Properties

Label 1805.2.a.w.1.5
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1805,2,Mod(1,1805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1805, 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("1805.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.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: \(14.4129975648\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 29x^{14} + 339x^{12} - 2038x^{10} + 6639x^{8} - 11261x^{6} + 8701x^{4} - 2592x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.91539\) of defining polynomial
Character \(\chi\) \(=\) 1805.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.91539 q^{2} +1.67190 q^{3} +1.66872 q^{4} +1.00000 q^{5} -3.20235 q^{6} -4.22760 q^{7} +0.634526 q^{8} -0.204741 q^{9} +O(q^{10})\) \(q-1.91539 q^{2} +1.67190 q^{3} +1.66872 q^{4} +1.00000 q^{5} -3.20235 q^{6} -4.22760 q^{7} +0.634526 q^{8} -0.204741 q^{9} -1.91539 q^{10} +2.92847 q^{11} +2.78994 q^{12} -2.06406 q^{13} +8.09751 q^{14} +1.67190 q^{15} -4.55281 q^{16} +3.59719 q^{17} +0.392159 q^{18} +1.66872 q^{20} -7.06814 q^{21} -5.60917 q^{22} +8.02202 q^{23} +1.06087 q^{24} +1.00000 q^{25} +3.95348 q^{26} -5.35802 q^{27} -7.05470 q^{28} -2.00502 q^{29} -3.20235 q^{30} -8.46117 q^{31} +7.45136 q^{32} +4.89612 q^{33} -6.89003 q^{34} -4.22760 q^{35} -0.341655 q^{36} -2.20105 q^{37} -3.45091 q^{39} +0.634526 q^{40} +9.76141 q^{41} +13.5383 q^{42} +10.5308 q^{43} +4.88681 q^{44} -0.204741 q^{45} -15.3653 q^{46} +3.58295 q^{47} -7.61186 q^{48} +10.8726 q^{49} -1.91539 q^{50} +6.01416 q^{51} -3.44435 q^{52} +3.24204 q^{53} +10.2627 q^{54} +2.92847 q^{55} -2.68252 q^{56} +3.84039 q^{58} -2.08653 q^{59} +2.78994 q^{60} -4.93614 q^{61} +16.2065 q^{62} +0.865563 q^{63} -5.16665 q^{64} -2.06406 q^{65} -9.37798 q^{66} +10.1043 q^{67} +6.00272 q^{68} +13.4120 q^{69} +8.09751 q^{70} -2.85201 q^{71} -0.129913 q^{72} +1.82820 q^{73} +4.21588 q^{74} +1.67190 q^{75} -12.3804 q^{77} +6.60984 q^{78} +12.7131 q^{79} -4.55281 q^{80} -8.34386 q^{81} -18.6969 q^{82} +17.2197 q^{83} -11.7948 q^{84} +3.59719 q^{85} -20.1706 q^{86} -3.35219 q^{87} +1.85819 q^{88} -2.46468 q^{89} +0.392159 q^{90} +8.72603 q^{91} +13.3865 q^{92} -14.1463 q^{93} -6.86276 q^{94} +12.4579 q^{96} +18.0240 q^{97} -20.8253 q^{98} -0.599577 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 26 q^{4} + 16 q^{5} - 2 q^{6} + 22 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 26 q^{4} + 16 q^{5} - 2 q^{6} + 22 q^{7} + 18 q^{9} + 12 q^{11} + 42 q^{16} + 22 q^{17} + 26 q^{20} + 42 q^{23} - 14 q^{24} + 16 q^{25} - 26 q^{26} + 46 q^{28} - 2 q^{30} + 22 q^{35} - 8 q^{36} - 38 q^{39} + 74 q^{42} + 88 q^{43} - 48 q^{44} + 18 q^{45} + 32 q^{47} + 30 q^{49} - 22 q^{54} + 12 q^{55} - 2 q^{58} + 20 q^{61} + 6 q^{62} - 6 q^{63} + 24 q^{64} - 24 q^{66} + 84 q^{68} + 44 q^{73} - 122 q^{74} + 4 q^{77} + 42 q^{80} - 36 q^{81} - 50 q^{82} + 56 q^{83} + 22 q^{85} + 34 q^{87} + 6 q^{92} - 58 q^{93} - 96 q^{96} + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.91539 −1.35439 −0.677193 0.735805i \(-0.736804\pi\)
−0.677193 + 0.735805i \(0.736804\pi\)
\(3\) 1.67190 0.965274 0.482637 0.875821i \(-0.339679\pi\)
0.482637 + 0.875821i \(0.339679\pi\)
\(4\) 1.66872 0.834361
\(5\) 1.00000 0.447214
\(6\) −3.20235 −1.30735
\(7\) −4.22760 −1.59788 −0.798942 0.601408i \(-0.794607\pi\)
−0.798942 + 0.601408i \(0.794607\pi\)
\(8\) 0.634526 0.224339
\(9\) −0.204741 −0.0682469
\(10\) −1.91539 −0.605700
\(11\) 2.92847 0.882967 0.441484 0.897269i \(-0.354452\pi\)
0.441484 + 0.897269i \(0.354452\pi\)
\(12\) 2.78994 0.805387
\(13\) −2.06406 −0.572468 −0.286234 0.958160i \(-0.592403\pi\)
−0.286234 + 0.958160i \(0.592403\pi\)
\(14\) 8.09751 2.16415
\(15\) 1.67190 0.431683
\(16\) −4.55281 −1.13820
\(17\) 3.59719 0.872448 0.436224 0.899838i \(-0.356316\pi\)
0.436224 + 0.899838i \(0.356316\pi\)
\(18\) 0.392159 0.0924327
\(19\) 0 0
\(20\) 1.66872 0.373138
\(21\) −7.06814 −1.54240
\(22\) −5.60917 −1.19588
\(23\) 8.02202 1.67271 0.836354 0.548190i \(-0.184683\pi\)
0.836354 + 0.548190i \(0.184683\pi\)
\(24\) 1.06087 0.216548
\(25\) 1.00000 0.200000
\(26\) 3.95348 0.775342
\(27\) −5.35802 −1.03115
\(28\) −7.05470 −1.33321
\(29\) −2.00502 −0.372322 −0.186161 0.982519i \(-0.559605\pi\)
−0.186161 + 0.982519i \(0.559605\pi\)
\(30\) −3.20235 −0.584666
\(31\) −8.46117 −1.51967 −0.759836 0.650115i \(-0.774721\pi\)
−0.759836 + 0.650115i \(0.774721\pi\)
\(32\) 7.45136 1.31723
\(33\) 4.89612 0.852305
\(34\) −6.89003 −1.18163
\(35\) −4.22760 −0.714595
\(36\) −0.341655 −0.0569426
\(37\) −2.20105 −0.361851 −0.180925 0.983497i \(-0.557909\pi\)
−0.180925 + 0.983497i \(0.557909\pi\)
\(38\) 0 0
\(39\) −3.45091 −0.552588
\(40\) 0.634526 0.100327
\(41\) 9.76141 1.52448 0.762238 0.647297i \(-0.224101\pi\)
0.762238 + 0.647297i \(0.224101\pi\)
\(42\) 13.5383 2.08900
\(43\) 10.5308 1.60593 0.802967 0.596024i \(-0.203254\pi\)
0.802967 + 0.596024i \(0.203254\pi\)
\(44\) 4.88681 0.736714
\(45\) −0.204741 −0.0305209
\(46\) −15.3653 −2.26549
\(47\) 3.58295 0.522628 0.261314 0.965254i \(-0.415844\pi\)
0.261314 + 0.965254i \(0.415844\pi\)
\(48\) −7.61186 −1.09868
