Properties

Label 1875.2.a.i.1.6
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
Analytic rank $0$
Dimension $6$
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] = [1875,2,Mod(1,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, 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("1875.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.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: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.46840000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 \) 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 1.6
Root \(-2.38719\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.38719 q^{2} +1.00000 q^{3} +3.69868 q^{4} +2.38719 q^{6} +3.31671 q^{7} +4.05506 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.38719 q^{2} +1.00000 q^{3} +3.69868 q^{4} +2.38719 q^{6} +3.31671 q^{7} +4.05506 q^{8} +1.00000 q^{9} -4.36655 q^{11} +3.69868 q^{12} +5.85502 q^{13} +7.91762 q^{14} +2.28286 q^{16} +0.407830 q^{17} +2.38719 q^{18} -6.64447 q^{19} +3.31671 q^{21} -10.4238 q^{22} +4.61860 q^{23} +4.05506 q^{24} +13.9771 q^{26} +1.00000 q^{27} +12.2674 q^{28} +3.30132 q^{29} -8.77495 q^{31} -2.66052 q^{32} -4.36655 q^{33} +0.973567 q^{34} +3.69868 q^{36} +3.09911 q^{37} -15.8616 q^{38} +5.85502 q^{39} +2.89349 q^{41} +7.91762 q^{42} +1.33270 q^{43} -16.1505 q^{44} +11.0255 q^{46} -11.0609 q^{47} +2.28286 q^{48} +4.00057 q^{49} +0.407830 q^{51} +21.6558 q^{52} +2.70021 q^{53} +2.38719 q^{54} +13.4495 q^{56} -6.64447 q^{57} +7.88089 q^{58} -5.80518 q^{59} +11.2366 q^{61} -20.9475 q^{62} +3.31671 q^{63} -10.9169 q^{64} -10.4238 q^{66} -0.0418484 q^{67} +1.50843 q^{68} +4.61860 q^{69} -16.2062 q^{71} +4.05506 q^{72} +5.35799 q^{73} +7.39817 q^{74} -24.5757 q^{76} -14.4826 q^{77} +13.9771 q^{78} -3.55370 q^{79} +1.00000 q^{81} +6.90732 q^{82} +4.98952 q^{83} +12.2674 q^{84} +3.18140 q^{86} +3.30132 q^{87} -17.7066 q^{88} -1.94025 q^{89} +19.4194 q^{91} +17.0827 q^{92} -8.77495 q^{93} -26.4044 q^{94} -2.66052 q^{96} +5.99258 q^{97} +9.55012 q^{98} -4.36655 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} + 6 q^{3} + 11 q^{4} - q^{6} + 2 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} + 6 q^{3} + 11 q^{4} - q^{6} + 2 q^{7} - 6 q^{8} + 6 q^{9} + 11 q^{12} - 4 q^{14} + 17 q^{16} - 2 q^{17} - q^{18} - 2 q^{19} + 2 q^{21} - 9 q^{22} + q^{23} - 6 q^{24} + 37 q^{26} + 6 q^{27} + 44 q^{28} + 31 q^{29} - 2 q^{31} - 33 q^{32} + 37 q^{34} + 11 q^{36} + 22 q^{37} - 27 q^{38} + 33 q^{41} - 4 q^{42} + 3 q^{43} - 11 q^{44} - 12 q^{46} + 6 q^{47} + 17 q^{48} + 4 q^{49} - 2 q^{51} + 33 q^{52} - 14 q^{53} - q^{54} - 30 q^{56} - 2 q^{57} + q^{58} - 8 q^{59} + 34 q^{61} - 31 q^{62} + 2 q^{63} + 12 q^{64} - 9 q^{66} - 2 q^{67} - 27 q^{68} + q^{69} - 3 q^{71} - 6 q^{72} + 36 q^{73} + 36 q^{74} + 27 q^{76} - 16 q^{77} + 37 q^{78} + 25 q^{79} + 6 q^{81} - 36 q^{82} + 12 q^{83} + 44 q^{84} - 30 q^{86} + 31 q^{87} - 56 q^{88} + 18 q^{89} + 28 q^{91} - 3 q^{92} - 2 q^{93} - 50 q^{94} - 33 q^{96} - 7 q^{97} - 15 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.38719 1.68800 0.843999 0.536345i \(-0.180195\pi\)
0.843999 + 0.536345i \(0.180195\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.69868 1.84934
\(5\) 0 0
\(6\) 2.38719 0.974566
\(7\) 3.31671 1.25360 0.626799 0.779181i \(-0.284365\pi\)
0.626799 + 0.779181i \(0.284365\pi\)
\(8\) 4.05506 1.43368
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.36655 −1.31656 −0.658282 0.752771i \(-0.728717\pi\)
−0.658282 + 0.752771i \(0.728717\pi\)
\(12\) 3.69868 1.06772
\(13\) 5.85502 1.62389 0.811946 0.583733i \(-0.198409\pi\)
0.811946 + 0.583733i \(0.198409\pi\)
\(14\) 7.91762 2.11607
\(15\) 0 0
\(16\) 2.28286 0.570714
\(17\) 0.407830 0.0989132 0.0494566 0.998776i \(-0.484251\pi\)
0.0494566 + 0.998776i \(0.484251\pi\)
\(18\) 2.38719 0.562666
\(19\) −6.64447 −1.52435 −0.762173 0.647374i \(-0.775867\pi\)
−0.762173 + 0.647374i \(0.775867\pi\)
\(20\) 0 0
\(21\) 3.31671 0.723766
\(22\) −10.4238 −2.22236
\(23\) 4.61860 0.963045 0.481523 0.876434i \(-0.340084\pi\)
0.481523 + 0.876434i \(0.340084\pi\)
\(24\) 4.05506 0.827736
\(25\) 0 0
\(26\) 13.9771 2.74113
\(27\) 1.00000 0.192450
\(28\) 12.2674 2.31833
\(29\) 3.30132 0.613040 0.306520 0.951864i \(-0.400835\pi\)
0.306520 + 0.951864i \(0.400835\pi\)
\(30\) 0 0
\(31\) −8.77495 −1.57603 −0.788014 0.615658i \(-0.788890\pi\)
−0.788014 + 0.615658i \(0.788890\pi\)
\(32\) −2.66052 −0.470318
\(33\) −4.36655 −0.760119
\(34\) 0.973567 0.166965
\(35\) 0 0
\(36\) 3.69868 0.616446
\(37\) 3.09911 0.509491 0.254745 0.967008i \(-0.418008\pi\)
0.254745 + 0.967008i \(0.418008\pi\)
\(38\) −15.8616 −2.57309
\(39\) 5.85502 0.937554
\(40\) 0 0
\(41\) 2.89349 0.451888 0.225944 0.974140i \(-0.427453\pi\)
0.225944 + 0.974140i \(0.427453\pi\)
\(42\) 7.91762 1.22172
\(43\) 1.33270 0.203234 0.101617 0.994824i \(-0.467598\pi\)
