Properties

Label 6027.2.a.bg.1.12
Level $6027$
Weight $2$
Character 6027.1
Self dual yes
Analytic conductor $48.126$
Analytic rank $0$
Dimension $12$
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] = [6027,2,Mod(1,6027)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6027.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6027.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: \(48.1258372982\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 15 x^{10} + 30 x^{9} + 74 x^{8} - 149 x^{7} - 140 x^{6} + 278 x^{5} + 126 x^{4} - 211 x^{3} - 64 x^{2} + 53 x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 861)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(-2.48589\) of defining polynomial
Character \(\chi\) \(=\) 6027.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.48589 q^{2} +1.00000 q^{3} +4.17966 q^{4} +1.76717 q^{5} +2.48589 q^{6} +5.41840 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.48589 q^{2} +1.00000 q^{3} +4.17966 q^{4} +1.76717 q^{5} +2.48589 q^{6} +5.41840 q^{8} +1.00000 q^{9} +4.39300 q^{10} +0.293529 q^{11} +4.17966 q^{12} +6.33387 q^{13} +1.76717 q^{15} +5.11024 q^{16} +1.00183 q^{17} +2.48589 q^{18} -5.87506 q^{19} +7.38618 q^{20} +0.729681 q^{22} +0.838269 q^{23} +5.41840 q^{24} -1.87710 q^{25} +15.7453 q^{26} +1.00000 q^{27} +2.34260 q^{29} +4.39300 q^{30} -1.39480 q^{31} +1.86671 q^{32} +0.293529 q^{33} +2.49044 q^{34} +4.17966 q^{36} -0.691089 q^{37} -14.6048 q^{38} +6.33387 q^{39} +9.57525 q^{40} -1.00000 q^{41} -0.408550 q^{43} +1.22685 q^{44} +1.76717 q^{45} +2.08385 q^{46} +11.0390 q^{47} +5.11024 q^{48} -4.66628 q^{50} +1.00183 q^{51} +26.4734 q^{52} -9.65969 q^{53} +2.48589 q^{54} +0.518716 q^{55} -5.87506 q^{57} +5.82344 q^{58} -1.49818 q^{59} +7.38618 q^{60} +12.8700 q^{61} -3.46733 q^{62} -5.58005 q^{64} +11.1930 q^{65} +0.729681 q^{66} -6.48266 q^{67} +4.18730 q^{68} +0.838269 q^{69} -11.1061 q^{71} +5.41840 q^{72} +4.74670 q^{73} -1.71797 q^{74} -1.87710 q^{75} -24.5558 q^{76} +15.7453 q^{78} +1.80292 q^{79} +9.03067 q^{80} +1.00000 q^{81} -2.48589 q^{82} +11.4808 q^{83} +1.77040 q^{85} -1.01561 q^{86} +2.34260 q^{87} +1.59046 q^{88} -15.8082 q^{89} +4.39300 q^{90} +3.50368 q^{92} -1.39480 q^{93} +27.4418 q^{94} -10.3822 q^{95} +1.86671 q^{96} +9.27612 q^{97} +0.293529 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} + 12 q^{3} + 10 q^{4} + 12 q^{5} - 2 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} + 12 q^{3} + 10 q^{4} + 12 q^{5} - 2 q^{6} + 12 q^{9} + 11 q^{10} + 10 q^{11} + 10 q^{12} + 15 q^{13} + 12 q^{15} + 14 q^{16} + 8 q^{17} - 2 q^{18} + 2 q^{19} + 16 q^{20} - 7 q^{22} + 5 q^{23} + 20 q^{25} + 12 q^{27} + 20 q^{29} + 11 q^{30} + 10 q^{31} + 3 q^{32} + 10 q^{33} - 23 q^{34} + 10 q^{36} - 17 q^{37} + 6 q^{38} + 15 q^{39} + 39 q^{40} - 12 q^{41} + 12 q^{43} + 20 q^{44} + 12 q^{45} - 36 q^{46} + 34 q^{47} + 14 q^{48} + 59 q^{50} + 8 q^{51} + 26 q^{52} + 6 q^{53} - 2 q^{54} - q^{55} + 2 q^{57} - 11 q^{58} + 27 q^{59} + 16 q^{60} + 22 q^{61} - 45 q^{62} + 26 q^{64} - 7 q^{66} - 26 q^{67} + 33 q^{68} + 5 q^{69} + 50 q^{71} + 21 q^{73} - 35 q^{74} + 20 q^{75} - 24 q^{76} - 10 q^{79} + 22 q^{80} + 12 q^{81} + 2 q^{82} + 8 q^{83} + 8 q^{85} - 17 q^{86} + 20 q^{87} - 46 q^{88} + 11 q^{89} + 11 q^{90} + 63 q^{92} + 10 q^{93} + 10 q^{94} + 35 q^{95} + 3 q^{96} + 32 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.48589 1.75779 0.878896 0.477014i \(-0.158281\pi\)
0.878896 + 0.477014i \(0.158281\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.17966 2.08983
\(5\) 1.76717 0.790303 0.395152 0.918616i \(-0.370692\pi\)
0.395152 + 0.918616i \(0.370692\pi\)
\(6\) 2.48589 1.01486
\(7\) 0 0
\(8\) 5.41840 1.91569
\(9\) 1.00000 0.333333
\(10\) 4.39300 1.38919
\(11\) 0.293529 0.0885023 0.0442511 0.999020i \(-0.485910\pi\)
0.0442511 + 0.999020i \(0.485910\pi\)
\(12\) 4.17966 1.20656
\(13\) 6.33387 1.75670 0.878350 0.478019i \(-0.158645\pi\)
0.878350 + 0.478019i \(0.158645\pi\)
\(14\) 0 0
\(15\) 1.76717 0.456282
\(16\) 5.11024 1.27756
\(17\) 1.00183 0.242979 0.121490 0.992593i \(-0.461233\pi\)
0.121490 + 0.992593i \(0.461233\pi\)
\(18\) 2.48589 0.585930
\(19\) −5.87506 −1.34783 −0.673916 0.738808i \(-0.735389\pi\)
−0.673916 + 0.738808i \(0.735389\pi\)
\(20\) 7.38618 1.65160
\(21\) 0 0
\(22\) 0.729681 0.155569
\(23\) 0.838269 0.174791 0.0873956 0.996174i \(-0.472146\pi\)
0.0873956 + 0.996174i \(0.472146\pi\)
\(24\) 5.41840 1.10603
\(25\) −1.87710 −0.375421
\(26\) 15.7453 3.08791
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.34260 0.435009 0.217505 0.976059i \(-0.430208\pi\)
0.217505 + 0.976059i \(0.430208\pi\)
\(30\) 4.39300 0.802048
\(31\) −1.39480 −0.250514 −0.125257 0.992124i \(-0.539976\pi\)
−0.125257 + 0.992124i \(0.539976\pi\)
\(32\) 1.86671 0.329990
\(33\) 0.293529 0.0510968
\(34\) 2.49044 0.427106
\(35\) 0 0
\(36\) 4.17966 0.696610
\(37\) −0.691089 −0.113614 −0.0568072 0.998385i \(-0.518092\pi\)
−0.0568072 + 0.998385i \(0.518092\pi\)
\(38\) −14.6048 −2.36921
\(39\) 6.33387 1.01423
