Properties

Label 1334.2.a.h.1.5
Level $1334$
Weight $2$
Character 1334.1
Self dual yes
Analytic conductor $10.652$
Analytic rank $1$
Dimension $5$
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] = [1334,2,Mod(1,1334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1334, 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("1334.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1334 = 2 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1334.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: \(10.6520436296\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.179024.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 8x^{3} + 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.568906\) of defining polynomial
Character \(\chi\) \(=\) 1334.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.01243 q^{3} +1.00000 q^{4} -3.34851 q^{5} +2.01243 q^{6} -2.94661 q^{7} +1.00000 q^{8} +1.04987 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.01243 q^{3} +1.00000 q^{4} -3.34851 q^{5} +2.01243 q^{6} -2.94661 q^{7} +1.00000 q^{8} +1.04987 q^{9} -3.34851 q^{10} -4.10744 q^{11} +2.01243 q^{12} +0.449700 q^{13} -2.94661 q^{14} -6.73865 q^{15} +1.00000 q^{16} +1.49340 q^{17} +1.04987 q^{18} -7.57782 q^{19} -3.34851 q^{20} -5.92985 q^{21} -4.10744 q^{22} -1.00000 q^{23} +2.01243 q^{24} +6.21255 q^{25} +0.449700 q^{26} -3.92449 q^{27} -2.94661 q^{28} +1.00000 q^{29} -6.73865 q^{30} -1.22039 q^{31} +1.00000 q^{32} -8.26593 q^{33} +1.49340 q^{34} +9.86677 q^{35} +1.04987 q^{36} +7.68734 q^{37} -7.57782 q^{38} +0.904990 q^{39} -3.34851 q^{40} +5.92160 q^{41} -5.92985 q^{42} -8.61445 q^{43} -4.10744 q^{44} -3.51552 q^{45} -1.00000 q^{46} -7.72271 q^{47} +2.01243 q^{48} +1.68252 q^{49} +6.21255 q^{50} +3.00536 q^{51} +0.449700 q^{52} +13.3982 q^{53} -3.92449 q^{54} +13.7538 q^{55} -2.94661 q^{56} -15.2498 q^{57} +1.00000 q^{58} -7.90148 q^{59} -6.73865 q^{60} -3.37271 q^{61} -1.22039 q^{62} -3.09357 q^{63} +1.00000 q^{64} -1.50583 q^{65} -8.26593 q^{66} +1.93008 q^{67} +1.49340 q^{68} -2.01243 q^{69} +9.86677 q^{70} -5.38155 q^{71} +1.04987 q^{72} -4.60202 q^{73} +7.68734 q^{74} +12.5023 q^{75} -7.57782 q^{76} +12.1030 q^{77} +0.904990 q^{78} +9.25601 q^{79} -3.34851 q^{80} -11.0474 q^{81} +5.92160 q^{82} +3.54863 q^{83} -5.92985 q^{84} -5.00066 q^{85} -8.61445 q^{86} +2.01243 q^{87} -4.10744 q^{88} +4.12612 q^{89} -3.51552 q^{90} -1.32509 q^{91} -1.00000 q^{92} -2.45595 q^{93} -7.72271 q^{94} +25.3744 q^{95} +2.01243 q^{96} -8.83222 q^{97} +1.68252 q^{98} -4.31229 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} - 3 q^{3} + 5 q^{4} - 5 q^{5} - 3 q^{6} - 6 q^{7} + 5 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{2} - 3 q^{3} + 5 q^{4} - 5 q^{5} - 3 q^{6} - 6 q^{7} + 5 q^{8} + 2 q^{9} - 5 q^{10} - 9 q^{11} - 3 q^{12} - 5 q^{13} - 6 q^{14} - 3 q^{15} + 5 q^{16} - 6 q^{17} + 2 q^{18} - 10 q^{19} - 5 q^{20} - 2 q^{21} - 9 q^{22} - 5 q^{23} - 3 q^{24} - 4 q^{25} - 5 q^{26} - 9 q^{27} - 6 q^{28} + 5 q^{29} - 3 q^{30} - 15 q^{31} + 5 q^{32} - 15 q^{33} - 6 q^{34} - 2 q^{35} + 2 q^{36} - 10 q^{38} + 3 q^{39} - 5 q^{40} - 2 q^{41} - 2 q^{42} - 5 q^{43} - 9 q^{44} - 6 q^{45} - 5 q^{46} - 9 q^{47} - 3 q^{48} - 3 q^{49} - 4 q^{50} - 2 q^{51} - 5 q^{52} + 3 q^{53} - 9 q^{54} - 3 q^{55} - 6 q^{56} - 2 q^{57} + 5 q^{58} - 26 q^{59} - 3 q^{60} - 8 q^{61} - 15 q^{62} - 6 q^{63} + 5 q^{64} + 19 q^{65} - 15 q^{66} + 12 q^{67} - 6 q^{68} + 3 q^{69} - 2 q^{70} - 12 q^{71} + 2 q^{72} + 2 q^{73} + 24 q^{75} - 10 q^{76} + 36 q^{77} + 3 q^{78} - 25 q^{79} - 5 q^{80} + 9 q^{81} - 2 q^{82} - 16 q^{83} - 2 q^{84} + 4 q^{85} - 5 q^{86} - 3 q^{87} - 9 q^{88} - 20 q^{89} - 6 q^{90} - 32 q^{91} - 5 q^{92} + 11 q^{93} - 9 q^{94} + 20 q^{95} - 3 q^{96} - 4 q^{97} - 3 q^{98} + 20 q^{99}+O(q^{100}) \) 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
\(3\) 2.01243 1.16188 0.580938 0.813947i \(-0.302686\pi\)
0.580938 + 0.813947i \(0.302686\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.34851 −1.49750 −0.748751 0.662852i \(-0.769346\pi\)
−0.748751 + 0.662852i \(0.769346\pi\)
\(6\) 2.01243 0.821571
\(7\) −2.94661 −1.11371 −0.556857 0.830608i \(-0.687993\pi\)
−0.556857 + 0.830608i \(0.687993\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.04987 0.349958
\(10\) −3.34851 −1.05889
\(11\) −4.10744 −1.23844 −0.619220 0.785218i \(-0.712551\pi\)
−0.619220 + 0.785218i \(0.712551\pi\)
\(12\) 2.01243 0.580938
\(13\) 0.449700 0.124724 0.0623622 0.998054i \(-0.480137\pi\)
0.0623622 + 0.998054i \(0.480137\pi\)
\(14\) −2.94661 −0.787515
\(15\) −6.73865 −1.73991
\(16\) 1.00000 0.250000
\(17\) 1.49340 0.362202 0.181101 0.983464i \(-0.442034\pi\)
0.181101 + 0.983464i \(0.442034\pi\)
\(18\) 1.04987 0.247458
\(19\) −7.57782 −1.73847 −0.869236 0.494398i \(-0.835389\pi\)
−0.869236 + 0.494398i \(0.835389\pi\)
\(20\) −3.34851 −0.748751
\(21\) −5.92985 −1.29400
\(22\) −4.10744 −0.875709
\(23\) −1.00000 −0.208514
\(24\) 2.01243 0.410786
\(25\) 6.21255 1.24251
\(26\) 0.449700 0.0881935
\(27\) −3.92449 −0.755269
\(28\) −2.94661 −0.556857
\(29\) 1.00000 0.185695
\(30\) −6.73865 −1.23030
\(31\) −1.22039 −0.219189 −0.109594 0.993976i \(-0.534955\pi\)
−0.109594 + 0.993976i \(0.534955\pi\)
\(32\) 1.00000 0.176777
\(33\) −8.26593 −1.43891
\(34\) 1.49340 0.256116
\(35\) 9.86677 1.66779