0.101617 + 0.994824i \(0.467598\pi\)
\(44\) −16.1505 −2.43477
\(45\) 0 0
\(46\) 11.0255 1.62562
\(47\) −11.0609 −1.61339 −0.806696 0.590967i \(-0.798746\pi\)
−0.806696 + 0.590967i \(0.798746\pi\)
\(48\) 2.28286 0.329502
\(49\) 4.00057 0.571510
\(50\) 0 0
\(51\) 0.407830 0.0571076
\(52\) 21.6558 3.00312
\(53\) 2.70021 0.370903 0.185451 0.982653i \(-0.440625\pi\)
0.185451 + 0.982653i \(0.440625\pi\)
\(54\) 2.38719 0.324855
\(55\) 0 0
\(56\) 13.4495 1.79726
\(57\) −6.64447 −0.880081
\(58\) 7.88089 1.03481
\(59\) −5.80518 −0.755770 −0.377885 0.925852i \(-0.623349\pi\)
−0.377885 + 0.925852i \(0.623349\pi\)
\(60\) 0 0
\(61\) 11.2366 1.43870 0.719352 0.694646i \(-0.244439\pi\)
0.719352 + 0.694646i \(0.244439\pi\)
\(62\) −20.9475 −2.66033
\(63\) 3.31671 0.417866
\(64\) −10.9169 −1.36461
\(65\) 0 0
\(66\) −10.4238 −1.28308
\(67\) −0.0418484 −0.00511260 −0.00255630 0.999997i \(-0.500814\pi\)
−0.00255630 + 0.999997i \(0.500814\pi\)
\(68\) 1.50843 0.182924
\(69\) 4.61860 0.556014
\(70\) 0 0
\(71\) −16.2062 −1.92332 −0.961659 0.274250i \(-0.911571\pi\)
−0.961659 + 0.274250i \(0.911571\pi\)
\(72\) 4.05506 0.477894
\(73\) 5.35799 0.627105 0.313553 0.949571i \(-0.398481\pi\)
0.313553 + 0.949571i \(0.398481\pi\)
\(74\) 7.39817 0.860019
\(75\) 0 0
\(76\) −24.5757 −2.81903
\(77\) −14.4826 −1.65044
\(78\) 13.9771 1.58259
\(79\) −3.55370 −0.399822 −0.199911 0.979814i \(-0.564065\pi\)
−0.199911 + 0.979814i \(0.564065\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.90732 0.762786
\(83\) 4.98952 0.547671 0.273836 0.961776i \(-0.411708\pi\)
0.273836 + 0.961776i \(0.411708\pi\)
\(84\) 12.2674 1.33849
\(85\) 0 0
\(86\) 3.18140 0.343059
\(87\) 3.30132 0.353939
\(88\) −17.7066 −1.88753
\(89\) −1.94025 −0.205666 −0.102833 0.994699i \(-0.532791\pi\)
−0.102833 + 0.994699i \(0.532791\pi\)
\(90\) 0 0
\(91\) 19.4194 2.03571
\(92\) 17.0827 1.78100
\(93\) −8.77495 −0.909920
\(94\) −26.4044 −2.72340
\(95\) 0 0
\(96\) −2.66052 −0.271538
\(97\) 5.99258 0.608454 0.304227 0.952600i \(-0.401602\pi\)
0.304227 + 0.952600i \(0.401602\pi\)
\(98\) 9.55012 0.964708
\(99\) −4.36655 −0.438855
\(100\) 0 0
\(101\) 5.06181 0.503669 0.251834 0.967770i \(-0.418966\pi\)
0.251834 + 0.967770i \(0.418966\pi\)
\(102\) 0.973567 0.0963975
\(103\) 6.90359 0.680231 0.340115 0.940384i \(-0.389534\pi\)
0.340115 + 0.940384i \(0.389534\pi\)
\(104\) 23.7425 2.32814
\(105\) 0 0
\(106\) 6.44592 0.626083
\(107\) −17.2723 −1.66978 −0.834890 0.550417i \(-0.814469\pi\)
−0.834890 + 0.550417i \(0.814469\pi\)
\(108\) 3.69868 0.355905
\(109\) −6.54439 −0.626839 −0.313419 0.949615i \(-0.601475\pi\)
−0.313419 + 0.949615i \(0.601475\pi\)
\(110\) 0 0
\(111\) 3.09911 0.294155
\(112\) 7.57157 0.715446
\(113\) 8.66993 0.815599 0.407799 0.913072i \(-0.366296\pi\)
0.407799 + 0.913072i \(0.366296\pi\)
\(114\) −15.8616 −1.48558
\(115\) 0 0
\(116\) 12.2105 1.13372
\(117\) 5.85502 0.541297
\(118\) −13.8581 −1.27574
\(119\) 1.35265 0.123998
\(120\) 0 0
\(121\) 8.06676 0.733342
\(122\) 26.8240 2.42853
\(123\) 2.89349 0.260898
\(124\) −32.4557 −2.91461
\(125\) 0 0
\(126\) 7.91762 0.705358
\(127\) −3.24595 −0.288031 −0.144016 0.989575i \(-0.546002\pi\)
−0.144016 + 0.989575i \(0.546002\pi\)
\(128\) −20.7396 −1.83314
\(129\) 1.33270 0.117337
\(130\) 0 0
\(131\) −4.40192 −0.384598 −0.192299 0.981336i \(-0.561594\pi\)
−0.192299 + 0.981336i \(0.561594\pi\)
\(132\) −16.1505 −1.40572
\(133\) −22.0378 −1.91092
\(134\) −0.0999001 −0.00863005
\(135\) 0 0
\(136\) 1.65378 0.141810
\(137\) 1.06523 0.0910085 0.0455042 0.998964i \(-0.485511\pi\)
0.0455042 + 0.998964i \(0.485511\pi\)
\(138\) 11.0255 0.938551
\(139\) −3.77286 −0.320010 −0.160005 0.987116i \(-0.551151\pi\)
−0.160005 + 0.987116i \(0.551151\pi\)
\(140\) 0 0
\(141\) −11.0609 −0.931492
\(142\) −38.6872 −3.24656
\(143\) −25.5663 −2.13796
\(144\) 2.28286 0.190238
\(145\) 0 0
\(146\) 12.7905 1.05855
\(147\) 4.00057 0.329961
\(148\) 11.4626 0.942221
\(149\) −7.88089 −0.645627 −0.322814 0.946463i \(-0.604629\pi\)
−0.322814 + 0.946463i \(0.604629\pi\)
\(150\) 0 0
\(151\) 7.46432 0.607437 0.303719 0.952762i \(-0.401772\pi\)
0.303719 + 0.952762i \(0.401772\pi\)
\(152\) −26.9437 −2.18543
\(153\) 0.407830 0.0329711
\(154\) −34.5727 −2.78595
\(155\) 0 0
\(156\) 21.6558 1.73385
\(157\) 0.268958 0.0214652 0.0107326 0.999942i \(-0.496584\pi\)
0.0107326 + 0.999942i \(0.496584\pi\)
\(158\) −8.48336 −0.674900
\(159\) 2.70021 0.214141
\(160\) 0 0
\(161\) 15.3186 1.20727
\(162\) 2.38719 0.187555
\(163\) 10.3496 0.810645 0.405322 0.914174i \(-0.367159\pi\)
0.405322 + 0.914174i \(0.367159\pi\)
\(164\) 10.7021 0.835693
\(165\) 0 0
\(166\) 11.9109 0.924468
\(167\) −4.26896 −0.330342 −0.165171 0.986265i \(-0.552818\pi\)
−0.165171 + 0.986265i \(0.552818\pi\)
\(168\) 13.4495 1.03765
\(169\) 21.2813 1.63702
\(170\) 0 0
\(171\) −6.64447 −0.508115
\(172\) 4.92921 0.375849
\(173\) −23.5031 −1.78691 −0.893454 0.449154i \(-0.851725\pi\)
