.139443 0.990230i \(-0.544531\pi\)
−0.139443 + 0.990230i \(0.544531\pi\)
\(228\) 0 0
\(229\) 22.5758 1.49185 0.745927 0.666028i \(-0.232007\pi\)
0.745927 + 0.666028i \(0.232007\pi\)
\(230\) 0 0
\(231\) 5.66048 0.372433
\(232\) −1.63945 −0.107635
\(233\) 10.9789 0.719251 0.359626 0.933097i \(-0.382904\pi\)
0.359626 + 0.933097i \(0.382904\pi\)
\(234\) 3.66165 0.239370
\(235\) 0 0
\(236\) −5.79054 −0.376932
\(237\) 16.9004 1.09780
\(238\) −0.243560 −0.0157876
\(239\) −2.72559 −0.176304 −0.0881520 0.996107i \(-0.528096\pi\)
−0.0881520 + 0.996107i \(0.528096\pi\)
\(240\) 0 0
\(241\) −11.1540 −0.718493 −0.359246 0.933243i \(-0.616966\pi\)
−0.359246 + 0.933243i \(0.616966\pi\)
\(242\) 20.3623 1.30894
\(243\) 6.26765 0.402070
\(244\) −20.9325 −1.34006
\(245\) 0 0
\(246\) −46.3930 −2.95791
\(247\) 0 0
\(248\) −1.64825 −0.104664
\(249\) −18.2537 −1.15678
\(250\) 0 0
\(251\) −27.3683 −1.72747 −0.863737 0.503944i \(-0.831882\pi\)
−0.863737 + 0.503944i \(0.831882\pi\)
\(252\) −3.94983 −0.248816
\(253\) −3.73871 −0.235051
\(254\) 31.2868 1.96311
\(255\) 0 0
\(256\) 14.5196 0.907477
\(257\) −20.2628 −1.26396 −0.631980 0.774985i \(-0.717757\pi\)
−0.631980 + 0.774985i \(0.717757\pi\)
\(258\) 19.0322 1.18489
\(259\) 3.45517 0.214694
\(260\) 0 0
\(261\) 5.70933 0.353399
\(262\) −1.70261 −0.105188
\(263\) −9.56235 −0.589640 −0.294820 0.955553i \(-0.595260\pi\)
−0.294820 + 0.955553i \(0.595260\pi\)
\(264\) 0.321742 0.0198019
\(265\) 0 0
\(266\) 0 0
\(267\) −16.6236 −1.01735
\(268\) 19.5104 1.19179
\(269\) −4.57455 −0.278915 −0.139458 0.990228i \(-0.544536\pi\)
−0.139458 + 0.990228i \(0.544536\pi\)
\(270\) 0 0
\(271\) −22.1359 −1.34466 −0.672330 0.740252i \(-0.734706\pi\)
−0.672330 + 0.740252i \(0.734706\pi\)
\(272\) 0.148779 0.00902105
\(273\) −17.3907 −1.05253
\(274\) −10.2573 −0.619665
\(275\) 0 0
\(276\) 15.3957 0.926712
\(277\) −23.5542 −1.41523 −0.707617 0.706597i \(-0.750230\pi\)
−0.707617 + 0.706597i \(0.750230\pi\)
\(278\) −5.51981 −0.331056
\(279\) 5.73996 0.343643
\(280\) 0 0
\(281\) −10.3229 −0.615815 −0.307907 0.951416i \(-0.599629\pi\)
−0.307907 + 0.951416i \(0.599629\pi\)
\(282\) −30.7972 −1.83395
\(283\) 30.8729 1.83520 0.917602 0.397501i \(-0.130123\pi\)
0.917602 + 0.397501i \(0.130123\pi\)
\(284\) −26.2733 −1.55903
\(285\) 0 0
\(286\) 5.76214 0.340722
\(287\) 37.3370 2.20393
\(288\) 4.94017 0.291103
\(289\) −16.9985 −0.999911
\(290\) 0 0
\(291\) −14.8019 −0.867703
\(292\) −1.87223 −0.109564
\(293\) 6.36103 0.371616 0.185808 0.982586i \(-0.440510\pi\)
0.185808 + 0.982586i \(0.440510\pi\)
\(294\) 9.84194 0.573993
\(295\) 0 0
\(296\) 0.196392 0.0114151
\(297\) 4.37130 0.253648
\(298\) 13.4350 0.778270
\(299\) 11.4864 0.664276
\(300\) 0 0
\(301\) −15.3170 −0.882859
\(302\) −10.4934 −0.603826
\(303\) 5.78456 0.332314
\(304\) 0 0
\(305\) 0 0
\(306\) 0.0482109 0.00275604
\(307\) −5.88645 −0.335958 −0.167979 0.985791i \(-0.553724\pi\)
−0.167979 + 0.985791i \(0.553724\pi\)
\(308\) −6.21563 −0.354168
\(309\) 0.624469 0.0355248
\(310\) 0 0
\(311\) −25.3885 −1.43965 −0.719825 0.694156i \(-0.755778\pi\)
−0.719825 + 0.694156i \(0.755778\pi\)
\(312\) −0.988486 −0.0559620
\(313\) −6.03850 −0.341316 −0.170658 0.985330i \(-0.554589\pi\)
−0.170658 + 0.985330i \(0.554589\pi\)
\(314\) 23.5475 1.32886
\(315\) 0 0
\(316\) −18.5579 −1.04396
\(317\) 15.8757 0.891670 0.445835 0.895115i \(-0.352907\pi\)
0.445835 + 0.895115i \(0.352907\pi\)
\(318\) 23.2829 1.30564
\(319\) 8.98445 0.503033