\(49\) 10.8726 1.55323
\(50\) −1.91539 −0.270877
\(51\) 6.01416 0.842151
\(52\) −3.44435 −0.477645
\(53\) 3.24204 0.445328 0.222664 0.974895i \(-0.428525\pi\)
0.222664 + 0.974895i \(0.428525\pi\)
\(54\) 10.2627 1.39658
\(55\) 2.92847 0.394875
\(56\) −2.68252 −0.358467
\(57\) 0 0
\(58\) 3.84039 0.504268
\(59\) −2.08653 −0.271643 −0.135822 0.990733i \(-0.543367\pi\)
−0.135822 + 0.990733i \(0.543367\pi\)
\(60\) 2.78994 0.360180
\(61\) −4.93614 −0.632008 −0.316004 0.948758i \(-0.602341\pi\)
−0.316004 + 0.948758i \(0.602341\pi\)
\(62\) 16.2065 2.05822
\(63\) 0.865563 0.109051
\(64\) −5.16665 −0.645831
\(65\) −2.06406 −0.256015
\(66\) −9.37798 −1.15435
\(67\) 10.1043 1.23444 0.617218 0.786792i \(-0.288260\pi\)
0.617218 + 0.786792i \(0.288260\pi\)
\(68\) 6.00272 0.727937
\(69\) 13.4120 1.61462
\(70\) 8.09751 0.967838
\(71\) −2.85201 −0.338471 −0.169235 0.985576i \(-0.554130\pi\)
−0.169235 + 0.985576i \(0.554130\pi\)
\(72\) −0.129913 −0.0153104
\(73\) 1.82820 0.213975 0.106987 0.994260i \(-0.465880\pi\)
0.106987 + 0.994260i \(0.465880\pi\)
\(74\) 4.21588 0.490086
\(75\) 1.67190 0.193055
\(76\) 0 0
\(77\) −12.3804 −1.41088
\(78\) 6.60984 0.748417
\(79\) 12.7131 1.43033 0.715167 0.698954i \(-0.246351\pi\)
0.715167 + 0.698954i \(0.246351\pi\)
\(80\) −4.55281 −0.509020
\(81\) −8.34386 −0.927095
\(82\) −18.6969 −2.06473
\(83\) 17.2197 1.89010 0.945051 0.326922i \(-0.106012\pi\)
0.945051 + 0.326922i \(0.106012\pi\)
\(84\) −11.7948 −1.28691
\(85\) 3.59719 0.390170
\(86\) −20.1706 −2.17505
\(87\) −3.35219 −0.359393
\(88\) 1.85819 0.198084
\(89\) −2.46468 −0.261256 −0.130628 0.991431i \(-0.541699\pi\)
−0.130628 + 0.991431i \(0.541699\pi\)
\(90\) 0.392159 0.0413371
\(91\) 8.72603 0.914737
\(92\) 13.3865 1.39564
\(93\) −14.1463 −1.46690
\(94\) −6.86276 −0.707840
\(95\) 0 0
\(96\) 12.4579 1.27148
\(97\) 18.0240 1.83006 0.915029 0.403389i \(-0.132168\pi\)
0.915029 + 0.403389i \(0.132168\pi\)
\(98\) −20.8253 −2.10368
\(99\) −0.599577 −0.0602598
\(100\) 1.66872 0.166872
\(101\) 8.95602 0.891157 0.445579 0.895243i \(-0.352998\pi\)
0.445579 + 0.895243i \(0.352998\pi\)
\(102\) −11.5195 −1.14060
\(103\) −11.7838 −1.16110 −0.580548 0.814226i \(-0.697162\pi\)
−0.580548 + 0.814226i \(0.697162\pi\)
\(104\) −1.30970 −0.128427
\(105\) −7.06814 −0.689780
\(106\) −6.20976 −0.603146
\(107\) −13.4561 −1.30085 −0.650426 0.759569i \(-0.725410\pi\)
−0.650426 + 0.759569i \(0.725410\pi\)
\(108\) −8.94104 −0.860352
\(109\) 7.93487 0.760023 0.380012 0.924982i \(-0.375920\pi\)
0.380012 + 0.924982i \(0.375920\pi\)
\(110\) −5.60917 −0.534813
\(111\) −3.67995 −0.349285
\(112\) 19.2475 1.81872
\(113\) −8.25147 −0.776233 −0.388117 0.921610i \(-0.626874\pi\)
−0.388117 + 0.921610i \(0.626874\pi\)
\(114\) 0 0
\(115\) 8.02202 0.748058
\(116\) −3.34582 −0.310651
\(117\) 0.422597 0.0390691
\(118\) 3.99653 0.367910
\(119\) −15.2075 −1.39407
\(120\) 1.06087 0.0968433
\(121\) −2.42406 −0.220369
\(122\) 9.45464 0.855983
\(123\) 16.3201 1.47154
\(124\) −14.1193 −1.26796
\(125\) 1.00000 0.0894427
\(126\) −1.65789 −0.147697
\(127\) −2.06393 −0.183144 −0.0915721 0.995798i \(-0.529189\pi\)
−0.0915721 + 0.995798i \(0.529189\pi\)
\(128\) −5.00657 −0.442523
\(129\) 17.6065 1.55017
\(130\) 3.95348 0.346744
\(131\) −0.968020 −0.0845763 −0.0422881 0.999105i \(-0.513465\pi\)
−0.0422881 + 0.999105i \(0.513465\pi\)
\(132\) 8.17026 0.711130
\(133\) 0 0
\(134\) −19.3537 −1.67190
\(135\) −5.35802 −0.461145
\(136\) 2.28251 0.195724
\(137\) 19.4256 1.65964 0.829819 0.558033i \(-0.188444\pi\)
0.829819 + 0.558033i \(0.188444\pi\)
\(138\) −25.6893 −2.18682
\(139\) −10.2777 −0.871746 −0.435873 0.900008i \(-0.643560\pi\)
−0.435873 + 0.900008i \(0.643560\pi\)
\(140\) −7.05470 −0.596231
\(141\) 5.99035 0.504479
\(142\) 5.46271 0.458420
\(143\) −6.04454 −0.505470
\(144\) 0.932146 0.0776788
\(145\) −2.00502 −0.166508
\(146\) −3.50172 −0.289804
\(147\) 18.1780 1.49930
\(148\) −3.67295 −0.301914
\(149\) 10.1694 0.833114 0.416557 0.909110i \(-0.363237\pi\)
0.416557 + 0.909110i \(0.363237\pi\)
\(150\) −3.20235 −0.261471
\(151\) −2.79628 −0.227558 −0.113779 0.993506i \(-0.536296\pi\)
−0.113779 + 0.993506i \(0.536296\pi\)
\(152\) 0 0
\(153\) −0.736492 −0.0595419
\(154\) 23.7133 1.91088
\(155\) −8.46117 −0.679618
\(156\) −5.75861 −0.461058
\(157\) −15.8271 −1.26314 −0.631569 0.775320i \(-0.717589\pi\)
−0.631569 + 0.775320i \(0.717589\pi\)
\(158\) −24.3505 −1.93722
\(159\) 5.42037 0.429863
\(160\) 7.45136 0.589082
\(161\) −33.9139 −2.67279
\(162\) 15.9818 1.25565
\(163\) 18.3145 1.43450 0.717251 0.696815i \(-0.245400\pi\)
0.717251 + 0.696815i \(0.245400\pi\)
\(164\) 16.2891 1.27196
\(165\) 4.89612 0.381162
\(166\) −32.9824 −2.55993
\(167\) −5.46238 −0.422692 −0.211346 0.977411i \(-0.567785\pi\)
−0.211346 + 0.977411i \(0.567785\pi\)
\(168\) −4.48492 −0.346019
\(169\) −8.73965 −0.672281
\(170\) −6.89003 −0.528441
\(171\) 0 0
\(172\) 17.5730 1.33993
\(173\) 7.69398 0.584962 0.292481 0.956271i \(-0.405519\pi\)
0.292481 + 0.956271i \(0.405519\pi\)
\(174\) 6.42076 0.486756
\(175\) −4.22760 −0.319577
\(176\) −13.3328 −1.00500
\(177\) −3.48848 −0.262210
\(178\) 4.72082 0.353841
