Properties

Label 736.4.a.e.1.2
Level $736$
Weight $4$
Character 736.1
Self dual yes
Analytic conductor $43.425$
Analytic rank $1$
Dimension $8$
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] = [736,4,Mod(1,736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("736.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 736.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: \(43.4254057642\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 137x^{6} + 344x^{5} + 6175x^{4} - 7924x^{3} - 89643x^{2} + 45072x + 51084 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-6.05402\) of defining polynomial
Character \(\chi\) \(=\) 736.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.05402 q^{3} +11.6466 q^{5} -24.4224 q^{7} +37.8672 q^{9} -18.4129 q^{11} +51.7005 q^{13} -93.8018 q^{15} -10.4582 q^{17} +22.3482 q^{19} +196.698 q^{21} +23.0000 q^{23} +10.6430 q^{25} -87.5244 q^{27} -167.380 q^{29} +176.061 q^{31} +148.298 q^{33} -284.438 q^{35} -35.8712 q^{37} -416.397 q^{39} +412.361 q^{41} +192.021 q^{43} +441.023 q^{45} +591.953 q^{47} +253.454 q^{49} +84.2303 q^{51} -750.312 q^{53} -214.447 q^{55} -179.993 q^{57} -198.433 q^{59} -264.657 q^{61} -924.807 q^{63} +602.135 q^{65} +155.057 q^{67} -185.242 q^{69} -1082.55 q^{71} -224.277 q^{73} -85.7190 q^{75} +449.687 q^{77} -261.911 q^{79} -317.491 q^{81} +15.8477 q^{83} -121.802 q^{85} +1348.08 q^{87} +200.192 q^{89} -1262.65 q^{91} -1417.99 q^{93} +260.281 q^{95} -113.155 q^{97} -697.244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} - 12 q^{5} + 14 q^{7} + 90 q^{9} - 88 q^{11} - 30 q^{13} - 30 q^{15} + 58 q^{17} - 190 q^{19} - 66 q^{21} + 184 q^{23} + 28 q^{25} - 432 q^{27} + 190 q^{29} + 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −8.05402 −1.55000 −0.774998 0.631964i \(-0.782249\pi\)
−0.774998 + 0.631964i \(0.782249\pi\)
\(4\) 0 0
\(5\) 11.6466 1.04170 0.520851 0.853647i \(-0.325615\pi\)
0.520851 + 0.853647i \(0.325615\pi\)
\(6\) 0 0
\(7\) −24.4224 −1.31869 −0.659343 0.751842i \(-0.729165\pi\)
−0.659343 + 0.751842i \(0.729165\pi\)
\(8\) 0 0
\(9\) 37.8672 1.40249
\(10\) 0 0
\(11\) −18.4129 −0.504700 −0.252350 0.967636i \(-0.581203\pi\)
−0.252350 + 0.967636i \(0.581203\pi\)
\(12\) 0 0
\(13\) 51.7005 1.10301 0.551506 0.834171i \(-0.314053\pi\)
0.551506 + 0.834171i \(0.314053\pi\)
\(14\) 0 0
\(15\) −93.8018 −1.61463
\(16\) 0 0
\(17\) −10.4582 −0.149205 −0.0746023 0.997213i \(-0.523769\pi\)
−0.0746023 + 0.997213i \(0.523769\pi\)
\(18\) 0 0
\(19\) 22.3482 0.269844 0.134922 0.990856i \(-0.456922\pi\)
0.134922 + 0.990856i \(0.456922\pi\)
\(20\) 0 0
\(21\) 196.698 2.04396
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 10.6430 0.0851441
\(26\) 0 0
\(27\) −87.5244 −0.623854
\(28\) 0 0
\(29\) −167.380 −1.07178 −0.535891 0.844287i \(-0.680024\pi\)
−0.535891 + 0.844287i \(0.680024\pi\)
\(30\) 0 0
\(31\) 176.061 1.02005 0.510023 0.860161i \(-0.329637\pi\)
0.510023 + 0.860161i \(0.329637\pi\)
\(32\) 0 0
\(33\) 148.298 0.782283
\(34\) 0 0
\(35\) −284.438 −1.37368
\(36\) 0 0
\(37\) −35.8712 −0.159384 −0.0796918 0.996820i \(-0.525394\pi\)