\(36\) 1.04987 0.174979
\(37\) 7.68734 1.26379 0.631895 0.775054i \(-0.282277\pi\)
0.631895 + 0.775054i \(0.282277\pi\)
\(38\) −7.57782 −1.22929
\(39\) 0.904990 0.144914
\(40\) −3.34851 −0.529447
\(41\) 5.92160 0.924798 0.462399 0.886672i \(-0.346989\pi\)
0.462399 + 0.886672i \(0.346989\pi\)
\(42\) −5.92985 −0.914996
\(43\) −8.61445 −1.31369 −0.656846 0.754025i \(-0.728110\pi\)
−0.656846 + 0.754025i \(0.728110\pi\)
\(44\) −4.10744 −0.619220
\(45\) −3.51552 −0.524063
\(46\) −1.00000 −0.147442
\(47\) −7.72271 −1.12647 −0.563236 0.826296i \(-0.690444\pi\)
−0.563236 + 0.826296i \(0.690444\pi\)
\(48\) 2.01243 0.290469
\(49\) 1.68252 0.240360
\(50\) 6.21255 0.878587
\(51\) 3.00536 0.420834
\(52\) 0.449700 0.0623622
\(53\) 13.3982 1.84039 0.920194 0.391462i \(-0.128030\pi\)
0.920194 + 0.391462i \(0.128030\pi\)
\(54\) −3.92449 −0.534056
\(55\) 13.7538 1.85456
\(56\) −2.94661 −0.393758
\(57\) −15.2498 −2.01989
\(58\) 1.00000 0.131306
\(59\) −7.90148 −1.02868 −0.514342 0.857585i \(-0.671964\pi\)
−0.514342 + 0.857585i \(0.671964\pi\)
\(60\) −6.73865 −0.869956
\(61\) −3.37271 −0.431831 −0.215916 0.976412i \(-0.569274\pi\)
−0.215916 + 0.976412i \(0.569274\pi\)
\(62\) −1.22039 −0.154990
\(63\) −3.09357 −0.389753
\(64\) 1.00000 0.125000
\(65\) −1.50583 −0.186775
\(66\) −8.26593 −1.01747
\(67\) 1.93008 0.235797 0.117898 0.993026i \(-0.462384\pi\)
0.117898 + 0.993026i \(0.462384\pi\)
\(68\) 1.49340 0.181101
\(69\) −2.01243 −0.242268
\(70\) 9.86677 1.17930
\(71\) −5.38155 −0.638672 −0.319336 0.947641i \(-0.603460\pi\)
−0.319336 + 0.947641i \(0.603460\pi\)
\(72\) 1.04987 0.123729
\(73\) −4.60202 −0.538626 −0.269313 0.963053i \(-0.586797\pi\)
−0.269313 + 0.963053i \(0.586797\pi\)
\(74\) 7.68734 0.893635
\(75\) 12.5023 1.44364
\(76\) −7.57782 −0.869236
\(77\) 12.1030 1.37927
\(78\) 0.904990 0.102470
\(79\) 9.25601 1.04138 0.520691 0.853745i \(-0.325674\pi\)
0.520691 + 0.853745i \(0.325674\pi\)
\(80\) −3.34851 −0.374375
\(81\) −11.0474 −1.22749
\(82\) 5.92160 0.653931
\(83\) 3.54863 0.389513 0.194756 0.980852i \(-0.437608\pi\)
0.194756 + 0.980852i \(0.437608\pi\)
\(84\) −5.92985 −0.647000
\(85\) −5.00066 −0.542398
\(86\) −8.61445 −0.928920
\(87\) 2.01243 0.215755
\(88\) −4.10744 −0.437855
\(89\) 4.12612 0.437368 0.218684 0.975796i \(-0.429824\pi\)
0.218684 + 0.975796i \(0.429824\pi\)
\(90\) −3.51552 −0.370568
\(91\) −1.32509 −0.138907
\(92\) −1.00000 −0.104257
\(93\) −2.45595 −0.254671
\(94\) −7.72271 −0.796536
\(95\) 25.3744 2.60336
\(96\) 2.01243 0.205393
\(97\) −8.83222 −0.896776 −0.448388 0.893839i \(-0.648002\pi\)
−0.448388 + 0.893839i \(0.648002\pi\)
\(98\) 1.68252 0.169960
\(99\) −4.31229 −0.433402
\(100\) 6.21255 0.621255
\(101\) 17.2338 1.71483 0.857414 0.514627i \(-0.172069\pi\)
0.857414 + 0.514627i \(0.172069\pi\)
\(102\) 3.00536 0.297575
\(103\) −6.94133 −0.683950 −0.341975 0.939709i \(-0.611096\pi\)
−0.341975 + 0.939709i \(0.611096\pi\)
\(104\) 0.449700 0.0440967
\(105\) 19.8562 1.93777
\(106\) 13.3982 1.30135
\(107\) −4.72189 −0.456482 −0.228241 0.973605i \(-0.573297\pi\)
−0.228241 + 0.973605i \(0.573297\pi\)
\(108\) −3.92449 −0.377634
\(109\) 1.54601 0.148081 0.0740405 0.997255i \(-0.476411\pi\)
0.0740405 + 0.997255i \(0.476411\pi\)
\(110\) 13.7538 1.31138
\(111\) 15.4702 1.46837
\(112\) −2.94661 −0.278429
\(113\) −8.89955 −0.837200 −0.418600 0.908171i \(-0.637479\pi\)
−0.418600 + 0.908171i \(0.637479\pi\)
\(114\) −15.2498 −1.42828
\(115\) 3.34851 0.312251
\(116\) 1.00000 0.0928477
\(117\) 0.472129 0.0436483
\(118\) −7.90148 −0.727390
\(119\) −4.40046 −0.403390
\(120\) −6.73865 −0.615152
\(121\) 5.87106 0.533733
\(122\) −3.37271 −0.305351
\(123\) 11.9168 1.07450
\(124\) −1.22039 −0.109594
\(125\) −4.06023 −0.363158
\(126\) −3.09357 −0.275597
\(127\) −2.22191 −0.197162 −0.0985812 0.995129i \(-0.531430\pi\)
−0.0985812 + 0.995129i \(0.531430\pi\)
\(128\) 1.00000 0.0883883
\(129\) −17.3360 −1.52635
\(130\) −1.50583 −0.132070
\(131\) 14.5729 1.27324 0.636619 0.771179i \(-0.280332\pi\)
0.636619 + 0.771179i \(0.280332\pi\)
\(132\) −8.26593 −0.719457
\(133\) 22.3289 1.93616
\(134\) 1.93008 0.166734
\(135\) 13.1412 1.13102
\(136\) 1.49340 0.128058
\(137\) −7.57308 −0.647012 −0.323506 0.946226i \(-0.604862\pi\)
−0.323506 + 0.946226i \(0.604862\pi\)
\(138\) −2.01243 −0.171309
\(139\) −20.7952 −1.76382 −0.881912 0.471414i \(-0.843744\pi\)
−0.881912 + 0.471414i \(0.843744\pi\)
\(140\) 9.86677 0.833894
\(141\) −15.5414 −1.30882
\(142\) −5.38155 −0.451610
\(143\) −1.84712 −0.154464
\(144\) 1.04987 0.0874895
\(145\) −3.34851 −0.278079
\(146\) −4.60202 −0.380866
\(147\) 3.38596 0.279269
\(148\) 7.68734 0.631895
\(149\) 15.2059 1.24571 0.622857 0.782336i \(-0.285972\pi\)
0.622857 + 0.782336i \(0.285972\pi\)
\(150\) 12.5023 1.02081
\(151\) 7.25184 0.590146 0.295073 0.955475i \(-0.404656\pi\)
0.295073 + 0.955475i \(0.404656\pi\)
\(152\) −7.57782 −0.614643
\(153\) 1.56788 0.126756
\(154\) 12.1030 0.975290
\(155\) 4.08650 0.328236
\(156\) 0.904990 0.0724572
\(157\) −9.92692 −0.792255 −0.396127 0.918196i \(-0.629646\pi\)
−0.396127 + 0.918196i \(0.629646\pi\)
\(158\) 9.25601 0.736369
\(159\) 26.9630 2.13831
\(160\) −3.34851 −0.264723
\(161\) 2.94661 0.232226
\(162\) −11.0474 −0.867965
\(163\) 22.9592 1.79830 0.899152 0.437636i \(-0.144184\pi\)
0.899152 + 0.437636i \(0.144184\pi\)