\(40\) 9.57525 1.51398
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) −0.408550 −0.0623034 −0.0311517 0.999515i \(-0.509917\pi\)
−0.0311517 + 0.999515i \(0.509917\pi\)
\(44\) 1.22685 0.184955
\(45\) 1.76717 0.263434
\(46\) 2.08385 0.307247
\(47\) 11.0390 1.61021 0.805103 0.593135i \(-0.202110\pi\)
0.805103 + 0.593135i \(0.202110\pi\)
\(48\) 5.11024 0.737600
\(49\) 0 0
\(50\) −4.66628 −0.659911
\(51\) 1.00183 0.140284
\(52\) 26.4734 3.67120
\(53\) −9.65969 −1.32686 −0.663430 0.748238i \(-0.730900\pi\)
−0.663430 + 0.748238i \(0.730900\pi\)
\(54\) 2.48589 0.338287
\(55\) 0.518716 0.0699436
\(56\) 0 0
\(57\) −5.87506 −0.778171
\(58\) 5.82344 0.764655
\(59\) −1.49818 −0.195047 −0.0975233 0.995233i \(-0.531092\pi\)
−0.0975233 + 0.995233i \(0.531092\pi\)
\(60\) 7.38618 0.953552
\(61\) 12.8700 1.64783 0.823916 0.566712i \(-0.191785\pi\)
0.823916 + 0.566712i \(0.191785\pi\)
\(62\) −3.46733 −0.440351
\(63\) 0 0
\(64\) −5.58005 −0.697506
\(65\) 11.1930 1.38833
\(66\) 0.729681 0.0898175
\(67\) −6.48266 −0.791983 −0.395992 0.918254i \(-0.629599\pi\)
−0.395992 + 0.918254i \(0.629599\pi\)
\(68\) 4.18730 0.507785
\(69\) 0.838269 0.100916
\(70\) 0 0
\(71\) −11.1061 −1.31805 −0.659023 0.752122i \(-0.729030\pi\)
−0.659023 + 0.752122i \(0.729030\pi\)
\(72\) 5.41840 0.638565
\(73\) 4.74670 0.555559 0.277780 0.960645i \(-0.410402\pi\)
0.277780 + 0.960645i \(0.410402\pi\)
\(74\) −1.71797 −0.199710
\(75\) −1.87710 −0.216749
\(76\) −24.5558 −2.81674
\(77\) 0 0
\(78\) 15.7453 1.78281
\(79\) 1.80292 0.202844 0.101422 0.994844i \(-0.467661\pi\)
0.101422 + 0.994844i \(0.467661\pi\)
\(80\) 9.03067 1.00966
\(81\) 1.00000 0.111111
\(82\) −2.48589 −0.274521
\(83\) 11.4808 1.26018 0.630090 0.776522i \(-0.283018\pi\)
0.630090 + 0.776522i \(0.283018\pi\)
\(84\) 0 0
\(85\) 1.77040 0.192027
\(86\) −1.01561 −0.109516
\(87\) 2.34260 0.251153
\(88\) 1.59046 0.169543
\(89\) −15.8082 −1.67566 −0.837831 0.545929i \(-0.816177\pi\)
−0.837831 + 0.545929i \(0.816177\pi\)
\(90\) 4.39300 0.463063
\(91\) 0 0
\(92\) 3.50368 0.365284
\(93\) −1.39480 −0.144634
\(94\) 27.4418 2.83041
\(95\) −10.3822 −1.06520
\(96\) 1.86671 0.190520
\(97\) 9.27612 0.941848 0.470924 0.882174i \(-0.343921\pi\)
0.470924 + 0.882174i \(0.343921\pi\)
\(98\) 0 0
\(99\) 0.293529 0.0295008
\(100\) −7.84565 −0.784565
\(101\) −12.8057 −1.27421 −0.637107 0.770775i \(-0.719869\pi\)
−0.637107 + 0.770775i \(0.719869\pi\)
\(102\) 2.49044 0.246590
\(103\) 3.88886 0.383181 0.191590 0.981475i \(-0.438636\pi\)
0.191590 + 0.981475i \(0.438636\pi\)
\(104\) 34.3194 3.36530
\(105\) 0 0
\(106\) −24.0130 −2.33234
\(107\) −16.5858 −1.60341 −0.801703 0.597723i \(-0.796072\pi\)
−0.801703 + 0.597723i \(0.796072\pi\)
\(108\) 4.17966 0.402188
\(109\) 4.13025 0.395606 0.197803 0.980242i \(-0.436619\pi\)
0.197803 + 0.980242i \(0.436619\pi\)
\(110\) 1.28947 0.122946
\(111\) −0.691089 −0.0655953
\(112\) 0 0
\(113\) 2.33895 0.220030 0.110015 0.993930i \(-0.464910\pi\)
0.110015 + 0.993930i \(0.464910\pi\)
\(114\) −14.6048 −1.36786
\(115\) 1.48137 0.138138
\(116\) 9.79125 0.909095
\(117\) 6.33387 0.585566
\(118\) −3.72432 −0.342851
\(119\) 0 0
\(120\) 9.57525 0.874096
\(121\) −10.9138 −0.992167
\(122\) 31.9934 2.89654
\(123\) −1.00000 −0.0901670
\(124\) −5.82980 −0.523532
\(125\) −12.1530 −1.08700
\(126\) 0 0
\(127\) −11.9908 −1.06401 −0.532006 0.846741i \(-0.678562\pi\)
−0.532006 + 0.846741i \(0.678562\pi\)
\(128\) −17.6048 −1.55606
\(129\) −0.408550 −0.0359709
\(130\) 27.8247 2.44039
\(131\) 11.6217 1.01540 0.507698 0.861535i \(-0.330497\pi\)
0.507698 + 0.861535i \(0.330497\pi\)
\(132\) 1.22685 0.106784
\(133\) 0 0
\(134\) −16.1152 −1.39214
\(135\) 1.76717 0.152094
\(136\) 5.42831 0.465473
\(137\) 4.19260 0.358198 0.179099 0.983831i \(-0.442682\pi\)
0.179099 + 0.983831i \(0.442682\pi\)
\(138\) 2.08385 0.177389
\(139\) −13.9015 −1.17911 −0.589554 0.807729i \(-0.700696\pi\)
−0.589554 + 0.807729i \(0.700696\pi\)
\(140\) 0 0
\(141\) 11.0390 0.929653
\(142\) −27.6085 −2.31685
\(143\) 1.85917 0.155472
\(144\) 5.11024 0.425853
\(145\) 4.13977 0.343789
\(146\) 11.7998 0.976557
\(147\) 0 0
\(148\) −2.88852 −0.237435
\(149\) 19.7434 1.61744 0.808720 0.588194i \(-0.200161\pi\)
0.808720 + 0.588194i \(0.200161\pi\)
\(150\) −4.66628 −0.381000
\(151\) 0.568140 0.0462346 0.0231173 0.999733i \(-0.492641\pi\)
0.0231173 + 0.999733i \(0.492641\pi\)
\(152\) −31.8335 −2.58203
\(153\) 1.00183 0.0809930
\(154\) 0 0
\(155\) −2.46486 −0.197982
\(156\) 26.4734 2.11957
\(157\) −21.6089 −1.72458 −0.862290 0.506415i \(-0.830970\pi\)
−0.862290 + 0.506415i \(0.830970\pi\)
\(158\) 4.48185 0.356557
\(159\) −9.65969 −0.766063
\(160\) 3.29879 0.260792
\(161\) 0 0
\(162\) 2.48589 0.195310
\(163\) −5.82941 −0.456595 −0.228297 0.973591i \(-0.573316\pi\)
−0.228297 + 0.973591i \(0.573316\pi\)
\(164\) −4.17966 −0.326377
\(165\) 0.518716 0.0403820
\(166\) 28.5400 2.21513
\(167\) 14.9641 1.15795 0.578977 0.815344i \(-0.303452\pi\)
0.578977 + 0.815344i \(0.303452\pi\)
\(168\) 0 0
\(169\) 27.1179 2.08599