−0.0796918 + 0.996820i \(0.525394\pi\)
\(38\) 0 0
\(39\) −416.397 −1.70966
\(40\) 0 0
\(41\) 412.361 1.57073 0.785365 0.619033i \(-0.212475\pi\)
0.785365 + 0.619033i \(0.212475\pi\)
\(42\) 0 0
\(43\) 192.021 0.680999 0.340500 0.940245i \(-0.389404\pi\)
0.340500 + 0.940245i \(0.389404\pi\)
\(44\) 0 0
\(45\) 441.023 1.46097
\(46\) 0 0
\(47\) 591.953 1.83713 0.918567 0.395266i \(-0.129348\pi\)
0.918567 + 0.395266i \(0.129348\pi\)
\(48\) 0 0
\(49\) 253.454 0.738932
\(50\) 0 0
\(51\) 84.2303 0.231267
\(52\) 0 0
\(53\) −750.312 −1.94459 −0.972295 0.233756i \(-0.924898\pi\)
−0.972295 + 0.233756i \(0.924898\pi\)
\(54\) 0 0
\(55\) −214.447 −0.525747
\(56\) 0 0
\(57\) −179.993 −0.418257
\(58\) 0 0
\(59\) −198.433 −0.437860 −0.218930 0.975741i \(-0.570257\pi\)
−0.218930 + 0.975741i \(0.570257\pi\)
\(60\) 0 0
\(61\) −264.657 −0.555506 −0.277753 0.960653i \(-0.589590\pi\)
−0.277753 + 0.960653i \(0.589590\pi\)
\(62\) 0 0
\(63\) −924.807 −1.84944
\(64\) 0 0
\(65\) 602.135 1.14901
\(66\) 0 0
\(67\) 155.057 0.282734 0.141367 0.989957i \(-0.454850\pi\)
0.141367 + 0.989957i \(0.454850\pi\)
\(68\) 0 0
\(69\) −185.242 −0.323197
\(70\) 0 0
\(71\) −1082.55 −1.80952 −0.904758 0.425926i \(-0.859948\pi\)
−0.904758 + 0.425926i \(0.859948\pi\)
\(72\) 0 0
\(73\) −224.277 −0.359585 −0.179792 0.983705i \(-0.557543\pi\)
−0.179792 + 0.983705i \(0.557543\pi\)
\(74\) 0 0
\(75\) −85.7190 −0.131973
\(76\) 0 0
\(77\) 449.687 0.665541
\(78\) 0 0
\(79\) −261.911 −0.373004 −0.186502 0.982455i \(-0.559715\pi\)
−0.186502 + 0.982455i \(0.559715\pi\)
\(80\) 0 0
\(81\) −317.491 −0.435516
\(82\) 0 0
\(83\) 15.8477 0.0209579 0.0104790 0.999945i \(-0.496664\pi\)
0.0104790 + 0.999945i \(0.496664\pi\)
\(84\) 0 0
\(85\) −121.802 −0.155427
\(86\) 0 0
\(87\) 1348.08 1.66126
\(88\) 0 0
\(89\) 200.192 0.238431 0.119215 0.992868i \(-0.461962\pi\)
0.119215 + 0.992868i \(0.461962\pi\)
\(90\) 0 0
\(91\) −1262.65 −1.45453
\(92\) 0 0
\(93\) −1417.99 −1.58107
\(94\) 0 0
\(95\) 260.281 0.281097
\(96\) 0 0
\(97\) −113.155 −0.118445 −0.0592227 0.998245i \(-0.518862\pi\)
−0.0592227 + 0.998245i \(0.518862\pi\)
\(98\) 0 0
\(99\) −697.244 −0.707835
\(100\) 0 0
\(101\) −950.072 −0.935997 −0.467999 0.883729i \(-0.655025\pi\)
−0.467999 + 0.883729i \(0.655025\pi\)
\(102\) 0 0
\(103\) −385.330 −0.368618 −0.184309 0.982868i \(-0.559005\pi\)
−0.184309 + 0.982868i \(0.559005\pi\)
\(104\) 0 0
\(105\) 2290.87 2.12920
\(106\) 0 0
\(107\) −1014.00 −0.916139 −0.458070 0.888916i \(-0.651459\pi\)
−0.458070 + 0.888916i \(0.651459\pi\)
\(108\) 0 0
\(109\) −2065.44 −1.81499 −0.907493 0.420067i \(-0.862006\pi\)
−0.907493 + 0.420067i \(0.862006\pi\)
\(110\) 0 0
\(111\) 288.908 0.247044
\(112\) 0 0
\(113\) 1217.12 1.01325 0.506625 0.862166i \(-0.330893\pi\)
0.506625 + 0.862166i \(0.330893\pi\)
\(114\) 0 0
\(115\) 267.872 0.217210
\(116\) 0 0
\(117\) 1957.75 1.54696
\(118\) 0 0
\(119\) 255.414 0.196754
\(120\) 0 0
\(121\) −991.965 −0.745278
\(122\) 0 0
\(123\) −3321.16 −2.43462
\(124\) 0 0
\(125\) −1331.87 −0.953008
\(126\) 0 0
\(127\) 138.778 0.0969652 0.0484826 0.998824i \(-0.484561\pi\)
