Properties

Label 6422.2.a.bd.1.5
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $0$
Dimension $6$
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] = [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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.316645497.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 12x^{4} + 24x^{3} + 13x^{2} - 24x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 494)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.19765\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.19765 q^{3} +1.00000 q^{4} +3.12703 q^{5} +2.19765 q^{6} +2.17944 q^{7} +1.00000 q^{8} +1.82965 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.19765 q^{3} +1.00000 q^{4} +3.12703 q^{5} +2.19765 q^{6} +2.17944 q^{7} +1.00000 q^{8} +1.82965 q^{9} +3.12703 q^{10} +3.12703 q^{11} +2.19765 q^{12} +2.17944 q^{14} +6.87211 q^{15} +1.00000 q^{16} -7.10300 q^{17} +1.82965 q^{18} -1.00000 q^{19} +3.12703 q^{20} +4.78963 q^{21} +3.12703 q^{22} -0.644393 q^{23} +2.19765 q^{24} +4.77832 q^{25} -2.57201 q^{27} +2.17944 q^{28} +0.426223 q^{29} +6.87211 q^{30} -4.85159 q^{31} +1.00000 q^{32} +6.87211 q^{33} -7.10300 q^{34} +6.81517 q^{35} +1.82965 q^{36} +7.52510 q^{37} -1.00000 q^{38} +3.12703 q^{40} -0.117941 q^{41} +4.78963 q^{42} +2.42854 q^{43} +3.12703 q^{44} +5.72138 q^{45} -0.644393 q^{46} +6.12703 q^{47} +2.19765 q^{48} -2.25005 q^{49} +4.77832 q^{50} -15.6099 q^{51} +4.11436 q^{53} -2.57201 q^{54} +9.77832 q^{55} +2.17944 q^{56} -2.19765 q^{57} +0.426223 q^{58} -10.6322 q^{59} +6.87211 q^{60} +8.65802 q^{61} -4.85159 q^{62} +3.98761 q^{63} +1.00000 q^{64} +6.87211 q^{66} +1.75463 q^{67} -7.10300 q^{68} -1.41615 q^{69} +6.81517 q^{70} +10.2346 q^{71} +1.82965 q^{72} +13.3753 q^{73} +7.52510 q^{74} +10.5011 q^{75} -1.00000 q^{76} +6.81517 q^{77} +11.3371 q^{79} +3.12703 q^{80} -11.1413 q^{81} -0.117941 q^{82} -12.5884 q^{83} +4.78963 q^{84} -22.2113 q^{85} +2.42854 q^{86} +0.936688 q^{87} +3.12703 q^{88} +1.43889 q^{89} +5.72138 q^{90} -0.644393 q^{92} -10.6621 q^{93} +6.12703 q^{94} -3.12703 q^{95} +2.19765 q^{96} -11.9346 q^{97} -2.25005 q^{98} +5.72138 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9} + 2 q^{10} + 2 q^{11} + 2 q^{12} + q^{14} - 9 q^{15} + 6 q^{16} - 2 q^{17} + 10 q^{18} - 6 q^{19} + 2 q^{20} + q^{21} + 2 q^{22} + 8 q^{23} + 2 q^{24} + 16 q^{25} - 4 q^{27} + q^{28} + 20 q^{29} - 9 q^{30} + 3 q^{31} + 6 q^{32} - 9 q^{33} - 2 q^{34} + 7 q^{35} + 10 q^{36} - 9 q^{37} - 6 q^{38} + 2 q^{40} + 3 q^{41} + q^{42} + 13 q^{43} + 2 q^{44} + 38 q^{45} + 8 q^{46} + 20 q^{47} + 2 q^{48} - 7 q^{49} + 16 q^{50} + 2 q^{51} + 25 q^{53} - 4 q^{54} + 46 q^{55} + q^{56} - 2 q^{57} + 20 q^{58} - 9 q^{60} + 6 q^{61} + 3 q^{62} + 46 q^{63} + 6 q^{64} - 9 q^{66} - 32 q^{67} - 2 q^{68} - 29 q^{69} + 7 q^{70} + 39 q^{71} + 10 q^{72} + 7 q^{73} - 9 q^{74} - 15 q^{75} - 6 q^{76} + 7 q^{77} + 18 q^{79} + 2 q^{80} + 54 q^{81} + 3 q^{82} - 7 q^{83} + q^{84} - 2 q^{85} + 13 q^{86} + 12 q^{87} + 2 q^{88} + 9 q^{89} + 38 q^{90} + 8 q^{92} - 47 q^{93} + 20 q^{94} - 2 q^{95} + 2 q^{96} - 4 q^{97} - 7 q^{98} + 38 q^{99}+O(q^{100}) \) 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
\(3\) 2.19765 1.26881 0.634406 0.773000i \(-0.281245\pi\)
0.634406 + 0.773000i \(0.281245\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.12703 1.39845 0.699225 0.714901i \(-0.253528\pi\)
0.699225 + 0.714901i \(0.253528\pi\)
\(6\) 2.19765 0.897186
\(7\) 2.17944 0.823750 0.411875 0.911240i \(-0.364874\pi\)
0.411875 + 0.911240i \(0.364874\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.82965 0.609884
\(10\) 3.12703 0.988854
\(11\) 3.12703 0.942835 0.471418 0.881910i \(-0.343742\pi\)
0.471418 + 0.881910i \(0.343742\pi\)
\(12\) 2.19765 0.634406
\(13\) 0 0
\(14\) 2.17944 0.582479
\(15\) 6.87211 1.77437
\(16\) 1.00000 0.250000
\(17\) −7.10300 −1.72273 −0.861365 0.507986i \(-0.830390\pi\)
−0.861365 + 0.507986i \(0.830390\pi\)
\(18\) 1.82965 0.431253
\(19\) −1.00000 −0.229416
\(20\) 3.12703 0.699225
\(21\) 4.78963 1.04518
\(22\) 3.12703 0.666685
\(23\) −0.644393 −0.134365 −0.0671826 0.997741i \(-0.521401\pi\)
−0.0671826 + 0.997741i \(0.521401\pi\)
\(24\) 2.19765 0.448593
\(25\) 4.77832 0.955664
\(26\) 0 0
\(27\) −2.57201 −0.494984
\(28\) 2.17944 0.411875
\(29\) 0.426223 0.0791477 0.0395738 0.999217i \(-0.487400\pi\)
0.0395738 + 0.999217i \(0.487400\pi\)
\(30\) 6.87211 1.25467
\(31\) −4.85159 −0.871371 −0.435685 0.900099i \(-0.643494\pi\)
−0.435685 + 0.900099i \(0.643494\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.87211 1.19628
\(34\) −7.10300 −1.21815
\(35\) 6.81517 1.15197
\(36\) 1.82965 0.304942
\(37\) 7.52510 1.23712 0.618559 0.785738i \(-0.287717\pi\)
0.618559 + 0.785738i \(0.287717\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 3.12703 0.494427
\(41\) −0.117941 −0.0184193 −0.00920967 0.999958i \(-0.502932\pi\)
−0.00920967 + 0.999958i \(0.502932\pi\)
\(42\) 4.78963 0.739057
\(43\) 2.42854 0.370348 0.185174 0.982706i \(-0.440715\pi\)
0.185174 + 0.982706i \(0.440715\pi\)
\(44\) 3.12703 0.471418
\(45\) 5.72138 0.852893
\(46\) −0.644393 −0.0950106