\(170\) 4.40103 0.337544
\(171\) −5.87506 −0.449277
\(172\) −1.70760 −0.130203
\(173\) 2.55467 0.194228 0.0971142 0.995273i \(-0.469039\pi\)
0.0971142 + 0.995273i \(0.469039\pi\)
\(174\) 5.82344 0.441474
\(175\) 0 0
\(176\) 1.50000 0.113067
\(177\) −1.49818 −0.112610
\(178\) −39.2974 −2.94546
\(179\) 11.4969 0.859315 0.429658 0.902992i \(-0.358634\pi\)
0.429658 + 0.902992i \(0.358634\pi\)
\(180\) 7.38618 0.550533
\(181\) 10.4879 0.779561 0.389780 0.920908i \(-0.372551\pi\)
0.389780 + 0.920908i \(0.372551\pi\)
\(182\) 0 0
\(183\) 12.8700 0.951376
\(184\) 4.54208 0.334847
\(185\) −1.22127 −0.0897898
\(186\) −3.46733 −0.254237
\(187\) 0.294065 0.0215042
\(188\) 46.1393 3.36506
\(189\) 0 0
\(190\) −25.8092 −1.87239
\(191\) −5.06053 −0.366167 −0.183083 0.983097i \(-0.558608\pi\)
−0.183083 + 0.983097i \(0.558608\pi\)
\(192\) −5.58005 −0.402705
\(193\) 3.54687 0.255310 0.127655 0.991819i \(-0.459255\pi\)
0.127655 + 0.991819i \(0.459255\pi\)
\(194\) 23.0594 1.65557
\(195\) 11.1930 0.801550
\(196\) 0 0
\(197\) −17.6617 −1.25835 −0.629173 0.777265i \(-0.716606\pi\)
−0.629173 + 0.777265i \(0.716606\pi\)
\(198\) 0.729681 0.0518562
\(199\) −15.1642 −1.07496 −0.537481 0.843276i \(-0.680624\pi\)
−0.537481 + 0.843276i \(0.680624\pi\)
\(200\) −10.1709 −0.719191
\(201\) −6.48266 −0.457252
\(202\) −31.8336 −2.23980
\(203\) 0 0
\(204\) 4.18730 0.293170
\(205\) −1.76717 −0.123425
\(206\) 9.66728 0.673551
\(207\) 0.838269 0.0582638
\(208\) 32.3676 2.24429
\(209\) −1.72450 −0.119286
\(210\) 0 0
\(211\) 6.89154 0.474433 0.237217 0.971457i \(-0.423765\pi\)
0.237217 + 0.971457i \(0.423765\pi\)
\(212\) −40.3742 −2.77291
\(213\) −11.1061 −0.760975
\(214\) −41.2304 −2.81845
\(215\) −0.721979 −0.0492386
\(216\) 5.41840 0.368676
\(217\) 0 0
\(218\) 10.2674 0.695393
\(219\) 4.74670 0.320752
\(220\) 2.16806 0.146170
\(221\) 6.34545 0.426841
\(222\) −1.71797 −0.115303
\(223\) −17.7285 −1.18719 −0.593594 0.804764i \(-0.702292\pi\)
−0.593594 + 0.804764i \(0.702292\pi\)
\(224\) 0 0
\(225\) −1.87710 −0.125140
\(226\) 5.81438 0.386767
\(227\) 6.90430 0.458254 0.229127 0.973397i \(-0.426413\pi\)
0.229127 + 0.973397i \(0.426413\pi\)
\(228\) −24.5558 −1.62625
\(229\) 14.1972 0.938175 0.469088 0.883152i \(-0.344583\pi\)
0.469088 + 0.883152i \(0.344583\pi\)
\(230\) 3.68252 0.242818
\(231\) 0 0
\(232\) 12.6931 0.833344
\(233\) 1.41764 0.0928725 0.0464363 0.998921i \(-0.485214\pi\)
0.0464363 + 0.998921i \(0.485214\pi\)
\(234\) 15.7453 1.02930
\(235\) 19.5078 1.27255
\(236\) −6.26189 −0.407614
\(237\) 1.80292 0.117112
\(238\) 0 0
\(239\) 1.91363 0.123782 0.0618912 0.998083i \(-0.480287\pi\)
0.0618912 + 0.998083i \(0.480287\pi\)
\(240\) 9.03067 0.582928
\(241\) −8.74190 −0.563115 −0.281558 0.959544i \(-0.590851\pi\)
−0.281558 + 0.959544i \(0.590851\pi\)
\(242\) −27.1306 −1.74402
\(243\) 1.00000 0.0641500
\(244\) 53.7921 3.44369
\(245\) 0 0
\(246\) −2.48589 −0.158495
\(247\) −37.2119 −2.36774
\(248\) −7.55760 −0.479908
\(249\) 11.4808 0.727566
\(250\) −30.2111 −1.91072
\(251\) 16.3935 1.03475 0.517373 0.855760i \(-0.326910\pi\)
0.517373 + 0.855760i \(0.326910\pi\)
\(252\) 0 0
\(253\) 0.246056 0.0154694
\(254\) −29.8079 −1.87031
\(255\) 1.77040 0.110867
\(256\) −32.6036 −2.03772
\(257\) −5.79030 −0.361189 −0.180594 0.983558i \(-0.557802\pi\)
−0.180594 + 0.983558i \(0.557802\pi\)
\(258\) −1.01561 −0.0632293
\(259\) 0 0
\(260\) 46.7831 2.90136
\(261\) 2.34260 0.145003
\(262\) 28.8904 1.78485
\(263\) 7.57467 0.467074 0.233537 0.972348i \(-0.424970\pi\)
0.233537 + 0.972348i \(0.424970\pi\)
\(264\) 1.59046 0.0978859
\(265\) −17.0703 −1.04862
\(266\) 0 0
\(267\) −15.8082 −0.967444
\(268\) −27.0953 −1.65511
\(269\) −24.0489 −1.46629 −0.733144 0.680073i \(-0.761948\pi\)
−0.733144 + 0.680073i \(0.761948\pi\)
\(270\) 4.39300 0.267349
\(271\) 14.6082 0.887385 0.443693 0.896179i \(-0.353668\pi\)
0.443693 + 0.896179i \(0.353668\pi\)
\(272\) 5.11958 0.310420
\(273\) 0 0
\(274\) 10.4223 0.629637
\(275\) −0.550984 −0.0332256
\(276\) 3.50368 0.210897
\(277\) 13.2496 0.796094 0.398047 0.917365i \(-0.369688\pi\)
0.398047 + 0.917365i \(0.369688\pi\)
\(278\) −34.5576 −2.07263
\(279\) −1.39480 −0.0835047
\(280\) 0 0
\(281\) 15.2773 0.911366 0.455683 0.890142i \(-0.349395\pi\)
0.455683 + 0.890142i \(0.349395\pi\)
\(282\) 27.4418 1.63414
\(283\) 27.9553 1.66177 0.830885 0.556444i \(-0.187834\pi\)
0.830885 + 0.556444i \(0.187834\pi\)
\(284\) −46.4196 −2.75449
\(285\) −10.3822 −0.614991
\(286\) 4.62170 0.273287
\(287\) 0 0
\(288\) 1.86671 0.109997
\(289\) −15.9963 −0.940961
\(290\) 10.2910 0.604309
\(291\) 9.27612 0.543776
\(292\) 19.8396 1.16102
\(293\) 10.3302 0.603498 0.301749 0.953387i \(-0.402430\pi\)
0.301749 + 0.953387i \(0.402430\pi\)
\(294\) 0 0
\(295\) −2.64754 −0.154146
\(296\) −3.74460 −0.217650
\(297\) 0.293529 0.0170323
\(298\) 49.0799 2.84312
\(299\) 5.30949 0.307056
\(300\) −7.84565 −0.452969
\(301\) 0 0
\(302\) 1.41233 0.0812707
\(303\) −12.8057 −0.735668
\(304\) −30.0230 −1.72194
\(305\) 22.7435 1.30229