0.0484826 + 0.998824i \(0.484561\pi\)
\(128\) 0 0
\(129\) −1546.54 −1.05555
\(130\) 0 0
\(131\) −1370.77 −0.914234 −0.457117 0.889406i \(-0.651118\pi\)
−0.457117 + 0.889406i \(0.651118\pi\)
\(132\) 0 0
\(133\) −545.798 −0.355840
\(134\) 0 0
\(135\) −1019.36 −0.649871
\(136\) 0 0
\(137\) −1687.59 −1.05242 −0.526208 0.850356i \(-0.676387\pi\)
−0.526208 + 0.850356i \(0.676387\pi\)
\(138\) 0 0
\(139\) −2507.17 −1.52989 −0.764947 0.644093i \(-0.777235\pi\)
−0.764947 + 0.644093i \(0.777235\pi\)
\(140\) 0 0
\(141\) −4767.60 −2.84755
\(142\) 0 0
\(143\) −951.957 −0.556690
\(144\) 0 0
\(145\) −1949.40 −1.11648
\(146\) 0 0
\(147\) −2041.32 −1.14534
\(148\) 0 0
\(149\) 3399.54 1.86913 0.934567 0.355787i \(-0.115787\pi\)
0.934567 + 0.355787i \(0.115787\pi\)
\(150\) 0 0
\(151\) −985.734 −0.531244 −0.265622 0.964077i \(-0.585577\pi\)
−0.265622 + 0.964077i \(0.585577\pi\)
\(152\) 0 0
\(153\) −396.021 −0.209258
\(154\) 0 0
\(155\) 2050.50 1.06258
\(156\) 0 0
\(157\) 201.810 0.102587 0.0512936 0.998684i \(-0.483666\pi\)
0.0512936 + 0.998684i \(0.483666\pi\)
\(158\) 0 0
\(159\) 6043.03 3.01411
\(160\) 0 0
\(161\) −561.715 −0.274965
\(162\) 0 0
\(163\) −3446.39 −1.65609 −0.828043 0.560664i \(-0.810546\pi\)
−0.828043 + 0.560664i \(0.810546\pi\)
\(164\) 0 0
\(165\) 1727.16 0.814906
\(166\) 0 0
\(167\) 971.671 0.450240 0.225120 0.974331i \(-0.427723\pi\)
0.225120 + 0.974331i \(0.427723\pi\)
\(168\) 0 0
\(169\) 475.945 0.216634
\(170\) 0 0
\(171\) 846.265 0.378453
\(172\) 0 0
\(173\) 1589.84 0.698689 0.349345 0.936994i \(-0.386404\pi\)
0.349345 + 0.936994i \(0.386404\pi\)
\(174\) 0 0
\(175\) −259.928 −0.112278
\(176\) 0 0
\(177\) 1598.18 0.678681
\(178\) 0 0
\(179\) −2393.41 −0.999395 −0.499697 0.866200i \(-0.666555\pi\)
−0.499697 + 0.866200i \(0.666555\pi\)
\(180\) 0 0
\(181\) 4104.06 1.68537 0.842687 0.538404i \(-0.180973\pi\)
0.842687 + 0.538404i \(0.180973\pi\)
\(182\) 0 0
\(183\) 2131.55 0.861032
\(184\) 0 0
\(185\) −417.778 −0.166030
\(186\) 0 0
\(187\) 192.565 0.0753036
\(188\) 0 0
\(189\) 2137.55 0.822668
\(190\) 0 0
\(191\) −320.971 −0.121595 −0.0607976 0.998150i \(-0.519364\pi\)
−0.0607976 + 0.998150i \(0.519364\pi\)
\(192\) 0 0
\(193\) −688.697 −0.256858 −0.128429 0.991719i \(-0.540993\pi\)
−0.128429 + 0.991719i \(0.540993\pi\)
\(194\) 0 0
\(195\) −4849.60 −1.78096
\(196\) 0 0
\(197\) −1389.43 −0.502501 −0.251250 0.967922i \(-0.580842\pi\)
−0.251250 + 0.967922i \(0.580842\pi\)
\(198\) 0 0
\(199\) −2153.82 −0.767236 −0.383618 0.923492i \(-0.625322\pi\)
−0.383618 + 0.923492i \(0.625322\pi\)
\(200\) 0 0
\(201\) −1248.83 −0.438237
\(202\) 0 0
\(203\) 4087.82 1.41334
\(204\) 0 0
\(205\) 4802.59 1.63623
\(206\) 0 0
\(207\) 870.945 0.292439
\(208\) 0 0
\(209\) −411.496 −0.136190
\(210\) 0 0
\(211\) −2265.82 −0.739267 −0.369634 0.929178i \(-0.620517\pi\)
−0.369634 + 0.929178i \(0.620517\pi\)
\(212\) 0 0
\(213\) 8718.91 2.80474
\(214\) 0 0
\(215\) 2236.39 0.709399
\(216\) 0 0
\(217\) −4299.82 −1.34512
\(218\) 0 0
\(219\) 1806.33 0.557355
\(220\) 0 0
\(221\) −540.693 −0.164574
