Properties

Label 6422.2.a.bl.1.6
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $1$
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] = [6422,2,Mod(1,6422)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6422, 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("6422.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.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: \(51.2799281781\)
Analytic rank: \(1\)
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: \( x^{9} - x^{8} - 12x^{7} + 14x^{6} + 35x^{5} - 35x^{4} - 28x^{3} + 10x^{2} + 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-2.97314\) of defining polynomial
Character \(\chi\) \(=\) 6422.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.04618 q^{3} +1.00000 q^{4} +0.929245 q^{5} +1.04618 q^{6} -2.89421 q^{7} +1.00000 q^{8} -1.90551 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.04618 q^{3} +1.00000 q^{4} +0.929245 q^{5} +1.04618 q^{6} -2.89421 q^{7} +1.00000 q^{8} -1.90551 q^{9} +0.929245 q^{10} +3.97504 q^{11} +1.04618 q^{12} -2.89421 q^{14} +0.972156 q^{15} +1.00000 q^{16} -4.54056 q^{17} -1.90551 q^{18} -1.00000 q^{19} +0.929245 q^{20} -3.02786 q^{21} +3.97504 q^{22} -4.09689 q^{23} +1.04618 q^{24} -4.13650 q^{25} -5.13204 q^{27} -2.89421 q^{28} +1.64963 q^{29} +0.972156 q^{30} -7.79245 q^{31} +1.00000 q^{32} +4.15860 q^{33} -4.54056 q^{34} -2.68943 q^{35} -1.90551 q^{36} +2.85188 q^{37} -1.00000 q^{38} +0.929245 q^{40} -0.453243 q^{41} -3.02786 q^{42} +4.16877 q^{43} +3.97504 q^{44} -1.77069 q^{45} -4.09689 q^{46} -9.08685 q^{47} +1.04618 q^{48} +1.37645 q^{49} -4.13650 q^{50} -4.75023 q^{51} -3.63605 q^{53} -5.13204 q^{54} +3.69378 q^{55} -2.89421 q^{56} -1.04618 q^{57} +1.64963 q^{58} +9.02326 q^{59} +0.972156 q^{60} +4.87261 q^{61} -7.79245 q^{62} +5.51495 q^{63} +1.00000 q^{64} +4.15860 q^{66} -14.2461 q^{67} -4.54056 q^{68} -4.28608 q^{69} -2.68943 q^{70} -10.8788 q^{71} -1.90551 q^{72} -11.6900 q^{73} +2.85188 q^{74} -4.32752 q^{75} -1.00000 q^{76} -11.5046 q^{77} +9.60591 q^{79} +0.929245 q^{80} +0.347501 q^{81} -0.453243 q^{82} -4.33877 q^{83} -3.02786 q^{84} -4.21929 q^{85} +4.16877 q^{86} +1.72581 q^{87} +3.97504 q^{88} +6.23045 q^{89} -1.77069 q^{90} -4.09689 q^{92} -8.15229 q^{93} -9.08685 q^{94} -0.929245 q^{95} +1.04618 q^{96} -11.5845 q^{97} +1.37645 q^{98} -7.57447 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9} + q^{10} - 3 q^{11} + q^{12} - 13 q^{14} - 3 q^{15} + 9 q^{16} - 8 q^{17} - 2 q^{18} - 9 q^{19} + q^{20} - 24 q^{21} - 3 q^{22} - 10 q^{23} + q^{24} + 8 q^{25} + 10 q^{27} - 13 q^{28} - 20 q^{29} - 3 q^{30} - q^{31} + 9 q^{32} + 2 q^{33} - 8 q^{34} + 4 q^{35} - 2 q^{36} - 15 q^{37} - 9 q^{38} + q^{40} - 19 q^{41} - 24 q^{42} - 16 q^{43} - 3 q^{44} - 15 q^{45} - 10 q^{46} - 18 q^{47} + q^{48} + 18 q^{49} + 8 q^{50} - 11 q^{51} + 17 q^{53} + 10 q^{54} - 26 q^{55} - 13 q^{56} - q^{57} - 20 q^{58} - 24 q^{59} - 3 q^{60} + 6 q^{61} - q^{62} - q^{63} + 9 q^{64} + 2 q^{66} - 29 q^{67} - 8 q^{68} - 12 q^{69} + 4 q^{70} - 23 q^{71} - 2 q^{72} - 38 q^{73} - 15 q^{74} + 11 q^{75} - 9 q^{76} - 40 q^{77} - 20 q^{79} + q^{80} - 31 q^{81} - 19 q^{82} - 20 q^{83} - 24 q^{84} - 39 q^{85} - 16 q^{86} - 10 q^{87} - 3 q^{88} + 7 q^{89} - 15 q^{90} - 10 q^{92} - 11 q^{93} - 18 q^{94} - q^{95} + q^{96} - 28 q^{97} + 18 q^{98} - 25 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.04618 0.604011 0.302006 0.953306i \(-0.402344\pi\)
0.302006 + 0.953306i \(0.402344\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.929245 0.415571 0.207785 0.978174i \(-0.433374\pi\)
0.207785 + 0.978174i \(0.433374\pi\)
\(6\) 1.04618 0.427101
\(7\) −2.89421 −1.09391 −0.546954 0.837162i \(-0.684213\pi\)
−0.546954 + 0.837162i \(0.684213\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.90551 −0.635170
\(10\) 0.929245 0.293853
\(11\) 3.97504 1.19852 0.599259 0.800555i \(-0.295462\pi\)
0.599259 + 0.800555i \(0.295462\pi\)
\(12\) 1.04618 0.302006
\(13\) 0 0
\(14\) −2.89421 −0.773510
\(15\) 0.972156 0.251010
\(16\) 1.00000 0.250000
\(17\) −4.54056 −1.10125 −0.550623 0.834754i \(-0.685610\pi\)
−0.550623 + 0.834754i \(0.685610\pi\)
\(18\) −1.90551 −0.449133
\(19\) −1.00000 −0.229416
\(20\) 0.929245 0.207785
\(21\) −3.02786 −0.660733
\(22\) 3.97504 0.847480
\(23\) −4.09689 −0.854262 −0.427131 0.904190i \(-0.640476\pi\)
−0.427131 + 0.904190i \(0.640476\pi\)
\(24\) 1.04618 0.213550
\(25\) −4.13650 −0.827301
\(26\) 0 0
\(27\) −5.13204 −0.987662
\(28\) −2.89421 −0.546954
\(29\) 1.64963 0.306329 0.153164 0.988201i \(-0.451054\pi\)
0.153164 + 0.988201i \(0.451054\pi\)
\(30\) 0.972156 0.177491
\(31\) −7.79245 −1.39957 −0.699783 0.714356i \(-0.746720\pi\)
−0.699783 + 0.714356i \(0.746720\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.15860 0.723919
\(34\) −4.54056 −0.778699
\(35\) −2.68943 −0.454597
\(36\) −1.90551 −0.317585
\(37\) 2.85188 0.468847 0.234423 0.972135i \(-0.424680\pi\)
0.234423 + 0.972135i \(0.424680\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0.929245 0.146926
\(41\) −0.453243 −0.0707847 −0.0353923 0.999373i \(-0.511268\pi\)
−0.0353923 + 0.999373i \(0.511268\pi\)
\(42\) −3.02786 −0.467209