\(320\) 0 0
\(321\) 24.3029 1.35646
\(322\) −24.2647 −1.35222
\(323\) 0 0
\(324\) −21.8327 −1.21293
\(325\) 0 0
\(326\) −1.12563 −0.0623428
\(327\) −28.9665 −1.60185
\(328\) 2.12224 0.117181
\(329\) 24.7855 1.36647
\(330\) 0 0
\(331\) 14.1696 0.778829 0.389414 0.921063i \(-0.372677\pi\)
0.389414 + 0.921063i \(0.372677\pi\)
\(332\) 20.0440 1.10005
\(333\) −0.683928 −0.0374790
\(334\) −44.7916 −2.45089
\(335\) 0 0
\(336\) −22.4410 −1.22426
\(337\) −2.14993 −0.117114 −0.0585571 0.998284i \(-0.518650\pi\)
−0.0585571 + 0.998284i \(0.518650\pi\)
\(338\) 8.57806 0.466585
\(339\) −4.87261 −0.264644
\(340\) 0 0
\(341\) 9.03266 0.489146
\(342\) 0 0
\(343\) 13.7245 0.741051
\(344\) −0.870620 −0.0469407
\(345\) 0 0
\(346\) 23.1599 1.24509
\(347\) 11.0525 0.593329 0.296664 0.954982i \(-0.404126\pi\)
0.296664 + 0.954982i \(0.404126\pi\)
\(348\) −36.9972 −1.98326
\(349\) 35.3451 1.89198 0.945989 0.324199i \(-0.105095\pi\)
0.945989 + 0.324199i \(0.105095\pi\)
\(350\) 0 0
\(351\) −13.4299 −0.716835
\(352\) 7.77407 0.414360
\(353\) −22.1904 −1.18108 −0.590538 0.807010i \(-0.701084\pi\)
−0.590538 + 0.807010i \(0.701084\pi\)
\(354\) −10.6607 −0.566611
\(355\) 0 0
\(356\) 18.2539 0.967457
\(357\) −0.228973 −0.0121185
\(358\) 12.0612 0.637455
\(359\) −9.50861 −0.501845 −0.250923 0.968007i \(-0.580734\pi\)
−0.250923 + 0.968007i \(0.580734\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) −12.4892 −0.656419
\(363\) 19.1428 1.00474
\(364\) 19.0962 1.00091
\(365\) 0 0
\(366\) −38.5379 −2.01441
\(367\) 2.18321 0.113963 0.0569813 0.998375i \(-0.481852\pi\)
0.0569813 + 0.998375i \(0.481852\pi\)
\(368\) 14.8221 0.772657
\(369\) −7.39060 −0.384739
\(370\) 0 0
\(371\) −18.7380 −0.972828
\(372\) −37.1957 −1.92851
\(373\) −12.3407 −0.638977 −0.319488 0.947590i \(-0.603511\pi\)
−0.319488 + 0.947590i \(0.603511\pi\)
\(374\) 0.0758669 0.00392298
\(375\) 0 0
\(376\) 1.40881 0.0726538
\(377\) −27.6029 −1.42162
\(378\) 28.3702 1.45921
\(379\) 18.5014 0.950352 0.475176 0.879891i \(-0.342384\pi\)
0.475176 + 0.879891i \(0.342384\pi\)
\(380\) 0 0
\(381\) 29.4130 1.50688
\(382\) 37.7312 1.93050
\(383\) −11.4322 −0.584156 −0.292078 0.956394i \(-0.594347\pi\)
−0.292078 + 0.956394i \(0.594347\pi\)
\(384\) −2.66978 −0.136242
\(385\) 0 0
\(386\) 48.3302 2.45994
\(387\) 3.03190 0.154120
\(388\) 16.2536 0.825151
\(389\) 12.5237 0.634978 0.317489 0.948262i \(-0.397160\pi\)
0.317489 + 0.948262i \(0.397160\pi\)
\(390\) 0 0
\(391\) 0.151235 0.00764829
\(392\) −0.450216 −0.0227394
\(393\) −1.60065 −0.0807420
\(394\) −13.5479 −0.682534
\(395\) 0 0
\(396\) 1.23034 0.0618270
\(397\) −11.7504 −0.589734 −0.294867 0.955538i \(-0.595275\pi\)
−0.294867 + 0.955538i \(0.595275\pi\)
\(398\) −6.38602 −0.320102
\(399\) 0 0
\(400\) 0 0
\(401\) 14.6418 0.731178 0.365589 0.930776i \(-0.380868\pi\)
0.365589 + 0.930776i \(0.380868\pi\)
\(402\) 35.9198 1.79152
\(403\) −27.7510 −1.38237
\(404\) −6.35187 −0.316017
\(405\) 0 0
\(406\) 58.3101 2.89388
\(407\) −1.07626 −0.0533482
\(408\) −0.0130148 −0.000644330 0
\(409\) −12.7478 −0.630338 −0.315169 0.949035i \(-0.602061\pi\)
−0.315169 + 0.949035i \(0.602061\pi\)
\(410\) 0 0
\(411\) −9.64298 −0.475653
\(412\) −0.685713 −0.0337827
\(413\) 8.57972 0.422180
\(414\) 4.80302 0.236056
\(415\) 0 0
\(416\) −23.8842 −1.17102
\(417\) −5.18923 −0.254118
\(418\) 0 0
\(419\) 2.84601 0.139037 0.0695184 0.997581i \(-0.477854\pi\)
0.0695184 + 0.997581i \(0.477854\pi\)
\(420\) 0 0