\(306\) 2.49044 0.142369
\(307\) −19.9196 −1.13687 −0.568437 0.822727i \(-0.692452\pi\)
−0.568437 + 0.822727i \(0.692452\pi\)
\(308\) 0 0
\(309\) 3.88886 0.221229
\(310\) −6.12737 −0.348011
\(311\) −19.0288 −1.07903 −0.539513 0.841977i \(-0.681392\pi\)
−0.539513 + 0.841977i \(0.681392\pi\)
\(312\) 34.3194 1.94296
\(313\) 30.7830 1.73996 0.869979 0.493090i \(-0.164133\pi\)
0.869979 + 0.493090i \(0.164133\pi\)
\(314\) −53.7175 −3.03145
\(315\) 0 0
\(316\) 7.53557 0.423909
\(317\) −3.77693 −0.212134 −0.106067 0.994359i \(-0.533826\pi\)
−0.106067 + 0.994359i \(0.533826\pi\)
\(318\) −24.0130 −1.34658
\(319\) 0.687619 0.0384993
\(320\) −9.86091 −0.551242
\(321\) −16.5858 −0.925726
\(322\) 0 0
\(323\) −5.88580 −0.327495
\(324\) 4.17966 0.232203
\(325\) −11.8893 −0.659501
\(326\) −14.4913 −0.802598
\(327\) 4.13025 0.228403
\(328\) −5.41840 −0.299181
\(329\) 0 0
\(330\) 1.28947 0.0709831
\(331\) −10.2326 −0.562434 −0.281217 0.959644i \(-0.590738\pi\)
−0.281217 + 0.959644i \(0.590738\pi\)
\(332\) 47.9858 2.63356
\(333\) −0.691089 −0.0378715
\(334\) 37.1991 2.03544
\(335\) −11.4560 −0.625907
\(336\) 0 0
\(337\) 36.0266 1.96249 0.981247 0.192753i \(-0.0617416\pi\)
0.981247 + 0.192753i \(0.0617416\pi\)
\(338\) 67.4122 3.66674
\(339\) 2.33895 0.127034
\(340\) 7.39968 0.401304
\(341\) −0.409415 −0.0221711
\(342\) −14.6048 −0.789736
\(343\) 0 0
\(344\) −2.21369 −0.119354
\(345\) 1.48137 0.0797541
\(346\) 6.35065 0.341413
\(347\) 8.72186 0.468214 0.234107 0.972211i \(-0.424783\pi\)
0.234107 + 0.972211i \(0.424783\pi\)
\(348\) 9.79125 0.524866
\(349\) 25.6152 1.37115 0.685576 0.728001i \(-0.259550\pi\)
0.685576 + 0.728001i \(0.259550\pi\)
\(350\) 0 0
\(351\) 6.33387 0.338077
\(352\) 0.547932 0.0292049
\(353\) 3.43415 0.182781 0.0913906 0.995815i \(-0.470869\pi\)
0.0913906 + 0.995815i \(0.470869\pi\)
\(354\) −3.72432 −0.197945
\(355\) −19.6263 −1.04166
\(356\) −66.0728 −3.50185
\(357\) 0 0
\(358\) 28.5799 1.51050
\(359\) −34.2365 −1.80693 −0.903467 0.428658i \(-0.858987\pi\)
−0.903467 + 0.428658i \(0.858987\pi\)
\(360\) 9.57525 0.504660
\(361\) 15.5164 0.816651
\(362\) 26.0718 1.37031
\(363\) −10.9138 −0.572828
\(364\) 0 0
\(365\) 8.38823 0.439060
\(366\) 31.9934 1.67232
\(367\) −15.7607 −0.822700 −0.411350 0.911478i \(-0.634943\pi\)
−0.411350 + 0.911478i \(0.634943\pi\)
\(368\) 4.28376 0.223306
\(369\) −1.00000 −0.0520579
\(370\) −3.03595 −0.157832
\(371\) 0 0
\(372\) −5.82980 −0.302261
\(373\) −34.4113 −1.78175 −0.890876 0.454247i \(-0.849908\pi\)
−0.890876 + 0.454247i \(0.849908\pi\)
\(374\) 0.731015 0.0377999
\(375\) −12.1530 −0.627579
\(376\) 59.8138 3.08466
\(377\) 14.8377 0.764180
\(378\) 0 0
\(379\) 20.4556 1.05073 0.525366 0.850877i \(-0.323929\pi\)
0.525366 + 0.850877i \(0.323929\pi\)
\(380\) −43.3943 −2.22608
\(381\) −11.9908 −0.614308
\(382\) −12.5799 −0.643645
\(383\) −30.8234 −1.57500 −0.787502 0.616312i \(-0.788626\pi\)
−0.787502 + 0.616312i \(0.788626\pi\)
\(384\) −17.6048 −0.898392
\(385\) 0 0
\(386\) 8.81715 0.448781
\(387\) −0.408550 −0.0207678
\(388\) 38.7710 1.96830
\(389\) 24.5119 1.24280 0.621401 0.783493i \(-0.286564\pi\)
0.621401 + 0.783493i \(0.286564\pi\)
\(390\) 27.8247 1.40896
\(391\) 0.839802 0.0424706
\(392\) 0 0
\(393\) 11.6217 0.586239
\(394\) −43.9052 −2.21191
\(395\) 3.18606 0.160308
\(396\) 1.22685 0.0616516
\(397\) 13.5526 0.680188 0.340094 0.940391i \(-0.389541\pi\)
0.340094 + 0.940391i \(0.389541\pi\)
\(398\) −37.6966 −1.88956
\(399\) 0 0
\(400\) −9.59245 −0.479623
\(401\) 19.7195 0.984746 0.492373 0.870384i \(-0.336130\pi\)
0.492373 + 0.870384i \(0.336130\pi\)
\(402\) −16.1152 −0.803753
\(403\) −8.83450 −0.440078
\(404\) −53.5235 −2.66289
\(405\) 1.76717 0.0878115
\(406\) 0 0
\(407\) −0.202855 −0.0100551
\(408\) 5.42831 0.268741
\(409\) −15.1964 −0.751415 −0.375707 0.926738i \(-0.622600\pi\)
−0.375707 + 0.926738i \(0.622600\pi\)
\(410\) −4.39300 −0.216955
\(411\) 4.19260 0.206806
\(412\) 16.2541 0.800782
\(413\) 0 0
\(414\) 2.08385 0.102416
\(415\) 20.2885 0.995925
\(416\) 11.8235 0.579694
\(417\) −13.9015 −0.680758
\(418\) −4.28692 −0.209680
\(419\) −38.4758 −1.87966 −0.939832 0.341637i \(-0.889019\pi\)
−0.939832 + 0.341637i \(0.889019\pi\)
\(420\) 0 0
\(421\) −39.0584 −1.90359 −0.951795 0.306734i \(-0.900764\pi\)
−0.951795 + 0.306734i \(0.900764\pi\)
\(422\) 17.1316 0.833955
\(423\) 11.0390 0.536735
\(424\) −52.3401 −2.54186
\(425\) −1.88053 −0.0912193
\(426\) −27.6085 −1.33763
\(427\) 0 0
\(428\) −69.3228 −3.35084
\(429\) 1.85917 0.0897617
\(430\) −1.79476 −0.0865511
\(431\) 1.05223 0.0506840 0.0253420 0.999679i \(-0.491933\pi\)
0.0253420 + 0.999679i \(0.491933\pi\)
\(432\) 5.11024 0.245867
\(433\) 24.7423 1.18904 0.594520 0.804081i \(-0.297342\pi\)
0.594520 + 0.804081i \(0.297342\pi\)
\(434\) 0 0
\(435\) 4.13977 0.198487
\(436\) 17.2630 0.826749
\(437\) −4.92489 −0.235589
\(438\) 11.7998 0.563815
\(439\) −13.9212 −0.664421 −0.332210 0.943205i \(-0.607794\pi\)
−0.332210 + 0.943205i \(0.607794\pi\)
\(440\) 2.81061 0.133991
\(441\) 0 0