\(47\) 6.12703 0.893719 0.446860 0.894604i \(-0.352542\pi\)
0.446860 + 0.894604i \(0.352542\pi\)
\(48\) 2.19765 0.317203
\(49\) −2.25005 −0.321436
\(50\) 4.77832 0.675757
\(51\) −15.6099 −2.18582
\(52\) 0 0
\(53\) 4.11436 0.565151 0.282575 0.959245i \(-0.408811\pi\)
0.282575 + 0.959245i \(0.408811\pi\)
\(54\) −2.57201 −0.350006
\(55\) 9.77832 1.31851
\(56\) 2.17944 0.291240
\(57\) −2.19765 −0.291085
\(58\) 0.426223 0.0559658
\(59\) −10.6322 −1.38420 −0.692099 0.721803i \(-0.743314\pi\)
−0.692099 + 0.721803i \(0.743314\pi\)
\(60\) 6.87211 0.887186
\(61\) 8.65802 1.10855 0.554273 0.832335i \(-0.312996\pi\)
0.554273 + 0.832335i \(0.312996\pi\)
\(62\) −4.85159 −0.616152
\(63\) 3.98761 0.502392
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.87211 0.845898
\(67\) 1.75463 0.214362 0.107181 0.994240i \(-0.465818\pi\)
0.107181 + 0.994240i \(0.465818\pi\)
\(68\) −7.10300 −0.861365
\(69\) −1.41615 −0.170484
\(70\) 6.81517 0.814568
\(71\) 10.2346 1.21463 0.607313 0.794463i \(-0.292248\pi\)
0.607313 + 0.794463i \(0.292248\pi\)
\(72\) 1.82965 0.215627
\(73\) 13.3753 1.56546 0.782728 0.622363i \(-0.213827\pi\)
0.782728 + 0.622363i \(0.213827\pi\)
\(74\) 7.52510 0.874775
\(75\) 10.5011 1.21256
\(76\) −1.00000 −0.114708
\(77\) 6.81517 0.776660
\(78\) 0 0
\(79\) 11.3371 1.27552 0.637760 0.770235i \(-0.279861\pi\)
0.637760 + 0.770235i \(0.279861\pi\)
\(80\) 3.12703 0.349613
\(81\) −11.1413 −1.23793
\(82\) −0.117941 −0.0130244
\(83\) −12.5884 −1.38175 −0.690876 0.722973i \(-0.742775\pi\)
−0.690876 + 0.722973i \(0.742775\pi\)
\(84\) 4.78963 0.522592
\(85\) −22.2113 −2.40915
\(86\) 2.42854 0.261876
\(87\) 0.936688 0.100423
\(88\) 3.12703 0.333343
\(89\) 1.43889 0.152522 0.0762612 0.997088i \(-0.475702\pi\)
0.0762612 + 0.997088i \(0.475702\pi\)
\(90\) 5.72138 0.603086
\(91\) 0 0
\(92\) −0.644393 −0.0671826
\(93\) −10.6621 −1.10561
\(94\) 6.12703 0.631955
\(95\) −3.12703 −0.320827
\(96\) 2.19765 0.224296
\(97\) −11.9346 −1.21178 −0.605888 0.795550i \(-0.707182\pi\)
−0.605888 + 0.795550i \(0.707182\pi\)
\(98\) −2.25005 −0.227290
\(99\) 5.72138 0.575020
\(100\) 4.77832 0.477832
\(101\) −12.1844 −1.21239 −0.606197 0.795315i \(-0.707306\pi\)
−0.606197 + 0.795315i \(0.707306\pi\)
\(102\) −15.6099 −1.54561
\(103\) 10.5338 1.03792 0.518961 0.854798i \(-0.326319\pi\)
0.518961 + 0.854798i \(0.326319\pi\)
\(104\) 0 0
\(105\) 14.9773 1.46164
\(106\) 4.11436 0.399622
\(107\) 5.67809 0.548922 0.274461 0.961598i \(-0.411501\pi\)
0.274461 + 0.961598i \(0.411501\pi\)
\(108\) −2.57201 −0.247492
\(109\) −11.7186 −1.12243 −0.561217 0.827669i \(-0.689667\pi\)
−0.561217 + 0.827669i \(0.689667\pi\)
\(110\) 9.77832 0.932326
\(111\) 16.5375 1.56967
\(112\) 2.17944 0.205937
\(113\) −0.945828 −0.0889760 −0.0444880 0.999010i \(-0.514166\pi\)
−0.0444880 + 0.999010i \(0.514166\pi\)
\(114\) −2.19765 −0.205828
\(115\) −2.01504 −0.187903
\(116\) 0.426223 0.0395738
\(117\) 0 0
\(118\) −10.6322 −0.978775
\(119\) −15.4805 −1.41910
\(120\) 6.87211 0.627335
\(121\) −1.22168 −0.111062
\(122\) 8.65802 0.783860
\(123\) −0.259193 −0.0233707
\(124\) −4.85159 −0.435685
\(125\) −0.693196 −0.0620013
\(126\) 3.98761 0.355245
\(127\) 9.51474 0.844296 0.422148 0.906527i \(-0.361276\pi\)
0.422148 + 0.906527i \(0.361276\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.33707 0.469902
\(130\) 0 0
\(131\) 2.94454 0.257266 0.128633 0.991692i \(-0.458941\pi\)
0.128633 + 0.991692i \(0.458941\pi\)
\(132\) 6.87211 0.598140
\(133\) −2.17944 −0.188981
\(134\) 1.75463 0.151577
\(135\) −8.04276 −0.692210
\(136\) −7.10300 −0.609077
\(137\) −4.54962 −0.388701 −0.194350 0.980932i \(-0.562260\pi\)
−0.194350 + 0.980932i \(0.562260\pi\)
\(138\) −1.41615 −0.120551
\(139\) −14.9108 −1.26472 −0.632359 0.774675i \(-0.717913\pi\)
−0.632359 + 0.774675i \(0.717913\pi\)
\(140\) 6.81517 0.575987
\(141\) 13.4651 1.13396
\(142\) 10.2346 0.858870
\(143\) 0 0
\(144\) 1.82965 0.152471
\(145\) 1.33281 0.110684
\(146\) 13.3753 1.10695
\(147\) −4.94482 −0.407842
\(148\) 7.52510 0.618559
\(149\) 22.5037 1.84358 0.921788 0.387694i \(-0.126728\pi\)
0.921788 + 0.387694i \(0.126728\pi\)
\(150\) 10.5011 0.857408
\(151\) 9.30065 0.756876 0.378438 0.925627i \(-0.376461\pi\)
0.378438 + 0.925627i \(0.376461\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −12.9960 −1.05067
\(154\) 6.81517 0.549182
\(155\) −15.1711 −1.21857
\(156\) 0 0
\(157\) 2.58803 0.206547 0.103274 0.994653i \(-0.467068\pi\)
0.103274 + 0.994653i \(0.467068\pi\)
\(158\) 11.3371 0.901929
\(159\) 9.04191 0.717070
\(160\) 3.12703 0.247213
\(161\) −1.40441 −0.110683
\(162\) −11.1413 −0.875346
\(163\) −22.3172 −1.74802 −0.874008 0.485911i \(-0.838488\pi\)
−0.874008 + 0.485911i \(0.838488\pi\)
\(164\) −0.117941 −0.00920967
\(165\) 21.4893 1.67294
\(166\) −12.5884 −0.977046
\(167\) −23.2517 −1.79927 −0.899636 0.436640i \(-0.856168\pi\)
−0.899636 + 0.436640i \(0.856168\pi\)
\(168\) 4.78963 0.369528
\(169\) 0 0
\(170\) −22.2113 −1.70353
\(171\) −1.82965 −0.139917
\(172\) 2.42854 0.185174
\(173\) −21.7696 −1.65511 −0.827557 0.561381i \(-0.810270\pi\)
−0.827557 + 0.561381i \(0.810270\pi\)
\(174\) 0.936688 0.0710101