\(179\) 21.3618 1.59666 0.798328 0.602223i \(-0.205718\pi\)
0.798328 + 0.602223i \(0.205718\pi\)
\(180\) −0.341655 −0.0254655
\(181\) 2.16604 0.161000 0.0805002 0.996755i \(-0.474348\pi\)
0.0805002 + 0.996755i \(0.474348\pi\)
\(182\) −16.7138 −1.23891
\(183\) −8.25275 −0.610061
\(184\) 5.09018 0.375253
\(185\) −2.20105 −0.161825
\(186\) 27.0956 1.98675
\(187\) 10.5343 0.770343
\(188\) 5.97896 0.436060
\(189\) 22.6516 1.64766
\(190\) 0 0
\(191\) 9.80930 0.709776 0.354888 0.934909i \(-0.384519\pi\)
0.354888 + 0.934909i \(0.384519\pi\)
\(192\) −8.63813 −0.623403
\(193\) 3.82213 0.275123 0.137561 0.990493i \(-0.456074\pi\)
0.137561 + 0.990493i \(0.456074\pi\)
\(194\) −34.5230 −2.47860
\(195\) −3.45091 −0.247125
\(196\) 18.1434 1.29596
\(197\) 10.7279 0.764329 0.382165 0.924094i \(-0.375179\pi\)
0.382165 + 0.924094i \(0.375179\pi\)
\(198\) 1.14842 0.0816150
\(199\) 23.6191 1.67432 0.837159 0.546960i \(-0.184215\pi\)
0.837159 + 0.546960i \(0.184215\pi\)
\(200\) 0.634526 0.0448678
\(201\) 16.8934 1.19157
\(202\) −17.1543 −1.20697
\(203\) 8.47641 0.594928
\(204\) 10.0360 0.702658
\(205\) 9.76141 0.681766
\(206\) 22.5707 1.57257
\(207\) −1.64243 −0.114157
\(208\) 9.39728 0.651584
\(209\) 0 0
\(210\) 13.5383 0.934229
\(211\) −14.6711 −1.01000 −0.505001 0.863119i \(-0.668508\pi\)
−0.505001 + 0.863119i \(0.668508\pi\)
\(212\) 5.41006 0.371564
\(213\) −4.76828 −0.326717
\(214\) 25.7737 1.76186
\(215\) 10.5308 0.718195
\(216\) −3.39980 −0.231327
\(217\) 35.7705 2.42826
\(218\) −15.1984 −1.02936
\(219\) 3.05657 0.206544
\(220\) 4.88681 0.329468
\(221\) −7.42483 −0.499448
\(222\) 7.04854 0.473067
\(223\) −3.94457 −0.264148 −0.132074 0.991240i \(-0.542164\pi\)
−0.132074 + 0.991240i \(0.542164\pi\)
\(224\) −31.5014 −2.10478
\(225\) −0.204741 −0.0136494
\(226\) 15.8048 1.05132
\(227\) 27.9711 1.85651 0.928254 0.371948i \(-0.121310\pi\)
0.928254 + 0.371948i \(0.121310\pi\)
\(228\) 0 0
\(229\) −6.40408 −0.423193 −0.211597 0.977357i \(-0.567866\pi\)
−0.211597 + 0.977357i \(0.567866\pi\)
\(230\) −15.3653 −1.01316
\(231\) −20.6989 −1.36188
\(232\) −1.27223 −0.0835263
\(233\) 16.6633 1.09165 0.545824 0.837900i \(-0.316217\pi\)
0.545824 + 0.837900i \(0.316217\pi\)
\(234\) −0.809439 −0.0529147
\(235\) 3.58295 0.233726
\(236\) −3.48184 −0.226649
\(237\) 21.2550 1.38066
\(238\) 29.1283 1.88811
\(239\) −8.53879 −0.552328 −0.276164 0.961110i \(-0.589063\pi\)
−0.276164 + 0.961110i \(0.589063\pi\)
\(240\) −7.61186 −0.491343
\(241\) −5.34514 −0.344311 −0.172155 0.985070i \(-0.555073\pi\)
−0.172155 + 0.985070i \(0.555073\pi\)
\(242\) 4.64302 0.298464
\(243\) 2.12392 0.136250
\(244\) −8.23705 −0.527323
\(245\) 10.8726 0.694627
\(246\) −31.2594 −1.99303
\(247\) 0 0
\(248\) −5.36883 −0.340921
\(249\) 28.7896 1.82447
\(250\) −1.91539 −0.121140
\(251\) −9.73324 −0.614357 −0.307178 0.951652i \(-0.599385\pi\)
−0.307178 + 0.951652i \(0.599385\pi\)
\(252\) 1.44438 0.0909876
\(253\) 23.4923 1.47695
\(254\) 3.95323 0.248048
\(255\) 6.01416 0.376621
\(256\) 19.9228 1.24518
\(257\) −6.34018 −0.395490 −0.197745 0.980254i \(-0.563362\pi\)
−0.197745 + 0.980254i \(0.563362\pi\)
\(258\) −33.7233 −2.09952
\(259\) 9.30518 0.578196
\(260\) −3.44435 −0.213609
\(261\) 0.410508 0.0254098
\(262\) 1.85414 0.114549
\(263\) 13.5339 0.834533 0.417267 0.908784i \(-0.362988\pi\)
0.417267 + 0.908784i \(0.362988\pi\)
\(264\) 3.10671 0.191205
\(265\) 3.24204 0.199157
\(266\) 0 0
\(267\) −4.12070 −0.252183
\(268\) 16.8613 1.02997
\(269\) −13.5665 −0.827164 −0.413582 0.910467i \(-0.635722\pi\)
−0.413582 + 0.910467i \(0.635722\pi\)
\(270\) 10.2627 0.624568
\(271\) 1.81018 0.109960 0.0549802 0.998487i \(-0.482490\pi\)
0.0549802 + 0.998487i \(0.482490\pi\)
\(272\) −16.3773 −0.993022
\(273\) 14.5891 0.882971
\(274\) −37.2075 −2.24779
\(275\) 2.92847 0.176593
\(276\) 22.3810 1.34718
\(277\) −1.28308 −0.0770929 −0.0385465 0.999257i \(-0.512273\pi\)
−0.0385465 + 0.999257i \(0.512273\pi\)
\(278\) 19.6859 1.18068
\(279\) 1.73235 0.103713
\(280\) −2.68252 −0.160311
\(281\) −9.73693 −0.580856 −0.290428 0.956897i \(-0.593798\pi\)
−0.290428 + 0.956897i \(0.593798\pi\)
\(282\) −11.4739 −0.683259
\(283\) 7.58068 0.450625 0.225312 0.974287i \(-0.427660\pi\)
0.225312 + 0.974287i \(0.427660\pi\)
\(284\) −4.75921 −0.282407
\(285\) 0 0
\(286\) 11.5777 0.684602
\(287\) −41.2674 −2.43594
\(288\) −1.52560 −0.0898967
\(289\) −4.06020 −0.238835
\(290\) 3.84039 0.225515
\(291\) 30.1343 1.76651
\(292\) 3.05076 0.178532
\(293\) 1.18387 0.0691625 0.0345813 0.999402i \(-0.488990\pi\)
0.0345813 + 0.999402i \(0.488990\pi\)
\(294\) −34.8180 −2.03062
\(295\) −2.08653 −0.121483
\(296\) −1.39663 −0.0811772
\(297\) −15.6908 −0.910472
\(298\) −19.4785 −1.12836
\(299\) −16.5579 −0.957571
\(300\) 2.78994 0.161077
\(301\) −44.5201 −2.56610
\(302\) 5.35597 0.308201
\(303\) 14.9736 0.860211
\(304\) 0 0
\(305\) −4.93614 −0.282643
\(306\) 1.41067 0.0806426
\(307\) 13.4937 0.770126 0.385063 0.922890i \(-0.374180\pi\)
0.385063 + 0.922890i \(0.374180\pi\)
\(308\) −20.6595 −1.17718
\(309\) −19.7014 −1.12078
\(310\) 16.2065 0.920465
\(311\) 14.2694 0.809143 0.404571 0.914506i \(-0.367421\pi\)
0.404571 + 0.914506i \(0.367421\pi\)