\(442\) 15.7741 0.750298
\(443\) 7.20423 0.342283 0.171142 0.985246i \(-0.445254\pi\)
0.171142 + 0.985246i \(0.445254\pi\)
\(444\) −2.88852 −0.137083
\(445\) −27.9357 −1.32428
\(446\) −44.0712 −2.08683
\(447\) 19.7434 0.933830
\(448\) 0 0
\(449\) −26.5965 −1.25517 −0.627584 0.778549i \(-0.715956\pi\)
−0.627584 + 0.778549i \(0.715956\pi\)
\(450\) −4.66628 −0.219970
\(451\) −0.293529 −0.0138217
\(452\) 9.77601 0.459825
\(453\) 0.568140 0.0266935
\(454\) 17.1633 0.805515
\(455\) 0 0
\(456\) −31.8335 −1.49074
\(457\) 22.2276 1.03976 0.519881 0.854238i \(-0.325976\pi\)
0.519881 + 0.854238i \(0.325976\pi\)
\(458\) 35.2926 1.64912
\(459\) 1.00183 0.0467613
\(460\) 6.19161 0.288685
\(461\) 1.88110 0.0876114 0.0438057 0.999040i \(-0.486052\pi\)
0.0438057 + 0.999040i \(0.486052\pi\)
\(462\) 0 0
\(463\) −9.16637 −0.425997 −0.212999 0.977053i \(-0.568323\pi\)
−0.212999 + 0.977053i \(0.568323\pi\)
\(464\) 11.9712 0.555750
\(465\) −2.46486 −0.114305
\(466\) 3.52409 0.163250
\(467\) −33.7593 −1.56220 −0.781098 0.624409i \(-0.785340\pi\)
−0.781098 + 0.624409i \(0.785340\pi\)
\(468\) 26.4734 1.22373
\(469\) 0 0
\(470\) 48.4944 2.23688
\(471\) −21.6089 −0.995687
\(472\) −8.11775 −0.373650
\(473\) −0.119921 −0.00551399
\(474\) 4.48185 0.205858
\(475\) 11.0281 0.506004
\(476\) 0 0
\(477\) −9.65969 −0.442287
\(478\) 4.75707 0.217583
\(479\) −7.16053 −0.327173 −0.163587 0.986529i \(-0.552306\pi\)
−0.163587 + 0.986529i \(0.552306\pi\)
\(480\) 3.29879 0.150569
\(481\) −4.37727 −0.199586
\(482\) −21.7314 −0.989839
\(483\) 0 0
\(484\) −45.6161 −2.07346
\(485\) 16.3925 0.744345
\(486\) 2.48589 0.112762
\(487\) −10.9331 −0.495424 −0.247712 0.968834i \(-0.579679\pi\)
−0.247712 + 0.968834i \(0.579679\pi\)
\(488\) 69.7347 3.15674
\(489\) −5.82941 −0.263615
\(490\) 0 0
\(491\) 24.5154 1.10637 0.553183 0.833060i \(-0.313413\pi\)
0.553183 + 0.833060i \(0.313413\pi\)
\(492\) −4.17966 −0.188434
\(493\) 2.34688 0.105698
\(494\) −92.5047 −4.16199
\(495\) 0.518716 0.0233145
\(496\) −7.12778 −0.320047
\(497\) 0 0
\(498\) 28.5400 1.27891
\(499\) −37.4506 −1.67652 −0.838261 0.545270i \(-0.816427\pi\)
−0.838261 + 0.545270i \(0.816427\pi\)
\(500\) −50.7955 −2.27164
\(501\) 14.9641 0.668545
\(502\) 40.7524 1.81887
\(503\) −11.7824 −0.525352 −0.262676 0.964884i \(-0.584605\pi\)
−0.262676 + 0.964884i \(0.584605\pi\)
\(504\) 0 0
\(505\) −22.6299 −1.00702
\(506\) 0.611669 0.0271920
\(507\) 27.1179 1.20435
\(508\) −50.1175 −2.22360
\(509\) 44.4596 1.97064 0.985319 0.170723i \(-0.0546104\pi\)
0.985319 + 0.170723i \(0.0546104\pi\)
\(510\) 4.40103 0.194881
\(511\) 0 0
\(512\) −45.8394 −2.02583
\(513\) −5.87506 −0.259390
\(514\) −14.3941 −0.634894
\(515\) 6.87228 0.302829
\(516\) −1.70760 −0.0751730
\(517\) 3.24027 0.142507
\(518\) 0 0
\(519\) 2.55467 0.112138
\(520\) 60.6484 2.65961
\(521\) 10.0439 0.440031 0.220015 0.975496i \(-0.429389\pi\)
0.220015 + 0.975496i \(0.429389\pi\)
\(522\) 5.82344 0.254885
\(523\) −29.9705 −1.31052 −0.655259 0.755405i \(-0.727440\pi\)
−0.655259 + 0.755405i \(0.727440\pi\)
\(524\) 48.5749 2.12200
\(525\) 0 0
\(526\) 18.8298 0.821019
\(527\) −1.39735 −0.0608696
\(528\) 1.50000 0.0652793
\(529\) −22.2973 −0.969448
\(530\) −42.4350 −1.84326
\(531\) −1.49818 −0.0650155
\(532\) 0 0
\(533\) −6.33387 −0.274350
\(534\) −39.2974 −1.70056
\(535\) −29.3099 −1.26718
\(536\) −35.1257 −1.51720
\(537\) 11.4969 0.496126
\(538\) −59.7830 −2.57743
\(539\) 0 0
\(540\) 7.38618 0.317851
\(541\) 5.69166 0.244704 0.122352 0.992487i \(-0.460956\pi\)
0.122352 + 0.992487i \(0.460956\pi\)
\(542\) 36.3144 1.55984
\(543\) 10.4879 0.450080
\(544\) 1.87012 0.0801807
\(545\) 7.29886 0.312649
\(546\) 0 0
\(547\) −39.2276 −1.67725 −0.838625 0.544710i \(-0.816640\pi\)
−0.838625 + 0.544710i \(0.816640\pi\)
\(548\) 17.5236 0.748573
\(549\) 12.8700 0.549277
\(550\) −1.36969 −0.0584036
\(551\) −13.7629 −0.586319
\(552\) 4.54208 0.193324
\(553\) 0 0
\(554\) 32.9372 1.39937
\(555\) −1.22127 −0.0518402
\(556\) −58.1035 −2.46414
\(557\) −14.4005 −0.610170 −0.305085 0.952325i \(-0.598685\pi\)
−0.305085 + 0.952325i \(0.598685\pi\)
\(558\) −3.46733 −0.146784
\(559\) −2.58771 −0.109448
\(560\) 0 0
\(561\) 0.294065 0.0124155
\(562\) 37.9777 1.60199
\(563\) 5.77006 0.243179 0.121590 0.992580i \(-0.461201\pi\)
0.121590 + 0.992580i \(0.461201\pi\)
\(564\) 46.1393 1.94282
\(565\) 4.13333 0.173890
\(566\) 69.4939 2.92105
\(567\) 0 0
\(568\) −60.1771 −2.52497
\(569\) 16.3956 0.687340 0.343670 0.939091i \(-0.388330\pi\)
0.343670 + 0.939091i \(0.388330\pi\)
\(570\) −25.8092 −1.08103
\(571\) 7.21318 0.301862 0.150931 0.988544i \(-0.451773\pi\)
0.150931 + 0.988544i \(0.451773\pi\)
\(572\) 7.77071 0.324910
\(573\) −5.06053 −0.211407
\(574\) 0 0
\(575\) −1.57352 −0.0656203
\(576\) −5.58005 −0.232502
\(577\) −47.7149 −1.98640 −0.993199 0.116433i \(-0.962854\pi\)
−0.993199 + 0.116433i \(0.962854\pi\)
\(578\) −39.7652 −1.65401
\(579\) 3.54687 0.147403
\(580\) 17.3028 0.718461
\(581\) 0 0
\(582\) 23.0594 0.955845
\(583\) −2.83540 −0.117430
\(584\) 25.7195 1.06428