−0.893454 + 0.449154i \(0.851725\pi\)
\(174\) 7.88089 0.597448
\(175\) 0 0
\(176\) −9.96820 −0.751382
\(177\) −5.80518 −0.436344
\(178\) −4.63175 −0.347164
\(179\) 19.6897 1.47168 0.735840 0.677155i \(-0.236788\pi\)
0.735840 + 0.677155i \(0.236788\pi\)
\(180\) 0 0
\(181\) −12.8221 −0.953060 −0.476530 0.879158i \(-0.658106\pi\)
−0.476530 + 0.879158i \(0.658106\pi\)
\(182\) 46.3578 3.43627
\(183\) 11.2366 0.830636
\(184\) 18.7287 1.38070
\(185\) 0 0
\(186\) −20.9475 −1.53594
\(187\) −1.78081 −0.130226
\(188\) −40.9105 −2.98371
\(189\) 3.31671 0.241255
\(190\) 0 0
\(191\) −19.0583 −1.37901 −0.689507 0.724279i \(-0.742173\pi\)
−0.689507 + 0.724279i \(0.742173\pi\)
\(192\) −10.9169 −0.787857
\(193\) 16.9842 1.22255 0.611275 0.791418i \(-0.290657\pi\)
0.611275 + 0.791418i \(0.290657\pi\)
\(194\) 14.3054 1.02707
\(195\) 0 0
\(196\) 14.7968 1.05692
\(197\) −0.469462 −0.0334478 −0.0167239 0.999860i \(-0.505324\pi\)
−0.0167239 + 0.999860i \(0.505324\pi\)
\(198\) −10.4238 −0.740786
\(199\) 7.02491 0.497983 0.248991 0.968506i \(-0.419901\pi\)
0.248991 + 0.968506i \(0.419901\pi\)
\(200\) 0 0
\(201\) −0.0418484 −0.00295176
\(202\) 12.0835 0.850192
\(203\) 10.9495 0.768507
\(204\) 1.50843 0.105611
\(205\) 0 0
\(206\) 16.4802 1.14823
\(207\) 4.61860 0.321015
\(208\) 13.3662 0.926777
\(209\) 29.0134 2.00690
\(210\) 0 0
\(211\) 15.7531 1.08449 0.542243 0.840222i \(-0.317575\pi\)
0.542243 + 0.840222i \(0.317575\pi\)
\(212\) 9.98721 0.685925
\(213\) −16.2062 −1.11043
\(214\) −41.2323 −2.81859
\(215\) 0 0
\(216\) 4.05506 0.275912
\(217\) −29.1040 −1.97571
\(218\) −15.6227 −1.05810
\(219\) 5.35799 0.362059
\(220\) 0 0
\(221\) 2.38785 0.160624
\(222\) 7.39817 0.496532
\(223\) −2.27697 −0.152477 −0.0762385 0.997090i \(-0.524291\pi\)
−0.0762385 + 0.997090i \(0.524291\pi\)
\(224\) −8.82417 −0.589590
\(225\) 0 0
\(226\) 20.6968 1.37673
\(227\) −20.2674 −1.34520 −0.672599 0.740007i \(-0.734822\pi\)
−0.672599 + 0.740007i \(0.734822\pi\)
\(228\) −24.5757 −1.62757
\(229\) 19.6358 1.29757 0.648786 0.760971i \(-0.275277\pi\)
0.648786 + 0.760971i \(0.275277\pi\)
\(230\) 0 0
\(231\) −14.4826 −0.952884
\(232\) 13.3871 0.878905
\(233\) −17.1468 −1.12333 −0.561663 0.827366i \(-0.689838\pi\)
−0.561663 + 0.827366i \(0.689838\pi\)
\(234\) 13.9771 0.913709
\(235\) 0 0
\(236\) −21.4715 −1.39768
\(237\) −3.55370 −0.230838
\(238\) 3.22904 0.209308
\(239\) 21.7654 1.40789 0.703945 0.710255i \(-0.251420\pi\)
0.703945 + 0.710255i \(0.251420\pi\)
\(240\) 0 0
\(241\) 3.90067 0.251264 0.125632 0.992077i \(-0.459904\pi\)
0.125632 + 0.992077i \(0.459904\pi\)
\(242\) 19.2569 1.23788
\(243\) 1.00000 0.0641500
\(244\) 41.5607 2.66065
\(245\) 0 0
\(246\) 6.90732 0.440395
\(247\) −38.9035 −2.47537
\(248\) −35.5830 −2.25952
\(249\) 4.98952 0.316198
\(250\) 0 0
\(251\) −21.8389 −1.37846 −0.689230 0.724543i \(-0.742051\pi\)
−0.689230 + 0.724543i \(0.742051\pi\)
\(252\) 12.2674 0.772776
\(253\) −20.1674 −1.26791
\(254\) −7.74869 −0.486196
\(255\) 0 0
\(256\) −27.6757 −1.72973
\(257\) 17.1453 1.06949 0.534747 0.845012i \(-0.320407\pi\)
0.534747 + 0.845012i \(0.320407\pi\)
\(258\) 3.18140 0.198065
\(259\) 10.2789 0.638697
\(260\) 0 0
\(261\) 3.30132 0.204347
\(262\) −10.5082 −0.649201
\(263\) 17.6266 1.08690 0.543451 0.839441i \(-0.317117\pi\)
0.543451 + 0.839441i \(0.317117\pi\)
\(264\) −17.7066 −1.08977
\(265\) 0 0
\(266\) −52.6084 −3.22563
\(267\) −1.94025 −0.118742
\(268\) −0.154784 −0.00945492
\(269\) 1.94772 0.118755 0.0593773 0.998236i \(-0.481088\pi\)
0.0593773 + 0.998236i \(0.481088\pi\)
\(270\) 0 0
\(271\) −13.8707 −0.842583 −0.421292 0.906925i \(-0.638423\pi\)
−0.421292 + 0.906925i \(0.638423\pi\)
\(272\) 0.931016 0.0564512
\(273\) 19.4194 1.17532
\(274\) 2.54290 0.153622
\(275\) 0 0
\(276\) 17.0827 1.02826
\(277\) −26.4814 −1.59111 −0.795556 0.605880i \(-0.792821\pi\)
−0.795556 + 0.605880i \(0.792821\pi\)
\(278\) −9.00654 −0.540177
\(279\) −8.77495 −0.525342
\(280\) 0 0
\(281\) −21.6050 −1.28885 −0.644425 0.764668i \(-0.722903\pi\)
−0.644425 + 0.764668i \(0.722903\pi\)
\(282\) −26.4044 −1.57236
\(283\) −2.05287 −0.122030 −0.0610151 0.998137i \(-0.519434\pi\)
−0.0610151 + 0.998137i \(0.519434\pi\)
\(284\) −59.9413 −3.55686
\(285\) 0 0
\(286\) −61.0315 −3.60887
\(287\) 9.59688 0.566486
\(288\) −2.66052 −0.156773
\(289\) −16.8337 −0.990216
\(290\) 0 0
\(291\) 5.99258 0.351291
\(292\) 19.8175 1.15973
\(293\) 11.9787 0.699805 0.349902 0.936786i \(-0.386215\pi\)
0.349902 + 0.936786i \(0.386215\pi\)
\(294\) 9.55012 0.556974
\(295\) 0 0
\(296\) 12.5671 0.730447
\(297\) −4.36655 −0.253373
\(298\) −18.8132 −1.08982
\(299\) 27.0420 1.56388
\(300\) 0 0
\(301\) 4.42016 0.254774
\(302\) 17.8187 1.02535
\(303\) 5.06181 0.290793
\(304\) −15.1684 −0.869965
\(305\) 0 0
\(306\) 0.973567 0.0556551
\(307\) 7.07969 0.404059 0.202030 0.979379i \(-0.435246\pi\)
0.202030 + 0.979379i \(0.435246\pi\)