\(43\) 4.16877 0.635732 0.317866 0.948136i \(-0.397034\pi\)
0.317866 + 0.948136i \(0.397034\pi\)
\(44\) 3.97504 0.599259
\(45\) −1.77069 −0.263958
\(46\) −4.09689 −0.604054
\(47\) −9.08685 −1.32545 −0.662727 0.748862i \(-0.730601\pi\)
−0.662727 + 0.748862i \(0.730601\pi\)
\(48\) 1.04618 0.151003
\(49\) 1.37645 0.196636
\(50\) −4.13650 −0.584990
\(51\) −4.75023 −0.665166
\(52\) 0 0
\(53\) −3.63605 −0.499450 −0.249725 0.968317i \(-0.580340\pi\)
−0.249725 + 0.968317i \(0.580340\pi\)
\(54\) −5.13204 −0.698382
\(55\) 3.69378 0.498069
\(56\) −2.89421 −0.386755
\(57\) −1.04618 −0.138570
\(58\) 1.64963 0.216607
\(59\) 9.02326 1.17473 0.587364 0.809323i \(-0.300166\pi\)
0.587364 + 0.809323i \(0.300166\pi\)
\(60\) 0.972156 0.125505
\(61\) 4.87261 0.623874 0.311937 0.950103i \(-0.399022\pi\)
0.311937 + 0.950103i \(0.399022\pi\)
\(62\) −7.79245 −0.989642
\(63\) 5.51495 0.694818
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.15860 0.511888
\(67\) −14.2461 −1.74043 −0.870217 0.492669i \(-0.836021\pi\)
−0.870217 + 0.492669i \(0.836021\pi\)
\(68\) −4.54056 −0.550623
\(69\) −4.28608 −0.515984
\(70\) −2.68943 −0.321448
\(71\) −10.8788 −1.29107 −0.645535 0.763731i \(-0.723366\pi\)
−0.645535 + 0.763731i \(0.723366\pi\)
\(72\) −1.90551 −0.224567
\(73\) −11.6900 −1.36821 −0.684105 0.729384i \(-0.739807\pi\)
−0.684105 + 0.729384i \(0.739807\pi\)
\(74\) 2.85188 0.331525
\(75\) −4.32752 −0.499699
\(76\) −1.00000 −0.114708
\(77\) −11.5046 −1.31107
\(78\) 0 0
\(79\) 9.60591 1.08075 0.540375 0.841425i \(-0.318283\pi\)
0.540375 + 0.841425i \(0.318283\pi\)
\(80\) 0.929245 0.103893
\(81\) 0.347501 0.0386112
\(82\) −0.453243 −0.0500523
\(83\) −4.33877 −0.476242 −0.238121 0.971235i \(-0.576531\pi\)
−0.238121 + 0.971235i \(0.576531\pi\)
\(84\) −3.02786 −0.330367
\(85\) −4.21929 −0.457646
\(86\) 4.16877 0.449530
\(87\) 1.72581 0.185026
\(88\) 3.97504 0.423740
\(89\) 6.23045 0.660427 0.330213 0.943906i \(-0.392879\pi\)
0.330213 + 0.943906i \(0.392879\pi\)
\(90\) −1.77069 −0.186647
\(91\) 0 0
\(92\) −4.09689 −0.427131
\(93\) −8.15229 −0.845354
\(94\) −9.08685 −0.937237
\(95\) −0.929245 −0.0953385
\(96\) 1.04618 0.106775
\(97\) −11.5845 −1.17623 −0.588116 0.808777i \(-0.700130\pi\)
−0.588116 + 0.808777i \(0.700130\pi\)
\(98\) 1.37645 0.139043
\(99\) −7.57447 −0.761263
\(100\) −4.13650 −0.413650
\(101\) 9.68495 0.963689 0.481844 0.876257i \(-0.339967\pi\)
0.481844 + 0.876257i \(0.339967\pi\)
\(102\) −4.75023 −0.470343
\(103\) −0.0784942 −0.00773427 −0.00386713 0.999993i \(-0.501231\pi\)
−0.00386713 + 0.999993i \(0.501231\pi\)
\(104\) 0 0
\(105\) −2.81362 −0.274582
\(106\) −3.63605 −0.353165
\(107\) 1.97833 0.191253 0.0956264 0.995417i \(-0.469515\pi\)
0.0956264 + 0.995417i \(0.469515\pi\)
\(108\) −5.13204 −0.493831
\(109\) 17.0468 1.63279 0.816395 0.577494i \(-0.195969\pi\)
0.816395 + 0.577494i \(0.195969\pi\)
\(110\) 3.69378 0.352188
\(111\) 2.98358 0.283189
\(112\) −2.89421 −0.273477
\(113\) 7.23739 0.680836 0.340418 0.940274i \(-0.389431\pi\)
0.340418 + 0.940274i \(0.389431\pi\)
\(114\) −1.04618 −0.0979836
\(115\) −3.80702 −0.355006
\(116\) 1.64963 0.153164
\(117\) 0 0
\(118\) 9.02326 0.830659
\(119\) 13.1413 1.20466
\(120\) 0.972156 0.0887453
\(121\) 4.80091 0.436446
\(122\) 4.87261 0.441145
\(123\) −0.474173 −0.0427547
\(124\) −7.79245 −0.699783
\(125\) −8.49005 −0.759373
\(126\) 5.51495 0.491311
\(127\) −10.6484 −0.944893 −0.472447 0.881359i \(-0.656629\pi\)
−0.472447 + 0.881359i \(0.656629\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.36128 0.383989
\(130\) 0 0
\(131\) 17.6085 1.53846 0.769232 0.638970i \(-0.220639\pi\)
0.769232 + 0.638970i \(0.220639\pi\)
\(132\) 4.15860 0.361959
\(133\) 2.89421 0.250960
\(134\) −14.2461 −1.23067
\(135\) −4.76892 −0.410443
\(136\) −4.54056 −0.389349
\(137\) −10.1912 −0.870689 −0.435345 0.900264i \(-0.643374\pi\)
−0.435345 + 0.900264i \(0.643374\pi\)
\(138\) −4.28608 −0.364856
\(139\) 18.1509 1.53954 0.769771 0.638320i \(-0.220370\pi\)
0.769771 + 0.638320i \(0.220370\pi\)
\(140\) −2.68943 −0.227298
\(141\) −9.50647 −0.800589
\(142\) −10.8788 −0.912924
\(143\) 0 0
\(144\) −1.90551 −0.158793
\(145\) 1.53291 0.127301
\(146\) −11.6900 −0.967470
\(147\) 1.44002 0.118771
\(148\) 2.85188 0.234423
\(149\) −2.69811 −0.221038 −0.110519 0.993874i \(-0.535251\pi\)
−0.110519 + 0.993874i \(0.535251\pi\)
\(150\) −4.32752 −0.353341
\(151\) 5.24220 0.426604 0.213302 0.976986i \(-0.431578\pi\)
0.213302 + 0.976986i \(0.431578\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 8.65208 0.699479
\(154\) −11.5046 −0.927066
\(155\) −7.24109 −0.581619
\(156\) 0 0
\(157\) 2.20220 0.175755 0.0878774 0.996131i \(-0.471992\pi\)
0.0878774 + 0.996131i \(0.471992\pi\)
\(158\) 9.60591 0.764205
\(159\) −3.80396 −0.301674
\(160\) 0.929245 0.0734632
\(161\) 11.8573 0.934484
\(162\) 0.347501 0.0273022
\(163\) −21.2249 −1.66246 −0.831232 0.555926i \(-0.812364\pi\)
−0.831232 + 0.555926i \(0.812364\pi\)
\(164\) −0.453243 −0.0353923
\(165\) 3.86435 0.300840
\(166\) −4.33877 −0.336754
\(167\) −16.6096 −1.28529 −0.642645 0.766164i \(-0.722163\pi\)
−0.642645 + 0.766164i \(0.722163\pi\)
\(168\) −3.02786 −0.233605