\(312\) −2.18969 −0.123967
\(313\) −28.9109 −1.63414 −0.817071 0.576537i \(-0.804404\pi\)
−0.817071 + 0.576537i \(0.804404\pi\)
\(314\) 30.3150 1.71078
\(315\) 0.865563 0.0487689
\(316\) 21.2146 1.19341
\(317\) −14.9871 −0.841758 −0.420879 0.907117i \(-0.638278\pi\)
−0.420879 + 0.907117i \(0.638278\pi\)
\(318\) −10.3821 −0.582201
\(319\) −5.87163 −0.328748
\(320\) −5.16665 −0.288824
\(321\) −22.4973 −1.25568
\(322\) 64.9584 3.61999
\(323\) 0 0
\(324\) −13.9236 −0.773533
\(325\) −2.06406 −0.114494
\(326\) −35.0794 −1.94287
\(327\) 13.2663 0.733630
\(328\) 6.19386 0.341999
\(329\) −15.1473 −0.835098
\(330\) −9.37798 −0.516241
\(331\) 15.2205 0.836594 0.418297 0.908310i \(-0.362627\pi\)
0.418297 + 0.908310i \(0.362627\pi\)
\(332\) 28.7348 1.57703
\(333\) 0.450645 0.0246952
\(334\) 10.4626 0.572488
\(335\) 10.1043 0.552056
\(336\) 32.1799 1.75556
\(337\) 6.15921 0.335513 0.167757 0.985828i \(-0.446348\pi\)
0.167757 + 0.985828i \(0.446348\pi\)
\(338\) 16.7398 0.910528
\(339\) −13.7957 −0.749277
\(340\) 6.00272 0.325543
\(341\) −24.7783 −1.34182
\(342\) 0 0
\(343\) −16.3720 −0.884003
\(344\) 6.68207 0.360273
\(345\) 13.4120 0.722080
\(346\) −14.7370 −0.792265
\(347\) 8.18354 0.439315 0.219658 0.975577i \(-0.429506\pi\)
0.219658 + 0.975577i \(0.429506\pi\)
\(348\) −5.59388 −0.299863
\(349\) −10.7999 −0.578104 −0.289052 0.957313i \(-0.593340\pi\)
−0.289052 + 0.957313i \(0.593340\pi\)
\(350\) 8.09751 0.432830
\(351\) 11.0593 0.590300
\(352\) 21.8211 1.16307
\(353\) −8.84321 −0.470677 −0.235338 0.971914i \(-0.575620\pi\)
−0.235338 + 0.971914i \(0.575620\pi\)
\(354\) 6.68180 0.355134
\(355\) −2.85201 −0.151369
\(356\) −4.11287 −0.217981
\(357\) −25.4255 −1.34566
\(358\) −40.9162 −2.16249
\(359\) −3.39178 −0.179011 −0.0895055 0.995986i \(-0.528529\pi\)
−0.0895055 + 0.995986i \(0.528529\pi\)
\(360\) −0.129913 −0.00684703
\(361\) 0 0
\(362\) −4.14881 −0.218057
\(363\) −4.05279 −0.212716
\(364\) 14.5613 0.763221
\(365\) 1.82820 0.0956923
\(366\) 15.8072 0.826258
\(367\) −12.2285 −0.638323 −0.319161 0.947700i \(-0.603401\pi\)
−0.319161 + 0.947700i \(0.603401\pi\)
\(368\) −36.5227 −1.90388
\(369\) −1.99856 −0.104041
\(370\) 4.21588 0.219173
\(371\) −13.7060 −0.711582
\(372\) −23.6062 −1.22392
\(373\) 13.7436 0.711619 0.355810 0.934558i \(-0.384205\pi\)
0.355810 + 0.934558i \(0.384205\pi\)
\(374\) −20.1773 −1.04334
\(375\) 1.67190 0.0863367
\(376\) 2.27348 0.117246
\(377\) 4.13848 0.213142
\(378\) −43.3866 −2.23157
\(379\) −22.5080 −1.15616 −0.578080 0.815980i \(-0.696198\pi\)
−0.578080 + 0.815980i \(0.696198\pi\)
\(380\) 0 0
\(381\) −3.45069 −0.176784
\(382\) −18.7886 −0.961311
\(383\) 30.7611 1.57182 0.785909 0.618342i \(-0.212195\pi\)
0.785909 + 0.618342i \(0.212195\pi\)
\(384\) −8.37050 −0.427155
\(385\) −12.3804 −0.630964
\(386\) −7.32087 −0.372622
\(387\) −2.15609 −0.109600
\(388\) 30.0770 1.52693
\(389\) −21.0616 −1.06786 −0.533932 0.845528i \(-0.679286\pi\)
−0.533932 + 0.845528i \(0.679286\pi\)
\(390\) 6.60984 0.334702
\(391\) 28.8568 1.45935
\(392\) 6.89897 0.348450
\(393\) −1.61844 −0.0816393
\(394\) −20.5481 −1.03520
\(395\) 12.7131 0.639665
\(396\) −1.00053 −0.0502784
\(397\) −22.0757 −1.10795 −0.553974 0.832534i \(-0.686889\pi\)
−0.553974 + 0.832534i \(0.686889\pi\)
\(398\) −45.2399 −2.26767
\(399\) 0 0
\(400\) −4.55281 −0.227641
\(401\) −28.7899 −1.43770 −0.718850 0.695165i \(-0.755331\pi\)
−0.718850 + 0.695165i \(0.755331\pi\)
\(402\) −32.3575 −1.61384
\(403\) 17.4644 0.869963
\(404\) 14.9451 0.743547
\(405\) −8.34386 −0.414610
\(406\) −16.2356 −0.805761
\(407\) −6.44572 −0.319503
\(408\) 3.81614 0.188927
\(409\) −18.3912 −0.909385 −0.454693 0.890648i \(-0.650251\pi\)
−0.454693 + 0.890648i \(0.650251\pi\)
\(410\) −18.6969 −0.923375
\(411\) 32.4776 1.60200
\(412\) −19.6640 −0.968774
\(413\) 8.82104 0.434055
\(414\) 3.14590 0.154613
\(415\) 17.2197 0.845279
\(416\) −15.3801 −0.754070
\(417\) −17.1834 −0.841473
\(418\) 0 0
\(419\) −22.2339 −1.08619 −0.543097 0.839670i \(-0.682749\pi\)
−0.543097 + 0.839670i \(0.682749\pi\)
\(420\) −11.7948 −0.575526
\(421\) 26.7601 1.30421 0.652103 0.758130i \(-0.273887\pi\)
0.652103 + 0.758130i \(0.273887\pi\)
\(422\) 28.1009 1.36793
\(423\) −0.733577 −0.0356677
\(424\) 2.05715 0.0999043
\(425\) 3.59719 0.174490
\(426\) 9.13311 0.442501
\(427\) 20.8681 1.00988
\(428\) −22.4545 −1.08538
\(429\) −10.1059 −0.487917
\(430\) −20.1706 −0.972714
\(431\) −3.80319 −0.183193 −0.0915966 0.995796i \(-0.529197\pi\)
−0.0915966 + 0.995796i \(0.529197\pi\)
\(432\) 24.3940 1.17366
\(433\) −31.2833 −1.50338 −0.751691 0.659516i \(-0.770761\pi\)
−0.751691 + 0.659516i \(0.770761\pi\)
\(434\) −68.5145 −3.28880
\(435\) −3.35219 −0.160725
\(436\) 13.2411 0.634134
\(437\) 0 0
\(438\) −5.85453 −0.279740
\(439\) 4.16432 0.198752 0.0993760 0.995050i \(-0.468315\pi\)
0.0993760 + 0.995050i \(0.468315\pi\)
\(440\) 1.85819 0.0885858
\(441\) −2.22607 −0.106003
\(442\) 14.2214 0.676445
\(443\) −6.61530 −0.314302 −0.157151 0.987575i \(-0.550231\pi\)
−0.157151 + 0.987575i \(0.550231\pi\)
\(444\) −6.14081 −0.291430
\(445\) −2.46468 −0.116837
\(446\) 7.55539 0.357758
\(447\) 17.0023 0.804183