\(308\) −53.5664 −3.05223
\(309\) 6.90359 0.392732
\(310\) 0 0
\(311\) −11.4130 −0.647171 −0.323586 0.946199i \(-0.604888\pi\)
−0.323586 + 0.946199i \(0.604888\pi\)
\(312\) 23.7425 1.34415
\(313\) −11.9077 −0.673064 −0.336532 0.941672i \(-0.609254\pi\)
−0.336532 + 0.941672i \(0.609254\pi\)
\(314\) 0.642054 0.0362332
\(315\) 0 0
\(316\) −13.1440 −0.739407
\(317\) −12.7719 −0.717340 −0.358670 0.933465i \(-0.616770\pi\)
−0.358670 + 0.933465i \(0.616770\pi\)
\(318\) 6.44592 0.361469
\(319\) −14.4154 −0.807107
\(320\) 0 0
\(321\) −17.2723 −0.964048
\(322\) 36.5683 2.03787
\(323\) −2.70981 −0.150778
\(324\) 3.69868 0.205482
\(325\) 0 0
\(326\) 24.7065 1.36837
\(327\) −6.54439 −0.361906
\(328\) 11.7333 0.647863
\(329\) −36.6857 −2.02255
\(330\) 0 0
\(331\) 8.66041 0.476019 0.238010 0.971263i \(-0.423505\pi\)
0.238010 + 0.971263i \(0.423505\pi\)
\(332\) 18.4546 1.01283
\(333\) 3.09911 0.169830
\(334\) −10.1908 −0.557617
\(335\) 0 0
\(336\) 7.57157 0.413063
\(337\) 15.2564 0.831069 0.415534 0.909577i \(-0.363595\pi\)
0.415534 + 0.909577i \(0.363595\pi\)
\(338\) 50.8025 2.76329
\(339\) 8.66993 0.470886
\(340\) 0 0
\(341\) 38.3163 2.07494
\(342\) −15.8616 −0.857697
\(343\) −9.94824 −0.537155
\(344\) 5.40416 0.291373
\(345\) 0 0
\(346\) −56.1064 −3.01630
\(347\) 21.8540 1.17319 0.586593 0.809882i \(-0.300469\pi\)
0.586593 + 0.809882i \(0.300469\pi\)
\(348\) 12.2105 0.654553
\(349\) 7.29614 0.390553 0.195277 0.980748i \(-0.437440\pi\)
0.195277 + 0.980748i \(0.437440\pi\)
\(350\) 0 0
\(351\) 5.85502 0.312518
\(352\) 11.6173 0.619203
\(353\) 30.5320 1.62506 0.812528 0.582923i \(-0.198091\pi\)
0.812528 + 0.582923i \(0.198091\pi\)
\(354\) −13.8581 −0.736548
\(355\) 0 0
\(356\) −7.17637 −0.380347
\(357\) 1.35265 0.0715900
\(358\) 47.0031 2.48419
\(359\) 3.06747 0.161895 0.0809473 0.996718i \(-0.474205\pi\)
0.0809473 + 0.996718i \(0.474205\pi\)
\(360\) 0 0
\(361\) 25.1489 1.32363
\(362\) −30.6088 −1.60876
\(363\) 8.06676 0.423395
\(364\) 71.8261 3.76471
\(365\) 0 0
\(366\) 26.8240 1.40211
\(367\) 31.6035 1.64969 0.824845 0.565358i \(-0.191262\pi\)
0.824845 + 0.565358i \(0.191262\pi\)
\(368\) 10.5436 0.549623
\(369\) 2.89349 0.150629
\(370\) 0 0
\(371\) 8.95582 0.464963
\(372\) −32.4557 −1.68275
\(373\) 22.4904 1.16451 0.582254 0.813007i \(-0.302171\pi\)
0.582254 + 0.813007i \(0.302171\pi\)
\(374\) −4.25113 −0.219821
\(375\) 0 0
\(376\) −44.8525 −2.31309
\(377\) 19.3293 0.995511
\(378\) 7.91762 0.407238
\(379\) 30.1612 1.54928 0.774638 0.632404i \(-0.217932\pi\)
0.774638 + 0.632404i \(0.217932\pi\)
\(380\) 0 0
\(381\) −3.24595 −0.166295
\(382\) −45.4959 −2.32777
\(383\) −16.7725 −0.857034 −0.428517 0.903534i \(-0.640964\pi\)
−0.428517 + 0.903534i \(0.640964\pi\)
\(384\) −20.7396 −1.05836
\(385\) 0 0
\(386\) 40.5445 2.06366
\(387\) 1.33270 0.0677447
\(388\) 22.1646 1.12524
\(389\) 22.4835 1.13996 0.569980 0.821659i \(-0.306951\pi\)
0.569980 + 0.821659i \(0.306951\pi\)
\(390\) 0 0
\(391\) 1.88360 0.0952579
\(392\) 16.2226 0.819363
\(393\) −4.40192 −0.222048
\(394\) −1.12069 −0.0564598
\(395\) 0 0
\(396\) −16.1505 −0.811591
\(397\) −3.65800 −0.183590 −0.0917948 0.995778i \(-0.529260\pi\)
−0.0917948 + 0.995778i \(0.529260\pi\)
\(398\) 16.7698 0.840594
\(399\) −22.0378 −1.10327
\(400\) 0 0
\(401\) −25.3774 −1.26729 −0.633643 0.773626i \(-0.718441\pi\)
−0.633643 + 0.773626i \(0.718441\pi\)
\(402\) −0.0999001 −0.00498256
\(403\) −51.3775 −2.55930
\(404\) 18.7220 0.931454
\(405\) 0 0
\(406\) 26.1386 1.29724
\(407\) −13.5324 −0.670777
\(408\) 1.65378 0.0818741
\(409\) −27.8011 −1.37468 −0.687339 0.726337i \(-0.741221\pi\)
−0.687339 + 0.726337i \(0.741221\pi\)
\(410\) 0 0
\(411\) 1.06523 0.0525438
\(412\) 25.5341 1.25798
\(413\) −19.2541 −0.947433
\(414\) 11.0255 0.541873
\(415\) 0 0
\(416\) −15.5774 −0.763745
\(417\) −3.77286 −0.184758
\(418\) 69.2605 3.38764
\(419\) −0.119817 −0.00585344 −0.00292672 0.999996i \(-0.500932\pi\)
−0.00292672 + 0.999996i \(0.500932\pi\)
\(420\) 0 0
\(421\) −0.0393633 −0.00191845 −0.000959225 1.00000i \(-0.500305\pi\)
−0.000959225 1.00000i \(0.500305\pi\)
\(422\) 37.6056 1.83061
\(423\) −11.0609 −0.537797
\(424\) 10.9495 0.531756
\(425\) 0 0
\(426\) −38.6872 −1.87440
\(427\) 37.2687 1.80356
\(428\) −63.8848 −3.08799
\(429\) −25.5663 −1.23435
\(430\) 0 0
\(431\) 16.3869 0.789328 0.394664 0.918825i \(-0.370861\pi\)
0.394664 + 0.918825i \(0.370861\pi\)
\(432\) 2.28286 0.109834
\(433\) 17.5610 0.843930 0.421965 0.906612i \(-0.361340\pi\)
0.421965 + 0.906612i \(0.361340\pi\)
\(434\) −69.4767 −3.33499
\(435\) 0 0
\(436\) −24.2056 −1.15924
\(437\) −30.6882 −1.46801
\(438\) 12.7905 0.611156
\(439\) −31.1577 −1.48708 −0.743538 0.668694i \(-0.766854\pi\)
−0.743538 + 0.668694i \(0.766854\pi\)
\(440\) 0 0
\(441\) 4.00057 0.190503
\(442\) 5.70026 0.271134
\(443\) 13.1787 0.626141 0.313071 0.949730i \(-0.398642\pi\)
0.313071 + 0.949730i \(0.398642\pi\)