\(222\) 0 0
\(223\) 3832.94 1.15100 0.575499 0.817803i \(-0.304808\pi\)
0.575499 + 0.817803i \(0.304808\pi\)
\(224\) 0 0
\(225\) 403.021 0.119414
\(226\) 0 0
\(227\) 551.580 0.161276 0.0806381 0.996743i \(-0.474304\pi\)
0.0806381 + 0.996743i \(0.474304\pi\)
\(228\) 0 0
\(229\) 3462.68 0.999216 0.499608 0.866252i \(-0.333477\pi\)
0.499608 + 0.866252i \(0.333477\pi\)
\(230\) 0 0
\(231\) −3621.79 −1.03159
\(232\) 0 0
\(233\) 2949.98 0.829441 0.414721 0.909949i \(-0.363879\pi\)
0.414721 + 0.909949i \(0.363879\pi\)
\(234\) 0 0
\(235\) 6894.24 1.91375
\(236\) 0 0
\(237\) 2109.44 0.578154
\(238\) 0 0
\(239\) 1063.71 0.287890 0.143945 0.989586i \(-0.454021\pi\)
0.143945 + 0.989586i \(0.454021\pi\)
\(240\) 0 0
\(241\) 160.174 0.0428121 0.0214061 0.999771i \(-0.493186\pi\)
0.0214061 + 0.999771i \(0.493186\pi\)
\(242\) 0 0
\(243\) 4920.24 1.29890
\(244\) 0 0
\(245\) 2951.87 0.769747
\(246\) 0 0
\(247\) 1155.42 0.297641
\(248\) 0 0
\(249\) −127.637 −0.0324847
\(250\) 0 0
\(251\) 2892.16 0.727297 0.363648 0.931536i \(-0.381531\pi\)
0.363648 + 0.931536i \(0.381531\pi\)
\(252\) 0 0
\(253\) −423.497 −0.105237
\(254\) 0 0
\(255\) 980.995 0.240911
\(256\) 0 0
\(257\) 5780.85 1.40311 0.701555 0.712615i \(-0.252489\pi\)
0.701555 + 0.712615i \(0.252489\pi\)
\(258\) 0 0
\(259\) 876.062 0.210177
\(260\) 0 0
\(261\) −6338.20 −1.50316
\(262\) 0 0
\(263\) 3054.14 0.716071 0.358035 0.933708i \(-0.383447\pi\)
0.358035 + 0.933708i \(0.383447\pi\)
\(264\) 0 0
\(265\) −8738.58 −2.02569
\(266\) 0 0
\(267\) −1612.35 −0.369567
\(268\) 0 0
\(269\) −4171.83 −0.945579 −0.472789 0.881175i \(-0.656753\pi\)
−0.472789 + 0.881175i \(0.656753\pi\)
\(270\) 0 0
\(271\) −1333.89 −0.298996 −0.149498 0.988762i \(-0.547766\pi\)
−0.149498 + 0.988762i \(0.547766\pi\)
\(272\) 0 0
\(273\) 10169.4 2.25451
\(274\) 0 0
\(275\) −195.969 −0.0429722
\(276\) 0 0
\(277\) −4171.55 −0.904852 −0.452426 0.891802i \(-0.649441\pi\)
−0.452426 + 0.891802i \(0.649441\pi\)
\(278\) 0 0
\(279\) 6666.92 1.43060
\(280\) 0 0
\(281\) 4657.91 0.988852 0.494426 0.869220i \(-0.335378\pi\)
0.494426 + 0.869220i \(0.335378\pi\)
\(282\) 0 0
\(283\) 4182.24 0.878474 0.439237 0.898371i \(-0.355249\pi\)
0.439237 + 0.898371i \(0.355249\pi\)
\(284\) 0 0
\(285\) −2096.31 −0.435700
\(286\) 0 0
\(287\) −10070.8 −2.07130
\(288\) 0 0
\(289\) −4803.63 −0.977738
\(290\) 0 0
\(291\) 911.356 0.183590
\(292\) 0 0
\(293\) 678.395 0.135264 0.0676318 0.997710i \(-0.478456\pi\)
0.0676318 + 0.997710i \(0.478456\pi\)
\(294\) 0 0
\(295\) −2311.06 −0.456120
\(296\) 0 0
\(297\) 1611.58 0.314859
\(298\) 0 0
\(299\) 1189.11 0.229994
\(300\) 0 0
\(301\) −4689.62 −0.898024
\(302\) 0 0
\(303\) 7651.90 1.45079
\(304\) 0 0
\(305\) −3082.35 −0.578672
\(306\) 0 0
\(307\) −7676.72 −1.42715 −0.713573 0.700581i \(-0.752924\pi\)
−0.713573 + 0.700581i \(0.752924\pi\)
\(308\) 0 0
\(309\) 3103.45 0.571357
\(310\) 0 0
\(311\) −4718.78 −0.860377 −0.430188 0.902739i \(-0.641553\pi\)
−0.430188 + 0.902739i \(0.641553\pi\)
\(312\) 0 0
\(313\) −7640.96 −1.37985 −0.689924 0.723882i \(-0.742356\pi\)
−0.689924 + 0.723882i \(0.742356\pi\)