\(175\) 10.4141 0.787228
\(176\) 3.12703 0.235709
\(177\) −23.3659 −1.75629
\(178\) 1.43889 0.107850
\(179\) −24.6147 −1.83979 −0.919893 0.392170i \(-0.871725\pi\)
−0.919893 + 0.392170i \(0.871725\pi\)
\(180\) 5.72138 0.426446
\(181\) −3.87802 −0.288251 −0.144126 0.989559i \(-0.546037\pi\)
−0.144126 + 0.989559i \(0.546037\pi\)
\(182\) 0 0
\(183\) 19.0273 1.40654
\(184\) −0.644393 −0.0475053
\(185\) 23.5312 1.73005
\(186\) −10.6621 −0.781781
\(187\) −22.2113 −1.62425
\(188\) 6.12703 0.446860
\(189\) −5.60554 −0.407743
\(190\) −3.12703 −0.226859
\(191\) 14.5288 1.05127 0.525634 0.850711i \(-0.323828\pi\)
0.525634 + 0.850711i \(0.323828\pi\)
\(192\) 2.19765 0.158602
\(193\) −25.5087 −1.83616 −0.918080 0.396396i \(-0.870261\pi\)
−0.918080 + 0.396396i \(0.870261\pi\)
\(194\) −11.9346 −0.856855
\(195\) 0 0
\(196\) −2.25005 −0.160718
\(197\) −5.12980 −0.365483 −0.182742 0.983161i \(-0.558497\pi\)
−0.182742 + 0.983161i \(0.558497\pi\)
\(198\) 5.72138 0.406601
\(199\) −6.58770 −0.466989 −0.233495 0.972358i \(-0.575016\pi\)
−0.233495 + 0.972358i \(0.575016\pi\)
\(200\) 4.77832 0.337878
\(201\) 3.85605 0.271985
\(202\) −12.1844 −0.857291
\(203\) 0.928927 0.0651979
\(204\) −15.6099 −1.09291
\(205\) −0.368806 −0.0257585
\(206\) 10.5338 0.733922
\(207\) −1.17902 −0.0819472
\(208\) 0 0
\(209\) −3.12703 −0.216301
\(210\) 14.9773 1.03353
\(211\) −4.98682 −0.343307 −0.171654 0.985157i \(-0.554911\pi\)
−0.171654 + 0.985157i \(0.554911\pi\)
\(212\) 4.11436 0.282575
\(213\) 22.4921 1.54113
\(214\) 5.67809 0.388146
\(215\) 7.59411 0.517914
\(216\) −2.57201 −0.175003
\(217\) −10.5737 −0.717791
\(218\) −11.7186 −0.793681
\(219\) 29.3941 1.98627
\(220\) 9.77832 0.659254
\(221\) 0 0
\(222\) 16.5375 1.10992
\(223\) −22.9805 −1.53889 −0.769445 0.638713i \(-0.779467\pi\)
−0.769445 + 0.638713i \(0.779467\pi\)
\(224\) 2.17944 0.145620
\(225\) 8.74267 0.582844
\(226\) −0.945828 −0.0629155
\(227\) 5.49722 0.364863 0.182432 0.983219i \(-0.441603\pi\)
0.182432 + 0.983219i \(0.441603\pi\)
\(228\) −2.19765 −0.145543
\(229\) −1.52318 −0.100655 −0.0503274 0.998733i \(-0.516026\pi\)
−0.0503274 + 0.998733i \(0.516026\pi\)
\(230\) −2.01504 −0.132868
\(231\) 14.9773 0.985436
\(232\) 0.426223 0.0279829
\(233\) 25.9377 1.69924 0.849618 0.527399i \(-0.176833\pi\)
0.849618 + 0.527399i \(0.176833\pi\)
\(234\) 0 0
\(235\) 19.1594 1.24982
\(236\) −10.6322 −0.692099
\(237\) 24.9149 1.61839
\(238\) −15.4805 −1.00345
\(239\) −5.55924 −0.359597 −0.179799 0.983703i \(-0.557545\pi\)
−0.179799 + 0.983703i \(0.557545\pi\)
\(240\) 6.87211 0.443593
\(241\) 11.9583 0.770302 0.385151 0.922853i \(-0.374149\pi\)
0.385151 + 0.922853i \(0.374149\pi\)
\(242\) −1.22168 −0.0785325
\(243\) −16.7687 −1.07571
\(244\) 8.65802 0.554273
\(245\) −7.03599 −0.449513
\(246\) −0.259193 −0.0165256
\(247\) 0 0
\(248\) −4.85159 −0.308076
\(249\) −27.6648 −1.75318
\(250\) −0.693196 −0.0438416
\(251\) −22.4814 −1.41902 −0.709508 0.704697i \(-0.751083\pi\)
−0.709508 + 0.704697i \(0.751083\pi\)
\(252\) 3.98761 0.251196
\(253\) −2.01504 −0.126684
\(254\) 9.51474 0.597008
\(255\) −48.8126 −3.05676
\(256\) 1.00000 0.0625000
\(257\) 4.85032 0.302555 0.151277 0.988491i \(-0.451661\pi\)
0.151277 + 0.988491i \(0.451661\pi\)
\(258\) 5.33707 0.332271
\(259\) 16.4005 1.01908
\(260\) 0 0
\(261\) 0.779840 0.0482709
\(262\) 2.94454 0.181914
\(263\) −10.1214 −0.624109 −0.312055 0.950064i \(-0.601017\pi\)
−0.312055 + 0.950064i \(0.601017\pi\)
\(264\) 6.87211 0.422949
\(265\) 12.8657 0.790335
\(266\) −2.17944 −0.133630
\(267\) 3.16218 0.193522
\(268\) 1.75463 0.107181
\(269\) 7.99747 0.487614 0.243807 0.969824i \(-0.421604\pi\)
0.243807 + 0.969824i \(0.421604\pi\)
\(270\) −8.04276 −0.489467
\(271\) 9.26475 0.562794 0.281397 0.959591i \(-0.409202\pi\)
0.281397 + 0.959591i \(0.409202\pi\)
\(272\) −7.10300 −0.430683
\(273\) 0 0
\(274\) −4.54962 −0.274853
\(275\) 14.9420 0.901034
\(276\) −1.41615 −0.0852421
\(277\) 14.1772 0.851823 0.425912 0.904765i \(-0.359953\pi\)
0.425912 + 0.904765i \(0.359953\pi\)
\(278\) −14.9108 −0.894291
\(279\) −8.87672 −0.531435
\(280\) 6.81517 0.407284
\(281\) −27.3738 −1.63299 −0.816493 0.577355i \(-0.804085\pi\)
−0.816493 + 0.577355i \(0.804085\pi\)
\(282\) 13.4651 0.801832
\(283\) 23.5136 1.39774 0.698868 0.715250i \(-0.253687\pi\)
0.698868 + 0.715250i \(0.253687\pi\)
\(284\) 10.2346 0.607313
\(285\) −6.87211 −0.407069
\(286\) 0 0
\(287\) −0.257046 −0.0151729
\(288\) 1.82965 0.107813
\(289\) 33.4526 1.96780
\(290\) 1.33281 0.0782655
\(291\) −26.2281 −1.53752
\(292\) 13.3753 0.782728
\(293\) 11.1201 0.649645 0.324823 0.945775i \(-0.394695\pi\)
0.324823 + 0.945775i \(0.394695\pi\)
\(294\) −4.94482 −0.288388
\(295\) −33.2473 −1.93573
\(296\) 7.52510 0.437387
\(297\) −8.04276 −0.466688
\(298\) 22.5037 1.30361
\(299\) 0 0
\(300\) 10.5011 0.606279
\(301\) 5.29284 0.305074
\(302\) 9.30065 0.535192
\(303\) −26.7770 −1.53830
\(304\) −1.00000 −0.0573539
\(305\) 27.0739 1.55025
\(306\) −12.9960 −0.742933
\(307\) −10.2579 −0.585450 −0.292725 0.956197i \(-0.594562\pi\)
−0.292725 + 0.956197i \(0.594562\pi\)
\(308\) 6.81517 0.388330
\(309\) 23.1495 1.31693