\(444\) 11.4626 0.543991
\(445\) 0 0
\(446\) −5.43555 −0.257381
\(447\) −7.88089 −0.372753
\(448\) −36.2081 −1.71067
\(449\) −29.2119 −1.37859 −0.689297 0.724479i \(-0.742080\pi\)
−0.689297 + 0.724479i \(0.742080\pi\)
\(450\) 0 0
\(451\) −12.6346 −0.594939
\(452\) 32.0673 1.50832
\(453\) 7.46432 0.350704
\(454\) −48.3822 −2.27069
\(455\) 0 0
\(456\) −26.9437 −1.26176
\(457\) −35.8001 −1.67466 −0.837330 0.546698i \(-0.815885\pi\)
−0.837330 + 0.546698i \(0.815885\pi\)
\(458\) 46.8745 2.19030
\(459\) 0.407830 0.0190359
\(460\) 0 0
\(461\) 6.01315 0.280060 0.140030 0.990147i \(-0.455280\pi\)
0.140030 + 0.990147i \(0.455280\pi\)
\(462\) −34.5727 −1.60847
\(463\) 41.2139 1.91537 0.957687 0.287813i \(-0.0929281\pi\)
0.957687 + 0.287813i \(0.0929281\pi\)
\(464\) 7.53644 0.349871
\(465\) 0 0
\(466\) −40.9328 −1.89617
\(467\) 29.9663 1.38667 0.693337 0.720614i \(-0.256140\pi\)
0.693337 + 0.720614i \(0.256140\pi\)
\(468\) 21.6558 1.00104
\(469\) −0.138799 −0.00640914
\(470\) 0 0
\(471\) 0.268958 0.0123929
\(472\) −23.5404 −1.08353
\(473\) −5.81928 −0.267571
\(474\) −8.48336 −0.389653
\(475\) 0 0
\(476\) 5.00303 0.229313
\(477\) 2.70021 0.123634
\(478\) 51.9582 2.37651
\(479\) 8.33937 0.381036 0.190518 0.981684i \(-0.438983\pi\)
0.190518 + 0.981684i \(0.438983\pi\)
\(480\) 0 0
\(481\) 18.1454 0.827357
\(482\) 9.31165 0.424134
\(483\) 15.3186 0.697019
\(484\) 29.8363 1.35620
\(485\) 0 0
\(486\) 2.38719 0.108285
\(487\) 4.91110 0.222543 0.111272 0.993790i \(-0.464508\pi\)
0.111272 + 0.993790i \(0.464508\pi\)
\(488\) 45.5653 2.06264
\(489\) 10.3496 0.468026
\(490\) 0 0
\(491\) 4.33484 0.195629 0.0978144 0.995205i \(-0.468815\pi\)
0.0978144 + 0.995205i \(0.468815\pi\)
\(492\) 10.7021 0.482488
\(493\) 1.34638 0.0606378
\(494\) −92.8701 −4.17842
\(495\) 0 0
\(496\) −20.0319 −0.899461
\(497\) −53.7511 −2.41107
\(498\) 11.9109 0.533742
\(499\) 36.1803 1.61965 0.809826 0.586671i \(-0.199562\pi\)
0.809826 + 0.586671i \(0.199562\pi\)
\(500\) 0 0
\(501\) −4.26896 −0.190723
\(502\) −52.1336 −2.32684
\(503\) 37.3990 1.66754 0.833769 0.552113i \(-0.186178\pi\)
0.833769 + 0.552113i \(0.186178\pi\)
\(504\) 13.4495 0.599087
\(505\) 0 0
\(506\) −48.1433 −2.14023
\(507\) 21.2813 0.945135
\(508\) −12.0057 −0.532667
\(509\) −26.2959 −1.16555 −0.582773 0.812635i \(-0.698032\pi\)
−0.582773 + 0.812635i \(0.698032\pi\)
\(510\) 0 0
\(511\) 17.7709 0.786138
\(512\) −24.5878 −1.08664
\(513\) −6.64447 −0.293360
\(514\) 40.9291 1.80530
\(515\) 0 0
\(516\) 4.92921 0.216996
\(517\) 48.2978 2.12413
\(518\) 24.5376 1.07812
\(519\) −23.5031 −1.03167
\(520\) 0 0
\(521\) 7.02231 0.307653 0.153826 0.988098i \(-0.450840\pi\)
0.153826 + 0.988098i \(0.450840\pi\)
\(522\) 7.88089 0.344937
\(523\) −13.0796 −0.571933 −0.285967 0.958240i \(-0.592315\pi\)
−0.285967 + 0.958240i \(0.592315\pi\)
\(524\) −16.2813 −0.711252
\(525\) 0 0
\(526\) 42.0780 1.83469
\(527\) −3.57869 −0.155890
\(528\) −9.96820 −0.433810
\(529\) −1.66851 −0.0725437
\(530\) 0 0
\(531\) −5.80518 −0.251923
\(532\) −81.5106 −3.53393
\(533\) 16.9415 0.733817
\(534\) −4.63175 −0.200436
\(535\) 0 0
\(536\) −0.169698 −0.00732984
\(537\) 19.6897 0.849675
\(538\) 4.64958 0.200458
\(539\) −17.4687 −0.752430
\(540\) 0 0
\(541\) −2.08066 −0.0894546 −0.0447273 0.998999i \(-0.514242\pi\)
−0.0447273 + 0.998999i \(0.514242\pi\)
\(542\) −33.1119 −1.42228
\(543\) −12.8221 −0.550249
\(544\) −1.08504 −0.0465206
\(545\) 0 0
\(546\) 46.3578 1.98393
\(547\) 36.9553 1.58009 0.790046 0.613047i \(-0.210056\pi\)
0.790046 + 0.613047i \(0.210056\pi\)
\(548\) 3.93993 0.168305
\(549\) 11.2366 0.479568
\(550\) 0 0
\(551\) −21.9355 −0.934485
\(552\) 18.7287 0.797148
\(553\) −11.7866 −0.501217
\(554\) −63.2161 −2.68579
\(555\) 0 0
\(556\) −13.9546 −0.591807
\(557\) 2.67455 0.113324 0.0566621 0.998393i \(-0.481954\pi\)
0.0566621 + 0.998393i \(0.481954\pi\)
\(558\) −20.9475 −0.886777
\(559\) 7.80296 0.330030
\(560\) 0 0
\(561\) −1.78081 −0.0751858
\(562\) −51.5754 −2.17557
\(563\) −13.8746 −0.584745 −0.292372 0.956305i \(-0.594445\pi\)
−0.292372 + 0.956305i \(0.594445\pi\)
\(564\) −40.9105 −1.72264
\(565\) 0 0
\(566\) −4.90058 −0.205987
\(567\) 3.31671 0.139289
\(568\) −65.7170 −2.75742
\(569\) 33.3404 1.39770 0.698852 0.715266i \(-0.253695\pi\)
0.698852 + 0.715266i \(0.253695\pi\)
\(570\) 0 0
\(571\) 35.3794 1.48058 0.740291 0.672287i \(-0.234688\pi\)
0.740291 + 0.672287i \(0.234688\pi\)
\(572\) −94.5613 −3.95381
\(573\) −19.0583 −0.796174
\(574\) 22.9096 0.956227
\(575\) 0 0
\(576\) −10.9169 −0.454870
\(577\) −16.6041 −0.691239 −0.345620 0.938375i \(-0.612331\pi\)
−0.345620 + 0.938375i \(0.612331\pi\)
\(578\) −40.1852 −1.67148
\(579\) 16.9842 0.705839
\(580\) 0 0
\(581\) 16.5488 0.686560
\(582\) 14.3054 0.592979
\(583\) −11.7906 −0.488317
\(584\) 21.7270 0.899069
\(585\) 0 0
\(586\) 28.5955 1.18127
\(587\) 26.9696 1.11315 0.556576 0.830796i \(-0.312115\pi\)