\(448\) 21.8425 1.03196
\(449\) −5.49993 −0.259558 −0.129779 0.991543i \(-0.541427\pi\)
−0.129779 + 0.991543i \(0.541427\pi\)
\(450\) 0.392159 0.0184865
\(451\) 28.5860 1.34606
\(452\) −13.7694 −0.647659
\(453\) −4.67511 −0.219656
\(454\) −53.5756 −2.51443
\(455\) 8.72603 0.409083
\(456\) 0 0
\(457\) 35.2315 1.64806 0.824029 0.566547i \(-0.191721\pi\)
0.824029 + 0.566547i \(0.191721\pi\)
\(458\) 12.2663 0.573167
\(459\) −19.2738 −0.899625
\(460\) 13.3865 0.624150
\(461\) 5.62840 0.262141 0.131070 0.991373i \(-0.458159\pi\)
0.131070 + 0.991373i \(0.458159\pi\)
\(462\) 39.6464 1.84452
\(463\) −20.6970 −0.961869 −0.480935 0.876756i \(-0.659703\pi\)
−0.480935 + 0.876756i \(0.659703\pi\)
\(464\) 9.12846 0.423778
\(465\) −14.1463 −0.656017
\(466\) −31.9167 −1.47851
\(467\) −18.4881 −0.855527 −0.427764 0.903891i \(-0.640698\pi\)
−0.427764 + 0.903891i \(0.640698\pi\)
\(468\) 0.705198 0.0325978
\(469\) −42.7169 −1.97249
\(470\) −6.86276 −0.316555
\(471\) −26.4613 −1.21927
\(472\) −1.32396 −0.0609402
\(473\) 30.8392 1.41799
\(474\) −40.7117 −1.86995
\(475\) 0 0
\(476\) −25.3771 −1.16316
\(477\) −0.663777 −0.0303922
\(478\) 16.3551 0.748066
\(479\) −32.0991 −1.46664 −0.733322 0.679881i \(-0.762031\pi\)
−0.733322 + 0.679881i \(0.762031\pi\)
\(480\) 12.4579 0.568625
\(481\) 4.54311 0.207148
\(482\) 10.2380 0.466329
\(483\) −56.7008 −2.57998
\(484\) −4.04508 −0.183867
\(485\) 18.0240 0.818426
\(486\) −4.06814 −0.184535
\(487\) 3.54350 0.160571 0.0802856 0.996772i \(-0.474417\pi\)
0.0802856 + 0.996772i \(0.474417\pi\)
\(488\) −3.13211 −0.141784
\(489\) 30.6201 1.38469
\(490\) −20.8253 −0.940793
\(491\) −19.7922 −0.893207 −0.446604 0.894732i \(-0.647367\pi\)
−0.446604 + 0.894732i \(0.647367\pi\)
\(492\) 27.2338 1.22779
\(493\) −7.21243 −0.324832
\(494\) 0 0
\(495\) −0.599577 −0.0269490
\(496\) 38.5221 1.72969
\(497\) 12.0571 0.540837
\(498\) −55.1433 −2.47103
\(499\) −29.4530 −1.31850 −0.659249 0.751925i \(-0.729126\pi\)
−0.659249 + 0.751925i \(0.729126\pi\)
\(500\) 1.66872 0.0746275
\(501\) −9.13257 −0.408013
\(502\) 18.6430 0.832076
\(503\) 2.94849 0.131467 0.0657333 0.997837i \(-0.479061\pi\)
0.0657333 + 0.997837i \(0.479061\pi\)
\(504\) 0.549222 0.0244643
\(505\) 8.95602 0.398538
\(506\) −44.9969 −2.00035
\(507\) −14.6118 −0.648935
\(508\) −3.44413 −0.152808
\(509\) 1.57700 0.0698995 0.0349497 0.999389i \(-0.488873\pi\)
0.0349497 + 0.999389i \(0.488873\pi\)
\(510\) −11.5195 −0.510090
\(511\) −7.72890 −0.341907
\(512\) −28.1469 −1.24393
\(513\) 0 0
\(514\) 12.1439 0.535646
\(515\) −11.7838 −0.519258
\(516\) 29.3803 1.29340
\(517\) 10.4926 0.461463
\(518\) −17.8231 −0.783100
\(519\) 12.8636 0.564649
\(520\) −1.30970 −0.0574342
\(521\) 0.569552 0.0249525 0.0124763 0.999922i \(-0.496029\pi\)
0.0124763 + 0.999922i \(0.496029\pi\)
\(522\) −0.786284 −0.0344147
\(523\) 22.8370 0.998593 0.499296 0.866431i \(-0.333592\pi\)
0.499296 + 0.866431i \(0.333592\pi\)
\(524\) −1.61536 −0.0705672
\(525\) −7.06814 −0.308479
\(526\) −25.9226 −1.13028
\(527\) −30.4365 −1.32583
\(528\) −22.2911 −0.970096
\(529\) 41.3529 1.79795
\(530\) −6.20976 −0.269735
\(531\) 0.427198 0.0185388
\(532\) 0 0
\(533\) −20.1481 −0.872713
\(534\) 7.89276 0.341553
\(535\) −13.4561 −0.581759
\(536\) 6.41143 0.276932
\(537\) 35.7149 1.54121
\(538\) 25.9851 1.12030
\(539\) 31.8402 1.37145
\(540\) −8.94104 −0.384761
\(541\) −14.2449 −0.612434 −0.306217 0.951962i \(-0.599063\pi\)
−0.306217 + 0.951962i \(0.599063\pi\)
\(542\) −3.46720 −0.148929
\(543\) 3.62141 0.155409
\(544\) 26.8040 1.14921
\(545\) 7.93487 0.339893
\(546\) −27.9438 −1.19588
\(547\) 2.72363 0.116454 0.0582270 0.998303i \(-0.481455\pi\)
0.0582270 + 0.998303i \(0.481455\pi\)
\(548\) 32.4159 1.38474
\(549\) 1.01063 0.0431326
\(550\) −5.60917 −0.239176
\(551\) 0 0
\(552\) 8.51029 0.362222
\(553\) −53.7459 −2.28551
\(554\) 2.45760 0.104414
\(555\) −3.67995 −0.156205
\(556\) −17.1507 −0.727351
\(557\) 27.7979 1.17784 0.588918 0.808193i \(-0.299554\pi\)
0.588918 + 0.808193i \(0.299554\pi\)
\(558\) −3.31812 −0.140467
\(559\) −21.7362 −0.919345
\(560\) 19.2475 0.813354
\(561\) 17.6123 0.743591
\(562\) 18.6500 0.786704
\(563\) −7.95268 −0.335165 −0.167583 0.985858i \(-0.553596\pi\)
−0.167583 + 0.985858i \(0.553596\pi\)
\(564\) 9.99624 0.420917
\(565\) −8.25147 −0.347142
\(566\) −14.5200 −0.610320
\(567\) 35.2745 1.48139
\(568\) −1.80967 −0.0759321
\(569\) −6.92573 −0.290342 −0.145171 0.989407i \(-0.546373\pi\)
−0.145171 + 0.989407i \(0.546373\pi\)
\(570\) 0 0
\(571\) 15.9111 0.665857 0.332929 0.942952i \(-0.391963\pi\)
0.332929 + 0.942952i \(0.391963\pi\)
\(572\) −10.0867 −0.421745
\(573\) 16.4002 0.685128
\(574\) 79.0431 3.29920
\(575\) 8.02202 0.334541
\(576\) 1.05782 0.0440760
\(577\) −29.5901 −1.23185 −0.615925 0.787805i \(-0.711218\pi\)
−0.615925 + 0.787805i \(0.711218\pi\)
\(578\) 7.77687 0.323475
\(579\) 6.39023 0.265569
\(580\) −3.34582 −0.138927
\(581\) −72.7979 −3.02016
\(582\) −57.7190 −2.39253
\(583\) 9.49421 0.393210
\(584\) 1.16004 0.0480028
\(585\) 0.422597 0.0174723
\(586\) −2.26758 −0.0936727
\(587\) −9.03206 −0.372793 −0.186397 0.982475i \(-0.559681\pi\)