\(585\) 11.1930 0.462775
\(586\) 25.6798 1.06082
\(587\) 10.7984 0.445698 0.222849 0.974853i \(-0.428464\pi\)
0.222849 + 0.974853i \(0.428464\pi\)
\(588\) 0 0
\(589\) 8.19455 0.337651
\(590\) −6.58151 −0.270956
\(591\) −17.6617 −0.726506
\(592\) −3.53163 −0.145149
\(593\) −27.3645 −1.12373 −0.561863 0.827230i \(-0.689915\pi\)
−0.561863 + 0.827230i \(0.689915\pi\)
\(594\) 0.729681 0.0299392
\(595\) 0 0
\(596\) 82.5206 3.38018
\(597\) −15.1642 −0.620630
\(598\) 13.1988 0.539740
\(599\) 25.8885 1.05777 0.528887 0.848692i \(-0.322610\pi\)
0.528887 + 0.848692i \(0.322610\pi\)
\(600\) −10.1709 −0.415225
\(601\) 15.1127 0.616459 0.308230 0.951312i \(-0.400263\pi\)
0.308230 + 0.951312i \(0.400263\pi\)
\(602\) 0 0
\(603\) −6.48266 −0.263994
\(604\) 2.37463 0.0966224
\(605\) −19.2866 −0.784113
\(606\) −31.8336 −1.29315
\(607\) −12.5002 −0.507369 −0.253685 0.967287i \(-0.581643\pi\)
−0.253685 + 0.967287i \(0.581643\pi\)
\(608\) −10.9670 −0.444771
\(609\) 0 0
\(610\) 56.5378 2.28915
\(611\) 69.9197 2.82865
\(612\) 4.18730 0.169262
\(613\) 9.60492 0.387939 0.193970 0.981008i \(-0.437864\pi\)
0.193970 + 0.981008i \(0.437864\pi\)
\(614\) −49.5181 −1.99839
\(615\) −1.76717 −0.0712593
\(616\) 0 0
\(617\) 3.72064 0.149787 0.0748936 0.997192i \(-0.476138\pi\)
0.0748936 + 0.997192i \(0.476138\pi\)
\(618\) 9.66728 0.388875
\(619\) 34.8001 1.39874 0.699368 0.714762i \(-0.253465\pi\)
0.699368 + 0.714762i \(0.253465\pi\)
\(620\) −10.3023 −0.413749
\(621\) 0.838269 0.0336386
\(622\) −47.3036 −1.89670
\(623\) 0 0
\(624\) 32.3676 1.29574
\(625\) −12.0910 −0.483639
\(626\) 76.5232 3.05848
\(627\) −1.72450 −0.0688699
\(628\) −90.3180 −3.60408
\(629\) −0.692353 −0.0276059
\(630\) 0 0
\(631\) −19.9796 −0.795377 −0.397688 0.917521i \(-0.630187\pi\)
−0.397688 + 0.917521i \(0.630187\pi\)
\(632\) 9.76892 0.388587
\(633\) 6.89154 0.273914
\(634\) −9.38905 −0.372887
\(635\) −21.1898 −0.840892
\(636\) −40.3742 −1.60094
\(637\) 0 0
\(638\) 1.70935 0.0676737
\(639\) −11.1061 −0.439349
\(640\) −31.1107 −1.22976
\(641\) 19.8441 0.783794 0.391897 0.920009i \(-0.371819\pi\)
0.391897 + 0.920009i \(0.371819\pi\)
\(642\) −41.2304 −1.62723
\(643\) −45.1521 −1.78062 −0.890312 0.455352i \(-0.849513\pi\)
−0.890312 + 0.455352i \(0.849513\pi\)
\(644\) 0 0
\(645\) −0.721979 −0.0284279
\(646\) −14.6315 −0.575668
\(647\) 24.8685 0.977683 0.488841 0.872373i \(-0.337420\pi\)
0.488841 + 0.872373i \(0.337420\pi\)
\(648\) 5.41840 0.212855
\(649\) −0.439759 −0.0172621
\(650\) −29.5556 −1.15927
\(651\) 0 0
\(652\) −24.3649 −0.954205
\(653\) −26.2667 −1.02789 −0.513947 0.857822i \(-0.671817\pi\)
−0.513947 + 0.857822i \(0.671817\pi\)
\(654\) 10.2674 0.401485
\(655\) 20.5376 0.802470
\(656\) −5.11024 −0.199521
\(657\) 4.74670 0.185186
\(658\) 0 0
\(659\) 7.06739 0.275307 0.137653 0.990480i \(-0.456044\pi\)
0.137653 + 0.990480i \(0.456044\pi\)
\(660\) 2.16806 0.0843915
\(661\) 21.1380 0.822174 0.411087 0.911596i \(-0.365149\pi\)
0.411087 + 0.911596i \(0.365149\pi\)
\(662\) −25.4371 −0.988641
\(663\) 6.34545 0.246437
\(664\) 62.2075 2.41412
\(665\) 0 0
\(666\) −1.71797 −0.0665701
\(667\) 1.96373 0.0760358
\(668\) 62.5447 2.41993
\(669\) −17.7285 −0.685424
\(670\) −28.4783 −1.10021
\(671\) 3.77771 0.145837
\(672\) 0 0
\(673\) 43.2461 1.66701 0.833506 0.552510i \(-0.186330\pi\)
0.833506 + 0.552510i \(0.186330\pi\)
\(674\) 89.5583 3.44966
\(675\) −1.87710 −0.0722497
\(676\) 113.344 4.35937
\(677\) −5.05319 −0.194210 −0.0971049 0.995274i \(-0.530958\pi\)
−0.0971049 + 0.995274i \(0.530958\pi\)
\(678\) 5.81438 0.223300
\(679\) 0 0
\(680\) 9.59275 0.367865
\(681\) 6.90430 0.264573
\(682\) −1.01776 −0.0389721
\(683\) 4.29790 0.164455 0.0822273 0.996614i \(-0.473797\pi\)
0.0822273 + 0.996614i \(0.473797\pi\)
\(684\) −24.5558 −0.938913
\(685\) 7.40904 0.283085
\(686\) 0 0
\(687\) 14.1972 0.541656
\(688\) −2.08779 −0.0795963
\(689\) −61.1832 −2.33090
\(690\) 3.68252 0.140191
\(691\) 1.35475 0.0515370 0.0257685 0.999668i \(-0.491797\pi\)
0.0257685 + 0.999668i \(0.491797\pi\)
\(692\) 10.6777 0.405904
\(693\) 0 0
\(694\) 21.6816 0.823023
\(695\) −24.5663 −0.931853
\(696\) 12.6931 0.481132
\(697\) −1.00183 −0.0379469
\(698\) 63.6767 2.41020
\(699\) 1.41764 0.0536200
\(700\) 0 0
\(701\) −43.4004 −1.63921 −0.819605 0.572929i \(-0.805807\pi\)
−0.819605 + 0.572929i \(0.805807\pi\)
\(702\) 15.7453 0.594269
\(703\) 4.06019 0.153133
\(704\) −1.63791 −0.0617309
\(705\) 19.5078 0.734708
\(706\) 8.53692 0.321291
\(707\) 0 0
\(708\) −6.26189 −0.235336
\(709\) 40.2824 1.51284 0.756419 0.654087i \(-0.226947\pi\)
0.756419 + 0.654087i \(0.226947\pi\)
\(710\) −48.7889 −1.83102
\(711\) 1.80292 0.0676146
\(712\) −85.6550 −3.21006
\(713\) −1.16922 −0.0437877
\(714\) 0 0
\(715\) 3.28548 0.122870
\(716\) 48.0530 1.79582
\(717\) 1.91363 0.0714658
\(718\) −85.1083 −3.17621
\(719\) −31.2329 −1.16479 −0.582394 0.812906i \(-0.697884\pi\)
−0.582394 + 0.812906i \(0.697884\pi\)
\(720\) 9.03067 0.336553
\(721\) 0 0
\(722\) 38.5720 1.43550
\(723\) −8.74190 −0.325115