\(164\) 5.92160 0.462399
\(165\) 27.6786 2.15478
\(166\) 3.54863 0.275427
\(167\) −3.46124 −0.267838 −0.133919 0.990992i \(-0.542756\pi\)
−0.133919 + 0.990992i \(0.542756\pi\)
\(168\) −5.92985 −0.457498
\(169\) −12.7978 −0.984444
\(170\) −5.00066 −0.383533
\(171\) −7.95576 −0.608392
\(172\) −8.61445 −0.656846
\(173\) 2.40357 0.182740 0.0913701 0.995817i \(-0.470875\pi\)
0.0913701 + 0.995817i \(0.470875\pi\)
\(174\) 2.01243 0.152562
\(175\) −18.3060 −1.38380
\(176\) −4.10744 −0.309610
\(177\) −15.9012 −1.19520
\(178\) 4.12612 0.309266
\(179\) −3.62288 −0.270787 −0.135393 0.990792i \(-0.543230\pi\)
−0.135393 + 0.990792i \(0.543230\pi\)
\(180\) −3.51552 −0.262031
\(181\) −3.44626 −0.256159 −0.128079 0.991764i \(-0.540881\pi\)
−0.128079 + 0.991764i \(0.540881\pi\)
\(182\) −1.32509 −0.0982224
\(183\) −6.78734 −0.501735
\(184\) −1.00000 −0.0737210
\(185\) −25.7412 −1.89253
\(186\) −2.45595 −0.180079
\(187\) −6.13404 −0.448566
\(188\) −7.72271 −0.563236
\(189\) 11.5640 0.841154
\(190\) 25.3744 1.84086
\(191\) −7.43775 −0.538177 −0.269088 0.963115i \(-0.586722\pi\)
−0.269088 + 0.963115i \(0.586722\pi\)
\(192\) 2.01243 0.145235
\(193\) −16.4333 −1.18290 −0.591449 0.806342i \(-0.701444\pi\)
−0.591449 + 0.806342i \(0.701444\pi\)
\(194\) −8.83222 −0.634117
\(195\) −3.03037 −0.217010
\(196\) 1.68252 0.120180
\(197\) −2.39554 −0.170675 −0.0853375 0.996352i \(-0.527197\pi\)
−0.0853375 + 0.996352i \(0.527197\pi\)
\(198\) −4.31229 −0.306461
\(199\) 19.1929 1.36055 0.680273 0.732959i \(-0.261861\pi\)
0.680273 + 0.732959i \(0.261861\pi\)
\(200\) 6.21255 0.439293
\(201\) 3.88415 0.273967
\(202\) 17.2338 1.21257
\(203\) −2.94661 −0.206812
\(204\) 3.00536 0.210417
\(205\) −19.8286 −1.38489
\(206\) −6.94133 −0.483625
\(207\) −1.04987 −0.0729713
\(208\) 0.449700 0.0311811
\(209\) 31.1254 2.15299
\(210\) 19.8562 1.37021
\(211\) 12.9238 0.889711 0.444855 0.895602i \(-0.353255\pi\)
0.444855 + 0.895602i \(0.353255\pi\)
\(212\) 13.3982 0.920194
\(213\) −10.8300 −0.742059
\(214\) −4.72189 −0.322782
\(215\) 28.8456 1.96725
\(216\) −3.92449 −0.267028
\(217\) 3.59602 0.244114
\(218\) 1.54601 0.104709
\(219\) −9.26124 −0.625817
\(220\) 13.7538 0.927282
\(221\) 0.671581 0.0451755
\(222\) 15.4702 1.03829
\(223\) −5.01497 −0.335827 −0.167914 0.985802i \(-0.553703\pi\)
−0.167914 + 0.985802i \(0.553703\pi\)
\(224\) −2.94661 −0.196879
\(225\) 6.52239 0.434826
\(226\) −8.89955 −0.591990
\(227\) −3.85327 −0.255751 −0.127875 0.991790i \(-0.540816\pi\)
−0.127875 + 0.991790i \(0.540816\pi\)
\(228\) −15.2498 −1.00995
\(229\) −23.5676 −1.55739 −0.778697 0.627400i \(-0.784119\pi\)
−0.778697 + 0.627400i \(0.784119\pi\)
\(230\) 3.34851 0.220794
\(231\) 24.3565 1.60254
\(232\) 1.00000 0.0656532
\(233\) −21.4376 −1.40442 −0.702212 0.711968i \(-0.747804\pi\)
−0.702212 + 0.711968i \(0.747804\pi\)
\(234\) 0.472129 0.0308640
\(235\) 25.8596 1.68689
\(236\) −7.90148 −0.514342
\(237\) 18.6271 1.20996
\(238\) −4.40046 −0.285240
\(239\) −28.6339 −1.85217 −0.926086 0.377312i \(-0.876849\pi\)
−0.926086 + 0.377312i \(0.876849\pi\)
\(240\) −6.73865 −0.434978
\(241\) −12.4373 −0.801160 −0.400580 0.916262i \(-0.631191\pi\)
−0.400580 + 0.916262i \(0.631191\pi\)
\(242\) 5.87106 0.377406
\(243\) −10.4586 −0.670921
\(244\) −3.37271 −0.215916
\(245\) −5.63395 −0.359940
\(246\) 11.9168 0.759787
\(247\) −3.40775 −0.216830
\(248\) −1.22039 −0.0774950
\(249\) 7.14137 0.452566
\(250\) −4.06023 −0.256791
\(251\) 12.5035 0.789215 0.394608 0.918850i \(-0.370881\pi\)
0.394608 + 0.918850i \(0.370881\pi\)
\(252\) −3.09357 −0.194877
\(253\) 4.10744 0.258233
\(254\) −2.22191 −0.139415
\(255\) −10.0635 −0.630200
\(256\) 1.00000 0.0625000
\(257\) −21.1475 −1.31914 −0.659571 0.751642i \(-0.729262\pi\)
−0.659571 + 0.751642i \(0.729262\pi\)
\(258\) −17.3360 −1.07929
\(259\) −22.6516 −1.40750
\(260\) −1.50583 −0.0933875
\(261\) 1.04987 0.0649856
\(262\) 14.5729 0.900315
\(263\) −13.8786 −0.855792 −0.427896 0.903828i \(-0.640745\pi\)
−0.427896 + 0.903828i \(0.640745\pi\)
\(264\) −8.26593 −0.508733
\(265\) −44.8642 −2.75598
\(266\) 22.3289 1.36907
\(267\) 8.30353 0.508168
\(268\) 1.93008 0.117898
\(269\) 18.7274 1.14183 0.570915 0.821009i \(-0.306589\pi\)
0.570915 + 0.821009i \(0.306589\pi\)
\(270\) 13.1412 0.799749
\(271\) −29.6416 −1.80060 −0.900300 0.435271i \(-0.856653\pi\)
−0.900300 + 0.435271i \(0.856653\pi\)
\(272\) 1.49340 0.0905505
\(273\) −2.66666 −0.161393
\(274\) −7.57308 −0.457507
\(275\) −25.5177 −1.53877
\(276\) −2.01243 −0.121134
\(277\) −28.1416 −1.69086 −0.845432 0.534083i \(-0.820657\pi\)
−0.845432 + 0.534083i \(0.820657\pi\)
\(278\) −20.7952 −1.24721
\(279\) −1.28126 −0.0767069
\(280\) 9.86677 0.589652
\(281\) 25.4697 1.51940 0.759698 0.650276i \(-0.225347\pi\)
0.759698 + 0.650276i \(0.225347\pi\)
\(282\) −15.5414 −0.925477
\(283\) −20.7801 −1.23525 −0.617625 0.786472i \(-0.711905\pi\)
−0.617625 + 0.786472i \(0.711905\pi\)
\(284\) −5.38155 −0.319336
\(285\) 51.0643 3.02479
\(286\) −1.84712 −0.109222
\(287\) −17.4487 −1.02996
\(288\) 1.04987 0.0618644
\(289\) −14.7698 −0.868810
\(290\) −3.34851 −0.196632
\(291\) −17.7742 −1.04194
\(292\) −4.60202 −0.269313
\(293\) 21.7496 1.27062 0.635312 0.772256i \(-0.280872\pi\)
0.635312 + 0.772256i \(0.280872\pi\)
\(294\) 3.38596 0.197473
\(295\) 26.4582 1.54046
\(296\) 7.68734 0.446817