−0.186397 + 0.982475i \(0.559681\pi\)
\(588\) 30.3340 1.25095
\(589\) 0 0
\(590\) 3.99653 0.164534
\(591\) 17.9360 0.737787
\(592\) 10.0210 0.411860
\(593\) 22.6342 0.929474 0.464737 0.885449i \(-0.346149\pi\)
0.464737 + 0.885449i \(0.346149\pi\)
\(594\) 30.0540 1.23313
\(595\) −15.2075 −0.623447
\(596\) 16.9700 0.695118
\(597\) 39.4889 1.61617
\(598\) 31.7149 1.29692
\(599\) 5.73330 0.234256 0.117128 0.993117i \(-0.462631\pi\)
0.117128 + 0.993117i \(0.462631\pi\)
\(600\) 1.06087 0.0433097
\(601\) 2.62512 0.107081 0.0535404 0.998566i \(-0.482949\pi\)
0.0535404 + 0.998566i \(0.482949\pi\)
\(602\) 85.2734 3.47548
\(603\) −2.06876 −0.0842464
\(604\) −4.66622 −0.189866
\(605\) −2.42406 −0.0985519
\(606\) −28.6803 −1.16506
\(607\) −26.1933 −1.06315 −0.531577 0.847010i \(-0.678401\pi\)
−0.531577 + 0.847010i \(0.678401\pi\)
\(608\) 0 0
\(609\) 14.1717 0.574268
\(610\) 9.45464 0.382807
\(611\) −7.39544 −0.299187
\(612\) −1.22900 −0.0496794
\(613\) −22.4575 −0.907048 −0.453524 0.891244i \(-0.649833\pi\)
−0.453524 + 0.891244i \(0.649833\pi\)
\(614\) −25.8457 −1.04305
\(615\) 16.3201 0.658091
\(616\) −7.85569 −0.316515
\(617\) 2.85697 0.115017 0.0575086 0.998345i \(-0.481684\pi\)
0.0575086 + 0.998345i \(0.481684\pi\)
\(618\) 37.7360 1.51796
\(619\) 9.16843 0.368510 0.184255 0.982878i \(-0.441013\pi\)
0.184255 + 0.982878i \(0.441013\pi\)
\(620\) −14.1193 −0.567047
\(621\) −42.9821 −1.72481
\(622\) −27.3315 −1.09589
\(623\) 10.4197 0.417456
\(624\) 15.7113 0.628957
\(625\) 1.00000 0.0400000
\(626\) 55.3757 2.21326
\(627\) 0 0
\(628\) −26.4110 −1.05391
\(629\) −7.91761 −0.315696
\(630\) −1.65789 −0.0660520
\(631\) 15.8291 0.630148 0.315074 0.949067i \(-0.397971\pi\)
0.315074 + 0.949067i \(0.397971\pi\)
\(632\) 8.06678 0.320879
\(633\) −24.5287 −0.974927
\(634\) 28.7061 1.14007
\(635\) −2.06393 −0.0819046
\(636\) 9.04509 0.358661
\(637\) −22.4418 −0.889176
\(638\) 11.2465 0.445252
\(639\) 0.583922 0.0230996
\(640\) −5.00657 −0.197902
\(641\) 26.7996 1.05852 0.529260 0.848460i \(-0.322470\pi\)
0.529260 + 0.848460i \(0.322470\pi\)
\(642\) 43.0912 1.70067
\(643\) 9.64700 0.380440 0.190220 0.981741i \(-0.439080\pi\)
0.190220 + 0.981741i \(0.439080\pi\)
\(644\) −56.5929 −2.23007
\(645\) 17.6065 0.693255
\(646\) 0 0
\(647\) −7.93652 −0.312017 −0.156008 0.987756i \(-0.549863\pi\)
−0.156008 + 0.987756i \(0.549863\pi\)
\(648\) −5.29439 −0.207983
\(649\) −6.11035 −0.239852
\(650\) 3.95348 0.155068
\(651\) 59.8048 2.34393
\(652\) 30.5618 1.19689
\(653\) −12.3813 −0.484517 −0.242258 0.970212i \(-0.577888\pi\)
−0.242258 + 0.970212i \(0.577888\pi\)
\(654\) −25.4102 −0.993618
\(655\) −0.968020 −0.0378237
\(656\) −44.4418 −1.73516
\(657\) −0.374307 −0.0146031
\(658\) 29.0130 1.13105
\(659\) −20.5666 −0.801162 −0.400581 0.916261i \(-0.631192\pi\)
−0.400581 + 0.916261i \(0.631192\pi\)
\(660\) 8.17026 0.318027
\(661\) −14.8343 −0.576989 −0.288495 0.957482i \(-0.593155\pi\)
−0.288495 + 0.957482i \(0.593155\pi\)
\(662\) −29.1532 −1.13307
\(663\) −12.4136 −0.482104
\(664\) 10.9263 0.424023
\(665\) 0 0
\(666\) −0.863162 −0.0334468
\(667\) −16.0843 −0.622786
\(668\) −9.11520 −0.352678
\(669\) −6.59494 −0.254975
\(670\) −19.3537 −0.747697
\(671\) −14.4554 −0.558043
\(672\) −52.6673 −2.03168
\(673\) −21.7573 −0.838683 −0.419342 0.907828i \(-0.637739\pi\)
−0.419342 + 0.907828i \(0.637739\pi\)
\(674\) −11.7973 −0.454415
\(675\) −5.35802 −0.206230
\(676\) −14.5841 −0.560925
\(677\) 10.5635 0.405989 0.202995 0.979180i \(-0.434933\pi\)
0.202995 + 0.979180i \(0.434933\pi\)
\(678\) 26.4241 1.01481
\(679\) −76.1982 −2.92422
\(680\) 2.28251 0.0875304
\(681\) 46.7650 1.79204
\(682\) 47.4601 1.81734
\(683\) −35.8085 −1.37018 −0.685088 0.728461i \(-0.740236\pi\)
−0.685088 + 0.728461i \(0.740236\pi\)
\(684\) 0 0
\(685\) 19.4256 0.742212
\(686\) 31.3587 1.19728
\(687\) −10.7070 −0.408498
\(688\) −47.9448 −1.82788
\(689\) −6.69176 −0.254936
\(690\) −25.6893 −0.977975
\(691\) 20.4006 0.776075 0.388038 0.921644i \(-0.373153\pi\)
0.388038 + 0.921644i \(0.373153\pi\)
\(692\) 12.8391 0.488070
\(693\) 2.53478 0.0962882
\(694\) −15.6747 −0.595002
\(695\) −10.2777 −0.389857
\(696\) −2.12705 −0.0806257
\(697\) 35.1137 1.33003
\(698\) 20.6860 0.782976
\(699\) 27.8594 1.05374
\(700\) −7.05470 −0.266643
\(701\) 47.5390 1.79552 0.897761 0.440483i \(-0.145193\pi\)
0.897761 + 0.440483i \(0.145193\pi\)
\(702\) −21.1828 −0.799494
\(703\) 0 0
\(704\) −15.1304 −0.570248
\(705\) 5.99035 0.225610
\(706\) 16.9382 0.637478
\(707\) −37.8625 −1.42397
\(708\) −5.82131 −0.218778
\(709\) 26.0530 0.978440 0.489220 0.872160i \(-0.337282\pi\)
0.489220 + 0.872160i \(0.337282\pi\)
\(710\) 5.46271 0.205012
\(711\) −2.60289 −0.0976158
\(712\) −1.56390 −0.0586097
\(713\) −67.8757 −2.54197
\(714\) 48.6997 1.82254
\(715\) −6.04454 −0.226053
\(716\) 35.6469 1.33219
\(717\) −14.2760 −0.533148
\(718\) 6.49658 0.242450
\(719\) 9.90811 0.369510 0.184755 0.982785i \(-0.440851\pi\)
0.184755 + 0.982785i \(0.440851\pi\)
\(720\) 0.932146 0.0347390
\(721\) 49.8174 1.85530
\(722\) 0 0
\(723\) −8.93655 −0.332354
\(724\) 3.61452 0.134333
\(725\) −2.00502 −0.0744644
\(726\) 7.76267 0.288100