0.556576 + 0.830796i \(0.312115\pi\)
\(588\) 14.7968 0.610210
\(589\) 58.3049 2.40241
\(590\) 0 0
\(591\) −0.469462 −0.0193111
\(592\) 7.07482 0.290773
\(593\) −12.7895 −0.525201 −0.262600 0.964905i \(-0.584580\pi\)
−0.262600 + 0.964905i \(0.584580\pi\)
\(594\) −10.4238 −0.427693
\(595\) 0 0
\(596\) −29.1489 −1.19398
\(597\) 7.02491 0.287511
\(598\) 64.5545 2.63983
\(599\) 31.1978 1.27471 0.637354 0.770571i \(-0.280029\pi\)
0.637354 + 0.770571i \(0.280029\pi\)
\(600\) 0 0
\(601\) 19.2310 0.784450 0.392225 0.919869i \(-0.371705\pi\)
0.392225 + 0.919869i \(0.371705\pi\)
\(602\) 10.5518 0.430058
\(603\) −0.0418484 −0.00170420
\(604\) 27.6081 1.12336
\(605\) 0 0
\(606\) 12.0835 0.490858
\(607\) 8.01104 0.325158 0.162579 0.986696i \(-0.448019\pi\)
0.162579 + 0.986696i \(0.448019\pi\)
\(608\) 17.6777 0.716926
\(609\) 10.9495 0.443698
\(610\) 0 0
\(611\) −64.7616 −2.61997
\(612\) 1.50843 0.0609747
\(613\) −11.3203 −0.457224 −0.228612 0.973518i \(-0.573419\pi\)
−0.228612 + 0.973518i \(0.573419\pi\)
\(614\) 16.9006 0.682052
\(615\) 0 0
\(616\) −58.7278 −2.36621
\(617\) 9.71325 0.391041 0.195520 0.980700i \(-0.437360\pi\)
0.195520 + 0.980700i \(0.437360\pi\)
\(618\) 16.4802 0.662930
\(619\) 20.4652 0.822567 0.411284 0.911507i \(-0.365081\pi\)
0.411284 + 0.911507i \(0.365081\pi\)
\(620\) 0 0
\(621\) 4.61860 0.185338
\(622\) −27.2450 −1.09242
\(623\) −6.43526 −0.257823
\(624\) 13.3662 0.535075
\(625\) 0 0
\(626\) −28.4260 −1.13613
\(627\) 29.0134 1.15868
\(628\) 0.994789 0.0396964
\(629\) 1.26391 0.0503954
\(630\) 0 0
\(631\) 2.46405 0.0980924 0.0490462 0.998797i \(-0.484382\pi\)
0.0490462 + 0.998797i \(0.484382\pi\)
\(632\) −14.4105 −0.573218
\(633\) 15.7531 0.626128
\(634\) −30.4889 −1.21087
\(635\) 0 0
\(636\) 9.98721 0.396019
\(637\) 23.4234 0.928070
\(638\) −34.4123 −1.36240
\(639\) −16.2062 −0.641106
\(640\) 0 0
\(641\) 31.7217 1.25293 0.626465 0.779449i \(-0.284501\pi\)
0.626465 + 0.779449i \(0.284501\pi\)
\(642\) −41.2323 −1.62731
\(643\) 11.5574 0.455779 0.227889 0.973687i \(-0.426818\pi\)
0.227889 + 0.973687i \(0.426818\pi\)
\(644\) 56.6584 2.23266
\(645\) 0 0
\(646\) −6.46883 −0.254513
\(647\) −6.06222 −0.238330 −0.119165 0.992874i \(-0.538022\pi\)
−0.119165 + 0.992874i \(0.538022\pi\)
\(648\) 4.05506 0.159298
\(649\) 25.3486 0.995021
\(650\) 0 0
\(651\) −29.1040 −1.14067
\(652\) 38.2799 1.49916
\(653\) −26.3569 −1.03143 −0.515713 0.856762i \(-0.672473\pi\)
−0.515713 + 0.856762i \(0.672473\pi\)
\(654\) −15.6227 −0.610896
\(655\) 0 0
\(656\) 6.60543 0.257899
\(657\) 5.35799 0.209035
\(658\) −87.5756 −3.41405
\(659\) 16.2879 0.634487 0.317243 0.948344i \(-0.397243\pi\)
0.317243 + 0.948344i \(0.397243\pi\)
\(660\) 0 0
\(661\) −20.4308 −0.794665 −0.397332 0.917675i \(-0.630064\pi\)
−0.397332 + 0.917675i \(0.630064\pi\)
\(662\) 20.6740 0.803519
\(663\) 2.38785 0.0927365
\(664\) 20.2328 0.785186
\(665\) 0 0
\(666\) 7.39817 0.286673
\(667\) 15.2475 0.590386
\(668\) −15.7895 −0.610914
\(669\) −2.27697 −0.0880326
\(670\) 0 0
\(671\) −49.0653 −1.89415
\(672\) −8.82417 −0.340400
\(673\) 20.7479 0.799772 0.399886 0.916565i \(-0.369050\pi\)
0.399886 + 0.916565i \(0.369050\pi\)
\(674\) 36.4199 1.40284
\(675\) 0 0
\(676\) 78.7126 3.02741
\(677\) −20.1892 −0.775935 −0.387967 0.921673i \(-0.626823\pi\)
−0.387967 + 0.921673i \(0.626823\pi\)
\(678\) 20.6968 0.794855
\(679\) 19.8756 0.762757
\(680\) 0 0
\(681\) −20.2674 −0.776650
\(682\) 91.4682 3.50250
\(683\) 6.69431 0.256151 0.128075 0.991764i \(-0.459120\pi\)
0.128075 + 0.991764i \(0.459120\pi\)
\(684\) −24.5757 −0.939677
\(685\) 0 0
\(686\) −23.7484 −0.906716
\(687\) 19.6358 0.749154
\(688\) 3.04235 0.115989
\(689\) 15.8098 0.602306
\(690\) 0 0
\(691\) −7.86690 −0.299271 −0.149635 0.988741i \(-0.547810\pi\)
−0.149635 + 0.988741i \(0.547810\pi\)
\(692\) −86.9304 −3.30460
\(693\) −14.4826 −0.550148
\(694\) 52.1697 1.98034
\(695\) 0 0
\(696\) 13.3871 0.507436
\(697\) 1.18005 0.0446977
\(698\) 17.4173 0.659254
\(699\) −17.1468 −0.648553
\(700\) 0 0
\(701\) 11.5944 0.437916 0.218958 0.975734i \(-0.429734\pi\)
0.218958 + 0.975734i \(0.429734\pi\)
\(702\) 13.9771 0.527530
\(703\) −20.5919 −0.776640
\(704\) 47.6691 1.79660
\(705\) 0 0
\(706\) 72.8857 2.74309
\(707\) 16.7885 0.631398
\(708\) −21.4715 −0.806948
\(709\) 37.6176 1.41276 0.706379 0.707834i \(-0.250327\pi\)
0.706379 + 0.707834i \(0.250327\pi\)
\(710\) 0 0
\(711\) −3.55370 −0.133274
\(712\) −7.86785 −0.294860
\(713\) −40.5280 −1.51779
\(714\) 3.22904 0.120844
\(715\) 0 0
\(716\) 72.8260 2.72163
\(717\) 21.7654 0.812845
\(718\) 7.32263 0.273278
\(719\) −21.0855 −0.786355 −0.393177 0.919463i \(-0.628624\pi\)
−0.393177 + 0.919463i \(0.628624\pi\)
\(720\) 0 0
\(721\) 22.8972 0.852737
\(722\) 60.0353 2.23428
\(723\) 3.90067 0.145068
\(724\) −47.4248 −1.76253
\(725\) 0 0
\(726\) 19.2569 0.714690
\(727\) −26.1113 −0.968416 −0.484208 0.874953i \(-0.660892\pi\)