\(421\) −11.1505 −0.543442 −0.271721 0.962376i \(-0.587593\pi\)
−0.271721 + 0.962376i \(0.587593\pi\)
\(422\) −23.1637 −1.12759
\(423\) −4.90613 −0.238544
\(424\) −1.06507 −0.0517242
\(425\) 0 0
\(426\) −48.3707 −2.34357
\(427\) 31.0152 1.50093
\(428\) −26.6864 −1.28993
\(429\) 5.41705 0.261538
\(430\) 0 0
\(431\) 14.9363 0.719455 0.359727 0.933057i \(-0.382870\pi\)
0.359727 + 0.933057i \(0.382870\pi\)
\(432\) −17.3300 −0.833791
\(433\) −38.7567 −1.86253 −0.931265 0.364343i \(-0.881294\pi\)
−0.931265 + 0.364343i \(0.881294\pi\)
\(434\) 58.6230 2.81399
\(435\) 0 0
\(436\) 31.8074 1.52330
\(437\) 0 0
\(438\) −3.44688 −0.164698
\(439\) 31.4285 1.50000 0.749999 0.661438i \(-0.230054\pi\)
0.749999 + 0.661438i \(0.230054\pi\)
\(440\) 0 0
\(441\) 1.56786 0.0746600
\(442\) −0.233085 −0.0110867
\(443\) −10.6508 −0.506036 −0.253018 0.967462i \(-0.581423\pi\)
−0.253018 + 0.967462i \(0.581423\pi\)
\(444\) 4.43194 0.210331
\(445\) 0 0
\(446\) 15.1909 0.719309
\(447\) 12.6304 0.597398
\(448\) 26.8393 1.26804
\(449\) −4.10400 −0.193680 −0.0968399 0.995300i \(-0.530873\pi\)
−0.0968399 + 0.995300i \(0.530873\pi\)
\(450\) 0 0
\(451\) −11.6302 −0.547644
\(452\) 5.35049 0.251666
\(453\) −9.86495 −0.463496
\(454\) 8.49452 0.398668
\(455\) 0 0
\(456\) 0 0
\(457\) −9.25947 −0.433140 −0.216570 0.976267i \(-0.569487\pi\)
−0.216570 + 0.976267i \(0.569487\pi\)
\(458\) −45.6397 −2.13260
\(459\) −0.176824 −0.00825343
\(460\) 0 0
\(461\) 6.64611 0.309540 0.154770 0.987951i \(-0.450536\pi\)
0.154770 + 0.987951i \(0.450536\pi\)
\(462\) −11.4433 −0.532392
\(463\) −22.2948 −1.03613 −0.518064 0.855342i \(-0.673347\pi\)
−0.518064 + 0.855342i \(0.673347\pi\)
\(464\) −35.6189 −1.65357
\(465\) 0 0
\(466\) −22.1951 −1.02817
\(467\) 15.4739 0.716048 0.358024 0.933712i \(-0.383451\pi\)
0.358024 + 0.933712i \(0.383451\pi\)
\(468\) −3.77997 −0.174729
\(469\) −28.9082 −1.33486
\(470\) 0 0
\(471\) 22.1372 1.02003
\(472\) 0.487671 0.0224469
\(473\) 4.77113 0.219377
\(474\) −34.1662 −1.56931
\(475\) 0 0
\(476\) 0.251429 0.0115242
\(477\) 3.70906 0.169826
\(478\) 5.51011 0.252026
\(479\) 32.2057 1.47152 0.735758 0.677245i \(-0.236826\pi\)
0.735758 + 0.677245i \(0.236826\pi\)
\(480\) 0 0
\(481\) 3.30658 0.150767
\(482\) 22.5491 1.02709
\(483\) −22.8115 −1.03796
\(484\) −21.0202 −0.955465
\(485\) 0 0
\(486\) −12.6708 −0.574759
\(487\) −3.65139 −0.165460 −0.0827301 0.996572i \(-0.526364\pi\)
−0.0827301 + 0.996572i \(0.526364\pi\)
\(488\) 1.76291 0.0798030
\(489\) −1.05822 −0.0478542
\(490\) 0 0
\(491\) 4.54862 0.205276 0.102638 0.994719i \(-0.467272\pi\)
0.102638 + 0.994719i \(0.467272\pi\)
\(492\) 47.8921 2.15914
\(493\) −0.363431 −0.0163681
\(494\) 0 0
\(495\) 0 0
\(496\) −35.8100 −1.60792
\(497\) 38.9286 1.74619
\(498\) 36.9021 1.65362
\(499\) −21.8234 −0.976950 −0.488475 0.872578i \(-0.662447\pi\)
−0.488475 + 0.872578i \(0.662447\pi\)
\(500\) 0 0
\(501\) −42.1091 −1.88130
\(502\) 55.3283 2.46942
\(503\) 24.7388 1.10305 0.551524 0.834159i \(-0.314046\pi\)
0.551524 + 0.834159i \(0.314046\pi\)
\(504\) 0.332650 0.0148174
\(505\) 0 0
\(506\) 7.55825 0.336005
\(507\) 8.06433 0.358149
\(508\) −32.2977 −1.43298
\(509\) 17.3491 0.768987 0.384494 0.923128i \(-0.374376\pi\)
0.384494 + 0.923128i \(0.374376\pi\)
\(510\) 0 0
\(511\) 2.77404 0.122716
\(512\) −32.1626 −1.42140
\(513\) 0 0
\(514\) 40.9637 1.80683
\(515\) 0 0
\(516\) −19.6471 −0.864916
\(517\) −7.72050 −0.339547
\(518\) −6.98504 −0.306905
\(519\) 21.7729 0.955725
\(520\) 0 0