\(169\) 0 0
\(170\) −4.21929 −0.323605
\(171\) 1.90551 0.145718
\(172\) 4.16877 0.317866
\(173\) −5.22082 −0.396932 −0.198466 0.980108i \(-0.563596\pi\)
−0.198466 + 0.980108i \(0.563596\pi\)
\(174\) 1.72581 0.130833
\(175\) 11.9719 0.904992
\(176\) 3.97504 0.299630
\(177\) 9.43994 0.709550
\(178\) 6.23045 0.466992
\(179\) −3.64551 −0.272478 −0.136239 0.990676i \(-0.543502\pi\)
−0.136239 + 0.990676i \(0.543502\pi\)
\(180\) −1.77069 −0.131979
\(181\) −14.4527 −1.07426 −0.537131 0.843499i \(-0.680492\pi\)
−0.537131 + 0.843499i \(0.680492\pi\)
\(182\) 0 0
\(183\) 5.09762 0.376827
\(184\) −4.09689 −0.302027
\(185\) 2.65010 0.194839
\(186\) −8.15229 −0.597755
\(187\) −18.0489 −1.31986
\(188\) −9.08685 −0.662727
\(189\) 14.8532 1.08041
\(190\) −0.929245 −0.0674145
\(191\) −7.34389 −0.531386 −0.265693 0.964058i \(-0.585601\pi\)
−0.265693 + 0.964058i \(0.585601\pi\)
\(192\) 1.04618 0.0755014
\(193\) 9.57969 0.689561 0.344781 0.938683i \(-0.387953\pi\)
0.344781 + 0.938683i \(0.387953\pi\)
\(194\) −11.5845 −0.831722
\(195\) 0 0
\(196\) 1.37645 0.0983182
\(197\) 1.25654 0.0895246 0.0447623 0.998998i \(-0.485747\pi\)
0.0447623 + 0.998998i \(0.485747\pi\)
\(198\) −7.57447 −0.538294
\(199\) 5.97439 0.423513 0.211757 0.977322i \(-0.432082\pi\)
0.211757 + 0.977322i \(0.432082\pi\)
\(200\) −4.13650 −0.292495
\(201\) −14.9039 −1.05124
\(202\) 9.68495 0.681431
\(203\) −4.77438 −0.335096
\(204\) −4.75023 −0.332583
\(205\) −0.421174 −0.0294160
\(206\) −0.0784942 −0.00546895
\(207\) 7.80668 0.542601
\(208\) 0 0
\(209\) −3.97504 −0.274959
\(210\) −2.81362 −0.194158
\(211\) −2.93090 −0.201772 −0.100886 0.994898i \(-0.532168\pi\)
−0.100886 + 0.994898i \(0.532168\pi\)
\(212\) −3.63605 −0.249725
\(213\) −11.3811 −0.779821
\(214\) 1.97833 0.135236
\(215\) 3.87381 0.264192
\(216\) −5.13204 −0.349191
\(217\) 22.5530 1.53100
\(218\) 17.0468 1.15456
\(219\) −12.2298 −0.826414
\(220\) 3.69378 0.249035
\(221\) 0 0
\(222\) 2.98358 0.200245
\(223\) 21.7936 1.45941 0.729705 0.683762i \(-0.239657\pi\)
0.729705 + 0.683762i \(0.239657\pi\)
\(224\) −2.89421 −0.193378
\(225\) 7.88215 0.525477
\(226\) 7.23739 0.481424
\(227\) −18.0721 −1.19949 −0.599743 0.800192i \(-0.704731\pi\)
−0.599743 + 0.800192i \(0.704731\pi\)
\(228\) −1.04618 −0.0692849
\(229\) −20.7571 −1.37167 −0.685833 0.727759i \(-0.740562\pi\)
−0.685833 + 0.727759i \(0.740562\pi\)
\(230\) −3.80702 −0.251027
\(231\) −12.0359 −0.791901
\(232\) 1.64963 0.108304
\(233\) −26.8846 −1.76127 −0.880636 0.473794i \(-0.842884\pi\)
−0.880636 + 0.473794i \(0.842884\pi\)
\(234\) 0 0
\(235\) −8.44390 −0.550820
\(236\) 9.02326 0.587364
\(237\) 10.0495 0.652785
\(238\) 13.1413 0.851826
\(239\) 12.0208 0.777559 0.388779 0.921331i \(-0.372897\pi\)
0.388779 + 0.921331i \(0.372897\pi\)
\(240\) 0.972156 0.0627524
\(241\) −16.9086 −1.08918 −0.544588 0.838704i \(-0.683314\pi\)
−0.544588 + 0.838704i \(0.683314\pi\)
\(242\) 4.80091 0.308614
\(243\) 15.7597 1.01098
\(244\) 4.87261 0.311937
\(245\) 1.27906 0.0817163
\(246\) −0.474173 −0.0302322
\(247\) 0 0
\(248\) −7.79245 −0.494821
\(249\) −4.53913 −0.287656
\(250\) −8.49005 −0.536958
\(251\) 24.5896 1.55208 0.776041 0.630682i \(-0.217225\pi\)
0.776041 + 0.630682i \(0.217225\pi\)
\(252\) 5.51495 0.347409
\(253\) −16.2853 −1.02385
\(254\) −10.6484 −0.668140
\(255\) −4.41413 −0.276423
\(256\) 1.00000 0.0625000
\(257\) −20.6988 −1.29115 −0.645577 0.763695i \(-0.723383\pi\)
−0.645577 + 0.763695i \(0.723383\pi\)
\(258\) 4.36128 0.271521
\(259\) −8.25395 −0.512876
\(260\) 0 0
\(261\) −3.14339 −0.194571
\(262\) 17.6085 1.08786
\(263\) −24.3236 −1.49986 −0.749930 0.661517i \(-0.769913\pi\)
−0.749930 + 0.661517i \(0.769913\pi\)
\(264\) 4.15860 0.255944
\(265\) −3.37878 −0.207557
\(266\) 2.89421 0.177455
\(267\) 6.51817 0.398905
\(268\) −14.2461 −0.870217
\(269\) −0.385169 −0.0234842 −0.0117421 0.999931i \(-0.503738\pi\)
−0.0117421 + 0.999931i \(0.503738\pi\)
\(270\) −4.76892 −0.290227
\(271\) 21.8256 1.32581 0.662906 0.748702i \(-0.269323\pi\)
0.662906 + 0.748702i \(0.269323\pi\)
\(272\) −4.54056 −0.275312
\(273\) 0 0
\(274\) −10.1912 −0.615670
\(275\) −16.4428 −0.991535
\(276\) −4.28608 −0.257992
\(277\) −12.9100 −0.775687 −0.387843 0.921725i \(-0.626780\pi\)
−0.387843 + 0.921725i \(0.626780\pi\)
\(278\) 18.1509 1.08862
\(279\) 14.8486 0.888962
\(280\) −2.68943 −0.160724
\(281\) −6.07209 −0.362230 −0.181115 0.983462i \(-0.557971\pi\)
−0.181115 + 0.983462i \(0.557971\pi\)
\(282\) −9.50647 −0.566102
\(283\) 12.3131 0.731940 0.365970 0.930627i \(-0.380737\pi\)
0.365970 + 0.930627i \(0.380737\pi\)
\(284\) −10.8788 −0.645535
\(285\) −0.972156 −0.0575855
\(286\) 0 0
\(287\) 1.31178 0.0774320
\(288\) −1.90551 −0.112283
\(289\) 3.61665 0.212744
\(290\) 1.53291 0.0900156
\(291\) −12.1195 −0.710458
\(292\) −11.6900 −0.684105
\(293\) −7.44576 −0.434986 −0.217493 0.976062i \(-0.569788\pi\)
−0.217493 + 0.976062i \(0.569788\pi\)
\(294\) 1.44002 0.0839835
\(295\) 8.38482 0.488183
\(296\) 2.85188 0.165762
\(297\) −20.4000 −1.18373
\(298\) −2.69811 −0.156297
\(299\) 0 0
\(300\) −4.32752 −0.249850
\(301\) −12.0653 −0.695433
\(302\) 5.24220 0.301655
\(303\) 10.1322 0.582079
\(304\) −1.00000 −0.0573539