\(727\) −19.2073 −0.712358 −0.356179 0.934418i \(-0.615921\pi\)
−0.356179 + 0.934418i \(0.615921\pi\)
\(728\) 5.53689 0.205211
\(729\) 28.5826 1.05861
\(730\) −3.50172 −0.129604
\(731\) 37.8814 1.40109
\(732\) −13.7716 −0.509011
\(733\) 20.2835 0.749189 0.374595 0.927189i \(-0.377782\pi\)
0.374595 + 0.927189i \(0.377782\pi\)
\(734\) 23.4224 0.864535
\(735\) 18.1780 0.670505
\(736\) 59.7750 2.20334
\(737\) 29.5901 1.08997
\(738\) 3.82802 0.140911
\(739\) 52.0382 1.91425 0.957127 0.289668i \(-0.0935449\pi\)
0.957127 + 0.289668i \(0.0935449\pi\)
\(740\) −3.67295 −0.135020
\(741\) 0 0
\(742\) 26.2524 0.963757
\(743\) 10.4897 0.384831 0.192416 0.981314i \(-0.438368\pi\)
0.192416 + 0.981314i \(0.438368\pi\)
\(744\) −8.97617 −0.329082
\(745\) 10.1694 0.372580
\(746\) −26.3245 −0.963807
\(747\) −3.52556 −0.128994
\(748\) 17.5788 0.642744
\(749\) 56.8872 2.07861
\(750\) −3.20235 −0.116933
\(751\) 45.0715 1.64468 0.822341 0.568996i \(-0.192668\pi\)
0.822341 + 0.568996i \(0.192668\pi\)
\(752\) −16.3125 −0.594856
\(753\) −16.2730 −0.593022
\(754\) −7.92680 −0.288677
\(755\) −2.79628 −0.101767
\(756\) 37.7992 1.37474
\(757\) −46.0535 −1.67384 −0.836922 0.547322i \(-0.815647\pi\)
−0.836922 + 0.547322i \(0.815647\pi\)
\(758\) 43.1117 1.56589
\(759\) 39.2768 1.42566
\(760\) 0 0
\(761\) −8.48523 −0.307589 −0.153795 0.988103i \(-0.549149\pi\)
−0.153795 + 0.988103i \(0.549149\pi\)
\(762\) 6.60942 0.239434
\(763\) −33.5455 −1.21443
\(764\) 16.3690 0.592210
\(765\) −0.736492 −0.0266279
\(766\) −58.9195 −2.12885
\(767\) 4.30673 0.155507
\(768\) 33.3090 1.20194
\(769\) −3.92677 −0.141603 −0.0708015 0.997490i \(-0.522556\pi\)
−0.0708015 + 0.997490i \(0.522556\pi\)
\(770\) 23.7133 0.854569
\(771\) −10.6002 −0.381756
\(772\) 6.37807 0.229552
\(773\) −31.1834 −1.12159 −0.560795 0.827955i \(-0.689504\pi\)
−0.560795 + 0.827955i \(0.689504\pi\)
\(774\) 4.12975 0.148441
\(775\) −8.46117 −0.303934
\(776\) 11.4367 0.410553
\(777\) 15.5574 0.558117
\(778\) 40.3411 1.44630
\(779\) 0 0
\(780\) −5.75861 −0.206191
\(781\) −8.35202 −0.298859
\(782\) −55.2720 −1.97652
\(783\) 10.7429 0.383920
\(784\) −49.5010 −1.76789
\(785\) −15.8271 −0.564892
\(786\) 3.09994 0.110571
\(787\) −18.7949 −0.669966 −0.334983 0.942224i \(-0.608731\pi\)
−0.334983 + 0.942224i \(0.608731\pi\)
\(788\) 17.9018 0.637727
\(789\) 22.6273 0.805553
\(790\) −24.3505 −0.866353
\(791\) 34.8840 1.24033
\(792\) −0.380447 −0.0135186
\(793\) 10.1885 0.361804
\(794\) 42.2836 1.50059
\(795\) 5.42037 0.192241
\(796\) 39.4138 1.39699
\(797\) 7.77327 0.275344 0.137672 0.990478i \(-0.456038\pi\)
0.137672 + 0.990478i \(0.456038\pi\)
\(798\) 0 0
\(799\) 12.8886 0.455965
\(800\) 7.45136 0.263445
\(801\) 0.504620 0.0178299
\(802\) 55.1439 1.94720
\(803\) 5.35383 0.188933
\(804\) 28.1904 0.994198
\(805\) −33.9139 −1.19531
\(806\) −33.4511 −1.17827
\(807\) −22.6819 −0.798439
\(808\) 5.68283 0.199921
\(809\) 23.0942 0.811950 0.405975 0.913884i \(-0.366932\pi\)
0.405975 + 0.913884i \(0.366932\pi\)
\(810\) 15.9818 0.561542
\(811\) −9.79395 −0.343912 −0.171956 0.985105i \(-0.555009\pi\)
−0.171956 + 0.985105i \(0.555009\pi\)
\(812\) 14.1448 0.496384
\(813\) 3.02644 0.106142
\(814\) 12.3461 0.432730
\(815\) 18.3145 0.641529
\(816\) −27.3813 −0.958538
\(817\) 0 0
\(818\) 35.2263 1.23166
\(819\) −1.78657 −0.0624280
\(820\) 16.2891 0.568839
\(821\) −30.3878 −1.06054 −0.530270 0.847829i \(-0.677910\pi\)
−0.530270 + 0.847829i \(0.677910\pi\)
\(822\) −62.2074 −2.16973
\(823\) 24.7436 0.862507 0.431254 0.902231i \(-0.358071\pi\)
0.431254 + 0.902231i \(0.358071\pi\)
\(824\) −7.47715 −0.260479
\(825\) 4.89612 0.170461
\(826\) −16.8957 −0.587878
\(827\) −17.3269 −0.602514 −0.301257 0.953543i \(-0.597406\pi\)
−0.301257 + 0.953543i \(0.597406\pi\)
\(828\) −2.74077 −0.0952483
\(829\) −45.9820 −1.59702 −0.798510 0.601981i \(-0.794378\pi\)
−0.798510 + 0.601981i \(0.794378\pi\)
\(830\) −32.9824 −1.14483
\(831\) −2.14519 −0.0744158
\(832\) 10.6643 0.369717
\(833\) 39.1110 1.35511
\(834\) 32.9129 1.13968
\(835\) −5.46238 −0.189034
\(836\) 0 0
\(837\) 45.3351 1.56701
\(838\) 42.5865 1.47113
\(839\) −11.6248 −0.401334 −0.200667 0.979660i \(-0.564311\pi\)
−0.200667 + 0.979660i \(0.564311\pi\)
\(840\) −4.48492 −0.154744
\(841\) −24.9799 −0.861376
\(842\) −51.2560 −1.76640
\(843\) −16.2792 −0.560685
\(844\) −24.4820 −0.842706
\(845\) −8.73965 −0.300653
\(846\) 1.40509 0.0483079
\(847\) 10.2480 0.352124
\(848\) −14.7604 −0.506873
\(849\) 12.6742 0.434976
\(850\) −6.89003 −0.236326
\(851\) −17.6569 −0.605271
\(852\) −7.95693 −0.272600
\(853\) −8.52849 −0.292010 −0.146005 0.989284i \(-0.546642\pi\)
−0.146005 + 0.989284i \(0.546642\pi\)
\(854\) −39.9705 −1.36776
\(855\) 0 0
\(856\) −8.53826 −0.291832
\(857\) 27.9866 0.956005 0.478002 0.878359i \(-0.341361\pi\)
0.478002 + 0.878359i \(0.341361\pi\)
\(858\) 19.3567 0.660828
\(859\) −43.4377 −1.48208 −0.741038 0.671464i \(-0.765666\pi\)
−0.741038 + 0.671464i \(0.765666\pi\)
\(860\) 17.5730 0.599234
\(861\) −68.9950 −2.35134
\(862\) 7.28460 0.248114
\(863\) −24.1508 −0.822101 −0.411051 0.911613i \(-0.634838\pi\)
−0.411051 + 0.911613i \(0.634838\pi\)