\(521\) −0.456675 −0.0200073 −0.0100036 0.999950i \(-0.503184\pi\)
−0.0100036 + 0.999950i \(0.503184\pi\)
\(522\) −11.5421 −0.505183
\(523\) −37.7965 −1.65273 −0.826363 0.563138i \(-0.809594\pi\)
−0.826363 + 0.563138i \(0.809594\pi\)
\(524\) 1.75763 0.0767824
\(525\) 0 0
\(526\) 19.3314 0.842890
\(527\) −0.365381 −0.0159163
\(528\) 6.99019 0.304209
\(529\) −7.93318 −0.344921
\(530\) 0 0
\(531\) −1.69830 −0.0736998
\(532\) 0 0
\(533\) 35.7313 1.54769
\(534\) 33.6066 1.45430
\(535\) 0 0
\(536\) −1.64314 −0.0709729
\(537\) 11.3389 0.489309
\(538\) 9.24800 0.398710
\(539\) 2.46726 0.106272
\(540\) 0 0
\(541\) −21.1299 −0.908444 −0.454222 0.890889i \(-0.650083\pi\)
−0.454222 + 0.890889i \(0.650083\pi\)
\(542\) 44.7503 1.92219
\(543\) −11.7413 −0.503866
\(544\) −0.314470 −0.0134828
\(545\) 0 0
\(546\) 35.1573 1.50459
\(547\) −12.5424 −0.536276 −0.268138 0.963381i \(-0.586408\pi\)
−0.268138 + 0.963381i \(0.586408\pi\)
\(548\) 10.5887 0.452327
\(549\) −6.13925 −0.262017
\(550\) 0 0
\(551\) 0 0
\(552\) −1.29660 −0.0551872
\(553\) 27.4969 1.16929
\(554\) 47.6175 2.02308
\(555\) 0 0
\(556\) 5.69816 0.241656
\(557\) 32.0252 1.35695 0.678475 0.734623i \(-0.262641\pi\)
0.678475 + 0.734623i \(0.262641\pi\)
\(558\) −11.6040 −0.491237
\(559\) −14.6583 −0.619981
\(560\) 0 0
\(561\) 0.0713233 0.00301127
\(562\) 20.8690 0.880307
\(563\) 19.5807 0.825228 0.412614 0.910906i \(-0.364616\pi\)
0.412614 + 0.910906i \(0.364616\pi\)
\(564\) 31.7923 1.33870
\(565\) 0 0
\(566\) −62.4132 −2.62342
\(567\) 32.3491 1.35853
\(568\) 2.21270 0.0928430
\(569\) 18.6525 0.781953 0.390977 0.920401i \(-0.372137\pi\)
0.390977 + 0.920401i \(0.372137\pi\)
\(570\) 0 0
\(571\) −0.460181 −0.0192580 −0.00962899 0.999954i \(-0.503065\pi\)
−0.00962899 + 0.999954i \(0.503065\pi\)
\(572\) −5.94832 −0.248712
\(573\) 35.4716 1.48185
\(574\) −75.4812 −3.15052
\(575\) 0 0
\(576\) −5.31265 −0.221361
\(577\) −11.7478 −0.489069 −0.244534 0.969641i \(-0.578635\pi\)
−0.244534 + 0.969641i \(0.578635\pi\)
\(578\) 34.3644 1.42937
\(579\) 45.4358 1.88825
\(580\) 0 0
\(581\) −29.6987 −1.23211
\(582\) 29.9238 1.24038
\(583\) 5.83673 0.241733
\(584\) 0.157676 0.00652470
\(585\) 0 0
\(586\) −12.8596 −0.531224
\(587\) 4.90039 0.202261 0.101130 0.994873i \(-0.467754\pi\)
0.101130 + 0.994873i \(0.467754\pi\)
\(588\) −10.1599 −0.418989
\(589\) 0 0
\(590\) 0 0
\(591\) −12.7365 −0.523911
\(592\) 4.26683 0.175366
\(593\) 18.4976 0.759606 0.379803 0.925067i \(-0.375992\pi\)
0.379803 + 0.925067i \(0.375992\pi\)
\(594\) −8.83709 −0.362590
\(595\) 0 0
\(596\) −13.8691 −0.568101
\(597\) −6.00357 −0.245710
\(598\) −23.2211 −0.949583
\(599\) 21.4258 0.875433 0.437717 0.899113i \(-0.355787\pi\)
0.437717 + 0.899113i \(0.355787\pi\)
\(600\) 0 0
\(601\) −30.0390 −1.22532 −0.612659 0.790348i \(-0.709900\pi\)
−0.612659 + 0.790348i \(0.709900\pi\)
\(602\) 30.9652 1.26205
\(603\) 5.72217 0.233025
\(604\) 10.8324 0.440766
\(605\) 0 0
\(606\) −11.6942 −0.475043
\(607\) 29.3882 1.19283 0.596415 0.802676i \(-0.296591\pi\)
0.596415 + 0.802676i \(0.296591\pi\)
\(608\) 0 0
\(609\) 54.8180 2.22134
\(610\) 0 0
\(611\) 23.7196 0.959593
\(612\) −0.0497687 −0.00201178
\(613\) 5.51121 0.222596 0.111298 0.993787i \(-0.464499\pi\)
0.111298 + 0.993787i \(0.464499\pi\)
\(614\) 11.9002 0.480251
\(615\) 0 0
\(616\) 0.523472 0.0210913
\(617\) −45.9066 −1.84813 −0.924065 0.382234i \(-0.875155\pi\)
−0.924065 + 0.382234i \(0.875155\pi\)
\(618\) −1.26244 −0.0507827