\(297\) 16.1196 0.935355
\(298\) 15.2059 0.880853
\(299\) −0.449700 −0.0260068
\(300\) 12.5023 0.721821
\(301\) 25.3834 1.46308
\(302\) 7.25184 0.417296
\(303\) 34.6818 1.99242
\(304\) −7.57782 −0.434618
\(305\) 11.2936 0.646668
\(306\) 1.56788 0.0896297
\(307\) −30.9335 −1.76547 −0.882734 0.469873i \(-0.844300\pi\)
−0.882734 + 0.469873i \(0.844300\pi\)
\(308\) 12.1030 0.689634
\(309\) −13.9689 −0.794665
\(310\) 4.08650 0.232098
\(311\) 19.0093 1.07792 0.538960 0.842331i \(-0.318817\pi\)
0.538960 + 0.842331i \(0.318817\pi\)
\(312\) 0.904990 0.0512350
\(313\) 23.3187 1.31805 0.659026 0.752120i \(-0.270969\pi\)
0.659026 + 0.752120i \(0.270969\pi\)
\(314\) −9.92692 −0.560209
\(315\) 10.3589 0.583656
\(316\) 9.25601 0.520691
\(317\) 0.472925 0.0265621 0.0132811 0.999912i \(-0.495772\pi\)
0.0132811 + 0.999912i \(0.495772\pi\)
\(318\) 26.9630 1.51201
\(319\) −4.10744 −0.229972
\(320\) −3.34851 −0.187188
\(321\) −9.50247 −0.530376
\(322\) 2.94661 0.164208
\(323\) −11.3167 −0.629678
\(324\) −11.0474 −0.613744
\(325\) 2.79378 0.154971
\(326\) 22.9592 1.27159
\(327\) 3.11124 0.172052
\(328\) 5.92160 0.326966
\(329\) 22.7558 1.25457
\(330\) 27.6786 1.52366
\(331\) 11.1247 0.611471 0.305735 0.952116i \(-0.401098\pi\)
0.305735 + 0.952116i \(0.401098\pi\)
\(332\) 3.54863 0.194756
\(333\) 8.07074 0.442274
\(334\) −3.46124 −0.189390
\(335\) −6.46290 −0.353106
\(336\) −5.92985 −0.323500
\(337\) −2.61571 −0.142487 −0.0712433 0.997459i \(-0.522697\pi\)
−0.0712433 + 0.997459i \(0.522697\pi\)
\(338\) −12.7978 −0.696107
\(339\) −17.9097 −0.972723
\(340\) −5.00066 −0.271199
\(341\) 5.01269 0.271452
\(342\) −7.95576 −0.430198
\(343\) 15.6685 0.846022
\(344\) −8.61445 −0.464460
\(345\) 6.73865 0.362797
\(346\) 2.40357 0.129217
\(347\) 26.9380 1.44611 0.723054 0.690792i \(-0.242738\pi\)
0.723054 + 0.690792i \(0.242738\pi\)
\(348\) 2.01243 0.107878
\(349\) −19.0367 −1.01901 −0.509506 0.860467i \(-0.670172\pi\)
−0.509506 + 0.860467i \(0.670172\pi\)
\(350\) −18.3060 −0.978495
\(351\) −1.76485 −0.0942005
\(352\) −4.10744 −0.218927
\(353\) 18.8385 1.00267 0.501335 0.865253i \(-0.332842\pi\)
0.501335 + 0.865253i \(0.332842\pi\)
\(354\) −15.9012 −0.845138
\(355\) 18.0202 0.956413
\(356\) 4.12612 0.218684
\(357\) −8.85563 −0.468689
\(358\) −3.62288 −0.191475
\(359\) −16.6130 −0.876803 −0.438401 0.898779i \(-0.644455\pi\)
−0.438401 + 0.898779i \(0.644455\pi\)
\(360\) −3.51552 −0.185284
\(361\) 38.4234 2.02228
\(362\) −3.44626 −0.181132
\(363\) 11.8151 0.620132
\(364\) −1.32509 −0.0694537
\(365\) 15.4099 0.806592
\(366\) −6.78734 −0.354780
\(367\) 10.2844 0.536843 0.268421 0.963302i \(-0.413498\pi\)
0.268421 + 0.963302i \(0.413498\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 6.21693 0.323641
\(370\) −25.7412 −1.33822
\(371\) −39.4794 −2.04967
\(372\) −2.45595 −0.127335
\(373\) 2.86104 0.148139 0.0740695 0.997253i \(-0.476401\pi\)
0.0740695 + 0.997253i \(0.476401\pi\)
\(374\) −6.13404 −0.317184
\(375\) −8.17092 −0.421945
\(376\) −7.72271 −0.398268
\(377\) 0.449700 0.0231607
\(378\) 11.5640 0.594786
\(379\) 4.57836 0.235175 0.117587 0.993063i \(-0.462484\pi\)
0.117587 + 0.993063i \(0.462484\pi\)
\(380\) 25.3744 1.30168
\(381\) −4.47143 −0.229078
\(382\) −7.43775 −0.380549
\(383\) −13.4638 −0.687966 −0.343983 0.938976i \(-0.611776\pi\)
−0.343983 + 0.938976i \(0.611776\pi\)
\(384\) 2.01243 0.102696
\(385\) −40.5272 −2.06546
\(386\) −16.4333 −0.836435
\(387\) −9.04409 −0.459737
\(388\) −8.83222 −0.448388
\(389\) −3.92455 −0.198982 −0.0994912 0.995038i \(-0.531722\pi\)
−0.0994912 + 0.995038i \(0.531722\pi\)
\(390\) −3.03037 −0.153449
\(391\) −1.49340 −0.0755244
\(392\) 1.68252 0.0849802
\(393\) 29.3269 1.47935
\(394\) −2.39554 −0.120685
\(395\) −30.9939 −1.55947
\(396\) −4.31229 −0.216701
\(397\) 30.7736 1.54448 0.772242 0.635328i \(-0.219135\pi\)
0.772242 + 0.635328i \(0.219135\pi\)
\(398\) 19.1929 0.962051
\(399\) 44.9353 2.24958
\(400\) 6.21255 0.310627
\(401\) 2.60010 0.129843 0.0649213 0.997890i \(-0.479320\pi\)
0.0649213 + 0.997890i \(0.479320\pi\)
\(402\) 3.88415 0.193724
\(403\) −0.548811 −0.0273382
\(404\) 17.2338 0.857414
\(405\) 36.9923 1.83816
\(406\) −2.94661 −0.146238
\(407\) −31.5753 −1.56513
\(408\) 3.00536 0.148787
\(409\) 1.80606 0.0893037 0.0446518 0.999003i \(-0.485782\pi\)
0.0446518 + 0.999003i \(0.485782\pi\)
\(410\) −19.8286 −0.979262
\(411\) −15.2403 −0.751749
\(412\) −6.94133 −0.341975
\(413\) 23.2826 1.14566
\(414\) −1.04987 −0.0515985
\(415\) −11.8826 −0.583296
\(416\) 0.449700 0.0220484
\(417\) −41.8488 −2.04935
\(418\) 31.1254 1.52240
\(419\) 16.3847 0.800444 0.400222 0.916418i \(-0.368933\pi\)
0.400222 + 0.916418i \(0.368933\pi\)
\(420\) 19.8562 0.968883
\(421\) −29.9884 −1.46155 −0.730773 0.682621i \(-0.760840\pi\)
−0.730773 + 0.682621i \(0.760840\pi\)
\(422\) 12.9238 0.629120
\(423\) −8.10787 −0.394218
\(424\) 13.3982 0.650676
\(425\) 9.27780 0.450040
\(426\) −10.8300 −0.524715
\(427\) 9.93807 0.480937
\(428\) −4.72189 −0.228241
\(429\) −3.71719 −0.179468
\(430\) 28.8456 1.39106
\(431\) −20.2190 −0.973916 −0.486958 0.873425i \(-0.661894\pi\)
−0.486958 + 0.873425i \(0.661894\pi\)
\(432\) −3.92449 −0.188817
\(433\) 29.7211 1.42831 0.714153 0.699989i \(-0.246812\pi\)
0.714153 + 0.699989i \(0.246812\pi\)
\(434\) 3.59602 0.172615
\(435\) −6.73865 −0.323094