\(305\) 4.52785 0.259264
\(306\) 8.65208 0.494606
\(307\) 18.8708 1.07701 0.538506 0.842622i \(-0.318989\pi\)
0.538506 + 0.842622i \(0.318989\pi\)
\(308\) −11.5046 −0.655535
\(309\) −0.0821190 −0.00467159
\(310\) −7.24109 −0.411266
\(311\) 5.01837 0.284566 0.142283 0.989826i \(-0.454556\pi\)
0.142283 + 0.989826i \(0.454556\pi\)
\(312\) 0 0
\(313\) −22.5284 −1.27338 −0.636689 0.771121i \(-0.719696\pi\)
−0.636689 + 0.771121i \(0.719696\pi\)
\(314\) 2.20220 0.124277
\(315\) 5.12474 0.288746
\(316\) 9.60591 0.540375
\(317\) 17.2007 0.966086 0.483043 0.875597i \(-0.339532\pi\)
0.483043 + 0.875597i \(0.339532\pi\)
\(318\) −3.80396 −0.213316
\(319\) 6.55734 0.367141
\(320\) 0.929245 0.0519464
\(321\) 2.06969 0.115519
\(322\) 11.8573 0.660780
\(323\) 4.54056 0.252643
\(324\) 0.347501 0.0193056
\(325\) 0 0
\(326\) −21.2249 −1.17554
\(327\) 17.8340 0.986224
\(328\) −0.453243 −0.0250262
\(329\) 26.2993 1.44992
\(330\) 3.86435 0.212726
\(331\) −7.57755 −0.416500 −0.208250 0.978076i \(-0.566777\pi\)
−0.208250 + 0.978076i \(0.566777\pi\)
\(332\) −4.33877 −0.238121
\(333\) −5.43429 −0.297797
\(334\) −16.6096 −0.908837
\(335\) −13.2381 −0.723273
\(336\) −3.02786 −0.165183
\(337\) 12.4953 0.680665 0.340333 0.940305i \(-0.389460\pi\)
0.340333 + 0.940305i \(0.389460\pi\)
\(338\) 0 0
\(339\) 7.57160 0.411233
\(340\) −4.21929 −0.228823
\(341\) −30.9753 −1.67740
\(342\) 1.90551 0.103038
\(343\) 16.2757 0.878807
\(344\) 4.16877 0.224765
\(345\) −3.98282 −0.214428
\(346\) −5.22082 −0.280673
\(347\) −9.64219 −0.517620 −0.258810 0.965928i \(-0.583330\pi\)
−0.258810 + 0.965928i \(0.583330\pi\)
\(348\) 1.72581 0.0925130
\(349\) 30.2434 1.61889 0.809446 0.587195i \(-0.199768\pi\)
0.809446 + 0.587195i \(0.199768\pi\)
\(350\) 11.9719 0.639926
\(351\) 0 0
\(352\) 3.97504 0.211870
\(353\) 11.1133 0.591500 0.295750 0.955265i \(-0.404431\pi\)
0.295750 + 0.955265i \(0.404431\pi\)
\(354\) 9.43994 0.501727
\(355\) −10.1090 −0.536531
\(356\) 6.23045 0.330213
\(357\) 13.7482 0.727631
\(358\) −3.64551 −0.192671
\(359\) −29.5541 −1.55981 −0.779904 0.625899i \(-0.784732\pi\)
−0.779904 + 0.625899i \(0.784732\pi\)
\(360\) −1.77069 −0.0933233
\(361\) 1.00000 0.0526316
\(362\) −14.4527 −0.759618
\(363\) 5.02261 0.263618
\(364\) 0 0
\(365\) −10.8629 −0.568588
\(366\) 5.09762 0.266457
\(367\) 30.4669 1.59036 0.795179 0.606375i \(-0.207377\pi\)
0.795179 + 0.606375i \(0.207377\pi\)
\(368\) −4.09689 −0.213565
\(369\) 0.863659 0.0449603
\(370\) 2.65010 0.137772
\(371\) 10.5235 0.546353
\(372\) −8.15229 −0.422677
\(373\) 28.9669 1.49985 0.749924 0.661524i \(-0.230090\pi\)
0.749924 + 0.661524i \(0.230090\pi\)
\(374\) −18.0489 −0.933285
\(375\) −8.88211 −0.458670
\(376\) −9.08685 −0.468618
\(377\) 0 0
\(378\) 14.8532 0.763966
\(379\) −32.9190 −1.69094 −0.845468 0.534026i \(-0.820679\pi\)
−0.845468 + 0.534026i \(0.820679\pi\)
\(380\) −0.929245 −0.0476692
\(381\) −11.1401 −0.570726
\(382\) −7.34389 −0.375746
\(383\) −17.1752 −0.877610 −0.438805 0.898582i \(-0.644598\pi\)
−0.438805 + 0.898582i \(0.644598\pi\)
\(384\) 1.04618 0.0533876
\(385\) −10.6906 −0.544842
\(386\) 9.57969 0.487593
\(387\) −7.94364 −0.403798
\(388\) −11.5845 −0.588116
\(389\) 1.10030 0.0557876 0.0278938 0.999611i \(-0.491120\pi\)
0.0278938 + 0.999611i \(0.491120\pi\)
\(390\) 0 0
\(391\) 18.6022 0.940753
\(392\) 1.37645 0.0695214
\(393\) 18.4217 0.929250
\(394\) 1.25654 0.0633035
\(395\) 8.92624 0.449128
\(396\) −7.57447 −0.380631
\(397\) 23.3055 1.16967 0.584836 0.811152i \(-0.301159\pi\)
0.584836 + 0.811152i \(0.301159\pi\)
\(398\) 5.97439 0.299469
\(399\) 3.02786 0.151583
\(400\) −4.13650 −0.206825
\(401\) 27.1581 1.35621 0.678106 0.734964i \(-0.262801\pi\)
0.678106 + 0.734964i \(0.262801\pi\)
\(402\) −14.9039 −0.743340
\(403\) 0 0
\(404\) 9.68495 0.481844
\(405\) 0.322913 0.0160457
\(406\) −4.77438 −0.236948
\(407\) 11.3363 0.561921
\(408\) −4.75023 −0.235172
\(409\) −4.47035 −0.221044 −0.110522 0.993874i \(-0.535252\pi\)
−0.110522 + 0.993874i \(0.535252\pi\)
\(410\) −0.421174 −0.0208003
\(411\) −10.6618 −0.525906
\(412\) −0.0784942 −0.00386713
\(413\) −26.1152 −1.28505
\(414\) 7.80668 0.383677
\(415\) −4.03178 −0.197912
\(416\) 0 0
\(417\) 18.9891 0.929901
\(418\) −3.97504 −0.194425
\(419\) 7.62391 0.372453 0.186226 0.982507i \(-0.440374\pi\)
0.186226 + 0.982507i \(0.440374\pi\)
\(420\) −2.81362 −0.137291
\(421\) −31.6349 −1.54179 −0.770896 0.636962i \(-0.780191\pi\)
−0.770896 + 0.636962i \(0.780191\pi\)
\(422\) −2.93090 −0.142674
\(423\) 17.3151 0.841888
\(424\) −3.63605 −0.176582
\(425\) 18.7820 0.911062
\(426\) −11.3811 −0.551417
\(427\) −14.1024 −0.682461
\(428\) 1.97833 0.0956264
\(429\) 0 0
\(430\) 3.87381 0.186812
\(431\) −5.99194 −0.288621 −0.144311 0.989532i \(-0.546096\pi\)
−0.144311 + 0.989532i \(0.546096\pi\)
\(432\) −5.13204 −0.246915
\(433\) 22.3035 1.07184 0.535919 0.844270i \(-0.319965\pi\)
0.535919 + 0.844270i \(0.319965\pi\)
\(434\) 22.5530 1.08258
\(435\) 1.60370 0.0768914
\(436\) 17.0468 0.816395
\(437\) 4.09689 0.195981
\(438\) −12.2298 −0.584363
\(439\) −8.13560 −0.388291 −0.194146 0.980973i \(-0.562193\pi\)
−0.194146 + 0.980973i \(0.562193\pi\)