\(314\) 0 0
\(315\) −10770.8 −1.92657
\(316\) 0 0
\(317\) −8981.33 −1.59130 −0.795649 0.605758i \(-0.792870\pi\)
−0.795649 + 0.605758i \(0.792870\pi\)
\(318\) 0 0
\(319\) 3081.95 0.540928
\(320\) 0 0
\(321\) 8166.76 1.42001
\(322\) 0 0
\(323\) −233.722 −0.0402620
\(324\) 0 0
\(325\) 550.249 0.0939149
\(326\) 0 0
\(327\) 16635.1 2.81322
\(328\) 0 0
\(329\) −14456.9 −2.42260
\(330\) 0 0
\(331\) −3408.01 −0.565925 −0.282962 0.959131i \(-0.591317\pi\)
−0.282962 + 0.959131i \(0.591317\pi\)
\(332\) 0 0
\(333\) −1358.34 −0.223534
\(334\) 0 0
\(335\) 1805.88 0.294525
\(336\) 0 0
\(337\) 7728.87 1.24931 0.624657 0.780900i \(-0.285239\pi\)
0.624657 + 0.780900i \(0.285239\pi\)
\(338\) 0 0
\(339\) −9802.73 −1.57053
\(340\) 0 0
\(341\) −3241.79 −0.514817
\(342\) 0 0
\(343\) 2186.94 0.344267
\(344\) 0 0
\(345\) −2157.44 −0.336675
\(346\) 0 0
\(347\) −1300.42 −0.201183 −0.100591 0.994928i \(-0.532073\pi\)
−0.100591 + 0.994928i \(0.532073\pi\)
\(348\) 0 0
\(349\) −1959.24 −0.300503 −0.150251 0.988648i \(-0.548008\pi\)
−0.150251 + 0.988648i \(0.548008\pi\)
\(350\) 0 0
\(351\) −4525.06 −0.688119
\(352\) 0 0
\(353\) −12393.2 −1.86862 −0.934311 0.356460i \(-0.883984\pi\)
−0.934311 + 0.356460i \(0.883984\pi\)
\(354\) 0 0
\(355\) −12608.1 −1.88498
\(356\) 0 0
\(357\) −2057.11 −0.304968
\(358\) 0 0
\(359\) 10544.6 1.55021 0.775104 0.631834i \(-0.217698\pi\)
0.775104 + 0.631834i \(0.217698\pi\)
\(360\) 0 0
\(361\) −6359.56 −0.927184
\(362\) 0 0
\(363\) 7989.30 1.15518
\(364\) 0 0
\(365\) −2612.07 −0.374580
\(366\) 0 0
\(367\) 6179.93 0.878991 0.439496 0.898245i \(-0.355157\pi\)
0.439496 + 0.898245i \(0.355157\pi\)
\(368\) 0 0
\(369\) 15614.9 2.20293
\(370\) 0 0
\(371\) 18324.4 2.56430
\(372\) 0 0
\(373\) −10557.8 −1.46558 −0.732792 0.680452i \(-0.761783\pi\)
−0.732792 + 0.680452i \(0.761783\pi\)
\(374\) 0 0
\(375\) 10726.9 1.47716
\(376\) 0 0
\(377\) −8653.63 −1.18219
\(378\) 0 0
\(379\) 6308.79 0.855041 0.427521 0.904006i \(-0.359387\pi\)
0.427521 + 0.904006i \(0.359387\pi\)
\(380\) 0 0
\(381\) −1117.72 −0.150296
\(382\) 0 0
\(383\) −7532.19 −1.00490 −0.502450 0.864606i \(-0.667568\pi\)
−0.502450 + 0.864606i \(0.667568\pi\)
\(384\) 0 0
\(385\) 5237.32 0.693295
\(386\) 0 0
\(387\) 7271.30 0.955093
\(388\) 0 0
\(389\) −5798.06 −0.755716 −0.377858 0.925864i \(-0.623339\pi\)
−0.377858 + 0.925864i \(0.623339\pi\)
\(390\) 0 0
\(391\) −240.538 −0.0311113
\(392\) 0 0
\(393\) 11040.2 1.41706
\(394\) 0 0
\(395\) −3050.37 −0.388559
\(396\) 0 0
\(397\) 13688.8 1.73053 0.865264 0.501317i \(-0.167151\pi\)
0.865264 + 0.501317i \(0.167151\pi\)
\(398\) 0 0
\(399\) 4395.86 0.551550
\(400\) 0 0
\(401\) −6132.31 −0.763673 −0.381836 0.924230i \(-0.624708\pi\)
−0.381836 + 0.924230i \(0.624708\pi\)
\(402\) 0 0
\(403\) 9102.43 1.12512
\(404\) 0 0
\(405\) −3697.69 −0.453678
\(406\) 0 0
\(407\) 660.494 0.0804409
\(408\) 0 0
\(409\) −6991.14 −0.845207 −0.422604 0.906315i \(-0.638884\pi\)
−0.422604 + 0.906315i \(0.638884\pi\)
\(410\) 0 0
\(411\) 13591.9 1.63124
\(412\) 0 0
\(413\) 4846.20 0.577400
\(414\) 0 0
\(415\) 184.571 0.0218319