\(619\) −0.478605 −0.0192368 −0.00961839 0.999954i \(-0.503062\pi\)
−0.00961839 + 0.999954i \(0.503062\pi\)
\(620\) 0 0
\(621\) −17.6161 −0.706910
\(622\) 51.3258 2.05798
\(623\) −27.0465 −1.08359
\(624\) −21.4759 −0.859725
\(625\) 0 0
\(626\) 12.2075 0.487912
\(627\) 0 0
\(628\) −24.3083 −0.970008
\(629\) 0.0435359 0.00173589
\(630\) 0 0
\(631\) 16.8542 0.670954 0.335477 0.942048i \(-0.391103\pi\)
0.335477 + 0.942048i \(0.391103\pi\)
\(632\) 1.56292 0.0621698
\(633\) −21.7765 −0.865536
\(634\) −32.0947 −1.27464
\(635\) 0 0
\(636\) −24.0352 −0.953056
\(637\) −7.58013 −0.300336
\(638\) −18.1631 −0.719085
\(639\) −7.70566 −0.304831
\(640\) 0 0
\(641\) −22.9012 −0.904542 −0.452271 0.891880i \(-0.649386\pi\)
−0.452271 + 0.891880i \(0.649386\pi\)
\(642\) −49.1312 −1.93905
\(643\) 8.58168 0.338429 0.169214 0.985579i \(-0.445877\pi\)
0.169214 + 0.985579i \(0.445877\pi\)
\(644\) 25.0487 0.987057
\(645\) 0 0
\(646\) 0 0
\(647\) 27.2647 1.07188 0.535942 0.844255i \(-0.319957\pi\)
0.535942 + 0.844255i \(0.319957\pi\)
\(648\) 1.83872 0.0722318
\(649\) −2.67251 −0.104905
\(650\) 0 0
\(651\) 55.1121 2.16001
\(652\) 1.16200 0.0455074
\(653\) 26.2598 1.02763 0.513813 0.857902i \(-0.328233\pi\)
0.513813 + 0.857902i \(0.328233\pi\)
\(654\) 58.5592 2.28985
\(655\) 0 0
\(656\) 46.1079 1.80021
\(657\) −0.549102 −0.0214225
\(658\) −50.1069 −1.95337
\(659\) 23.0115 0.896399 0.448200 0.893934i \(-0.352065\pi\)
0.448200 + 0.893934i \(0.352065\pi\)
\(660\) 0 0
\(661\) −30.5998 −1.19019 −0.595097 0.803654i \(-0.702886\pi\)
−0.595097 + 0.803654i \(0.702886\pi\)
\(662\) −28.6454 −1.11334
\(663\) −0.219126 −0.00851015
\(664\) −1.68807 −0.0655100
\(665\) 0 0
\(666\) 1.38264 0.0535763
\(667\) −36.2069 −1.40194
\(668\) 46.2389 1.78904
\(669\) 14.2811 0.552140
\(670\) 0 0
\(671\) −9.66099 −0.372959
\(672\) 47.4330 1.82977
\(673\) −37.8667 −1.45966 −0.729828 0.683631i \(-0.760400\pi\)
−0.729828 + 0.683631i \(0.760400\pi\)
\(674\) 4.34634 0.167415
\(675\) 0 0
\(676\) −8.85523 −0.340586
\(677\) −43.6943 −1.67931 −0.839654 0.543122i \(-0.817242\pi\)
−0.839654 + 0.543122i \(0.817242\pi\)
\(678\) 9.85056 0.378309
\(679\) −24.0826 −0.924206
\(680\) 0 0
\(681\) 7.98579 0.306016
\(682\) −18.2606 −0.699234
\(683\) 15.0878 0.577320 0.288660 0.957432i \(-0.406790\pi\)
0.288660 + 0.957432i \(0.406790\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −27.7456 −1.05933
\(687\) −42.9064 −1.63698
\(688\) −18.9152 −0.721134
\(689\) −17.9322 −0.683161
\(690\) 0 0
\(691\) 10.7207 0.407834 0.203917 0.978988i \(-0.434633\pi\)
0.203917 + 0.978988i \(0.434633\pi\)
\(692\) −23.9083 −0.908856
\(693\) −1.82297 −0.0692489
\(694\) −22.3439 −0.848164
\(695\) 0 0
\(696\) 3.11585 0.118106
\(697\) 0.470454 0.0178197
\(698\) −71.4542 −2.70458
\(699\) −20.8659 −0.789220
\(700\) 0 0
\(701\) 24.5411 0.926904 0.463452 0.886122i \(-0.346611\pi\)
0.463452 + 0.886122i \(0.346611\pi\)
\(702\) 27.1501 1.02472
\(703\) 0 0
\(704\) −8.36023 −0.315088
\(705\) 0 0
\(706\) 44.8605 1.68835
\(707\) 9.41144 0.353954
\(708\) 11.0052 0.413600
\(709\) −42.8046 −1.60756 −0.803781 0.594925i \(-0.797182\pi\)
−0.803781 + 0.594925i \(0.797182\pi\)
\(710\) 0 0
\(711\) −5.44282 −0.204122
\(712\) −1.53732 −0.0576136
\(713\) −36.4012 −1.36323
\(714\) 0.462896 0.0173235
\(715\) 0 0
\(716\) −12.4509 −0.465313
\(717\) 5.18011 0.193455
\(718\) 19.2228 0.717388
\(719\) 21.2712 0.793283 0.396641 0.917974i \(-0.370176\pi\)
0.396641 + 0.917974i \(0.370176\pi\)