\(440\) 3.69378 0.176094
\(441\) −2.62285 −0.124898
\(442\) 0 0
\(443\) 9.37320 0.445334 0.222667 0.974895i \(-0.428524\pi\)
0.222667 + 0.974895i \(0.428524\pi\)
\(444\) 2.98358 0.141594
\(445\) 5.78961 0.274454
\(446\) 21.7936 1.03196
\(447\) −2.82271 −0.133509
\(448\) −2.89421 −0.136739
\(449\) 23.5049 1.10926 0.554632 0.832096i \(-0.312859\pi\)
0.554632 + 0.832096i \(0.312859\pi\)
\(450\) 7.88215 0.371568
\(451\) −1.80166 −0.0848367
\(452\) 7.23739 0.340418
\(453\) 5.48428 0.257674
\(454\) −18.0721 −0.848165
\(455\) 0 0
\(456\) −1.04618 −0.0489918
\(457\) 7.02736 0.328726 0.164363 0.986400i \(-0.447443\pi\)
0.164363 + 0.986400i \(0.447443\pi\)
\(458\) −20.7571 −0.969915
\(459\) 23.3023 1.08766
\(460\) −3.80702 −0.177503
\(461\) 2.09583 0.0976127 0.0488064 0.998808i \(-0.484458\pi\)
0.0488064 + 0.998808i \(0.484458\pi\)
\(462\) −12.0359 −0.559959
\(463\) −22.3671 −1.03949 −0.519743 0.854323i \(-0.673972\pi\)
−0.519743 + 0.854323i \(0.673972\pi\)
\(464\) 1.64963 0.0765822
\(465\) −7.57548 −0.351304
\(466\) −26.8846 −1.24541
\(467\) 36.8013 1.70296 0.851480 0.524387i \(-0.175705\pi\)
0.851480 + 0.524387i \(0.175705\pi\)
\(468\) 0 0
\(469\) 41.2311 1.90388
\(470\) −8.44390 −0.389488
\(471\) 2.30390 0.106158
\(472\) 9.02326 0.415329
\(473\) 16.5710 0.761936
\(474\) 10.0495 0.461589
\(475\) 4.13650 0.189796
\(476\) 13.1413 0.602332
\(477\) 6.92854 0.317236
\(478\) 12.0208 0.549817
\(479\) −18.0660 −0.825459 −0.412729 0.910854i \(-0.635424\pi\)
−0.412729 + 0.910854i \(0.635424\pi\)
\(480\) 0.972156 0.0443726
\(481\) 0 0
\(482\) −16.9086 −0.770164
\(483\) 12.4048 0.564439
\(484\) 4.80091 0.218223
\(485\) −10.7649 −0.488808
\(486\) 15.7597 0.714873
\(487\) 19.7205 0.893622 0.446811 0.894628i \(-0.352560\pi\)
0.446811 + 0.894628i \(0.352560\pi\)
\(488\) 4.87261 0.220573
\(489\) −22.2050 −1.00415
\(490\) 1.27906 0.0577822
\(491\) −26.6062 −1.20072 −0.600360 0.799730i \(-0.704976\pi\)
−0.600360 + 0.799730i \(0.704976\pi\)
\(492\) −0.474173 −0.0213774
\(493\) −7.49024 −0.337343
\(494\) 0 0
\(495\) −7.03854 −0.316359
\(496\) −7.79245 −0.349891
\(497\) 31.4854 1.41231
\(498\) −4.53913 −0.203403
\(499\) −5.10306 −0.228444 −0.114222 0.993455i \(-0.536438\pi\)
−0.114222 + 0.993455i \(0.536438\pi\)
\(500\) −8.49005 −0.379686
\(501\) −17.3766 −0.776330
\(502\) 24.5896 1.09749
\(503\) 27.0694 1.20697 0.603483 0.797376i \(-0.293779\pi\)
0.603483 + 0.797376i \(0.293779\pi\)
\(504\) 5.51495 0.245655
\(505\) 8.99969 0.400481
\(506\) −16.2853 −0.723970
\(507\) 0 0
\(508\) −10.6484 −0.472447
\(509\) 9.16648 0.406297 0.203148 0.979148i \(-0.434883\pi\)
0.203148 + 0.979148i \(0.434883\pi\)
\(510\) −4.41413 −0.195461
\(511\) 33.8333 1.49670
\(512\) 1.00000 0.0441942
\(513\) 5.13204 0.226585
\(514\) −20.6988 −0.912984
\(515\) −0.0729404 −0.00321414
\(516\) 4.36128 0.191995
\(517\) −36.1205 −1.58858
\(518\) −8.25395 −0.362658
\(519\) −5.46191 −0.239751
\(520\) 0 0
\(521\) −30.1666 −1.32162 −0.660811 0.750553i \(-0.729787\pi\)
−0.660811 + 0.750553i \(0.729787\pi\)
\(522\) −3.14339 −0.137582
\(523\) −9.39066 −0.410625 −0.205312 0.978697i \(-0.565821\pi\)
−0.205312 + 0.978697i \(0.565821\pi\)
\(524\) 17.6085 0.769232
\(525\) 12.5248 0.546625
\(526\) −24.3236 −1.06056
\(527\) 35.3821 1.54127
\(528\) 4.15860 0.180980
\(529\) −6.21545 −0.270237
\(530\) −3.37878 −0.146765
\(531\) −17.1939 −0.746153
\(532\) 2.89421 0.125480
\(533\) 0 0
\(534\) 6.51817 0.282069
\(535\) 1.83836 0.0794791
\(536\) −14.2461 −0.615336
\(537\) −3.81385 −0.164580
\(538\) −0.385169 −0.0166058
\(539\) 5.47145 0.235672
\(540\) −4.76892 −0.205222
\(541\) 13.6802 0.588160 0.294080 0.955781i \(-0.404987\pi\)
0.294080 + 0.955781i \(0.404987\pi\)
\(542\) 21.8256 0.937491
\(543\) −15.1201 −0.648866
\(544\) −4.54056 −0.194675
\(545\) 15.8407 0.678540
\(546\) 0 0
\(547\) −6.13964 −0.262512 −0.131256 0.991349i \(-0.541901\pi\)
−0.131256 + 0.991349i \(0.541901\pi\)
\(548\) −10.1912 −0.435345
\(549\) −9.28481 −0.396266
\(550\) −16.4428 −0.701121
\(551\) −1.64963 −0.0702766
\(552\) −4.28608 −0.182428
\(553\) −27.8015 −1.18224
\(554\) −12.9100 −0.548494
\(555\) 2.77248 0.117685
\(556\) 18.1509 0.769771
\(557\) 12.9201 0.547444 0.273722 0.961809i \(-0.411745\pi\)
0.273722 + 0.961809i \(0.411745\pi\)
\(558\) 14.8486 0.628591
\(559\) 0 0
\(560\) −2.68943 −0.113649
\(561\) −18.8823 −0.797213
\(562\) −6.07209 −0.256136
\(563\) −39.3705 −1.65927 −0.829634 0.558307i \(-0.811451\pi\)
−0.829634 + 0.558307i \(0.811451\pi\)
\(564\) −9.50647 −0.400294
\(565\) 6.72530 0.282936
\(566\) 12.3131 0.517560
\(567\) −1.00574 −0.0422371
\(568\) −10.8788 −0.456462
\(569\) 27.7960 1.16527 0.582634 0.812734i \(-0.302022\pi\)
0.582634 + 0.812734i \(0.302022\pi\)
\(570\) −0.972156 −0.0407191
\(571\) −4.79328 −0.200592 −0.100296 0.994958i \(-0.531979\pi\)
−0.100296 + 0.994958i \(0.531979\pi\)
\(572\) 0 0
\(573\) −7.68303 −0.320963
\(574\) 1.31178 0.0547527
\(575\) 16.9468 0.706731
\(576\) −1.90551 −0.0793963
\(577\) 19.1405 0.796829 0.398415 0.917205i \(-0.369561\pi\)
0.398415 + 0.917205i \(0.369561\pi\)
\(578\) 3.61665 0.150433
\(579\) 10.0221 0.416503
\(580\) 1.53291 0.0636506
\(581\) 12.5573 0.520965