\(310\) −15.1711 −0.861658
\(311\) 13.8693 0.786455 0.393228 0.919441i \(-0.371358\pi\)
0.393228 + 0.919441i \(0.371358\pi\)
\(312\) 0 0
\(313\) −3.37717 −0.190889 −0.0954446 0.995435i \(-0.530427\pi\)
−0.0954446 + 0.995435i \(0.530427\pi\)
\(314\) 2.58803 0.146051
\(315\) 12.4694 0.702570
\(316\) 11.3371 0.637760
\(317\) 6.12834 0.344202 0.172101 0.985079i \(-0.444944\pi\)
0.172101 + 0.985079i \(0.444944\pi\)
\(318\) 9.04191 0.507045
\(319\) 1.33281 0.0746232
\(320\) 3.12703 0.174806
\(321\) 12.4784 0.696479
\(322\) −1.40441 −0.0782649
\(323\) 7.10300 0.395221
\(324\) −11.1413 −0.618963
\(325\) 0 0
\(326\) −22.3172 −1.23603
\(327\) −25.7533 −1.42416
\(328\) −0.117941 −0.00651222
\(329\) 13.3535 0.736201
\(330\) 21.4893 1.18295
\(331\) 0.0943337 0.00518505 0.00259253 0.999997i \(-0.499175\pi\)
0.00259253 + 0.999997i \(0.499175\pi\)
\(332\) −12.5884 −0.690876
\(333\) 13.7683 0.754499
\(334\) −23.2517 −1.27228
\(335\) 5.48677 0.299774
\(336\) 4.78963 0.261296
\(337\) −9.25406 −0.504101 −0.252050 0.967714i \(-0.581105\pi\)
−0.252050 + 0.967714i \(0.581105\pi\)
\(338\) 0 0
\(339\) −2.07860 −0.112894
\(340\) −22.2113 −1.20458
\(341\) −15.1711 −0.821559
\(342\) −1.82965 −0.0989363
\(343\) −20.1599 −1.08853
\(344\) 2.42854 0.130938
\(345\) −4.42834 −0.238414
\(346\) −21.7696 −1.17034
\(347\) −29.1677 −1.56580 −0.782902 0.622145i \(-0.786261\pi\)
−0.782902 + 0.622145i \(0.786261\pi\)
\(348\) 0.936688 0.0502117
\(349\) −29.5303 −1.58072 −0.790361 0.612641i \(-0.790107\pi\)
−0.790361 + 0.612641i \(0.790107\pi\)
\(350\) 10.4141 0.556654
\(351\) 0 0
\(352\) 3.12703 0.166671
\(353\) −29.0884 −1.54822 −0.774110 0.633052i \(-0.781802\pi\)
−0.774110 + 0.633052i \(0.781802\pi\)
\(354\) −23.3659 −1.24188
\(355\) 32.0040 1.69859
\(356\) 1.43889 0.0762612
\(357\) −34.0208 −1.80057
\(358\) −24.6147 −1.30092
\(359\) −19.0423 −1.00502 −0.502508 0.864573i \(-0.667589\pi\)
−0.502508 + 0.864573i \(0.667589\pi\)
\(360\) 5.72138 0.301543
\(361\) 1.00000 0.0526316
\(362\) −3.87802 −0.203824
\(363\) −2.68482 −0.140916
\(364\) 0 0
\(365\) 41.8249 2.18921
\(366\) 19.0273 0.994571
\(367\) 1.02770 0.0536456 0.0268228 0.999640i \(-0.491461\pi\)
0.0268228 + 0.999640i \(0.491461\pi\)
\(368\) −0.644393 −0.0335913
\(369\) −0.215792 −0.0112337
\(370\) 23.5312 1.22333
\(371\) 8.96699 0.465543
\(372\) −10.6621 −0.552803
\(373\) 15.9433 0.825512 0.412756 0.910842i \(-0.364566\pi\)
0.412756 + 0.910842i \(0.364566\pi\)
\(374\) −22.2113 −1.14852
\(375\) −1.52340 −0.0786681
\(376\) 6.12703 0.315977
\(377\) 0 0
\(378\) −5.60554 −0.288318
\(379\) 4.79302 0.246201 0.123100 0.992394i \(-0.460716\pi\)
0.123100 + 0.992394i \(0.460716\pi\)
\(380\) −3.12703 −0.160413
\(381\) 20.9100 1.07125
\(382\) 14.5288 0.743359
\(383\) −21.8340 −1.11567 −0.557833 0.829953i \(-0.688367\pi\)
−0.557833 + 0.829953i \(0.688367\pi\)
\(384\) 2.19765 0.112148
\(385\) 21.3112 1.08612
\(386\) −25.5087 −1.29836
\(387\) 4.44338 0.225870
\(388\) −11.9346 −0.605888
\(389\) 16.3944 0.831231 0.415615 0.909540i \(-0.363566\pi\)
0.415615 + 0.909540i \(0.363566\pi\)
\(390\) 0 0
\(391\) 4.57712 0.231475
\(392\) −2.25005 −0.113645
\(393\) 6.47106 0.326422
\(394\) −5.12980 −0.258436
\(395\) 35.4514 1.78375
\(396\) 5.72138 0.287510
\(397\) 8.32544 0.417842 0.208921 0.977933i \(-0.433005\pi\)
0.208921 + 0.977933i \(0.433005\pi\)
\(398\) −6.58770 −0.330211
\(399\) −4.78963 −0.239782
\(400\) 4.77832 0.238916
\(401\) 35.8203 1.78878 0.894390 0.447289i \(-0.147610\pi\)
0.894390 + 0.447289i \(0.147610\pi\)
\(402\) 3.85605 0.192322
\(403\) 0 0
\(404\) −12.1844 −0.606197
\(405\) −34.8393 −1.73118
\(406\) 0.928927 0.0461019
\(407\) 23.5312 1.16640
\(408\) −15.6099 −0.772804
\(409\) 28.5608 1.41224 0.706121 0.708091i \(-0.250444\pi\)
0.706121 + 0.708091i \(0.250444\pi\)
\(410\) −0.368806 −0.0182140
\(411\) −9.99847 −0.493188
\(412\) 10.5338 0.518961
\(413\) −23.1723 −1.14023
\(414\) −1.17902 −0.0579454
\(415\) −39.3642 −1.93231
\(416\) 0 0
\(417\) −32.7687 −1.60469
\(418\) −3.12703 −0.152948
\(419\) −39.0336 −1.90692 −0.953459 0.301522i \(-0.902505\pi\)
−0.953459 + 0.301522i \(0.902505\pi\)
\(420\) 14.9773 0.730819
\(421\) 35.6361 1.73680 0.868398 0.495867i \(-0.165150\pi\)
0.868398 + 0.495867i \(0.165150\pi\)
\(422\) −4.98682 −0.242755
\(423\) 11.2103 0.545065
\(424\) 4.11436 0.199811
\(425\) −33.9404 −1.64635
\(426\) 22.4921 1.08974
\(427\) 18.8696 0.913164
\(428\) 5.67809 0.274461
\(429\) 0 0
\(430\) 7.59411 0.366220
\(431\) 28.5835 1.37682 0.688408 0.725323i \(-0.258310\pi\)
0.688408 + 0.725323i \(0.258310\pi\)
\(432\) −2.57201 −0.123746
\(433\) −24.3228 −1.16888 −0.584440 0.811437i \(-0.698686\pi\)
−0.584440 + 0.811437i \(0.698686\pi\)
\(434\) −10.5737 −0.507555
\(435\) 2.92905 0.140437
\(436\) −11.7186 −0.561217
\(437\) 0.644393 0.0308255
\(438\) 29.3941 1.40451
\(439\) −3.40022 −0.162283 −0.0811417 0.996703i \(-0.525857\pi\)
−0.0811417 + 0.996703i \(0.525857\pi\)
\(440\) 9.77832 0.466163
\(441\) −4.11682 −0.196039
\(442\) 0 0
\(443\) 16.5029 0.784075 0.392038 0.919949i \(-0.371770\pi\)
0.392038 + 0.919949i \(0.371770\pi\)
\(444\) 16.5375 0.784835
\(445\) 4.49947 0.213295