\(720\) 0 0
\(721\) 1.01601 0.0378381
\(722\) 0 0
\(723\) 21.1987 0.788388
\(724\) 12.8928 0.479156
\(725\) 0 0
\(726\) −38.6995 −1.43627
\(727\) 10.5343 0.390696 0.195348 0.980734i \(-0.437416\pi\)
0.195348 + 0.980734i \(0.437416\pi\)
\(728\) −1.60826 −0.0596061
\(729\) 19.4728 0.721215
\(730\) 0 0
\(731\) −0.192998 −0.00713828
\(732\) 39.7831 1.47043
\(733\) 25.0892 0.926692 0.463346 0.886178i \(-0.346649\pi\)
0.463346 + 0.886178i \(0.346649\pi\)
\(734\) −4.41362 −0.162910
\(735\) 0 0
\(736\) −31.3291 −1.15481
\(737\) 9.00466 0.331691
\(738\) 14.9410 0.549985
\(739\) 28.4679 1.04721 0.523604 0.851962i \(-0.324587\pi\)
0.523604 + 0.851962i \(0.324587\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 37.8811 1.39066
\(743\) −10.1611 −0.372774 −0.186387 0.982476i \(-0.559678\pi\)
−0.186387 + 0.982476i \(0.559678\pi\)
\(744\) 3.13257 0.114846
\(745\) 0 0
\(746\) 24.9482 0.913417
\(747\) 5.87865 0.215089
\(748\) −0.0783182 −0.00286360
\(749\) 39.5407 1.44478
\(750\) 0 0
\(751\) 25.6900 0.937440 0.468720 0.883347i \(-0.344715\pi\)
0.468720 + 0.883347i \(0.344715\pi\)
\(752\) 30.6079 1.11616
\(753\) 52.0148 1.89552
\(754\) 55.8024 2.03221
\(755\) 0 0
\(756\) −29.2869 −1.06515
\(757\) 31.5781 1.14773 0.573863 0.818952i \(-0.305444\pi\)
0.573863 + 0.818952i \(0.305444\pi\)
\(758\) −37.4027 −1.35853
\(759\) 7.10559 0.257917
\(760\) 0 0
\(761\) −3.42183 −0.124041 −0.0620207 0.998075i \(-0.519754\pi\)
−0.0620207 + 0.998075i \(0.519754\pi\)
\(762\) −59.4619 −2.15408
\(763\) −47.1283 −1.70616
\(764\) −38.9504 −1.40918
\(765\) 0 0
\(766\) 23.1115 0.835052
\(767\) 8.21074 0.296473
\(768\) −27.5952 −0.995757
\(769\) −28.5086 −1.02805 −0.514024 0.857776i \(-0.671846\pi\)
−0.514024 + 0.857776i \(0.671846\pi\)
\(770\) 0 0
\(771\) 38.5104 1.38692
\(772\) −49.8919 −1.79565
\(773\) 1.42585 0.0512844 0.0256422 0.999671i \(-0.491837\pi\)
0.0256422 + 0.999671i \(0.491837\pi\)
\(774\) −6.12934 −0.220315
\(775\) 0 0
\(776\) −1.36886 −0.0491391
\(777\) −6.56672 −0.235580
\(778\) −25.3182 −0.907701
\(779\) 0 0
\(780\) 0 0
\(781\) −12.1260 −0.433901
\(782\) −0.305740 −0.0109332
\(783\) 42.3331 1.51286
\(784\) −9.78144 −0.349337
\(785\) 0 0
\(786\) 3.23590 0.115421
\(787\) 24.3199 0.866912 0.433456 0.901175i \(-0.357294\pi\)
0.433456 + 0.901175i \(0.357294\pi\)
\(788\) 13.9857 0.498219
\(789\) 18.1737 0.647001
\(790\) 0 0
\(791\) −7.92771 −0.281877
\(792\) −0.103618 −0.00368189
\(793\) 29.6814 1.05402
\(794\) 23.7547 0.843024
\(795\) 0 0
\(796\) 6.59236 0.233660
\(797\) 33.5714 1.18916 0.594580 0.804037i \(-0.297319\pi\)
0.594580 + 0.804037i \(0.297319\pi\)
\(798\) 0 0
\(799\) 0.312303 0.0110485
\(800\) 0 0
\(801\) 5.35366 0.189162
\(802\) −29.6002 −1.04522
\(803\) −0.864091 −0.0304931
\(804\) −37.0804 −1.30773
\(805\) 0 0
\(806\) 56.1019 1.97610
\(807\) 8.69415 0.306049
\(808\) 0.534946 0.0188193
\(809\) 18.2407 0.641311 0.320655 0.947196i \(-0.396097\pi\)
0.320655 + 0.947196i \(0.396097\pi\)
\(810\) 0 0
\(811\) 36.5439 1.28323 0.641615 0.767027i \(-0.278265\pi\)
0.641615 + 0.767027i \(0.278265\pi\)
\(812\) −60.1942 −2.11240
\(813\) 42.0703 1.47547
\(814\) 2.17578 0.0762612
\(815\) 0 0
\(816\) −0.282761 −0.00989863
\(817\) 0 0
\(818\) 25.7712 0.901069
\(819\) 5.60070 0.195704
\(820\) 0 0
\(821\) 10.6240 0.370781 0.185390 0.982665i \(-0.440645\pi\)
0.185390 + 0.982665i \(0.440645\pi\)
\(822\) 19.4944 0.679946
\(823\) 37.0620 1.29190 0.645950 0.763380i \(-0.276462\pi\)