\(582\) −12.1195 −0.502369
\(583\) −14.4534 −0.598600
\(584\) −11.6900 −0.483735
\(585\) 0 0
\(586\) −7.44576 −0.307582
\(587\) −15.8721 −0.655110 −0.327555 0.944832i \(-0.606225\pi\)
−0.327555 + 0.944832i \(0.606225\pi\)
\(588\) 1.44002 0.0593853
\(589\) 7.79245 0.321082
\(590\) 8.38482 0.345198
\(591\) 1.31456 0.0540739
\(592\) 2.85188 0.117212
\(593\) 10.8277 0.444642 0.222321 0.974973i \(-0.428637\pi\)
0.222321 + 0.974973i \(0.428637\pi\)
\(594\) −20.4000 −0.837024
\(595\) 12.2115 0.500623
\(596\) −2.69811 −0.110519
\(597\) 6.25028 0.255807
\(598\) 0 0
\(599\) −0.831670 −0.0339811 −0.0169906 0.999856i \(-0.505409\pi\)
−0.0169906 + 0.999856i \(0.505409\pi\)
\(600\) −4.32752 −0.176670
\(601\) 35.6665 1.45487 0.727433 0.686179i \(-0.240713\pi\)
0.727433 + 0.686179i \(0.240713\pi\)
\(602\) −12.0653 −0.491745
\(603\) 27.1460 1.10547
\(604\) 5.24220 0.213302
\(605\) 4.46122 0.181374
\(606\) 10.1322 0.411592
\(607\) 16.8104 0.682311 0.341156 0.940007i \(-0.389182\pi\)
0.341156 + 0.940007i \(0.389182\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −4.99485 −0.202402
\(610\) 4.52785 0.183327
\(611\) 0 0
\(612\) 8.65208 0.349739
\(613\) 27.5552 1.11294 0.556471 0.830867i \(-0.312155\pi\)
0.556471 + 0.830867i \(0.312155\pi\)
\(614\) 18.8708 0.761563
\(615\) −0.440623 −0.0177676
\(616\) −11.5046 −0.463533
\(617\) −40.6943 −1.63829 −0.819145 0.573586i \(-0.805552\pi\)
−0.819145 + 0.573586i \(0.805552\pi\)
\(618\) −0.0821190 −0.00330331
\(619\) −1.84653 −0.0742183 −0.0371092 0.999311i \(-0.511815\pi\)
−0.0371092 + 0.999311i \(0.511815\pi\)
\(620\) −7.24109 −0.290809
\(621\) 21.0254 0.843721
\(622\) 5.01837 0.201218
\(623\) −18.0322 −0.722446
\(624\) 0 0
\(625\) 12.7932 0.511728
\(626\) −22.5284 −0.900414
\(627\) −4.15860 −0.166078
\(628\) 2.20220 0.0878774
\(629\) −12.9491 −0.516316
\(630\) 5.12474 0.204174
\(631\) −4.53563 −0.180560 −0.0902802 0.995916i \(-0.528776\pi\)
−0.0902802 + 0.995916i \(0.528776\pi\)
\(632\) 9.60591 0.382103
\(633\) −3.06625 −0.121872
\(634\) 17.2007 0.683126
\(635\) −9.89497 −0.392670
\(636\) −3.80396 −0.150837
\(637\) 0 0
\(638\) 6.55734 0.259608
\(639\) 20.7296 0.820049
\(640\) 0.929245 0.0367316
\(641\) −7.09568 −0.280263 −0.140131 0.990133i \(-0.544752\pi\)
−0.140131 + 0.990133i \(0.544752\pi\)
\(642\) 2.06969 0.0816842
\(643\) 31.5892 1.24576 0.622879 0.782318i \(-0.285963\pi\)
0.622879 + 0.782318i \(0.285963\pi\)
\(644\) 11.8573 0.467242
\(645\) 4.05270 0.159575
\(646\) 4.54056 0.178646
\(647\) 3.99801 0.157178 0.0785891 0.996907i \(-0.474959\pi\)
0.0785891 + 0.996907i \(0.474959\pi\)
\(648\) 0.347501 0.0136511
\(649\) 35.8678 1.40793
\(650\) 0 0
\(651\) 23.5945 0.924740
\(652\) −21.2249 −0.831232
\(653\) 21.4463 0.839258 0.419629 0.907696i \(-0.362160\pi\)
0.419629 + 0.907696i \(0.362160\pi\)
\(654\) 17.8340 0.697366
\(655\) 16.3626 0.639341
\(656\) −0.453243 −0.0176962
\(657\) 22.2754 0.869045
\(658\) 26.2993 1.02525
\(659\) −21.7818 −0.848497 −0.424248 0.905546i \(-0.639462\pi\)
−0.424248 + 0.905546i \(0.639462\pi\)
\(660\) 3.86435 0.150420
\(661\) −10.0216 −0.389795 −0.194898 0.980824i \(-0.562437\pi\)
−0.194898 + 0.980824i \(0.562437\pi\)
\(662\) −7.57755 −0.294510
\(663\) 0 0
\(664\) −4.33877 −0.168377
\(665\) 2.68943 0.104292
\(666\) −5.43429 −0.210575
\(667\) −6.75836 −0.261685
\(668\) −16.6096 −0.642645
\(669\) 22.8000 0.881501
\(670\) −13.2381 −0.511431
\(671\) 19.3688 0.747724
\(672\) −3.02786 −0.116802
\(673\) −2.72873 −0.105185 −0.0525925 0.998616i \(-0.516748\pi\)
−0.0525925 + 0.998616i \(0.516748\pi\)
\(674\) 12.4953 0.481303
\(675\) 21.2287 0.817093
\(676\) 0 0
\(677\) 25.2469 0.970315 0.485158 0.874427i \(-0.338762\pi\)
0.485158 + 0.874427i \(0.338762\pi\)
\(678\) 7.57160 0.290786
\(679\) 33.5281 1.28669
\(680\) −4.21929 −0.161802
\(681\) −18.9066 −0.724504
\(682\) −30.9753 −1.18610
\(683\) −42.1086 −1.61124 −0.805620 0.592432i \(-0.798168\pi\)
−0.805620 + 0.592432i \(0.798168\pi\)
\(684\) 1.90551 0.0728590
\(685\) −9.47008 −0.361833
\(686\) 16.2757 0.621410
\(687\) −21.7156 −0.828502
\(688\) 4.16877 0.158933
\(689\) 0 0
\(690\) −3.98282 −0.151623
\(691\) −39.6111 −1.50688 −0.753438 0.657519i \(-0.771606\pi\)
−0.753438 + 0.657519i \(0.771606\pi\)
\(692\) −5.22082 −0.198466
\(693\) 21.9221 0.832752
\(694\) −9.64219 −0.366013
\(695\) 16.8667 0.639789
\(696\) 1.72581 0.0654166
\(697\) 2.05798 0.0779514
\(698\) 30.2434 1.14473
\(699\) −28.1261 −1.06383
\(700\) 11.9719 0.452496
\(701\) 36.4844 1.37800 0.688998 0.724764i \(-0.258051\pi\)
0.688998 + 0.724764i \(0.258051\pi\)
\(702\) 0 0
\(703\) −2.85188 −0.107561
\(704\) 3.97504 0.149815
\(705\) −8.83383 −0.332701
\(706\) 11.1133 0.418253
\(707\) −28.0303 −1.05419
\(708\) 9.43994 0.354775
\(709\) 38.2462 1.43637 0.718183 0.695854i \(-0.244974\pi\)
0.718183 + 0.695854i \(0.244974\pi\)
\(710\) −10.1090 −0.379385
\(711\) −18.3042 −0.686460
\(712\) 6.23045 0.233496
\(713\) 31.9248 1.19559
\(714\) 13.7482 0.514512
\(715\) 0 0
\(716\) −3.64551 −0.136239
\(717\) 12.5759 0.469654
\(718\) −29.5541 −1.10295
\(719\) 36.6382 1.36637 0.683187 0.730244i \(-0.260593\pi\)
0.683187 + 0.730244i \(0.260593\pi\)
\(720\) −1.77069 −0.0659895