\(446\) −22.9805 −1.08816
\(447\) 49.4552 2.33915
\(448\) 2.17944 0.102969
\(449\) 7.72011 0.364335 0.182167 0.983268i \(-0.441689\pi\)
0.182167 + 0.983268i \(0.441689\pi\)
\(450\) 8.74267 0.412133
\(451\) −0.368806 −0.0173664
\(452\) −0.945828 −0.0444880
\(453\) 20.4395 0.960333
\(454\) 5.49722 0.257997
\(455\) 0 0
\(456\) −2.19765 −0.102914
\(457\) 35.4938 1.66033 0.830165 0.557518i \(-0.188246\pi\)
0.830165 + 0.557518i \(0.188246\pi\)
\(458\) −1.52318 −0.0711737
\(459\) 18.2690 0.852724
\(460\) −2.01504 −0.0939516
\(461\) −14.3955 −0.670467 −0.335233 0.942135i \(-0.608815\pi\)
−0.335233 + 0.942135i \(0.608815\pi\)
\(462\) 14.9773 0.696809
\(463\) 2.13703 0.0993164 0.0496582 0.998766i \(-0.484187\pi\)
0.0496582 + 0.998766i \(0.484187\pi\)
\(464\) 0.426223 0.0197869
\(465\) −33.3406 −1.54613
\(466\) 25.9377 1.20154
\(467\) 22.3298 1.03330 0.516651 0.856196i \(-0.327179\pi\)
0.516651 + 0.856196i \(0.327179\pi\)
\(468\) 0 0
\(469\) 3.82410 0.176580
\(470\) 19.1594 0.883758
\(471\) 5.68757 0.262070
\(472\) −10.6322 −0.489388
\(473\) 7.59411 0.349177
\(474\) 24.9149 1.14438
\(475\) −4.77832 −0.219244
\(476\) −15.4805 −0.709549
\(477\) 7.52785 0.344676
\(478\) −5.55924 −0.254274
\(479\) 6.43997 0.294250 0.147125 0.989118i \(-0.452998\pi\)
0.147125 + 0.989118i \(0.452998\pi\)
\(480\) 6.87211 0.313667
\(481\) 0 0
\(482\) 11.9583 0.544686
\(483\) −3.08641 −0.140436
\(484\) −1.22168 −0.0555308
\(485\) −37.3199 −1.69461
\(486\) −16.7687 −0.760643
\(487\) −34.7098 −1.57285 −0.786427 0.617684i \(-0.788071\pi\)
−0.786427 + 0.617684i \(0.788071\pi\)
\(488\) 8.65802 0.391930
\(489\) −49.0453 −2.21790
\(490\) −7.03599 −0.317853
\(491\) 27.6996 1.25006 0.625032 0.780599i \(-0.285086\pi\)
0.625032 + 0.780599i \(0.285086\pi\)
\(492\) −0.259193 −0.0116853
\(493\) −3.02746 −0.136350
\(494\) 0 0
\(495\) 17.8909 0.804137
\(496\) −4.85159 −0.217843
\(497\) 22.3057 1.00055
\(498\) −27.6648 −1.23969
\(499\) −10.0199 −0.448551 −0.224276 0.974526i \(-0.572002\pi\)
−0.224276 + 0.974526i \(0.572002\pi\)
\(500\) −0.693196 −0.0310007
\(501\) −51.0991 −2.28294
\(502\) −22.4814 −1.00340
\(503\) −0.0625293 −0.00278804 −0.00139402 0.999999i \(-0.500444\pi\)
−0.00139402 + 0.999999i \(0.500444\pi\)
\(504\) 3.98761 0.177622
\(505\) −38.1010 −1.69547
\(506\) −2.01504 −0.0895793
\(507\) 0 0
\(508\) 9.51474 0.422148
\(509\) −26.8041 −1.18807 −0.594035 0.804439i \(-0.702466\pi\)
−0.594035 + 0.804439i \(0.702466\pi\)
\(510\) −48.8126 −2.16146
\(511\) 29.1506 1.28954
\(512\) 1.00000 0.0441942
\(513\) 2.57201 0.113557
\(514\) 4.85032 0.213938
\(515\) 32.9394 1.45148
\(516\) 5.33707 0.234951
\(517\) 19.1594 0.842630
\(518\) 16.4005 0.720595
\(519\) −47.8420 −2.10003
\(520\) 0 0
\(521\) −21.2646 −0.931619 −0.465810 0.884885i \(-0.654237\pi\)
−0.465810 + 0.884885i \(0.654237\pi\)
\(522\) 0.779840 0.0341327
\(523\) −11.6367 −0.508838 −0.254419 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(524\) 2.94454 0.128633
\(525\) 22.8864 0.998845
\(526\) −10.1214 −0.441312
\(527\) 34.4608 1.50114
\(528\) 6.87211 0.299070
\(529\) −22.5848 −0.981946
\(530\) 12.8657 0.558852
\(531\) −19.4533 −0.844200
\(532\) −2.17944 −0.0944906
\(533\) 0 0
\(534\) 3.16218 0.136841
\(535\) 17.7556 0.767640
\(536\) 1.75463 0.0757883
\(537\) −54.0943 −2.33434
\(538\) 7.99747 0.344795
\(539\) −7.03599 −0.303061
\(540\) −8.04276 −0.346105
\(541\) 11.0025 0.473033 0.236517 0.971627i \(-0.423994\pi\)
0.236517 + 0.971627i \(0.423994\pi\)
\(542\) 9.26475 0.397955
\(543\) −8.52252 −0.365736
\(544\) −7.10300 −0.304539
\(545\) −36.6443 −1.56967
\(546\) 0 0
\(547\) −42.3946 −1.81266 −0.906332 0.422566i \(-0.861129\pi\)
−0.906332 + 0.422566i \(0.861129\pi\)
\(548\) −4.54962 −0.194350
\(549\) 15.8412 0.676084
\(550\) 14.9420 0.637127
\(551\) −0.426223 −0.0181577
\(552\) −1.41615 −0.0602753
\(553\) 24.7084 1.05071
\(554\) 14.1772 0.602330
\(555\) 51.7133 2.19511
\(556\) −14.9108 −0.632359
\(557\) 21.7486 0.921519 0.460760 0.887525i \(-0.347577\pi\)
0.460760 + 0.887525i \(0.347577\pi\)
\(558\) −8.87672 −0.375781
\(559\) 0 0
\(560\) 6.81517 0.287993
\(561\) −48.8126 −2.06087
\(562\) −27.3738 −1.15470
\(563\) −6.23601 −0.262817 −0.131408 0.991328i \(-0.541950\pi\)
−0.131408 + 0.991328i \(0.541950\pi\)
\(564\) 13.4651 0.566981
\(565\) −2.95763 −0.124429
\(566\) 23.5136 0.988349
\(567\) −24.2818 −1.01974
\(568\) 10.2346 0.429435
\(569\) −24.7089 −1.03585 −0.517925 0.855426i \(-0.673295\pi\)
−0.517925 + 0.855426i \(0.673295\pi\)
\(570\) −6.87211 −0.287841
\(571\) 12.5056 0.523342 0.261671 0.965157i \(-0.415727\pi\)
0.261671 + 0.965157i \(0.415727\pi\)
\(572\) 0 0
\(573\) 31.9292 1.33386
\(574\) −0.257046 −0.0107289
\(575\) −3.07912 −0.128408
\(576\) 1.82965 0.0762355
\(577\) −23.2185 −0.966599 −0.483300 0.875455i \(-0.660562\pi\)
−0.483300 + 0.875455i \(0.660562\pi\)
\(578\) 33.4526 1.39144
\(579\) −56.0592 −2.32974
\(580\) 1.33281 0.0553420
\(581\) −27.4355 −1.13822
\(582\) −26.2281 −1.08719
\(583\) 12.8657 0.532844
\(584\) 13.3753 0.553473
\(585\) 0 0
\(586\) 11.1201 0.459368
\(587\) −32.3213 −1.33404 −0.667022 0.745038i \(-0.732431\pi\)