\(436\) 1.54601 0.0740405
\(437\) 7.57782 0.362496
\(438\) −9.26124 −0.442519
\(439\) 28.3637 1.35373 0.676864 0.736108i \(-0.263339\pi\)
0.676864 + 0.736108i \(0.263339\pi\)
\(440\) 13.7538 0.655688
\(441\) 1.76644 0.0841160
\(442\) 0.671581 0.0319439
\(443\) 33.1054 1.57289 0.786443 0.617662i \(-0.211920\pi\)
0.786443 + 0.617662i \(0.211920\pi\)
\(444\) 15.4702 0.734185
\(445\) −13.8164 −0.654959
\(446\) −5.01497 −0.237466
\(447\) 30.6008 1.44737
\(448\) −2.94661 −0.139214
\(449\) −8.66957 −0.409142 −0.204571 0.978852i \(-0.565580\pi\)
−0.204571 + 0.978852i \(0.565580\pi\)
\(450\) 6.52239 0.307468
\(451\) −24.3226 −1.14531
\(452\) −8.89955 −0.418600
\(453\) 14.5938 0.685677
\(454\) −3.85327 −0.180843
\(455\) 4.43709 0.208014
\(456\) −15.2498 −0.714139
\(457\) 14.1426 0.661563 0.330781 0.943707i \(-0.392688\pi\)
0.330781 + 0.943707i \(0.392688\pi\)
\(458\) −23.5676 −1.10124
\(459\) −5.86083 −0.273560
\(460\) 3.34851 0.156125
\(461\) −37.5912 −1.75080 −0.875398 0.483402i \(-0.839401\pi\)
−0.875398 + 0.483402i \(0.839401\pi\)
\(462\) 24.3565 1.13317
\(463\) 24.3916 1.13357 0.566786 0.823865i \(-0.308187\pi\)
0.566786 + 0.823865i \(0.308187\pi\)
\(464\) 1.00000 0.0464238
\(465\) 8.22380 0.381369
\(466\) −21.4376 −0.993078
\(467\) −1.71268 −0.0792532 −0.0396266 0.999215i \(-0.512617\pi\)
−0.0396266 + 0.999215i \(0.512617\pi\)
\(468\) 0.472129 0.0218242
\(469\) −5.68720 −0.262611
\(470\) 25.8596 1.19281
\(471\) −19.9772 −0.920502
\(472\) −7.90148 −0.363695
\(473\) 35.3833 1.62693
\(474\) 18.6271 0.855570
\(475\) −47.0776 −2.16007
\(476\) −4.40046 −0.201695
\(477\) 14.0665 0.644059
\(478\) −28.6339 −1.30968
\(479\) −31.2291 −1.42689 −0.713446 0.700710i \(-0.752867\pi\)
−0.713446 + 0.700710i \(0.752867\pi\)
\(480\) −6.73865 −0.307576
\(481\) 3.45700 0.157626
\(482\) −12.4373 −0.566506
\(483\) 5.92985 0.269818
\(484\) 5.87106 0.266866
\(485\) 29.5748 1.34292
\(486\) −10.4586 −0.474412
\(487\) 2.28762 0.103662 0.0518310 0.998656i \(-0.483494\pi\)
0.0518310 + 0.998656i \(0.483494\pi\)
\(488\) −3.37271 −0.152675
\(489\) 46.2038 2.08941
\(490\) −5.63395 −0.254516
\(491\) 29.5094 1.33174 0.665870 0.746068i \(-0.268061\pi\)
0.665870 + 0.746068i \(0.268061\pi\)
\(492\) 11.9168 0.537251
\(493\) 1.49340 0.0672593
\(494\) −3.40775 −0.153322
\(495\) 14.4398 0.649020
\(496\) −1.22039 −0.0547972
\(497\) 15.8573 0.711299
\(498\) 7.14137 0.320012
\(499\) −24.7717 −1.10893 −0.554467 0.832206i \(-0.687078\pi\)
−0.554467 + 0.832206i \(0.687078\pi\)
\(500\) −4.06023 −0.181579
\(501\) −6.96549 −0.311195
\(502\) 12.5035 0.558059
\(503\) −2.51484 −0.112131 −0.0560655 0.998427i \(-0.517856\pi\)
−0.0560655 + 0.998427i \(0.517856\pi\)
\(504\) −3.09357 −0.137799
\(505\) −57.7077 −2.56796
\(506\) 4.10744 0.182598
\(507\) −25.7546 −1.14380
\(508\) −2.22191 −0.0985812
\(509\) 19.3823 0.859107 0.429554 0.903041i \(-0.358671\pi\)
0.429554 + 0.903041i \(0.358671\pi\)
\(510\) −10.0635 −0.445619
\(511\) 13.5604 0.599875
\(512\) 1.00000 0.0441942
\(513\) 29.7391 1.31301
\(514\) −21.1475 −0.932775
\(515\) 23.2431 1.02422
\(516\) −17.3360 −0.763174
\(517\) 31.7205 1.39507
\(518\) −22.6516 −0.995254
\(519\) 4.83702 0.212322
\(520\) −1.50583 −0.0660349
\(521\) 1.35159 0.0592141 0.0296071 0.999562i \(-0.490574\pi\)
0.0296071 + 0.999562i \(0.490574\pi\)
\(522\) 1.04987 0.0459517
\(523\) −11.3329 −0.495555 −0.247777 0.968817i \(-0.579700\pi\)
−0.247777 + 0.968817i \(0.579700\pi\)
\(524\) 14.5729 0.636619
\(525\) −36.8395 −1.60781
\(526\) −13.8786 −0.605136
\(527\) −1.82253 −0.0793907
\(528\) −8.26593 −0.359729
\(529\) 1.00000 0.0434783
\(530\) −44.8642 −1.94878
\(531\) −8.29556 −0.359996
\(532\) 22.3289 0.968081
\(533\) 2.66294 0.115345
\(534\) 8.30353 0.359329
\(535\) 15.8113 0.683582
\(536\) 1.93008 0.0833668
\(537\) −7.29079 −0.314621
\(538\) 18.7274 0.807396
\(539\) −6.91086 −0.297672
\(540\) 13.1412 0.565508
\(541\) −10.2323 −0.439921 −0.219961 0.975509i \(-0.570593\pi\)
−0.219961 + 0.975509i \(0.570593\pi\)
\(542\) −29.6416 −1.27322
\(543\) −6.93536 −0.297625
\(544\) 1.49340 0.0640289
\(545\) −5.17684 −0.221751
\(546\) −2.66666 −0.114122
\(547\) −35.6571 −1.52459 −0.762293 0.647232i \(-0.775926\pi\)
−0.762293 + 0.647232i \(0.775926\pi\)
\(548\) −7.57308 −0.323506
\(549\) −3.54092 −0.151123
\(550\) −25.5177 −1.08808
\(551\) −7.57782 −0.322826
\(552\) −2.01243 −0.0856547
\(553\) −27.2739 −1.15980
\(554\) −28.1416 −1.19562
\(555\) −51.8023 −2.19888
\(556\) −20.7952 −0.881912
\(557\) 10.9917 0.465732 0.232866 0.972509i \(-0.425190\pi\)
0.232866 + 0.972509i \(0.425190\pi\)
\(558\) −1.28126 −0.0542400
\(559\) −3.87392 −0.163849
\(560\) 9.86677 0.416947
\(561\) −12.3443 −0.521178
\(562\) 25.4697 1.07437
\(563\) 5.43294 0.228971 0.114486 0.993425i \(-0.463478\pi\)
0.114486 + 0.993425i \(0.463478\pi\)
\(564\) −15.5414 −0.654411
\(565\) 29.8003 1.25371
\(566\) −20.7801 −0.873454
\(567\) 32.5524 1.36707
\(568\) −5.38155 −0.225805
\(569\) −11.0418 −0.462894 −0.231447 0.972847i \(-0.574346\pi\)
−0.231447 + 0.972847i \(0.574346\pi\)
\(570\) 51.0643 2.13885
\(571\) 8.52720 0.356852 0.178426 0.983953i \(-0.442899\pi\)
0.178426 + 0.983953i \(0.442899\pi\)
\(572\) −1.84712 −0.0772318
\(573\) −14.9680 −0.625295
\(574\) −17.4487 −0.728293
\(575\) −6.21255 −0.259081
\(576\) 1.04987 0.0437448
\(577\) −35.8618 −1.49294 −0.746472 0.665416i \(-0.768254\pi\)