\(724\) 43.8359 1.62915
\(725\) −4.39729 −0.163311
\(726\) −27.1306 −1.00691
\(727\) 33.0217 1.22471 0.612354 0.790584i \(-0.290223\pi\)
0.612354 + 0.790584i \(0.290223\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 20.8522 0.771776
\(731\) −0.409297 −0.0151384
\(732\) 53.7921 1.98821
\(733\) 9.34705 0.345241 0.172621 0.984988i \(-0.444777\pi\)
0.172621 + 0.984988i \(0.444777\pi\)
\(734\) −39.1793 −1.44613
\(735\) 0 0
\(736\) 1.56480 0.0576794
\(737\) −1.90285 −0.0700923
\(738\) −2.48589 −0.0915070
\(739\) −48.8290 −1.79620 −0.898101 0.439789i \(-0.855053\pi\)
−0.898101 + 0.439789i \(0.855053\pi\)
\(740\) −5.10451 −0.187645
\(741\) −37.2119 −1.36701
\(742\) 0 0
\(743\) 14.2245 0.521845 0.260923 0.965360i \(-0.415973\pi\)
0.260923 + 0.965360i \(0.415973\pi\)
\(744\) −7.55760 −0.277075
\(745\) 34.8899 1.27827
\(746\) −85.5429 −3.13195
\(747\) 11.4808 0.420060
\(748\) 1.22909 0.0449401
\(749\) 0 0
\(750\) −30.2111 −1.10315
\(751\) −6.47294 −0.236201 −0.118100 0.993002i \(-0.537680\pi\)
−0.118100 + 0.993002i \(0.537680\pi\)
\(752\) 56.4120 2.05714
\(753\) 16.3935 0.597411
\(754\) 36.8849 1.34327
\(755\) 1.00400 0.0365393
\(756\) 0 0
\(757\) 28.8314 1.04790 0.523948 0.851751i \(-0.324459\pi\)
0.523948 + 0.851751i \(0.324459\pi\)
\(758\) 50.8503 1.84697
\(759\) 0.246056 0.00893128
\(760\) −56.2552 −2.04059
\(761\) −34.6531 −1.25617 −0.628087 0.778143i \(-0.716162\pi\)
−0.628087 + 0.778143i \(0.716162\pi\)
\(762\) −29.8079 −1.07982
\(763\) 0 0
\(764\) −21.1513 −0.765227
\(765\) 1.77040 0.0640090
\(766\) −76.6238 −2.76853
\(767\) −9.48929 −0.342638
\(768\) −32.6036 −1.17648
\(769\) −23.0565 −0.831438 −0.415719 0.909493i \(-0.636470\pi\)
−0.415719 + 0.909493i \(0.636470\pi\)
\(770\) 0 0
\(771\) −5.79030 −0.208532
\(772\) 14.8247 0.533554
\(773\) 13.9342 0.501180 0.250590 0.968093i \(-0.419375\pi\)
0.250590 + 0.968093i \(0.419375\pi\)
\(774\) −1.01561 −0.0365054
\(775\) 2.61819 0.0940481
\(776\) 50.2618 1.80429
\(777\) 0 0
\(778\) 60.9339 2.18459
\(779\) 5.87506 0.210496
\(780\) 46.7831 1.67510
\(781\) −3.25995 −0.116650
\(782\) 2.08766 0.0746545
\(783\) 2.34260 0.0837175
\(784\) 0 0
\(785\) −38.1867 −1.36294
\(786\) 28.8904 1.03049
\(787\) −7.78663 −0.277564 −0.138782 0.990323i \(-0.544319\pi\)
−0.138782 + 0.990323i \(0.544319\pi\)
\(788\) −73.8200 −2.62973
\(789\) 7.57467 0.269665
\(790\) 7.92020 0.281788
\(791\) 0 0
\(792\) 1.59046 0.0565144
\(793\) 81.5168 2.89475
\(794\) 33.6904 1.19563
\(795\) −17.0703 −0.605422
\(796\) −63.3813 −2.24649
\(797\) −12.2136 −0.432628 −0.216314 0.976324i \(-0.569404\pi\)
−0.216314 + 0.976324i \(0.569404\pi\)
\(798\) 0 0
\(799\) 11.0592 0.391246
\(800\) −3.50400 −0.123885
\(801\) −15.8082 −0.558554
\(802\) 49.0206 1.73098
\(803\) 1.39329 0.0491683
\(804\) −27.0953 −0.955578
\(805\) 0 0
\(806\) −21.9616 −0.773565
\(807\) −24.0489 −0.846562
\(808\) −69.3864 −2.44101
\(809\) 40.8740 1.43705 0.718527 0.695499i \(-0.244817\pi\)
0.718527 + 0.695499i \(0.244817\pi\)
\(810\) 4.39300 0.154354
\(811\) −22.6939 −0.796891 −0.398446 0.917192i \(-0.630450\pi\)
−0.398446 + 0.917192i \(0.630450\pi\)
\(812\) 0 0
\(813\) 14.6082 0.512332
\(814\) −0.504275 −0.0176748
\(815\) −10.3016 −0.360848
\(816\) 5.11958 0.179221
\(817\) 2.40026 0.0839745
\(818\) −37.7767 −1.32083
\(819\) 0 0
\(820\) −7.38618 −0.257937
\(821\) 23.5550 0.822077 0.411038 0.911618i \(-0.365166\pi\)
0.411038 + 0.911618i \(0.365166\pi\)
\(822\) 10.4223 0.363521
\(823\) 38.9259 1.35687 0.678436 0.734659i \(-0.262658\pi\)
0.678436 + 0.734659i \(0.262658\pi\)
\(824\) 21.0714 0.734057
\(825\) −0.550984 −0.0191828
\(826\) 0 0
\(827\) 27.6495 0.961468 0.480734 0.876867i \(-0.340370\pi\)
0.480734 + 0.876867i \(0.340370\pi\)
\(828\) 3.50368 0.121761
\(829\) 5.42158 0.188299 0.0941496 0.995558i \(-0.469987\pi\)
0.0941496 + 0.995558i \(0.469987\pi\)
\(830\) 50.4351 1.75063
\(831\) 13.2496 0.459625
\(832\) −35.3433 −1.22531
\(833\) 0 0
\(834\) −34.5576 −1.19663
\(835\) 26.4441 0.915135
\(836\) −7.20783 −0.249288
\(837\) −1.39480 −0.0482114
\(838\) −95.6466 −3.30406
\(839\) 21.8886 0.755680 0.377840 0.925871i \(-0.376667\pi\)
0.377840 + 0.925871i \(0.376667\pi\)
\(840\) 0 0
\(841\) −23.5122 −0.810767
\(842\) −97.0950 −3.34611
\(843\) 15.2773 0.526178
\(844\) 28.8043 0.991485
\(845\) 47.9220 1.64857
\(846\) 27.4418 0.943469
\(847\) 0 0
\(848\) −49.3634 −1.69514
\(849\) 27.9553 0.959424
\(850\) −4.67481 −0.160345
\(851\) −0.579319 −0.0198588
\(852\) −46.4196 −1.59031
\(853\) −12.9626 −0.443831 −0.221915 0.975066i \(-0.571231\pi\)
−0.221915 + 0.975066i \(0.571231\pi\)
\(854\) 0 0
\(855\) −10.3822 −0.355065
\(856\) −89.8683 −3.07163
\(857\) 42.7269 1.45952 0.729762 0.683702i \(-0.239631\pi\)
0.729762 + 0.683702i \(0.239631\pi\)
\(858\) 4.62170 0.157782
\(859\) 33.9106 1.15701 0.578507 0.815678i \(-0.303636\pi\)
0.578507 + 0.815678i \(0.303636\pi\)
\(860\) −3.01763 −0.102900
\(861\) 0 0
\(862\) 2.61572 0.0890919
\(863\) −20.2687 −0.689955 −0.344978 0.938611i \(-0.612113\pi\)
−0.344978 + 0.938611i \(0.612113\pi\)