−0.667022 + 0.745038i \(0.732431\pi\)
\(588\) −4.94482 −0.203921
\(589\) 4.85159 0.199906
\(590\) −33.2473 −1.36877
\(591\) −11.2735 −0.463729
\(592\) 7.52510 0.309280
\(593\) −19.5435 −0.802556 −0.401278 0.915956i \(-0.631434\pi\)
−0.401278 + 0.915956i \(0.631434\pi\)
\(594\) −8.04276 −0.329998
\(595\) −48.4081 −1.98454
\(596\) 22.5037 0.921788
\(597\) −14.4774 −0.592522
\(598\) 0 0
\(599\) 23.8796 0.975692 0.487846 0.872930i \(-0.337783\pi\)
0.487846 + 0.872930i \(0.337783\pi\)
\(600\) 10.5011 0.428704
\(601\) −1.52829 −0.0623402 −0.0311701 0.999514i \(-0.509923\pi\)
−0.0311701 + 0.999514i \(0.509923\pi\)
\(602\) 5.29284 0.215720
\(603\) 3.21035 0.130736
\(604\) 9.30065 0.378438
\(605\) −3.82023 −0.155314
\(606\) −26.7770 −1.08774
\(607\) 43.4059 1.76179 0.880896 0.473309i \(-0.156941\pi\)
0.880896 + 0.473309i \(0.156941\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 2.04145 0.0827238
\(610\) 27.0739 1.09619
\(611\) 0 0
\(612\) −12.9960 −0.525333
\(613\) 36.6253 1.47928 0.739641 0.673001i \(-0.234995\pi\)
0.739641 + 0.673001i \(0.234995\pi\)
\(614\) −10.2579 −0.413975
\(615\) −0.810506 −0.0326827
\(616\) 6.81517 0.274591
\(617\) 37.2515 1.49969 0.749844 0.661614i \(-0.230128\pi\)
0.749844 + 0.661614i \(0.230128\pi\)
\(618\) 23.1495 0.931209
\(619\) 16.3273 0.656249 0.328124 0.944635i \(-0.393584\pi\)
0.328124 + 0.944635i \(0.393584\pi\)
\(620\) −15.1711 −0.609284
\(621\) 1.65739 0.0665086
\(622\) 13.8693 0.556108
\(623\) 3.13598 0.125640
\(624\) 0 0
\(625\) −26.0593 −1.04237
\(626\) −3.37717 −0.134979
\(627\) −6.87211 −0.274446
\(628\) 2.58803 0.103274
\(629\) −53.4507 −2.13122
\(630\) 12.4694 0.496792
\(631\) −6.00903 −0.239216 −0.119608 0.992821i \(-0.538164\pi\)
−0.119608 + 0.992821i \(0.538164\pi\)
\(632\) 11.3371 0.450964
\(633\) −10.9593 −0.435592
\(634\) 6.12834 0.243387
\(635\) 29.7529 1.18071
\(636\) 9.04191 0.358535
\(637\) 0 0
\(638\) 1.33281 0.0527666
\(639\) 18.7258 0.740781
\(640\) 3.12703 0.123607
\(641\) −14.9927 −0.592178 −0.296089 0.955160i \(-0.595682\pi\)
−0.296089 + 0.955160i \(0.595682\pi\)
\(642\) 12.4784 0.492485
\(643\) 48.0012 1.89298 0.946492 0.322727i \(-0.104600\pi\)
0.946492 + 0.322727i \(0.104600\pi\)
\(644\) −1.40441 −0.0553417
\(645\) 16.6892 0.657135
\(646\) 7.10300 0.279464
\(647\) 37.2312 1.46371 0.731855 0.681460i \(-0.238655\pi\)
0.731855 + 0.681460i \(0.238655\pi\)
\(648\) −11.1413 −0.437673
\(649\) −33.2473 −1.30507
\(650\) 0 0
\(651\) −23.2373 −0.910742
\(652\) −22.3172 −0.874008
\(653\) 36.6613 1.43467 0.717334 0.696730i \(-0.245362\pi\)
0.717334 + 0.696730i \(0.245362\pi\)
\(654\) −25.7533 −1.00703
\(655\) 9.20767 0.359773
\(656\) −0.117941 −0.00460483
\(657\) 24.4721 0.954747
\(658\) 13.3535 0.520573
\(659\) 22.8356 0.889550 0.444775 0.895642i \(-0.353284\pi\)
0.444775 + 0.895642i \(0.353284\pi\)
\(660\) 21.4893 0.836470
\(661\) 44.8720 1.74532 0.872660 0.488329i \(-0.162393\pi\)
0.872660 + 0.488329i \(0.162393\pi\)
\(662\) 0.0943337 0.00366638
\(663\) 0 0
\(664\) −12.5884 −0.488523
\(665\) −6.81517 −0.264281
\(666\) 13.7683 0.533511
\(667\) −0.274655 −0.0106347
\(668\) −23.2517 −0.899636
\(669\) −50.5031 −1.95256
\(670\) 5.48677 0.211972
\(671\) 27.0739 1.04518
\(672\) 4.78963 0.184764
\(673\) −7.38846 −0.284804 −0.142402 0.989809i \(-0.545483\pi\)
−0.142402 + 0.989809i \(0.545483\pi\)
\(674\) −9.25406 −0.356453
\(675\) −12.2899 −0.473038
\(676\) 0 0
\(677\) 12.5549 0.482526 0.241263 0.970460i \(-0.422438\pi\)
0.241263 + 0.970460i \(0.422438\pi\)
\(678\) −2.07860 −0.0798280
\(679\) −26.0107 −0.998200
\(680\) −22.2113 −0.851764
\(681\) 12.0809 0.462943
\(682\) −15.1711 −0.580930
\(683\) 30.5047 1.16723 0.583616 0.812030i \(-0.301637\pi\)
0.583616 + 0.812030i \(0.301637\pi\)
\(684\) −1.82965 −0.0699585
\(685\) −14.2268 −0.543579
\(686\) −20.1599 −0.769709
\(687\) −3.34742 −0.127712
\(688\) 2.42854 0.0925871
\(689\) 0 0
\(690\) −4.42834 −0.168584
\(691\) −4.48550 −0.170637 −0.0853183 0.996354i \(-0.527191\pi\)
−0.0853183 + 0.996354i \(0.527191\pi\)
\(692\) −21.7696 −0.827557
\(693\) 12.4694 0.473673
\(694\) −29.1677 −1.10719
\(695\) −46.6266 −1.76865
\(696\) 0.936688 0.0355051
\(697\) 0.837737 0.0317315
\(698\) −29.5303 −1.11774
\(699\) 57.0019 2.15601
\(700\) 10.4141 0.393614
\(701\) −6.07918 −0.229607 −0.114804 0.993388i \(-0.536624\pi\)
−0.114804 + 0.993388i \(0.536624\pi\)
\(702\) 0 0
\(703\) −7.52510 −0.283814
\(704\) 3.12703 0.117854
\(705\) 42.1056 1.58579
\(706\) −29.0884 −1.09476
\(707\) −26.5551 −0.998709
\(708\) −23.3659 −0.878143
\(709\) −45.0758 −1.69286 −0.846429 0.532502i \(-0.821252\pi\)
−0.846429 + 0.532502i \(0.821252\pi\)
\(710\) 32.0040 1.20109
\(711\) 20.7429 0.777919
\(712\) 1.43889 0.0539248
\(713\) 3.12633 0.117082
\(714\) −34.0208 −1.27320
\(715\) 0 0
\(716\) −24.6147 −0.919893
\(717\) −12.2172 −0.456262
\(718\) −19.0423 −0.710654
\(719\) 24.2298 0.903621 0.451810 0.892114i \(-0.350778\pi\)
0.451810 + 0.892114i \(0.350778\pi\)
\(720\) 5.72138 0.213223
\(721\) 22.9577 0.854988
\(722\) 1.00000 0.0372161
\(723\) 26.2801 0.977369
\(724\) −3.87802 −0.144126
\(725\) 2.03663 0.0756386
\(726\) −2.68482 −0.0996430