0.645950 + 0.763380i \(0.276462\pi\)
\(824\) 0.0577498 0.00201181
\(825\) 0 0
\(826\) −17.3449 −0.603507
\(827\) 27.8197 0.967385 0.483693 0.875238i \(-0.339295\pi\)
0.483693 + 0.875238i \(0.339295\pi\)
\(828\) −4.95821 −0.172310
\(829\) 13.8198 0.479982 0.239991 0.970775i \(-0.422855\pi\)
0.239991 + 0.970775i \(0.422855\pi\)
\(830\) 0 0
\(831\) 44.7658 1.55291
\(832\) 25.6851 0.890469
\(833\) −0.0998033 −0.00345798
\(834\) 10.4906 0.363261
\(835\) 0 0
\(836\) 0 0
\(837\) 42.5602 1.47110
\(838\) −5.75355 −0.198753
\(839\) 6.42771 0.221909 0.110955 0.993825i \(-0.464609\pi\)
0.110955 + 0.993825i \(0.464609\pi\)
\(840\) 0 0
\(841\) 58.0084 2.00029
\(842\) 22.5421 0.776850
\(843\) 19.6192 0.675721
\(844\) 23.9122 0.823090
\(845\) 0 0
\(846\) 9.91831 0.340999
\(847\) 31.1452 1.07016
\(848\) −23.1397 −0.794622
\(849\) −58.6754 −2.01373
\(850\) 0 0
\(851\) 4.33727 0.148680
\(852\) 49.9337 1.71070
\(853\) −26.3952 −0.903753 −0.451876 0.892081i \(-0.649245\pi\)
−0.451876 + 0.892081i \(0.649245\pi\)
\(854\) −62.7009 −2.14558
\(855\) 0 0
\(856\) 2.24749 0.0768177
\(857\) 17.0307 0.581757 0.290878 0.956760i \(-0.406053\pi\)
0.290878 + 0.956760i \(0.406053\pi\)
\(858\) −10.9512 −0.373868
\(859\) 24.7212 0.843476 0.421738 0.906718i \(-0.361420\pi\)
0.421738 + 0.906718i \(0.361420\pi\)
\(860\) 0 0
\(861\) −70.9607 −2.41833
\(862\) −30.1954 −1.02846
\(863\) −1.03890 −0.0353646 −0.0176823 0.999844i \(-0.505629\pi\)
−0.0176823 + 0.999844i \(0.505629\pi\)
\(864\) 36.6300 1.24618
\(865\) 0 0
\(866\) 78.3513 2.66249
\(867\) 32.3064 1.09718
\(868\) −60.5172 −2.05409
\(869\) −8.56506 −0.290550
\(870\) 0 0
\(871\) −27.6650 −0.937391
\(872\) −2.67877 −0.0907147
\(873\) 4.76699 0.161338
\(874\) 0 0
\(875\) 0 0
\(876\) 3.55825 0.120222
\(877\) 21.2022 0.715948 0.357974 0.933732i \(-0.383468\pi\)
0.357974 + 0.933732i \(0.383468\pi\)
\(878\) −63.5364 −2.14425
\(879\) −12.0894 −0.407767
\(880\) 0 0
\(881\) 15.6819 0.528338 0.264169 0.964476i \(-0.414902\pi\)
0.264169 + 0.964476i \(0.414902\pi\)
\(882\) −3.16962 −0.106727
\(883\) −19.6406 −0.660960 −0.330480 0.943813i \(-0.607211\pi\)
−0.330480 + 0.943813i \(0.607211\pi\)
\(884\) 0.240617 0.00809281
\(885\) 0 0
\(886\) 21.5319 0.723378
\(887\) 24.7805 0.832049 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(888\) −0.373252 −0.0125255
\(889\) 47.8548 1.60500
\(890\) 0 0
\(891\) −10.0765 −0.337575
\(892\) −15.6817 −0.525063
\(893\) 0 0
\(894\) −25.5339 −0.853980
\(895\) 0 0
\(896\) −4.34372 −0.145114
\(897\) −21.8305 −0.728898
\(898\) 8.29672 0.276865
\(899\) 87.4752 2.91746
\(900\) 0 0
\(901\) −0.236103 −0.00786572
\(902\) 23.5118 0.782856
\(903\) 29.1107 0.968744
\(904\) −0.450611 −0.0149871
\(905\) 0 0
\(906\) 19.9432 0.662567
\(907\) −21.3228 −0.708010 −0.354005 0.935243i \(-0.615181\pi\)
−0.354005 + 0.935243i \(0.615181\pi\)
\(908\) −8.76899 −0.291009
\(909\) −1.86293 −0.0617895
\(910\) 0 0
\(911\) 34.0175 1.12705 0.563524 0.826100i \(-0.309445\pi\)
0.563524 + 0.826100i \(0.309445\pi\)
\(912\) 0 0
\(913\) 9.25091 0.306160
\(914\) 18.7191 0.619174
\(915\) 0 0
\(916\) 47.1144 1.55670
\(917\) −2.60424 −0.0859997
\(918\) 0.357470 0.0117983
\(919\) −41.7419 −1.37694 −0.688470 0.725265i \(-0.741717\pi\)
−0.688470 + 0.725265i \(0.741717\pi\)
\(920\) 0 0
\(921\) 11.1875 0.368640
\(922\) −13.4359 −0.442487
\(923\) 37.2545 1.22625
\(924\) 11.8131 0.388622
\(925\) 0 0
\(926\) 45.0716 1.48114
\(927\) −0.201112 −0.00660537
\(928\) 75.2867 2.47141