−0.746472 + 0.665416i \(0.768254\pi\)
\(578\) −14.7698 −0.614341
\(579\) −33.0710 −1.37438
\(580\) −3.34851 −0.139039
\(581\) −10.4564 −0.433806
\(582\) −17.7742 −0.736765
\(583\) −55.0324 −2.27921
\(584\) −4.60202 −0.190433
\(585\) −1.58093 −0.0653634
\(586\) 21.7496 0.898466
\(587\) −24.2605 −1.00134 −0.500669 0.865639i \(-0.666913\pi\)
−0.500669 + 0.865639i \(0.666913\pi\)
\(588\) 3.38596 0.139635
\(589\) 9.24791 0.381054
\(590\) 26.4582 1.08927
\(591\) −4.82085 −0.198303
\(592\) 7.68734 0.315948
\(593\) −33.7632 −1.38649 −0.693243 0.720704i \(-0.743819\pi\)
−0.693243 + 0.720704i \(0.743819\pi\)
\(594\) 16.1196 0.661396
\(595\) 14.7350 0.604077
\(596\) 15.2059 0.622857
\(597\) 38.6243 1.58079
\(598\) −0.449700 −0.0183896
\(599\) −7.63298 −0.311875 −0.155938 0.987767i \(-0.549840\pi\)
−0.155938 + 0.987767i \(0.549840\pi\)
\(600\) 12.5023 0.510405
\(601\) −25.3357 −1.03347 −0.516733 0.856147i \(-0.672852\pi\)
−0.516733 + 0.856147i \(0.672852\pi\)
\(602\) 25.3834 1.03455
\(603\) 2.02634 0.0825190
\(604\) 7.25184 0.295073
\(605\) −19.6593 −0.799265
\(606\) 34.6818 1.40885
\(607\) 5.59433 0.227067 0.113533 0.993534i \(-0.463783\pi\)
0.113533 + 0.993534i \(0.463783\pi\)
\(608\) −7.57782 −0.307321
\(609\) −5.92985 −0.240290
\(610\) 11.2936 0.457263
\(611\) −3.47290 −0.140499
\(612\) 1.56788 0.0633778
\(613\) 29.1101 1.17575 0.587873 0.808953i \(-0.299965\pi\)
0.587873 + 0.808953i \(0.299965\pi\)
\(614\) −30.9335 −1.24837
\(615\) −39.9036 −1.60907
\(616\) 12.1030 0.487645
\(617\) −11.1186 −0.447617 −0.223809 0.974633i \(-0.571849\pi\)
−0.223809 + 0.974633i \(0.571849\pi\)
\(618\) −13.9689 −0.561913
\(619\) −36.4026 −1.46315 −0.731573 0.681763i \(-0.761213\pi\)
−0.731573 + 0.681763i \(0.761213\pi\)
\(620\) 4.08650 0.164118
\(621\) 3.92449 0.157484
\(622\) 19.0093 0.762204
\(623\) −12.1581 −0.487103
\(624\) 0.904990 0.0362286
\(625\) −17.4670 −0.698680
\(626\) 23.3187 0.932004
\(627\) 62.6378 2.50151
\(628\) −9.92692 −0.396127
\(629\) 11.4803 0.457748
\(630\) 10.3589 0.412707
\(631\) 3.55998 0.141720 0.0708602 0.997486i \(-0.477426\pi\)
0.0708602 + 0.997486i \(0.477426\pi\)
\(632\) 9.25601 0.368184
\(633\) 26.0082 1.03373
\(634\) 0.472925 0.0187822
\(635\) 7.44009 0.295251
\(636\) 26.9630 1.06915
\(637\) 0.756631 0.0299788
\(638\) −4.10744 −0.162615
\(639\) −5.64995 −0.223509
\(640\) −3.34851 −0.132362
\(641\) −5.02032 −0.198291 −0.0991453 0.995073i \(-0.531611\pi\)
−0.0991453 + 0.995073i \(0.531611\pi\)
\(642\) −9.50247 −0.375033
\(643\) −8.07710 −0.318530 −0.159265 0.987236i \(-0.550912\pi\)
−0.159265 + 0.987236i \(0.550912\pi\)
\(644\) 2.94661 0.116113
\(645\) 58.0497 2.28571
\(646\) −11.3167 −0.445250
\(647\) −12.4312 −0.488720 −0.244360 0.969685i \(-0.578578\pi\)
−0.244360 + 0.969685i \(0.578578\pi\)
\(648\) −11.0474 −0.433982
\(649\) 32.4548 1.27396
\(650\) 2.79378 0.109581
\(651\) 7.23674 0.283630
\(652\) 22.9592 0.899152
\(653\) −8.09108 −0.316629 −0.158314 0.987389i \(-0.550606\pi\)
−0.158314 + 0.987389i \(0.550606\pi\)
\(654\) 3.11124 0.121659
\(655\) −48.7975 −1.90667
\(656\) 5.92160 0.231200
\(657\) −4.83154 −0.188496
\(658\) 22.7558 0.887114
\(659\) 1.58997 0.0619366 0.0309683 0.999520i \(-0.490141\pi\)
0.0309683 + 0.999520i \(0.490141\pi\)
\(660\) 27.6786 1.07739
\(661\) −43.2396 −1.68182 −0.840912 0.541171i \(-0.817981\pi\)
−0.840912 + 0.541171i \(0.817981\pi\)
\(662\) 11.1247 0.432375
\(663\) 1.35151 0.0524883
\(664\) 3.54863 0.137714
\(665\) −74.7686 −2.89940
\(666\) 8.07074 0.312735
\(667\) −1.00000 −0.0387202
\(668\) −3.46124 −0.133919
\(669\) −10.0923 −0.390190
\(670\) −6.46290 −0.249684
\(671\) 13.8532 0.534797
\(672\) −5.92985 −0.228749
\(673\) 11.8305 0.456030 0.228015 0.973658i \(-0.426776\pi\)
0.228015 + 0.973658i \(0.426776\pi\)
\(674\) −2.61571 −0.100753
\(675\) −24.3811 −0.938428
\(676\) −12.7978 −0.492222
\(677\) 7.36323 0.282992 0.141496 0.989939i \(-0.454809\pi\)
0.141496 + 0.989939i \(0.454809\pi\)
\(678\) −17.9097 −0.687819
\(679\) 26.0251 0.998753
\(680\) −5.00066 −0.191767
\(681\) −7.75444 −0.297151
\(682\) 5.01269 0.191946
\(683\) 9.49956 0.363490 0.181745 0.983346i \(-0.441825\pi\)
0.181745 + 0.983346i \(0.441825\pi\)
\(684\) −7.95576 −0.304196
\(685\) 25.3586 0.968901
\(686\) 15.6685 0.598228
\(687\) −47.4282 −1.80950
\(688\) −8.61445 −0.328423
\(689\) 6.02519 0.229541
\(690\) 6.73865 0.256536
\(691\) −23.5689 −0.896601 −0.448301 0.893883i \(-0.647971\pi\)
−0.448301 + 0.893883i \(0.647971\pi\)
\(692\) 2.40357 0.0913701
\(693\) 12.7067 0.482686
\(694\) 26.9380 1.02255
\(695\) 69.6330 2.64133
\(696\) 2.01243 0.0762810
\(697\) 8.84330 0.334964
\(698\) −19.0367 −0.720551
\(699\) −43.1417 −1.63177
\(700\) −18.3060 −0.691900
\(701\) −1.92411 −0.0726725 −0.0363363 0.999340i \(-0.511569\pi\)
−0.0363363 + 0.999340i \(0.511569\pi\)
\(702\) −1.76485 −0.0666098
\(703\) −58.2533 −2.19706
\(704\) −4.10744 −0.154805
\(705\) 52.0406 1.95996
\(706\) 18.8385 0.708995
\(707\) −50.7814 −1.90983
\(708\) −15.9012 −0.597602
\(709\) 43.0363 1.61626 0.808130 0.589004i \(-0.200480\pi\)
0.808130 + 0.589004i \(0.200480\pi\)
\(710\) 18.0202 0.676286
\(711\) 9.71765 0.364440
\(712\) 4.12612 0.154633
\(713\) 1.22039 0.0457040
\(714\) −8.85563 −0.331413
\(715\) 6.18510 0.231309
\(716\) −3.62288 −0.135393
\(717\) −57.6237 −2.15200