\(727\) −10.3596 −0.384218 −0.192109 0.981374i \(-0.561533\pi\)
−0.192109 + 0.981374i \(0.561533\pi\)
\(728\) 0 0
\(729\) −3.42764 −0.126950
\(730\) 41.8249 1.54801
\(731\) −17.2499 −0.638010
\(732\) 19.0273 0.703268
\(733\) −24.3843 −0.900654 −0.450327 0.892864i \(-0.648693\pi\)
−0.450327 + 0.892864i \(0.648693\pi\)
\(734\) 1.02770 0.0379332
\(735\) −15.4626 −0.570347
\(736\) −0.644393 −0.0237526
\(737\) 5.48677 0.202108
\(738\) −0.215792 −0.00794340
\(739\) −1.07875 −0.0396825 −0.0198413 0.999803i \(-0.506316\pi\)
−0.0198413 + 0.999803i \(0.506316\pi\)
\(740\) 23.5312 0.865024
\(741\) 0 0
\(742\) 8.96699 0.329188
\(743\) −3.38672 −0.124247 −0.0621233 0.998068i \(-0.519787\pi\)
−0.0621233 + 0.998068i \(0.519787\pi\)
\(744\) −10.6621 −0.390891
\(745\) 70.3698 2.57815
\(746\) 15.9433 0.583725
\(747\) −23.0323 −0.842708
\(748\) −22.2113 −0.812125
\(749\) 12.3750 0.452174
\(750\) −1.52340 −0.0556267
\(751\) 25.1897 0.919186 0.459593 0.888130i \(-0.347995\pi\)
0.459593 + 0.888130i \(0.347995\pi\)
\(752\) 6.12703 0.223430
\(753\) −49.4063 −1.80046
\(754\) 0 0
\(755\) 29.0834 1.05845
\(756\) −5.60554 −0.203871
\(757\) 42.8857 1.55871 0.779353 0.626585i \(-0.215548\pi\)
0.779353 + 0.626585i \(0.215548\pi\)
\(758\) 4.79302 0.174090
\(759\) −4.42834 −0.160739
\(760\) −3.12703 −0.113429
\(761\) 22.8112 0.826906 0.413453 0.910525i \(-0.364323\pi\)
0.413453 + 0.910525i \(0.364323\pi\)
\(762\) 20.9100 0.757491
\(763\) −25.5399 −0.924605
\(764\) 14.5288 0.525634
\(765\) −40.6389 −1.46930
\(766\) −21.8340 −0.788895
\(767\) 0 0
\(768\) 2.19765 0.0793008
\(769\) 16.1128 0.581042 0.290521 0.956869i \(-0.406171\pi\)
0.290521 + 0.956869i \(0.406171\pi\)
\(770\) 21.3112 0.768004
\(771\) 10.6593 0.383885
\(772\) −25.5087 −0.918080
\(773\) 33.3582 1.19981 0.599905 0.800071i \(-0.295205\pi\)
0.599905 + 0.800071i \(0.295205\pi\)
\(774\) 4.44338 0.159714
\(775\) −23.1824 −0.832738
\(776\) −11.9346 −0.428428
\(777\) 36.0424 1.29302
\(778\) 16.3944 0.587769
\(779\) 0.117941 0.00422569
\(780\) 0 0
\(781\) 32.0040 1.14519
\(782\) 4.57712 0.163678
\(783\) −1.09625 −0.0391768
\(784\) −2.25005 −0.0803591
\(785\) 8.09285 0.288846
\(786\) 6.47106 0.230815
\(787\) −14.1846 −0.505628 −0.252814 0.967515i \(-0.581356\pi\)
−0.252814 + 0.967515i \(0.581356\pi\)
\(788\) −5.12980 −0.182742
\(789\) −22.2432 −0.791877
\(790\) 35.4514 1.26130
\(791\) −2.06137 −0.0732940
\(792\) 5.72138 0.203300
\(793\) 0 0
\(794\) 8.32544 0.295459
\(795\) 28.2743 1.00279
\(796\) −6.58770 −0.233495
\(797\) −16.5599 −0.586581 −0.293291 0.956023i \(-0.594750\pi\)
−0.293291 + 0.956023i \(0.594750\pi\)
\(798\) −4.78963 −0.169551
\(799\) −43.5203 −1.53964
\(800\) 4.77832 0.168939
\(801\) 2.63268 0.0930210
\(802\) 35.8203 1.26486
\(803\) 41.8249 1.47597
\(804\) 3.85605 0.135992
\(805\) −4.39165 −0.154785
\(806\) 0 0
\(807\) 17.5756 0.618691
\(808\) −12.1844 −0.428646
\(809\) 22.3953 0.787378 0.393689 0.919244i \(-0.371199\pi\)
0.393689 + 0.919244i \(0.371199\pi\)
\(810\) −34.8393 −1.22413
\(811\) −28.1667 −0.989066 −0.494533 0.869159i \(-0.664661\pi\)
−0.494533 + 0.869159i \(0.664661\pi\)
\(812\) 0.928927 0.0325989
\(813\) 20.3607 0.714080
\(814\) 23.5312 0.824768
\(815\) −69.7865 −2.44452
\(816\) −15.6099 −0.546455
\(817\) −2.42854 −0.0849637
\(818\) 28.5608 0.998606
\(819\) 0 0
\(820\) −0.368806 −0.0128793
\(821\) −10.2960 −0.359334 −0.179667 0.983727i \(-0.557502\pi\)
−0.179667 + 0.983727i \(0.557502\pi\)
\(822\) −9.99847 −0.348737
\(823\) 18.1042 0.631074 0.315537 0.948913i \(-0.397815\pi\)
0.315537 + 0.948913i \(0.397815\pi\)
\(824\) 10.5338 0.366961
\(825\) 32.8371 1.14324
\(826\) −23.1723 −0.806266
\(827\) −0.466529 −0.0162228 −0.00811139 0.999967i \(-0.502582\pi\)
−0.00811139 + 0.999967i \(0.502582\pi\)
\(828\) −1.17902 −0.0409736
\(829\) −14.4832 −0.503023 −0.251511 0.967854i \(-0.580928\pi\)
−0.251511 + 0.967854i \(0.580928\pi\)
\(830\) −39.3642 −1.36635
\(831\) 31.1564 1.08080
\(832\) 0 0
\(833\) 15.9821 0.553748
\(834\) −32.7687 −1.13469
\(835\) −72.7089 −2.51619
\(836\) −3.12703 −0.108151
\(837\) 12.4783 0.431314
\(838\) −39.0336 −1.34839
\(839\) −0.938352 −0.0323955 −0.0161978 0.999869i \(-0.505156\pi\)
−0.0161978 + 0.999869i \(0.505156\pi\)
\(840\) 14.9773 0.516767
\(841\) −28.8183 −0.993736
\(842\) 35.6361 1.22810
\(843\) −60.1580 −2.07195
\(844\) −4.98682 −0.171654
\(845\) 0 0
\(846\) 11.2103 0.385419
\(847\) −2.66257 −0.0914871
\(848\) 4.11436 0.141288
\(849\) 51.6745 1.77346
\(850\) −33.9404 −1.16415
\(851\) −4.84912 −0.166226
\(852\) 22.4921 0.770566
\(853\) −16.5020 −0.565018 −0.282509 0.959265i \(-0.591167\pi\)
−0.282509 + 0.959265i \(0.591167\pi\)
\(854\) 18.8696 0.645705
\(855\) −5.72138 −0.195667
\(856\) 5.67809 0.194073
\(857\) −1.55202 −0.0530159 −0.0265079 0.999649i \(-0.508439\pi\)
−0.0265079 + 0.999649i \(0.508439\pi\)
\(858\) 0 0
\(859\) 18.1217 0.618306 0.309153 0.951012i \(-0.399955\pi\)
0.309153 + 0.951012i \(0.399955\pi\)
\(860\) 7.59411 0.258957
\(861\) −0.564896 −0.0192516
\(862\) 28.5835 0.973557
\(863\) 47.5374 1.61819 0.809096 0.587676i \(-0.199957\pi\)
0.809096 + 0.587676i \(0.199957\pi\)