\(929\) −22.9048 −0.751481 −0.375741 0.926725i \(-0.622612\pi\)
−0.375741 + 0.926725i \(0.622612\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 22.9123 0.750517
\(933\) 48.2520 1.57970
\(934\) −31.2824 −1.02359
\(935\) 0 0
\(936\) 0.318344 0.0104054
\(937\) −11.6675 −0.381160 −0.190580 0.981672i \(-0.561037\pi\)
−0.190580 + 0.981672i \(0.561037\pi\)
\(938\) 58.4413 1.90818
\(939\) 11.4765 0.374520
\(940\) 0 0
\(941\) −55.4401 −1.80730 −0.903648 0.428276i \(-0.859121\pi\)
−0.903648 + 0.428276i \(0.859121\pi\)
\(942\) −44.7530 −1.45813
\(943\) 46.8690 1.52627
\(944\) 10.5952 0.344844
\(945\) 0 0
\(946\) −9.64541 −0.313599
\(947\) 9.30476 0.302364 0.151182 0.988506i \(-0.451692\pi\)
0.151182 + 0.988506i \(0.451692\pi\)
\(948\) 35.2702 1.14552
\(949\) 2.65474 0.0861765
\(950\) 0 0
\(951\) −30.1726 −0.978413
\(952\) −0.0211751 −0.000686287 0
\(953\) −17.2460 −0.558652 −0.279326 0.960196i \(-0.590111\pi\)
−0.279326 + 0.960196i \(0.590111\pi\)
\(954\) −7.49829 −0.242766
\(955\) 0 0
\(956\) −5.68815 −0.183968
\(957\) −17.0754 −0.551968
\(958\) −65.1076 −2.10353
\(959\) −15.6891 −0.506626
\(960\) 0 0
\(961\) 56.9445 1.83692
\(962\) −6.68465 −0.215522
\(963\) −7.82680 −0.252215
\(964\) −23.2777 −0.749726
\(965\) 0 0
\(966\) 46.1161 1.48376
\(967\) −3.68636 −0.118545 −0.0592727 0.998242i \(-0.518878\pi\)
−0.0592727 + 0.998242i \(0.518878\pi\)
\(968\) 1.77029 0.0568994
\(969\) 0 0
\(970\) 0 0
\(971\) 22.7457 0.729944 0.364972 0.931019i \(-0.381079\pi\)
0.364972 + 0.931019i \(0.381079\pi\)
\(972\) 13.0802 0.419548
\(973\) −8.44285 −0.270665
\(974\) 7.38171 0.236525
\(975\) 0 0
\(976\) 38.3010 1.22599
\(977\) −45.0177 −1.44024 −0.720122 0.693848i \(-0.755914\pi\)
−0.720122 + 0.693848i \(0.755914\pi\)
\(978\) 2.13931 0.0684075
\(979\) 8.42476 0.269257
\(980\) 0 0
\(981\) 9.32873 0.297843
\(982\) −9.19557 −0.293442
\(983\) 42.3156 1.34966 0.674830 0.737974i \(-0.264217\pi\)
0.674830 + 0.737974i \(0.264217\pi\)
\(984\) −4.03341 −0.128580
\(985\) 0 0
\(986\) 0.734720 0.0233982
\(987\) −47.1061 −1.49940
\(988\) 0 0
\(989\) −19.2274 −0.611397
\(990\) 0 0
\(991\) −31.1589 −0.989796 −0.494898 0.868951i \(-0.664795\pi\)
−0.494898 + 0.868951i \(0.664795\pi\)
\(992\) 75.6906 2.40318
\(993\) −26.9299 −0.854594
\(994\) −78.6988 −2.49618
\(995\) 0 0
\(996\) −38.0945 −1.20707
\(997\) −38.8905 −1.23167 −0.615837 0.787873i \(-0.711182\pi\)
−0.615837 + 0.787873i \(0.711182\pi\)
\(998\) 44.1186 1.39655
\(999\) −5.07113 −0.160444
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 9025.2.a.cd.1.1 9
5.4 even 2 1805.2.a.u.1.9 9
19.2 odd 18 475.2.l.b.251.1 18
19.10 odd 18 475.2.l.b.176.1 18
19.18 odd 2 9025.2.a.ce.1.9 9
95.2 even 36 475.2.u.c.99.6 36
95.29 odd 18 95.2.k.b.81.3 yes 18
95.48 even 36 475.2.u.c.24.6 36
95.59 odd 18 95.2.k.b.61.3 18
95.67 even 36 475.2.u.c.24.1 36
95.78 even 36 475.2.u.c.99.1 36
95.94 odd 2 1805.2.a.t.1.1 9
285.29 even 18 855.2.bs.b.271.1 18
285.59 even 18 855.2.bs.b.631.1 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.k.b.61.3 18 95.59 odd 18
95.2.k.b.81.3 yes 18 95.29 odd 18
475.2.l.b.176.1 18 19.10 odd 18
475.2.l.b.251.1 18 19.2 odd 18
475.2.u.c.24.1 36 95.67 even 36
475.2.u.c.24.6 36 95.48 even 36
475.2.u.c.99.1 36 95.78 even 36
475.2.u.c.99.6 36 95.2 even 36
855.2.bs.b.271.1 18 285.29 even 18
855.2.bs.b.631.1 18 285.59 even 18
1805.2.a.t.1.1 9 95.94 odd 2
1805.2.a.u.1.9 9 5.4 even 2
9025.2.a.cd.1.1 9 1.1 even 1 trivial
9025.2.a.ce.1.9 9 19.18 odd 2