\(721\) 0.227179 0.00846058
\(722\) 1.00000 0.0372161
\(723\) −17.6894 −0.657875
\(724\) −14.4527 −0.537131
\(725\) −6.82370 −0.253426
\(726\) 5.02261 0.186406
\(727\) −25.0966 −0.930781 −0.465391 0.885105i \(-0.654086\pi\)
−0.465391 + 0.885105i \(0.654086\pi\)
\(728\) 0 0
\(729\) 15.4449 0.572034
\(730\) −10.8629 −0.402052
\(731\) −18.9285 −0.700097
\(732\) 5.09762 0.188413
\(733\) −10.8948 −0.402410 −0.201205 0.979549i \(-0.564486\pi\)
−0.201205 + 0.979549i \(0.564486\pi\)
\(734\) 30.4669 1.12455
\(735\) 1.33813 0.0493576
\(736\) −4.09689 −0.151014
\(737\) −56.6286 −2.08594
\(738\) 0.863659 0.0317917
\(739\) 7.22806 0.265888 0.132944 0.991124i \(-0.457557\pi\)
0.132944 + 0.991124i \(0.457557\pi\)
\(740\) 2.65010 0.0974195
\(741\) 0 0
\(742\) 10.5235 0.386330
\(743\) −53.9055 −1.97760 −0.988801 0.149239i \(-0.952318\pi\)
−0.988801 + 0.149239i \(0.952318\pi\)
\(744\) −8.15229 −0.298878
\(745\) −2.50720 −0.0918569
\(746\) 28.9669 1.06055
\(747\) 8.26757 0.302495
\(748\) −18.0489 −0.659932
\(749\) −5.72572 −0.209213
\(750\) −8.88211 −0.324329
\(751\) −15.8038 −0.576689 −0.288344 0.957527i \(-0.593105\pi\)
−0.288344 + 0.957527i \(0.593105\pi\)
\(752\) −9.08685 −0.331363
\(753\) 25.7251 0.937475
\(754\) 0 0
\(755\) 4.87129 0.177284
\(756\) 14.8532 0.540206
\(757\) −19.9728 −0.725925 −0.362963 0.931804i \(-0.618235\pi\)
−0.362963 + 0.931804i \(0.618235\pi\)
\(758\) −32.9190 −1.19567
\(759\) −17.0373 −0.618416
\(760\) −0.929245 −0.0337072
\(761\) 13.8528 0.502164 0.251082 0.967966i \(-0.419214\pi\)
0.251082 + 0.967966i \(0.419214\pi\)
\(762\) −11.1401 −0.403564
\(763\) −49.3371 −1.78612
\(764\) −7.34389 −0.265693
\(765\) 8.03990 0.290683
\(766\) −17.1752 −0.620564
\(767\) 0 0
\(768\) 1.04618 0.0377507
\(769\) 18.3743 0.662594 0.331297 0.943526i \(-0.392514\pi\)
0.331297 + 0.943526i \(0.392514\pi\)
\(770\) −10.6906 −0.385262
\(771\) −21.6546 −0.779872
\(772\) 9.57969 0.344781
\(773\) −26.3325 −0.947115 −0.473558 0.880763i \(-0.657030\pi\)
−0.473558 + 0.880763i \(0.657030\pi\)
\(774\) −7.94364 −0.285528
\(775\) 32.2335 1.15786
\(776\) −11.5845 −0.415861
\(777\) −8.63511 −0.309783
\(778\) 1.10030 0.0394478
\(779\) 0.453243 0.0162391
\(780\) 0 0
\(781\) −43.2434 −1.54737
\(782\) 18.6022 0.665213
\(783\) −8.46597 −0.302549
\(784\) 1.37645 0.0491591
\(785\) 2.04638 0.0730386
\(786\) 18.4217 0.657079
\(787\) −54.9301 −1.95805 −0.979023 0.203748i \(-0.934688\pi\)
−0.979023 + 0.203748i \(0.934688\pi\)
\(788\) 1.25654 0.0447623
\(789\) −25.4469 −0.905932
\(790\) 8.92624 0.317581
\(791\) −20.9465 −0.744773
\(792\) −7.57447 −0.269147
\(793\) 0 0
\(794\) 23.3055 0.827083
\(795\) −3.53481 −0.125367
\(796\) 5.97439 0.211757
\(797\) −25.0272 −0.886510 −0.443255 0.896396i \(-0.646176\pi\)
−0.443255 + 0.896396i \(0.646176\pi\)
\(798\) 3.02786 0.107185
\(799\) 41.2593 1.45965
\(800\) −4.13650 −0.146248
\(801\) −11.8722 −0.419483
\(802\) 27.1581 0.958986
\(803\) −46.4681 −1.63982
\(804\) −14.9039 −0.525621
\(805\) 11.0183 0.388344
\(806\) 0 0
\(807\) −0.402955 −0.0141847
\(808\) 9.68495 0.340716
\(809\) −16.4556 −0.578549 −0.289275 0.957246i \(-0.593414\pi\)
−0.289275 + 0.957246i \(0.593414\pi\)
\(810\) 0.322913 0.0113460
\(811\) −18.3589 −0.644667 −0.322334 0.946626i \(-0.604467\pi\)
−0.322334 + 0.946626i \(0.604467\pi\)
\(812\) −4.77438 −0.167548
\(813\) 22.8335 0.800806
\(814\) 11.3363 0.397338
\(815\) −19.7231 −0.690871
\(816\) −4.75023 −0.166291
\(817\) −4.16877 −0.145847
\(818\) −4.47035 −0.156302
\(819\) 0 0
\(820\) −0.421174 −0.0147080
\(821\) −56.5259 −1.97277 −0.986384 0.164460i \(-0.947412\pi\)
−0.986384 + 0.164460i \(0.947412\pi\)
\(822\) −10.6618 −0.371872
\(823\) −22.1382 −0.771690 −0.385845 0.922564i \(-0.626090\pi\)
−0.385845 + 0.922564i \(0.626090\pi\)
\(824\) −0.0784942 −0.00273448
\(825\) −17.2021 −0.598899
\(826\) −26.1152 −0.908665
\(827\) 14.6709 0.510157 0.255078 0.966920i \(-0.417899\pi\)
0.255078 + 0.966920i \(0.417899\pi\)
\(828\) 7.80668 0.271301
\(829\) 32.7680 1.13808 0.569040 0.822310i \(-0.307315\pi\)
0.569040 + 0.822310i \(0.307315\pi\)
\(830\) −4.03178 −0.139945
\(831\) −13.5062 −0.468524
\(832\) 0 0
\(833\) −6.24987 −0.216545
\(834\) 18.9891 0.657539
\(835\) −15.4344 −0.534129
\(836\) −3.97504 −0.137479
\(837\) 39.9912 1.38230
\(838\) 7.62391 0.263364
\(839\) −13.6785 −0.472234 −0.236117 0.971725i \(-0.575875\pi\)
−0.236117 + 0.971725i \(0.575875\pi\)
\(840\) −2.81362 −0.0970792
\(841\) −26.2787 −0.906163
\(842\) −31.6349 −1.09021
\(843\) −6.35249 −0.218791
\(844\) −2.93090 −0.100886
\(845\) 0 0
\(846\) 17.3151 0.595305
\(847\) −13.8948 −0.477432
\(848\) −3.63605 −0.124863
\(849\) 12.8817 0.442100
\(850\) 18.7820 0.644218
\(851\) −11.6839 −0.400518
\(852\) −11.3811 −0.389911
\(853\) −41.0799 −1.40655 −0.703274 0.710919i \(-0.748279\pi\)
−0.703274 + 0.710919i \(0.748279\pi\)
\(854\) −14.1024 −0.482573
\(855\) 1.77069 0.0605562
\(856\) 1.97833 0.0676181
\(857\) 11.4735 0.391929 0.195964 0.980611i \(-0.437216\pi\)
0.195964 + 0.980611i \(0.437216\pi\)
\(858\) 0 0
\(859\) −10.2630 −0.350170 −0.175085 0.984553i \(-0.556020\pi\)
−0.175085 + 0.984553i \(0.556020\pi\)
\(860\) 3.87381 0.132096