\(864\) −2.57201 −0.0875016
\(865\) −68.0743 −2.31460
\(866\) −24.3228 −0.826524
\(867\) 73.5170 2.49677
\(868\) −10.5737 −0.358896
\(869\) 35.4514 1.20260
\(870\) 2.92905 0.0993042
\(871\) 0 0
\(872\) −11.7186 −0.396841
\(873\) −21.8362 −0.739043
\(874\) 0.644393 0.0217969
\(875\) −1.51078 −0.0510736
\(876\) 29.3941 0.993135
\(877\) −53.0121 −1.79009 −0.895046 0.445973i \(-0.852858\pi\)
−0.895046 + 0.445973i \(0.852858\pi\)
\(878\) −3.40022 −0.114752
\(879\) 24.4381 0.824278
\(880\) 9.77832 0.329627
\(881\) −26.0499 −0.877645 −0.438822 0.898574i \(-0.644604\pi\)
−0.438822 + 0.898574i \(0.644604\pi\)
\(882\) −4.11682 −0.138620
\(883\) 36.9504 1.24348 0.621740 0.783224i \(-0.286426\pi\)
0.621740 + 0.783224i \(0.286426\pi\)
\(884\) 0 0
\(885\) −73.0658 −2.45608
\(886\) 16.5029 0.554425
\(887\) 42.5093 1.42732 0.713661 0.700491i \(-0.247036\pi\)
0.713661 + 0.700491i \(0.247036\pi\)
\(888\) 16.5375 0.554962
\(889\) 20.7368 0.695489
\(890\) 4.49947 0.150822
\(891\) −34.8393 −1.16716
\(892\) −22.9805 −0.769445
\(893\) −6.12703 −0.205033
\(894\) 49.4552 1.65403
\(895\) −76.9708 −2.57285
\(896\) 2.17944 0.0728099
\(897\) 0 0
\(898\) 7.72011 0.257623
\(899\) −2.06786 −0.0689669
\(900\) 8.74267 0.291422
\(901\) −29.2243 −0.973602
\(902\) −0.368806 −0.0122799
\(903\) 11.6318 0.387082
\(904\) −0.945828 −0.0314578
\(905\) −12.1267 −0.403105
\(906\) 20.4395 0.679058
\(907\) 4.12649 0.137018 0.0685090 0.997651i \(-0.478176\pi\)
0.0685090 + 0.997651i \(0.478176\pi\)
\(908\) 5.49722 0.182432
\(909\) −22.2932 −0.739419
\(910\) 0 0
\(911\) 14.4605 0.479097 0.239549 0.970884i \(-0.423001\pi\)
0.239549 + 0.970884i \(0.423001\pi\)
\(912\) −2.19765 −0.0727714
\(913\) −39.3642 −1.30276
\(914\) 35.4938 1.17403
\(915\) 59.4989 1.96697
\(916\) −1.52318 −0.0503274
\(917\) 6.41744 0.211923
\(918\) 18.2690 0.602967
\(919\) 59.7790 1.97193 0.985963 0.166965i \(-0.0533967\pi\)
0.985963 + 0.166965i \(0.0533967\pi\)
\(920\) −2.01504 −0.0664338
\(921\) −22.5433 −0.742826
\(922\) −14.3955 −0.474092
\(923\) 0 0
\(924\) 14.9773 0.492718
\(925\) 35.9573 1.18227
\(926\) 2.13703 0.0702273
\(927\) 19.2731 0.633012
\(928\) 0.426223 0.0139915
\(929\) 5.85030 0.191942 0.0959711 0.995384i \(-0.469404\pi\)
0.0959711 + 0.995384i \(0.469404\pi\)
\(930\) −33.3406 −1.09328
\(931\) 2.25005 0.0737425
\(932\) 25.9377 0.849618
\(933\) 30.4798 0.997864
\(934\) 22.3298 0.730654
\(935\) −69.4554 −2.27143
\(936\) 0 0
\(937\) −28.4140 −0.928243 −0.464122 0.885771i \(-0.653630\pi\)
−0.464122 + 0.885771i \(0.653630\pi\)
\(938\) 3.82410 0.124861
\(939\) −7.42184 −0.242203
\(940\) 19.1594 0.624911
\(941\) 5.81060 0.189420 0.0947101 0.995505i \(-0.469808\pi\)
0.0947101 + 0.995505i \(0.469808\pi\)
\(942\) 5.68757 0.185311
\(943\) 0.0760006 0.00247492
\(944\) −10.6322 −0.346049
\(945\) −17.5287 −0.570208
\(946\) 7.59411 0.246906
\(947\) 8.52929 0.277165 0.138582 0.990351i \(-0.455745\pi\)
0.138582 + 0.990351i \(0.455745\pi\)
\(948\) 24.9149 0.809197
\(949\) 0 0
\(950\) −4.77832 −0.155029
\(951\) 13.4679 0.436727
\(952\) −15.4805 −0.501727
\(953\) 37.1660 1.20392 0.601962 0.798525i \(-0.294386\pi\)
0.601962 + 0.798525i \(0.294386\pi\)
\(954\) 7.52785 0.243723
\(955\) 45.4321 1.47015
\(956\) −5.55924 −0.179799
\(957\) 2.92905 0.0946828
\(958\) 6.43997 0.208066
\(959\) −9.91562 −0.320192
\(960\) 6.87211 0.221796
\(961\) −7.46211 −0.240713
\(962\) 0 0
\(963\) 10.3889 0.334779
\(964\) 11.9583 0.385151
\(965\) −79.7666 −2.56778
\(966\) −3.08641 −0.0993035
\(967\) −38.7782 −1.24702 −0.623511 0.781815i \(-0.714294\pi\)
−0.623511 + 0.781815i \(0.714294\pi\)
\(968\) −1.22168 −0.0392662
\(969\) 15.6099 0.501462
\(970\) −37.3199 −1.19827
\(971\) 4.84313 0.155424 0.0777118 0.996976i \(-0.475239\pi\)
0.0777118 + 0.996976i \(0.475239\pi\)
\(972\) −16.7687 −0.537856
\(973\) −32.4972 −1.04181
\(974\) −34.7098 −1.11218
\(975\) 0 0
\(976\) 8.65802 0.277136
\(977\) −20.1405 −0.644352 −0.322176 0.946680i \(-0.604414\pi\)
−0.322176 + 0.946680i \(0.604414\pi\)
\(978\) −49.0453 −1.56830
\(979\) 4.49947 0.143804
\(980\) −7.03599 −0.224756
\(981\) −21.4409 −0.684555
\(982\) 27.6996 0.883929
\(983\) 11.3342 0.361506 0.180753 0.983529i \(-0.442147\pi\)
0.180753 + 0.983529i \(0.442147\pi\)
\(984\) −0.259193 −0.00826278
\(985\) −16.0410 −0.511110
\(986\) −3.02746 −0.0964140
\(987\) 29.3462 0.934101
\(988\) 0 0
\(989\) −1.56493 −0.0497619
\(990\) 17.8909 0.568611
\(991\) −52.9777 −1.68289 −0.841447 0.540340i \(-0.818296\pi\)
−0.841447 + 0.540340i \(0.818296\pi\)
\(992\) −4.85159 −0.154038
\(993\) 0.207312 0.00657885
\(994\) 22.3057 0.707494
\(995\) −20.5999 −0.653062
\(996\) −27.6648 −0.876592
\(997\) 54.6400 1.73047 0.865233 0.501369i \(-0.167170\pi\)
0.865233 + 0.501369i \(0.167170\pi\)
\(998\) −10.0199 −0.317174
\(999\) −19.3546 −0.612353
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.bd.1.5 6
13.3 even 3 494.2.g.f.191.2 12
13.9 even 3 494.2.g.f.419.2 yes 12
13.12 even 2 6422.2.a.bc.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
494.2.g.f.191.2 12 13.3 even 3
494.2.g.f.419.2 yes 12 13.9 even 3
6422.2.a.bc.1.5 6 13.12 even 2
6422.2.a.bd.1.5 6 1.1 even 1 trivial