\(718\) −16.6130 −0.619993
\(719\) 42.7397 1.59392 0.796961 0.604030i \(-0.206439\pi\)
0.796961 + 0.604030i \(0.206439\pi\)
\(720\) −3.51552 −0.131016
\(721\) 20.4534 0.761725
\(722\) 38.4234 1.42997
\(723\) −25.0293 −0.930849
\(724\) −3.44626 −0.128079
\(725\) 6.21255 0.230728
\(726\) 11.8151 0.438499
\(727\) −50.2135 −1.86231 −0.931157 0.364618i \(-0.881200\pi\)
−0.931157 + 0.364618i \(0.881200\pi\)
\(728\) −1.32509 −0.0491112
\(729\) 12.0949 0.447960
\(730\) 15.4099 0.570347
\(731\) −12.8648 −0.475822
\(732\) −6.78734 −0.250867
\(733\) 19.6376 0.725332 0.362666 0.931919i \(-0.381867\pi\)
0.362666 + 0.931919i \(0.381867\pi\)
\(734\) 10.2844 0.379605
\(735\) −11.3379 −0.418206
\(736\) −1.00000 −0.0368605
\(737\) −7.92769 −0.292020
\(738\) 6.21693 0.228848
\(739\) 44.2013 1.62597 0.812985 0.582285i \(-0.197841\pi\)
0.812985 + 0.582285i \(0.197841\pi\)
\(740\) −25.7412 −0.946264
\(741\) −6.85786 −0.251930
\(742\) −39.4794 −1.44933
\(743\) −30.0155 −1.10116 −0.550581 0.834781i \(-0.685594\pi\)
−0.550581 + 0.834781i \(0.685594\pi\)
\(744\) −2.45595 −0.0900396
\(745\) −50.9171 −1.86546
\(746\) 2.86104 0.104750
\(747\) 3.72562 0.136313
\(748\) −6.13404 −0.224283
\(749\) 13.9136 0.508391
\(750\) −8.17092 −0.298360
\(751\) 11.5227 0.420469 0.210235 0.977651i \(-0.432577\pi\)
0.210235 + 0.977651i \(0.432577\pi\)
\(752\) −7.72271 −0.281618
\(753\) 25.1625 0.916971
\(754\) 0.449700 0.0163771
\(755\) −24.2829 −0.883744
\(756\) 11.5640 0.420577
\(757\) −19.1922 −0.697554 −0.348777 0.937206i \(-0.613403\pi\)
−0.348777 + 0.937206i \(0.613403\pi\)
\(758\) 4.57836 0.166294
\(759\) 8.26593 0.300034
\(760\) 25.3744 0.920428
\(761\) −12.0269 −0.435973 −0.217987 0.975952i \(-0.569949\pi\)
−0.217987 + 0.975952i \(0.569949\pi\)
\(762\) −4.47143 −0.161983
\(763\) −4.55550 −0.164920
\(764\) −7.43775 −0.269088
\(765\) −5.25007 −0.189817
\(766\) −13.4638 −0.486466
\(767\) −3.55330 −0.128302
\(768\) 2.01243 0.0726173
\(769\) 42.2556 1.52378 0.761889 0.647708i \(-0.224272\pi\)
0.761889 + 0.647708i \(0.224272\pi\)
\(770\) −40.5272 −1.46050
\(771\) −42.5578 −1.53268
\(772\) −16.4333 −0.591449
\(773\) 6.53047 0.234885 0.117442 0.993080i \(-0.462530\pi\)
0.117442 + 0.993080i \(0.462530\pi\)
\(774\) −9.04409 −0.325083
\(775\) −7.58174 −0.272344
\(776\) −8.83222 −0.317058
\(777\) −45.5848 −1.63534
\(778\) −3.92455 −0.140702
\(779\) −44.8728 −1.60774
\(780\) −3.03037 −0.108505
\(781\) 22.1044 0.790957
\(782\) −1.49340 −0.0534038
\(783\) −3.92449 −0.140250
\(784\) 1.68252 0.0600901
\(785\) 33.2404 1.18640
\(786\) 29.3269 1.04606
\(787\) 31.6289 1.12745 0.563724 0.825963i \(-0.309368\pi\)
0.563724 + 0.825963i \(0.309368\pi\)
\(788\) −2.39554 −0.0853375
\(789\) −27.9297 −0.994325
\(790\) −30.9939 −1.10271
\(791\) 26.2235 0.932401
\(792\) −4.31229 −0.153231
\(793\) −1.51671 −0.0538599
\(794\) 30.7736 1.09212
\(795\) −90.2860 −3.20211
\(796\) 19.1929 0.680273
\(797\) −54.5784 −1.93327 −0.966634 0.256162i \(-0.917542\pi\)
−0.966634 + 0.256162i \(0.917542\pi\)
\(798\) 44.9353 1.59069
\(799\) −11.5331 −0.408011
\(800\) 6.21255 0.219647
\(801\) 4.33191 0.153060
\(802\) 2.60010 0.0918126
\(803\) 18.9025 0.667055
\(804\) 3.88415 0.136984
\(805\) −9.86677 −0.347758
\(806\) −0.548811 −0.0193310
\(807\) 37.6876 1.32667
\(808\) 17.2338 0.606284
\(809\) 33.0661 1.16254 0.581270 0.813710i \(-0.302556\pi\)
0.581270 + 0.813710i \(0.302556\pi\)
\(810\) 36.9923 1.29978
\(811\) 28.7021 1.00787 0.503933 0.863742i \(-0.331886\pi\)
0.503933 + 0.863742i \(0.331886\pi\)
\(812\) −2.94661 −0.103406
\(813\) −59.6516 −2.09207
\(814\) −31.5753 −1.10671
\(815\) −76.8793 −2.69296
\(816\) 3.00536 0.105209
\(817\) 65.2788 2.28381
\(818\) 1.80606 0.0631472
\(819\) −1.39118 −0.0486118
\(820\) −19.8286 −0.692443
\(821\) −2.02510 −0.0706766 −0.0353383 0.999375i \(-0.511251\pi\)
−0.0353383 + 0.999375i \(0.511251\pi\)
\(822\) −15.2403 −0.531567
\(823\) 7.59197 0.264639 0.132320 0.991207i \(-0.457757\pi\)
0.132320 + 0.991207i \(0.457757\pi\)
\(824\) −6.94133 −0.241813
\(825\) −51.3525 −1.78786
\(826\) 23.2826 0.810105
\(827\) 8.34687 0.290249 0.145124 0.989413i \(-0.453642\pi\)
0.145124 + 0.989413i \(0.453642\pi\)
\(828\) −1.04987 −0.0364856
\(829\) 3.08964 0.107308 0.0536538 0.998560i \(-0.482913\pi\)
0.0536538 + 0.998560i \(0.482913\pi\)
\(830\) −11.8826 −0.412452
\(831\) −56.6330 −1.96458
\(832\) 0.449700 0.0155906
\(833\) 2.51268 0.0870590
\(834\) −41.8488 −1.44911
\(835\) 11.5900 0.401088
\(836\) 31.1254 1.07650
\(837\) 4.78942 0.165547
\(838\) 16.3847 0.565999
\(839\) −17.4894 −0.603802 −0.301901 0.953339i \(-0.597621\pi\)
−0.301901 + 0.953339i \(0.597621\pi\)
\(840\) 19.8562 0.685104
\(841\) 1.00000 0.0344828
\(842\) −29.9884 −1.03347
\(843\) 51.2560 1.76535
\(844\) 12.9238 0.444855
\(845\) 42.8535 1.47421
\(846\) −8.10787 −0.278754
\(847\) −17.2997 −0.594426
\(848\) 13.3982 0.460097
\(849\) −41.8186 −1.43521
\(850\) 9.27780 0.318226
\(851\) −7.68734 −0.263519
\(852\) −10.8300 −0.371029
\(853\) −10.0668 −0.344679 −0.172340 0.985038i \(-0.555133\pi\)
−0.172340 + 0.985038i \(0.555133\pi\)
\(854\) 9.93807 0.340074
\(855\) 26.6400 0.911068
\(856\) −4.72189 −0.161391
\(857\) −13.0068 −0.444305 −0.222153 0.975012i \(-0.571308\pi\)
−0.222153 + 0.975012i \(0.571308\pi\)
\(858\) −3.71719 −0.126903