\(864\) −39.9245 −1.35826
\(865\) 7.69398 0.261603
\(866\) 59.9198 2.03616
\(867\) −6.78826 −0.230541
\(868\) 59.6910 2.02605
\(869\) 37.2299 1.26294
\(870\) 6.42076 0.217684
\(871\) −20.8559 −0.706674
\(872\) 5.03488 0.170503
\(873\) −3.69024 −0.124896
\(874\) 0 0
\(875\) −4.22760 −0.142919
\(876\) 5.10057 0.172332
\(877\) −45.4037 −1.53317 −0.766586 0.642141i \(-0.778046\pi\)
−0.766586 + 0.642141i \(0.778046\pi\)
\(878\) −7.97630 −0.269187
\(879\) 1.97932 0.0667607
\(880\) −13.3328 −0.449448
\(881\) −31.7657 −1.07021 −0.535107 0.844784i \(-0.679729\pi\)
−0.535107 + 0.844784i \(0.679729\pi\)
\(882\) 4.26380 0.143569
\(883\) −29.5408 −0.994126 −0.497063 0.867715i \(-0.665588\pi\)
−0.497063 + 0.867715i \(0.665588\pi\)
\(884\) −12.3900 −0.416720
\(885\) −3.48848 −0.117264
\(886\) 12.6709 0.425687
\(887\) −4.82481 −0.162001 −0.0810006 0.996714i \(-0.525812\pi\)
−0.0810006 + 0.996714i \(0.525812\pi\)
\(888\) −2.33502 −0.0783582
\(889\) 8.72548 0.292643
\(890\) 4.72082 0.158242
\(891\) −24.4348 −0.818595
\(892\) −6.58239 −0.220395
\(893\) 0 0
\(894\) −32.5661 −1.08917
\(895\) 21.3618 0.714046
\(896\) 21.1658 0.707100
\(897\) −27.6833 −0.924318
\(898\) 10.5345 0.351542
\(899\) 16.9648 0.565807
\(900\) −0.341655 −0.0113885
\(901\) 11.6622 0.388525
\(902\) −54.7534 −1.82309
\(903\) −74.4333 −2.47698
\(904\) −5.23577 −0.174139
\(905\) 2.16604 0.0720016
\(906\) 8.95466 0.297499
\(907\) −34.7324 −1.15327 −0.576635 0.817002i \(-0.695634\pi\)
−0.576635 + 0.817002i \(0.695634\pi\)
\(908\) 46.6760 1.54900
\(909\) −1.83366 −0.0608187
\(910\) −16.7138 −0.554056
\(911\) 49.1485 1.62836 0.814181 0.580611i \(-0.197186\pi\)
0.814181 + 0.580611i \(0.197186\pi\)
\(912\) 0 0
\(913\) 50.4273 1.66890
\(914\) −67.4820 −2.23211
\(915\) −8.25275 −0.272828
\(916\) −10.6866 −0.353096
\(917\) 4.09240 0.135143
\(918\) 36.9169 1.21844
\(919\) 38.1829 1.25954 0.629769 0.776783i \(-0.283150\pi\)
0.629769 + 0.776783i \(0.283150\pi\)
\(920\) 5.09018 0.167818
\(921\) 22.5602 0.743383
\(922\) −10.7806 −0.355040
\(923\) 5.88671 0.193764
\(924\) −34.5406 −1.13630
\(925\) −2.20105 −0.0723702
\(926\) 39.6428 1.30274
\(927\) 2.41263 0.0792412
\(928\) −14.9401 −0.490433
\(929\) −41.9415 −1.37606 −0.688028 0.725684i \(-0.741524\pi\)
−0.688028 + 0.725684i \(0.741524\pi\)
\(930\) 27.0956 0.888500
\(931\) 0 0
\(932\) 27.8064 0.910829
\(933\) 23.8570 0.781044
\(934\) 35.4119 1.15871
\(935\) 10.5343 0.344508
\(936\) 0.268149 0.00876472
\(937\) −27.1660 −0.887473 −0.443737 0.896157i \(-0.646347\pi\)
−0.443737 + 0.896157i \(0.646347\pi\)
\(938\) 81.8196 2.67151
\(939\) −48.3363 −1.57739
\(940\) 5.97896 0.195012
\(941\) 41.4889 1.35250 0.676250 0.736672i \(-0.263604\pi\)
0.676250 + 0.736672i \(0.263604\pi\)
\(942\) 50.6838 1.65137
\(943\) 78.3062 2.55000
\(944\) 9.49959 0.309185
\(945\) 22.6516 0.736855
\(946\) −59.0691 −1.92050
\(947\) 42.0065 1.36503 0.682513 0.730873i \(-0.260887\pi\)
0.682513 + 0.730873i \(0.260887\pi\)
\(948\) 35.4688 1.15197
\(949\) −3.77352 −0.122494
\(950\) 0 0
\(951\) −25.0569 −0.812527
\(952\) −9.64956 −0.312744
\(953\) 24.2497 0.785525 0.392762 0.919640i \(-0.371519\pi\)
0.392762 + 0.919640i \(0.371519\pi\)
\(954\) 1.27139 0.0411628
\(955\) 9.80930 0.317422
\(956\) −14.2489 −0.460841
\(957\) −9.81680 −0.317332
\(958\) 61.4823 1.98640
\(959\) −82.1235 −2.65191
\(960\) −8.63813 −0.278795
\(961\) 40.5914 1.30940
\(962\) −8.70183 −0.280558
\(963\) 2.75502 0.0887792
\(964\) −8.91955 −0.287279
\(965\) 3.82213 0.123039
\(966\) 108.604 3.49428
\(967\) −7.93704 −0.255238 −0.127619 0.991823i \(-0.540733\pi\)
−0.127619 + 0.991823i \(0.540733\pi\)
\(968\) −1.53813 −0.0494373
\(969\) 0 0
\(970\) −34.5230 −1.10847
\(971\) 43.3213 1.39025 0.695123 0.718891i \(-0.255350\pi\)
0.695123 + 0.718891i \(0.255350\pi\)
\(972\) 3.54424 0.113682
\(973\) 43.4502 1.39295
\(974\) −6.78719 −0.217475
\(975\) −3.45091 −0.110518
\(976\) 22.4733 0.719353
\(977\) −37.7002 −1.20614 −0.603068 0.797690i \(-0.706055\pi\)
−0.603068 + 0.797690i \(0.706055\pi\)
\(978\) −58.6494 −1.87540
\(979\) −7.21774 −0.230680
\(980\) 18.1434 0.579570
\(981\) −1.62459 −0.0518692
\(982\) 37.9097 1.20975
\(983\) 49.0724 1.56517 0.782584 0.622545i \(-0.213901\pi\)
0.782584 + 0.622545i \(0.213901\pi\)
\(984\) 10.3555 0.330123
\(985\) 10.7279 0.341818
\(986\) 13.8146 0.439947
\(987\) −25.3248 −0.806098
\(988\) 0 0
\(989\) 84.4784 2.68626
\(990\) 1.14842 0.0364993
\(991\) −48.5874 −1.54343 −0.771715 0.635969i \(-0.780601\pi\)
−0.771715 + 0.635969i \(0.780601\pi\)
\(992\) −63.0472 −2.00175
\(993\) 25.4472 0.807542
\(994\) −23.0942 −0.732502
\(995\) 23.6191 0.748777
\(996\) 48.0418 1.52226
\(997\) −21.7606 −0.689165 −0.344582 0.938756i \(-0.611979\pi\)
−0.344582 + 0.938756i \(0.611979\pi\)
\(998\) 56.4140 1.78576
\(999\) 11.7933 0.373123
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 1805.2.a.w.1.5 16
5.4 even 2 9025.2.a.cm.1.12 16
19.18 odd 2 inner 1805.2.a.w.1.12 yes 16
95.94 odd 2 9025.2.a.cm.1.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.w.1.5 16 1.1 even 1 trivial
1805.2.a.w.1.12 yes 16 19.18 odd 2 inner
9025.2.a.cm.1.5 16 95.94 odd 2
9025.2.a.cm.1.12 16 5.4 even 2