−0.484208 + 0.874953i \(0.660892\pi\)
\(728\) 78.7470 2.91856
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.543513 0.0201025
\(732\) 41.5607 1.53613
\(733\) −13.4461 −0.496642 −0.248321 0.968678i \(-0.579879\pi\)
−0.248321 + 0.968678i \(0.579879\pi\)
\(734\) 75.4436 2.78468
\(735\) 0 0
\(736\) −12.2879 −0.452937
\(737\) 0.182733 0.00673106
\(738\) 6.90732 0.254262
\(739\) −53.9014 −1.98279 −0.991397 0.130887i \(-0.958218\pi\)
−0.991397 + 0.130887i \(0.958218\pi\)
\(740\) 0 0
\(741\) −38.9035 −1.42916
\(742\) 21.3793 0.784857
\(743\) −4.33521 −0.159043 −0.0795216 0.996833i \(-0.525339\pi\)
−0.0795216 + 0.996833i \(0.525339\pi\)
\(744\) −35.5830 −1.30454
\(745\) 0 0
\(746\) 53.6888 1.96569
\(747\) 4.98952 0.182557
\(748\) −6.58664 −0.240831
\(749\) −57.2873 −2.09323
\(750\) 0 0
\(751\) −2.87653 −0.104966 −0.0524831 0.998622i \(-0.516714\pi\)
−0.0524831 + 0.998622i \(0.516714\pi\)
\(752\) −25.2503 −0.920785
\(753\) −21.8389 −0.795854
\(754\) 46.1428 1.68042
\(755\) 0 0
\(756\) 12.2674 0.446162
\(757\) 16.0480 0.583275 0.291637 0.956529i \(-0.405800\pi\)
0.291637 + 0.956529i \(0.405800\pi\)
\(758\) 72.0005 2.61518
\(759\) −20.1674 −0.732029
\(760\) 0 0
\(761\) −43.5182 −1.57753 −0.788767 0.614692i \(-0.789280\pi\)
−0.788767 + 0.614692i \(0.789280\pi\)
\(762\) −7.74869 −0.280706
\(763\) −21.7058 −0.785804
\(764\) −70.4907 −2.55026
\(765\) 0 0
\(766\) −40.0391 −1.44667
\(767\) −33.9895 −1.22729
\(768\) −27.6757 −0.998659
\(769\) −25.9575 −0.936052 −0.468026 0.883715i \(-0.655035\pi\)
−0.468026 + 0.883715i \(0.655035\pi\)
\(770\) 0 0
\(771\) 17.1453 0.617473
\(772\) 62.8191 2.26091
\(773\) 52.9965 1.90615 0.953076 0.302731i \(-0.0978984\pi\)
0.953076 + 0.302731i \(0.0978984\pi\)
\(774\) 3.18140 0.114353
\(775\) 0 0
\(776\) 24.3003 0.872329
\(777\) 10.2789 0.368752
\(778\) 53.6724 1.92425
\(779\) −19.2257 −0.688833
\(780\) 0 0
\(781\) 70.7650 2.53217
\(782\) 4.49652 0.160795
\(783\) 3.30132 0.117980
\(784\) 9.13272 0.326169
\(785\) 0 0
\(786\) −10.5082 −0.374816
\(787\) 13.1832 0.469929 0.234964 0.972004i \(-0.424503\pi\)
0.234964 + 0.972004i \(0.424503\pi\)
\(788\) −1.73639 −0.0618562
\(789\) 17.6266 0.627523
\(790\) 0 0
\(791\) 28.7557 1.02243
\(792\) −17.7066 −0.629178
\(793\) 65.7908 2.33630
\(794\) −8.73233 −0.309899
\(795\) 0 0
\(796\) 25.9829 0.920939
\(797\) −15.6197 −0.553280 −0.276640 0.960974i \(-0.589221\pi\)
−0.276640 + 0.960974i \(0.589221\pi\)
\(798\) −52.6084 −1.86232
\(799\) −4.51095 −0.159586
\(800\) 0 0
\(801\) −1.94025 −0.0685555
\(802\) −60.5806 −2.13918
\(803\) −23.3959 −0.825625
\(804\) −0.154784 −0.00545880
\(805\) 0 0
\(806\) −122.648 −4.32009
\(807\) 1.94772 0.0685630
\(808\) 20.5259 0.722100
\(809\) 11.7683 0.413750 0.206875 0.978367i \(-0.433671\pi\)
0.206875 + 0.978367i \(0.433671\pi\)
\(810\) 0 0
\(811\) −38.3050 −1.34507 −0.672536 0.740065i \(-0.734795\pi\)
−0.672536 + 0.740065i \(0.734795\pi\)
\(812\) 40.4988 1.42123
\(813\) −13.8707 −0.486466
\(814\) −32.3045 −1.13227
\(815\) 0 0
\(816\) 0.931016 0.0325921
\(817\) −8.85505 −0.309799
\(818\) −66.3666 −2.32045
\(819\) 19.4194 0.678569
\(820\) 0 0
\(821\) 22.4713 0.784254 0.392127 0.919911i \(-0.371739\pi\)
0.392127 + 0.919911i \(0.371739\pi\)
\(822\) 2.54290 0.0886938
\(823\) −19.8645 −0.692434 −0.346217 0.938155i \(-0.612534\pi\)
−0.346217 + 0.938155i \(0.612534\pi\)
\(824\) 27.9945 0.975235
\(825\) 0 0
\(826\) −45.9632 −1.59927
\(827\) −21.4039 −0.744288 −0.372144 0.928175i \(-0.621377\pi\)
−0.372144 + 0.928175i \(0.621377\pi\)
\(828\) 17.0827 0.593666
\(829\) −24.6676 −0.856743 −0.428371 0.903603i \(-0.640912\pi\)
−0.428371 + 0.903603i \(0.640912\pi\)
\(830\) 0 0
\(831\) −26.4814 −0.918629
\(832\) −63.9185 −2.21598
\(833\) 1.63155 0.0565299
\(834\) −9.00654 −0.311871
\(835\) 0 0
\(836\) 107.311 3.71143
\(837\) −8.77495 −0.303307
\(838\) −0.286026 −0.00988060
\(839\) −4.60054 −0.158828 −0.0794142 0.996842i \(-0.525305\pi\)
−0.0794142 + 0.996842i \(0.525305\pi\)
\(840\) 0 0
\(841\) −18.1013 −0.624182
\(842\) −0.0939677 −0.00323834
\(843\) −21.6050 −0.744117
\(844\) 58.2655 2.00558
\(845\) 0 0
\(846\) −26.4044 −0.907801
\(847\) 26.7551 0.919317
\(848\) 6.16419 0.211679
\(849\) −2.05287 −0.0704542
\(850\) 0 0
\(851\) 14.3136 0.490663
\(852\) −59.9413 −2.05356
\(853\) 27.1689 0.930245 0.465122 0.885246i \(-0.346010\pi\)
0.465122 + 0.885246i \(0.346010\pi\)
\(854\) 88.9674 3.04440
\(855\) 0 0
\(856\) −70.0404 −2.39393
\(857\) −7.66692 −0.261897 −0.130949 0.991389i \(-0.541802\pi\)
−0.130949 + 0.991389i \(0.541802\pi\)
\(858\) −61.0315 −2.08358
\(859\) 9.09944 0.310469 0.155234 0.987878i \(-0.450387\pi\)
0.155234 + 0.987878i \(0.450387\pi\)
\(860\) 0 0
\(861\) 9.59688 0.327061
\(862\) 39.1186 1.33238
\(863\) −21.7471 −0.740279 −0.370140 0.928976i \(-0.620690\pi\)
−0.370140 + 0.928976i \(0.620690\pi\)
\(864\) −2.66052 −0.0905127
\(865\) 0 0
\(866\) 41.9216 1.42455