\(416\) 0 0
\(417\) 20192.8 2.37133
\(418\) 0 0
\(419\) −12124.1 −1.41360 −0.706801 0.707412i \(-0.749863\pi\)
−0.706801 + 0.707412i \(0.749863\pi\)
\(420\) 0 0
\(421\) 11372.3 1.31651 0.658256 0.752794i \(-0.271294\pi\)
0.658256 + 0.752794i \(0.271294\pi\)
\(422\) 0 0
\(423\) 22415.6 2.57656
\(424\) 0 0
\(425\) −111.306 −0.0127039
\(426\) 0 0
\(427\) 6463.56 0.732538
\(428\) 0 0
\(429\) 7667.08 0.862867
\(430\) 0 0
\(431\) −12259.9 −1.37016 −0.685079 0.728469i \(-0.740232\pi\)
−0.685079 + 0.728469i \(0.740232\pi\)
\(432\) 0 0
\(433\) 4860.00 0.539391 0.269696 0.962946i \(-0.413077\pi\)
0.269696 + 0.962946i \(0.413077\pi\)
\(434\) 0 0
\(435\) 15700.5 1.73054
\(436\) 0 0
\(437\) 514.010 0.0562664
\(438\) 0 0
\(439\) 11660.7 1.26773 0.633866 0.773443i \(-0.281467\pi\)
0.633866 + 0.773443i \(0.281467\pi\)
\(440\) 0 0
\(441\) 9597.57 1.03634
\(442\) 0 0
\(443\) −15197.5 −1.62992 −0.814958 0.579520i \(-0.803240\pi\)
−0.814958 + 0.579520i \(0.803240\pi\)
\(444\) 0 0
\(445\) 2331.56 0.248374
\(446\) 0 0
\(447\) −27379.9 −2.89715
\(448\) 0 0
\(449\) −7803.19 −0.820168 −0.410084 0.912048i \(-0.634501\pi\)
−0.410084 + 0.912048i \(0.634501\pi\)
\(450\) 0 0
\(451\) −7592.76 −0.792747
\(452\) 0 0
\(453\) 7939.11 0.823426
\(454\) 0 0
\(455\) −14705.6 −1.51518
\(456\) 0 0
\(457\) 4159.89 0.425802 0.212901 0.977074i \(-0.431709\pi\)
0.212901 + 0.977074i \(0.431709\pi\)
\(458\) 0 0
\(459\) 915.345 0.0930820
\(460\) 0 0
\(461\) −1308.17 −0.132163 −0.0660817 0.997814i \(-0.521050\pi\)
−0.0660817 + 0.997814i \(0.521050\pi\)
\(462\) 0 0
\(463\) 19412.5 1.94855 0.974273 0.225372i \(-0.0723599\pi\)
0.974273 + 0.225372i \(0.0723599\pi\)
\(464\) 0 0
\(465\) −16514.8 −1.64700
\(466\) 0 0
\(467\) 3742.18 0.370808 0.185404 0.982662i \(-0.440641\pi\)
0.185404 + 0.982662i \(0.440641\pi\)
\(468\) 0 0
\(469\) −3786.86 −0.372838
\(470\) 0 0
\(471\) −1625.38 −0.159010
\(472\) 0 0
\(473\) −3535.67 −0.343700
\(474\) 0 0
\(475\) 237.853 0.0229756
\(476\) 0 0
\(477\) −28412.2 −2.72726
\(478\) 0 0
\(479\) −11322.5 −1.08004 −0.540021 0.841652i \(-0.681584\pi\)
−0.540021 + 0.841652i \(0.681584\pi\)
\(480\) 0 0
\(481\) −1854.56 −0.175802
\(482\) 0 0
\(483\) 4524.06 0.426195
\(484\) 0 0
\(485\) −1317.88 −0.123385
\(486\) 0 0
\(487\) 6688.99 0.622397 0.311198 0.950345i \(-0.399270\pi\)
0.311198 + 0.950345i \(0.399270\pi\)
\(488\) 0 0
\(489\) 27757.3 2.56693
\(490\) 0 0
\(491\) −2058.53 −0.189206 −0.0946030 0.995515i \(-0.530158\pi\)
−0.0946030 + 0.995515i \(0.530158\pi\)
\(492\) 0 0
\(493\) 1750.49 0.159915
\(494\) 0 0
\(495\) −8120.52 −0.737354
\(496\) 0 0
\(497\) 26438.6 2.38618
\(498\) 0 0
\(499\) 19788.5 1.77526 0.887632 0.460553i \(-0.152349\pi\)
0.887632 + 0.460553i \(0.152349\pi\)
\(500\) 0 0
\(501\) −7825.85 −0.697871
\(502\) 0 0
\(503\) −17381.5 −1.54076 −0.770381 0.637584i \(-0.779934\pi\)
−0.770381 + 0.637584i \(0.779934\pi\)
\(504\) 0 0
\(505\) −11065.1 −0.975030
\(506\) 0 0
\(507\) −3833.27 −0.335782
\(508\) 0 0
\(509\) −11595.2 −1.00972 −0.504862 0.863200i \(-0.668457\pi\)
−0.504862 + 0.863200i \(0.668457\pi\)