\(864\) 1.86671 0.0635066
\(865\) 4.51455 0.153499
\(866\) 61.5067 2.09008
\(867\) −15.9963 −0.543264
\(868\) 0 0
\(869\) 0.529208 0.0179521
\(870\) 10.2910 0.348898
\(871\) −41.0603 −1.39128
\(872\) 22.3793 0.757860
\(873\) 9.27612 0.313949
\(874\) −12.2427 −0.414117
\(875\) 0 0
\(876\) 19.8396 0.670318
\(877\) −5.45153 −0.184085 −0.0920425 0.995755i \(-0.529340\pi\)
−0.0920425 + 0.995755i \(0.529340\pi\)
\(878\) −34.6065 −1.16791
\(879\) 10.3302 0.348430
\(880\) 2.65076 0.0893572
\(881\) 52.0187 1.75256 0.876278 0.481806i \(-0.160019\pi\)
0.876278 + 0.481806i \(0.160019\pi\)
\(882\) 0 0
\(883\) −18.9846 −0.638882 −0.319441 0.947606i \(-0.603495\pi\)
−0.319441 + 0.947606i \(0.603495\pi\)
\(884\) 26.5218 0.892025
\(885\) −2.64754 −0.0889962
\(886\) 17.9089 0.601662
\(887\) −41.7572 −1.40207 −0.701035 0.713127i \(-0.747278\pi\)
−0.701035 + 0.713127i \(0.747278\pi\)
\(888\) −3.74460 −0.125661
\(889\) 0 0
\(890\) −69.4453 −2.32781
\(891\) 0.293529 0.00983359
\(892\) −74.0991 −2.48102
\(893\) −64.8549 −2.17029
\(894\) 49.0799 1.64148
\(895\) 20.3169 0.679120
\(896\) 0 0
\(897\) 5.30949 0.177279
\(898\) −66.1161 −2.20632
\(899\) −3.26746 −0.108976
\(900\) −7.84565 −0.261522
\(901\) −9.67735 −0.322399
\(902\) −0.729681 −0.0242957
\(903\) 0 0
\(904\) 12.6734 0.421510
\(905\) 18.5340 0.616090
\(906\) 1.41233 0.0469217
\(907\) 6.25865 0.207815 0.103908 0.994587i \(-0.466865\pi\)
0.103908 + 0.994587i \(0.466865\pi\)
\(908\) 28.8576 0.957673
\(909\) −12.8057 −0.424738
\(910\) 0 0
\(911\) 1.44814 0.0479789 0.0239895 0.999712i \(-0.492363\pi\)
0.0239895 + 0.999712i \(0.492363\pi\)
\(912\) −30.0230 −0.994161
\(913\) 3.36994 0.111529
\(914\) 55.2554 1.82769
\(915\) 22.7435 0.751876
\(916\) 59.3393 1.96063
\(917\) 0 0
\(918\) 2.49044 0.0821967
\(919\) 48.5596 1.60183 0.800916 0.598776i \(-0.204346\pi\)
0.800916 + 0.598776i \(0.204346\pi\)
\(920\) 8.02664 0.264630
\(921\) −19.9196 −0.656375
\(922\) 4.67620 0.154002
\(923\) −70.3443 −2.31541
\(924\) 0 0
\(925\) 1.29725 0.0426532
\(926\) −22.7866 −0.748814
\(927\) 3.88886 0.127727
\(928\) 4.37294 0.143549
\(929\) 12.0033 0.393814 0.196907 0.980422i \(-0.436910\pi\)
0.196907 + 0.980422i \(0.436910\pi\)
\(930\) −6.12737 −0.200924
\(931\) 0 0
\(932\) 5.92524 0.194088
\(933\) −19.0288 −0.622976
\(934\) −83.9221 −2.74601
\(935\) 0.519664 0.0169948
\(936\) 34.3194 1.12177
\(937\) 23.2643 0.760013 0.380007 0.924984i \(-0.375922\pi\)
0.380007 + 0.924984i \(0.375922\pi\)
\(938\) 0 0
\(939\) 30.7830 1.00456
\(940\) 81.5362 2.65942
\(941\) 29.8756 0.973918 0.486959 0.873425i \(-0.338106\pi\)
0.486959 + 0.873425i \(0.338106\pi\)
\(942\) −53.7175 −1.75021
\(943\) −0.838269 −0.0272978
\(944\) −7.65607 −0.249184
\(945\) 0 0
\(946\) −0.298112 −0.00969244
\(947\) −12.3349 −0.400830 −0.200415 0.979711i \(-0.564229\pi\)
−0.200415 + 0.979711i \(0.564229\pi\)
\(948\) 7.53557 0.244744
\(949\) 30.0650 0.975950
\(950\) 27.4147 0.889449
\(951\) −3.77693 −0.122475
\(952\) 0 0
\(953\) 33.1038 1.07234 0.536169 0.844111i \(-0.319871\pi\)
0.536169 + 0.844111i \(0.319871\pi\)
\(954\) −24.0130 −0.777448
\(955\) −8.94282 −0.289383
\(956\) 7.99832 0.258684
\(957\) 0.687619 0.0222276
\(958\) −17.8003 −0.575102
\(959\) 0 0
\(960\) −9.86091 −0.318259
\(961\) −29.0545 −0.937243
\(962\) −10.8814 −0.350831
\(963\) −16.5858 −0.534468
\(964\) −36.5382 −1.17682
\(965\) 6.26794 0.201772
\(966\) 0 0
\(967\) 0.206392 0.00663711 0.00331856 0.999994i \(-0.498944\pi\)
0.00331856 + 0.999994i \(0.498944\pi\)
\(968\) −59.1356 −1.90069
\(969\) −5.88580 −0.189079
\(970\) 40.7500 1.30840
\(971\) 19.7601 0.634133 0.317067 0.948403i \(-0.397302\pi\)
0.317067 + 0.948403i \(0.397302\pi\)
\(972\) 4.17966 0.134063
\(973\) 0 0
\(974\) −27.1784 −0.870852
\(975\) −11.8893 −0.380763
\(976\) 65.7687 2.10520
\(977\) 43.0318 1.37671 0.688354 0.725375i \(-0.258333\pi\)
0.688354 + 0.725375i \(0.258333\pi\)
\(978\) −14.4913 −0.463380
\(979\) −4.64015 −0.148300
\(980\) 0 0
\(981\) 4.13025 0.131869
\(982\) 60.9428 1.94476
\(983\) −58.2137 −1.85673 −0.928365 0.371670i \(-0.878785\pi\)
−0.928365 + 0.371670i \(0.878785\pi\)
\(984\) −5.41840 −0.172732
\(985\) −31.2113 −0.994475
\(986\) 5.83409 0.185795
\(987\) 0 0
\(988\) −155.533 −4.94817
\(989\) −0.342475 −0.0108901
\(990\) 1.28947 0.0409821
\(991\) 56.9979 1.81060 0.905300 0.424774i \(-0.139646\pi\)
0.905300 + 0.424774i \(0.139646\pi\)
\(992\) −2.60369 −0.0826672
\(993\) −10.2326 −0.324721
\(994\) 0 0
\(995\) −26.7978 −0.849547
\(996\) 47.9858 1.52049
\(997\) 48.8707 1.54775 0.773875 0.633338i \(-0.218316\pi\)
0.773875 + 0.633338i \(0.218316\pi\)
\(998\) −93.0983 −2.94697
\(999\) −0.691089 −0.0218651
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 6027.2.a.bg.1.12 12
7.2 even 3 861.2.i.e.739.1 yes 24
7.4 even 3 861.2.i.e.247.1 24
7.6 odd 2 6027.2.a.bf.1.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
861.2.i.e.247.1 24 7.4 even 3
861.2.i.e.739.1 yes 24 7.2 even 3
6027.2.a.bf.1.12 12 7.6 odd 2
6027.2.a.bg.1.12 12 1.1 even 1 trivial