\(867\) −16.8337 −0.571702
\(868\) −107.646 −3.65375
\(869\) 15.5174 0.526392
\(870\) 0 0
\(871\) −0.245023 −0.00830230
\(872\) −26.5379 −0.898687
\(873\) 5.99258 0.202818
\(874\) −73.2585 −2.47800
\(875\) 0 0
\(876\) 19.8175 0.669570
\(877\) 20.9781 0.708379 0.354190 0.935174i \(-0.384757\pi\)
0.354190 + 0.935174i \(0.384757\pi\)
\(878\) −74.3794 −2.51018
\(879\) 11.9787 0.404032
\(880\) 0 0
\(881\) 40.6001 1.36785 0.683927 0.729551i \(-0.260271\pi\)
0.683927 + 0.729551i \(0.260271\pi\)
\(882\) 9.55012 0.321569
\(883\) 44.2689 1.48977 0.744883 0.667195i \(-0.232505\pi\)
0.744883 + 0.667195i \(0.232505\pi\)
\(884\) 8.83189 0.297049
\(885\) 0 0
\(886\) 31.4602 1.05693
\(887\) −21.2137 −0.712285 −0.356142 0.934432i \(-0.615908\pi\)
−0.356142 + 0.934432i \(0.615908\pi\)
\(888\) 12.5671 0.421724
\(889\) −10.7659 −0.361076
\(890\) 0 0
\(891\) −4.36655 −0.146285
\(892\) −8.42177 −0.281982
\(893\) 73.4935 2.45937
\(894\) −18.8132 −0.629207
\(895\) 0 0
\(896\) −68.7873 −2.29802
\(897\) 27.0420 0.902907
\(898\) −69.7343 −2.32706
\(899\) −28.9689 −0.966168
\(900\) 0 0
\(901\) 1.10123 0.0366872
\(902\) −30.1612 −1.00426
\(903\) 4.42016 0.147094
\(904\) 35.1571 1.16931
\(905\) 0 0
\(906\) 17.8187 0.591988
\(907\) 42.8598 1.42314 0.711568 0.702617i \(-0.247985\pi\)
0.711568 + 0.702617i \(0.247985\pi\)
\(908\) −74.9627 −2.48772
\(909\) 5.06181 0.167890
\(910\) 0 0
\(911\) 52.6870 1.74560 0.872800 0.488078i \(-0.162302\pi\)
0.872800 + 0.488078i \(0.162302\pi\)
\(912\) −15.1684 −0.502274
\(913\) −21.7870 −0.721045
\(914\) −85.4617 −2.82682
\(915\) 0 0
\(916\) 72.6266 2.39965
\(917\) −14.5999 −0.482131
\(918\) 0.973567 0.0321325
\(919\) −17.5655 −0.579434 −0.289717 0.957112i \(-0.593561\pi\)
−0.289717 + 0.957112i \(0.593561\pi\)
\(920\) 0 0
\(921\) 7.07969 0.233284
\(922\) 14.3545 0.472741
\(923\) −94.8874 −3.12326
\(924\) −53.5664 −1.76221
\(925\) 0 0
\(926\) 98.3855 3.23315
\(927\) 6.90359 0.226744
\(928\) −8.78323 −0.288324
\(929\) −58.2546 −1.91127 −0.955636 0.294550i \(-0.904830\pi\)
−0.955636 + 0.294550i \(0.904830\pi\)
\(930\) 0 0
\(931\) −26.5816 −0.871178
\(932\) −63.4206 −2.07741
\(933\) −11.4130 −0.373644
\(934\) 71.5352 2.34070
\(935\) 0 0
\(936\) 23.7425 0.776048
\(937\) −21.6766 −0.708142 −0.354071 0.935218i \(-0.615203\pi\)
−0.354071 + 0.935218i \(0.615203\pi\)
\(938\) −0.331340 −0.0108186
\(939\) −11.9077 −0.388594
\(940\) 0 0
\(941\) −41.9457 −1.36739 −0.683695 0.729768i \(-0.739628\pi\)
−0.683695 + 0.729768i \(0.739628\pi\)
\(942\) 0.642054 0.0209193
\(943\) 13.3639 0.435188
\(944\) −13.2524 −0.431329
\(945\) 0 0
\(946\) −13.8917 −0.451659
\(947\) 40.0301 1.30080 0.650401 0.759591i \(-0.274601\pi\)
0.650401 + 0.759591i \(0.274601\pi\)
\(948\) −13.1440 −0.426897
\(949\) 31.3712 1.01835
\(950\) 0 0
\(951\) −12.7719 −0.414156
\(952\) 5.48510 0.177773
\(953\) 33.1969 1.07535 0.537677 0.843151i \(-0.319302\pi\)
0.537677 + 0.843151i \(0.319302\pi\)
\(954\) 6.44592 0.208694
\(955\) 0 0
\(956\) 80.5033 2.60366
\(957\) −14.4154 −0.465984
\(958\) 19.9077 0.643187
\(959\) 3.53305 0.114088
\(960\) 0 0
\(961\) 45.9997 1.48386
\(962\) 43.3164 1.39658
\(963\) −17.2723 −0.556593
\(964\) 14.4273 0.464673
\(965\) 0 0
\(966\) 36.5683 1.17657
\(967\) −20.5477 −0.660770 −0.330385 0.943846i \(-0.607179\pi\)
−0.330385 + 0.943846i \(0.607179\pi\)
\(968\) 32.7112 1.05138
\(969\) −2.70981 −0.0870517
\(970\) 0 0
\(971\) 20.4201 0.655312 0.327656 0.944797i \(-0.393741\pi\)
0.327656 + 0.944797i \(0.393741\pi\)
\(972\) 3.69868 0.118635
\(973\) −12.5135 −0.401164
\(974\) 11.7237 0.375653
\(975\) 0 0
\(976\) 25.6516 0.821088
\(977\) 27.7767 0.888655 0.444327 0.895864i \(-0.353443\pi\)
0.444327 + 0.895864i \(0.353443\pi\)
\(978\) 24.7065 0.790027
\(979\) 8.47221 0.270773
\(980\) 0 0
\(981\) −6.54439 −0.208946
\(982\) 10.3481 0.330221
\(983\) 15.7171 0.501298 0.250649 0.968078i \(-0.419356\pi\)
0.250649 + 0.968078i \(0.419356\pi\)
\(984\) 11.7333 0.374044
\(985\) 0 0
\(986\) 3.21406 0.102357
\(987\) −36.6857 −1.16772
\(988\) −143.891 −4.57780
\(989\) 6.15519 0.195724
\(990\) 0 0
\(991\) −6.18035 −0.196325 −0.0981627 0.995170i \(-0.531297\pi\)
−0.0981627 + 0.995170i \(0.531297\pi\)
\(992\) 23.3459 0.741233
\(993\) 8.66041 0.274830
\(994\) −128.314 −4.06988
\(995\) 0 0
\(996\) 18.4546 0.584757
\(997\) 60.9412 1.93003 0.965014 0.262200i \(-0.0844479\pi\)
0.965014 + 0.262200i \(0.0844479\pi\)
\(998\) 86.3692 2.73397
\(999\) 3.09911 0.0980515
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 1875.2.a.i.1.6 6
3.2 odd 2 5625.2.a.r.1.1 6
5.2 odd 4 1875.2.b.e.1249.11 12
5.3 odd 4 1875.2.b.e.1249.2 12
5.4 even 2 1875.2.a.l.1.1 yes 6
15.14 odd 2 5625.2.a.o.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.i.1.6 6 1.1 even 1 trivial
1875.2.a.l.1.1 yes 6 5.4 even 2
1875.2.b.e.1249.2 12 5.3 odd 4
1875.2.b.e.1249.11 12 5.2 odd 4
5625.2.a.o.1.6 6 15.14 odd 2
5625.2.a.r.1.1 6 3.2 odd 2