\(510\) 0 0
\(511\) 5477.39 0.474179
\(512\) 0 0
\(513\) −1956.02 −0.168343
\(514\) 0 0
\(515\) −4487.78 −0.383991
\(516\) 0 0
\(517\) −10899.6 −0.927201
\(518\) 0 0
\(519\) −12804.6 −1.08297
\(520\) 0 0
\(521\) −2597.34 −0.218409 −0.109205 0.994019i \(-0.534830\pi\)
−0.109205 + 0.994019i \(0.534830\pi\)
\(522\) 0 0
\(523\) −7487.17 −0.625987 −0.312993 0.949755i \(-0.601332\pi\)
−0.312993 + 0.949755i \(0.601332\pi\)
\(524\) 0 0
\(525\) 2093.46 0.174031
\(526\) 0 0
\(527\) −1841.27 −0.152196
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) −7514.08 −0.614093
\(532\) 0 0
\(533\) 21319.3 1.73253
\(534\) 0 0
\(535\) −11809.6 −0.954345
\(536\) 0 0
\(537\) 19276.5 1.54906
\(538\) 0 0
\(539\) −4666.82 −0.372939
\(540\) 0 0
\(541\) 10070.5 0.800302 0.400151 0.916449i \(-0.368958\pi\)
0.400151 + 0.916449i \(0.368958\pi\)
\(542\) 0 0
\(543\) −33054.2 −2.61232
\(544\) 0 0
\(545\) −24055.4 −1.89068
\(546\) 0 0
\(547\) −21474.4 −1.67857 −0.839284 0.543693i \(-0.817026\pi\)
−0.839284 + 0.543693i \(0.817026\pi\)
\(548\) 0 0
\(549\) −10021.8 −0.779091
\(550\) 0 0
\(551\) −3740.65 −0.289214
\(552\) 0 0
\(553\) 6396.50 0.491875
\(554\) 0 0
\(555\) 3364.79 0.257346
\(556\) 0 0
\(557\) 17358.6 1.32048 0.660242 0.751053i \(-0.270454\pi\)
0.660242 + 0.751053i \(0.270454\pi\)
\(558\) 0 0
\(559\) 9927.60 0.751150
\(560\) 0 0
\(561\) −1550.92 −0.116720
\(562\) 0 0
\(563\) −17977.9 −1.34579 −0.672895 0.739738i \(-0.734950\pi\)
−0.672895 + 0.739738i \(0.734950\pi\)
\(564\) 0 0
\(565\) 14175.3 1.05551
\(566\) 0 0
\(567\) 7753.89 0.574309
\(568\) 0 0
\(569\) −13529.5 −0.996810 −0.498405 0.866944i \(-0.666081\pi\)
−0.498405 + 0.866944i \(0.666081\pi\)
\(570\) 0 0
\(571\) −3866.06 −0.283344 −0.141672 0.989914i \(-0.545248\pi\)
−0.141672 + 0.989914i \(0.545248\pi\)
\(572\) 0 0
\(573\) 2585.11 0.188472
\(574\) 0 0
\(575\) 244.789 0.0177538
\(576\) 0 0
\(577\) 21500.0 1.55123 0.775614 0.631208i \(-0.217441\pi\)
0.775614 + 0.631208i \(0.217441\pi\)
\(578\) 0 0
\(579\) 5546.78 0.398128
\(580\) 0 0
\(581\) −387.038 −0.0276369
\(582\) 0 0
\(583\) 13815.4 0.981435
\(584\) 0 0
\(585\) 22801.1 1.61147
\(586\) 0 0
\(587\) 4927.12 0.346446 0.173223 0.984883i \(-0.444582\pi\)
0.173223 + 0.984883i \(0.444582\pi\)
\(588\) 0 0
\(589\) 3934.64 0.275253
\(590\) 0 0
\(591\) 11190.5 0.778874
\(592\) 0 0
\(593\) 17675.5 1.22402 0.612011 0.790849i \(-0.290361\pi\)
0.612011 + 0.790849i \(0.290361\pi\)
\(594\) 0 0
\(595\) 2974.70 0.204959
\(596\) 0 0
\(597\) 17346.9 1.18921
\(598\) 0 0
\(599\) −22302.6 −1.52130 −0.760651 0.649161i \(-0.775120\pi\)
−0.760651 + 0.649161i \(0.775120\pi\)
\(600\) 0 0
\(601\) −25526.8 −1.73255 −0.866273 0.499570i \(-0.833491\pi\)
−0.866273 + 0.499570i \(0.833491\pi\)
\(602\) 0 0
\(603\) 5871.56 0.396531
\(604\) 0 0
\(605\) −11553.0 −0.776358
\(606\) 0 0
\(607\) 9865.32 0.659672 0.329836 0.944038i \(-0.393006\pi\)
0.329836 + 0.944038i \(0.393006\pi\)
\(608\) 0 0
\(609\) −32923.3 −2.19068
\(610\) 0 0
\(611\) 30604.3 2.02638
\(612\) 0 0
\(613\) 3760.57 0.247778 0.123889 0.992296i \(-0.460463\pi\)