\(859\) −34.5801 −1.17986 −0.589928 0.807456i \(-0.700844\pi\)
−0.589928 + 0.807456i \(0.700844\pi\)
\(860\) 28.8456 0.983627
\(861\) −35.1142 −1.19669
\(862\) −20.2190 −0.688663
\(863\) 51.1433 1.74094 0.870470 0.492222i \(-0.163815\pi\)
0.870470 + 0.492222i \(0.163815\pi\)
\(864\) −3.92449 −0.133514
\(865\) −8.04839 −0.273654
\(866\) 29.7211 1.00997
\(867\) −29.7231 −1.00945
\(868\) 3.59602 0.122057
\(869\) −38.0185 −1.28969
\(870\) −6.73865 −0.228462
\(871\) 0.867958 0.0294096
\(872\) 1.54601 0.0523546
\(873\) −9.27272 −0.313834
\(874\) 7.57782 0.256324
\(875\) 11.9639 0.404454
\(876\) −9.26124 −0.312908
\(877\) 52.8468 1.78451 0.892255 0.451531i \(-0.149122\pi\)
0.892255 + 0.451531i \(0.149122\pi\)
\(878\) 28.3637 0.957230
\(879\) 43.7695 1.47631
\(880\) 13.7538 0.463641
\(881\) 0.405614 0.0136655 0.00683274 0.999977i \(-0.497825\pi\)
0.00683274 + 0.999977i \(0.497825\pi\)
\(882\) 1.76644 0.0594790
\(883\) −30.3038 −1.01980 −0.509901 0.860233i \(-0.670318\pi\)
−0.509901 + 0.860233i \(0.670318\pi\)
\(884\) 0.671581 0.0225877
\(885\) 53.2453 1.78982
\(886\) 33.1054 1.11220
\(887\) 30.8406 1.03553 0.517763 0.855524i \(-0.326765\pi\)
0.517763 + 0.855524i \(0.326765\pi\)
\(888\) 15.4702 0.519147
\(889\) 6.54710 0.219583
\(890\) −13.8164 −0.463126
\(891\) 45.3765 1.52017
\(892\) −5.01497 −0.167914
\(893\) 58.5213 1.95834
\(894\) 30.6008 1.02344
\(895\) 12.1313 0.405504
\(896\) −2.94661 −0.0984394
\(897\) −0.904990 −0.0302167
\(898\) −8.66957 −0.289307
\(899\) −1.22039 −0.0407024
\(900\) 6.52239 0.217413
\(901\) 20.0089 0.666593
\(902\) −24.3226 −0.809854
\(903\) 51.0824 1.69992
\(904\) −8.89955 −0.295995
\(905\) 11.5399 0.383598
\(906\) 14.5938 0.484847
\(907\) 8.62844 0.286503 0.143251 0.989686i \(-0.454244\pi\)
0.143251 + 0.989686i \(0.454244\pi\)
\(908\) −3.85327 −0.127875
\(909\) 18.0933 0.600118
\(910\) 4.43709 0.147088
\(911\) 36.6066 1.21283 0.606416 0.795148i \(-0.292607\pi\)
0.606416 + 0.795148i \(0.292607\pi\)
\(912\) −15.2498 −0.504973
\(913\) −14.5758 −0.482388
\(914\) 14.1426 0.467795
\(915\) 22.7275 0.751348
\(916\) −23.5676 −0.778697
\(917\) −42.9406 −1.41802
\(918\) −5.86083 −0.193436
\(919\) 45.0971 1.48762 0.743808 0.668394i \(-0.233018\pi\)
0.743808 + 0.668394i \(0.233018\pi\)
\(920\) 3.34851 0.110397
\(921\) −62.2515 −2.05126
\(922\) −37.5912 −1.23800
\(923\) −2.42008 −0.0796580
\(924\) 24.3565 0.801270
\(925\) 47.7579 1.57027
\(926\) 24.3916 0.801557
\(927\) −7.28752 −0.239354
\(928\) 1.00000 0.0328266
\(929\) −21.3839 −0.701583 −0.350792 0.936454i \(-0.614087\pi\)
−0.350792 + 0.936454i \(0.614087\pi\)
\(930\) 8.22380 0.269669
\(931\) −12.7499 −0.417860
\(932\) −21.4376 −0.702212
\(933\) 38.2549 1.25241
\(934\) −1.71268 −0.0560405
\(935\) 20.5399 0.671727
\(936\) 0.472129 0.0154320
\(937\) −34.5974 −1.13025 −0.565124 0.825006i \(-0.691172\pi\)
−0.565124 + 0.825006i \(0.691172\pi\)
\(938\) −5.68720 −0.185694
\(939\) 46.9273 1.53141
\(940\) 25.8596 0.843447
\(941\) −44.1629 −1.43967 −0.719835 0.694145i \(-0.755782\pi\)
−0.719835 + 0.694145i \(0.755782\pi\)
\(942\) −19.9772 −0.650894
\(943\) −5.92160 −0.192834
\(944\) −7.90148 −0.257171
\(945\) −38.7221 −1.25963
\(946\) 35.3833 1.15041
\(947\) 21.6888 0.704791 0.352395 0.935851i \(-0.385367\pi\)
0.352395 + 0.935851i \(0.385367\pi\)
\(948\) 18.6271 0.604979
\(949\) −2.06953 −0.0671798
\(950\) −47.0776 −1.52740
\(951\) 0.951728 0.0308619
\(952\) −4.40046 −0.142620
\(953\) 0.441114 0.0142891 0.00714454 0.999974i \(-0.497726\pi\)
0.00714454 + 0.999974i \(0.497726\pi\)
\(954\) 14.0665 0.455418
\(955\) 24.9054 0.805921
\(956\) −28.6339 −0.926086
\(957\) −8.26593 −0.267200
\(958\) −31.2291 −1.00897
\(959\) 22.3149 0.720587
\(960\) −6.73865 −0.217489
\(961\) −29.5106 −0.951956
\(962\) 3.45700 0.111458
\(963\) −4.95739 −0.159750
\(964\) −12.4373 −0.400580
\(965\) 55.0273 1.77139
\(966\) 5.92985 0.190790
\(967\) 43.3851 1.39517 0.697586 0.716501i \(-0.254258\pi\)
0.697586 + 0.716501i \(0.254258\pi\)
\(968\) 5.87106 0.188703
\(969\) −22.7741 −0.731609
\(970\) 29.5748 0.949590
\(971\) 7.68638 0.246668 0.123334 0.992365i \(-0.460641\pi\)
0.123334 + 0.992365i \(0.460641\pi\)
\(972\) −10.4586 −0.335460
\(973\) 61.2753 1.96440
\(974\) 2.28762 0.0733001
\(975\) 5.62229 0.180058
\(976\) −3.37271 −0.107958
\(977\) −1.22029 −0.0390405 −0.0195203 0.999809i \(-0.506214\pi\)
−0.0195203 + 0.999809i \(0.506214\pi\)
\(978\) 46.2038 1.47744
\(979\) −16.9478 −0.541654
\(980\) −5.63395 −0.179970
\(981\) 1.62312 0.0518221
\(982\) 29.5094 0.941682
\(983\) 24.2195 0.772483 0.386241 0.922398i \(-0.373773\pi\)
0.386241 + 0.922398i \(0.373773\pi\)
\(984\) 11.9168 0.379894
\(985\) 8.02149 0.255586
\(986\) 1.49340 0.0475595
\(987\) 45.7945 1.45765
\(988\) −3.40775 −0.108415
\(989\) 8.61445 0.273924
\(990\) 14.4398 0.458926
\(991\) −18.9870 −0.603143 −0.301571 0.953444i \(-0.597511\pi\)
−0.301571 + 0.953444i \(0.597511\pi\)
\(992\) −1.22039 −0.0387475
\(993\) 22.3878 0.710454
\(994\) 15.8573 0.502964
\(995\) −64.2676 −2.03742
\(996\) 7.14137 0.226283
\(997\) −13.0196 −0.412333 −0.206167 0.978517i \(-0.566099\pi\)
−0.206167 + 0.978517i \(0.566099\pi\)
\(998\) −24.7717 −0.784134
\(999\) −30.1689 −0.954502
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 1334.2.a.h.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1334.2.a.h.1.5 5 1.1 even 1 trivial