\(861\) 1.37236 0.0467698
\(862\) −5.99194 −0.204086
\(863\) −53.9995 −1.83816 −0.919082 0.394067i \(-0.871068\pi\)
−0.919082 + 0.394067i \(0.871068\pi\)
\(864\) −5.13204 −0.174596
\(865\) −4.85142 −0.164953
\(866\) 22.3035 0.757904
\(867\) 3.78366 0.128500
\(868\) 22.5530 0.765498
\(869\) 38.1838 1.29530
\(870\) 1.60370 0.0543705
\(871\) 0 0
\(872\) 17.0468 0.577278
\(873\) 22.0745 0.747107
\(874\) 4.09689 0.138580
\(875\) 24.5720 0.830685
\(876\) −12.2298 −0.413207
\(877\) 46.9344 1.58486 0.792431 0.609962i \(-0.208815\pi\)
0.792431 + 0.609962i \(0.208815\pi\)
\(878\) −8.13560 −0.274563
\(879\) −7.78960 −0.262737
\(880\) 3.69378 0.124517
\(881\) 36.7205 1.23714 0.618572 0.785728i \(-0.287712\pi\)
0.618572 + 0.785728i \(0.287712\pi\)
\(882\) −2.62285 −0.0883159
\(883\) −40.7395 −1.37099 −0.685497 0.728076i \(-0.740415\pi\)
−0.685497 + 0.728076i \(0.740415\pi\)
\(884\) 0 0
\(885\) 8.77202 0.294868
\(886\) 9.37320 0.314899
\(887\) 38.9664 1.30836 0.654182 0.756337i \(-0.273013\pi\)
0.654182 + 0.756337i \(0.273013\pi\)
\(888\) 2.98358 0.100122
\(889\) 30.8187 1.03363
\(890\) 5.78961 0.194068
\(891\) 1.38133 0.0462762
\(892\) 21.7936 0.729705
\(893\) 9.08685 0.304080
\(894\) −2.82271 −0.0944054
\(895\) −3.38757 −0.113234
\(896\) −2.89421 −0.0966888
\(897\) 0 0
\(898\) 23.5049 0.784368
\(899\) −12.8547 −0.428727
\(900\) 7.88215 0.262738
\(901\) 16.5097 0.550018
\(902\) −1.80166 −0.0599886
\(903\) −12.6225 −0.420049
\(904\) 7.23739 0.240712
\(905\) −13.4301 −0.446432
\(906\) 5.48428 0.182203
\(907\) 57.5459 1.91078 0.955391 0.295345i \(-0.0954347\pi\)
0.955391 + 0.295345i \(0.0954347\pi\)
\(908\) −18.0721 −0.599743
\(909\) −18.4548 −0.612106
\(910\) 0 0
\(911\) 14.3410 0.475140 0.237570 0.971370i \(-0.423649\pi\)
0.237570 + 0.971370i \(0.423649\pi\)
\(912\) −1.04618 −0.0346424
\(913\) −17.2468 −0.570785
\(914\) 7.02736 0.232445
\(915\) 4.73694 0.156598
\(916\) −20.7571 −0.685833
\(917\) −50.9628 −1.68294
\(918\) 23.3023 0.769091
\(919\) −9.88857 −0.326194 −0.163097 0.986610i \(-0.552148\pi\)
−0.163097 + 0.986610i \(0.552148\pi\)
\(920\) −3.80702 −0.125514
\(921\) 19.7422 0.650528
\(922\) 2.09583 0.0690226
\(923\) 0 0
\(924\) −12.0359 −0.395951
\(925\) −11.7968 −0.387877
\(926\) −22.3671 −0.735028
\(927\) 0.149572 0.00491258
\(928\) 1.64963 0.0541518
\(929\) −29.4749 −0.967041 −0.483521 0.875333i \(-0.660642\pi\)
−0.483521 + 0.875333i \(0.660642\pi\)
\(930\) −7.57548 −0.248410
\(931\) −1.37645 −0.0451115
\(932\) −26.8846 −0.880636
\(933\) 5.25011 0.171881
\(934\) 36.8013 1.20417
\(935\) −16.7718 −0.548497
\(936\) 0 0
\(937\) −11.6698 −0.381237 −0.190619 0.981664i \(-0.561049\pi\)
−0.190619 + 0.981664i \(0.561049\pi\)
\(938\) 41.2311 1.34624
\(939\) −23.5687 −0.769135
\(940\) −8.44390 −0.275410
\(941\) 14.3074 0.466407 0.233203 0.972428i \(-0.425079\pi\)
0.233203 + 0.972428i \(0.425079\pi\)
\(942\) 2.30390 0.0750650
\(943\) 1.85689 0.0604686
\(944\) 9.02326 0.293682
\(945\) 13.8023 0.448988
\(946\) 16.5710 0.538770
\(947\) 26.7816 0.870285 0.435142 0.900362i \(-0.356698\pi\)
0.435142 + 0.900362i \(0.356698\pi\)
\(948\) 10.0495 0.326392
\(949\) 0 0
\(950\) 4.13650 0.134206
\(951\) 17.9950 0.583527
\(952\) 13.1413 0.425913
\(953\) 37.3685 1.21048 0.605242 0.796042i \(-0.293076\pi\)
0.605242 + 0.796042i \(0.293076\pi\)
\(954\) 6.92854 0.224320
\(955\) −6.82427 −0.220828
\(956\) 12.0208 0.388779
\(957\) 6.86015 0.221757
\(958\) −18.0660 −0.583687
\(959\) 29.4954 0.952455
\(960\) 0.972156 0.0313762
\(961\) 29.7223 0.958783
\(962\) 0 0
\(963\) −3.76974 −0.121478
\(964\) −16.9086 −0.544588
\(965\) 8.90188 0.286562
\(966\) 12.4048 0.399119
\(967\) −23.2450 −0.747508 −0.373754 0.927528i \(-0.621930\pi\)
−0.373754 + 0.927528i \(0.621930\pi\)
\(968\) 4.80091 0.154307
\(969\) 4.75023 0.152599
\(970\) −10.7649 −0.345639
\(971\) −3.68714 −0.118326 −0.0591630 0.998248i \(-0.518843\pi\)
−0.0591630 + 0.998248i \(0.518843\pi\)
\(972\) 15.7597 0.505492
\(973\) −52.5326 −1.68412
\(974\) 19.7205 0.631886
\(975\) 0 0
\(976\) 4.87261 0.155968
\(977\) 29.2065 0.934399 0.467200 0.884152i \(-0.345263\pi\)
0.467200 + 0.884152i \(0.345263\pi\)
\(978\) −22.2050 −0.710039
\(979\) 24.7663 0.791533
\(980\) 1.27906 0.0408582
\(981\) −32.4829 −1.03710
\(982\) −26.6062 −0.849037
\(983\) −10.6377 −0.339291 −0.169646 0.985505i \(-0.554262\pi\)
−0.169646 + 0.985505i \(0.554262\pi\)
\(984\) −0.474173 −0.0151161
\(985\) 1.16763 0.0372038
\(986\) −7.49024 −0.238538
\(987\) 27.5137 0.875771
\(988\) 0 0
\(989\) −17.0790 −0.543081
\(990\) −7.03854 −0.223699
\(991\) 35.2488 1.11972 0.559858 0.828589i \(-0.310856\pi\)
0.559858 + 0.828589i \(0.310856\pi\)
\(992\) −7.79245 −0.247411
\(993\) −7.92747 −0.251571
\(994\) 31.4854 0.998656
\(995\) 5.55167 0.176000
\(996\) −4.53913 −0.143828
\(997\) 34.8962 1.10517 0.552586 0.833456i \(-0.313641\pi\)
0.552586 + 0.833456i \(0.313641\pi\)
\(998\) −5.10306 −0.161534
\(999\) −14.6360 −0.463062
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 6422.2.a.bl.1.6 yes 9
13.12 even 2 6422.2.a.bj.1.6 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6422.2.a.bj.1.6 9 13.12 even 2
6422.2.a.bl.1.6 yes 9 1.1 even 1 trivial