0.123889 + 0.992296i \(0.460463\pi\)
\(614\) 0 0
\(615\) −38680.2 −2.53615
\(616\) 0 0
\(617\) −4170.47 −0.272118 −0.136059 0.990701i \(-0.543444\pi\)
−0.136059 + 0.990701i \(0.543444\pi\)
\(618\) 0 0
\(619\) 24392.4 1.58387 0.791935 0.610606i \(-0.209074\pi\)
0.791935 + 0.610606i \(0.209074\pi\)
\(620\) 0 0
\(621\) −2013.06 −0.130083
\(622\) 0 0
\(623\) −4889.18 −0.314415
\(624\) 0 0
\(625\) −16842.1 −1.07789
\(626\) 0 0
\(627\) 3314.20 0.211094
\(628\) 0 0
\(629\) 375.148 0.0237808
\(630\) 0 0
\(631\) −10776.6 −0.679886 −0.339943 0.940446i \(-0.610408\pi\)
−0.339943 + 0.940446i \(0.610408\pi\)
\(632\) 0 0
\(633\) 18248.9 1.14586
\(634\) 0 0
\(635\) 1616.29 0.101009
\(636\) 0 0
\(637\) 13103.7 0.815050
\(638\) 0 0
\(639\) −40993.3 −2.53782
\(640\) 0 0
\(641\) 27979.4 1.72405 0.862027 0.506863i \(-0.169195\pi\)
0.862027 + 0.506863i \(0.169195\pi\)
\(642\) 0 0
\(643\) −18519.5 −1.13583 −0.567915 0.823087i \(-0.692250\pi\)
−0.567915 + 0.823087i \(0.692250\pi\)
\(644\) 0 0
\(645\) −18011.9 −1.09957
\(646\) 0 0
\(647\) 20048.6 1.21822 0.609112 0.793084i \(-0.291526\pi\)
0.609112 + 0.793084i \(0.291526\pi\)
\(648\) 0 0
\(649\) 3653.72 0.220988
\(650\) 0 0
\(651\) 34630.8 2.08493
\(652\) 0 0
\(653\) 30798.2 1.84568 0.922839 0.385185i \(-0.125862\pi\)
0.922839 + 0.385185i \(0.125862\pi\)
\(654\) 0 0
\(655\) −15964.8 −0.952360
\(656\) 0 0
\(657\) −8492.75 −0.504313
\(658\) 0 0
\(659\) −25007.5 −1.47823 −0.739116 0.673578i \(-0.764757\pi\)
−0.739116 + 0.673578i \(0.764757\pi\)
\(660\) 0 0
\(661\) −9196.16 −0.541133 −0.270567 0.962701i \(-0.587211\pi\)
−0.270567 + 0.962701i \(0.587211\pi\)
\(662\) 0 0
\(663\) 4354.75 0.255090
\(664\) 0 0
\(665\) −6356.68 −0.370679
\(666\) 0 0
\(667\) −3849.74 −0.223482
\(668\) 0 0
\(669\) −30870.5 −1.78404
\(670\) 0 0
\(671\) 4873.11 0.280364
\(672\) 0 0
\(673\) −8440.22 −0.483427 −0.241714 0.970348i \(-0.577709\pi\)
−0.241714 + 0.970348i \(0.577709\pi\)
\(674\) 0 0
\(675\) −931.522 −0.0531175
\(676\) 0 0
\(677\) 7557.24 0.429023 0.214511 0.976721i \(-0.431184\pi\)
0.214511 + 0.976721i \(0.431184\pi\)
\(678\) 0 0
\(679\) 2763.53 0.156192
\(680\) 0 0
\(681\) −4442.44 −0.249977
\(682\) 0 0
\(683\) −8410.59 −0.471189 −0.235595 0.971851i \(-0.575704\pi\)
−0.235595 + 0.971851i \(0.575704\pi\)
\(684\) 0 0
\(685\) −19654.7 −1.09630
\(686\) 0 0
\(687\) −27888.5 −1.54878
\(688\) 0 0
\(689\) −38791.5 −2.14491
\(690\) 0 0
\(691\) −15913.2 −0.876072 −0.438036 0.898958i \(-0.644326\pi\)
−0.438036 + 0.898958i \(0.644326\pi\)
\(692\) 0 0
\(693\) 17028.4 0.933412
\(694\) 0 0
\(695\) −29200.0 −1.59369
\(696\) 0 0
\(697\) −4312.54 −0.234360
\(698\) 0 0
\(699\) −23759.2 −1.28563
\(700\) 0 0
\(701\) −29374.3 −1.58267 −0.791335 0.611383i \(-0.790614\pi\)
−0.791335 + 0.611383i \(0.790614\pi\)
\(702\) 0 0
\(703\) −801.659 −0.0430087
\(704\) 0 0
\(705\) −55526.3 −2.96630
\(706\) 0 0
\(707\) 23203.0 1.23429
\(708\) 0 0
\(709\) 23319.2 1.23522 0.617609 0.786485i \(-0.288101\pi\)
0.617609 + 0.786485i \(0.288101\pi\)
\(710\) 0 0
\(711\) −9917.83 −0.523133
\(712\) 0 0
\(713\) 4049.39 0.212694
\(714\) 0 0
\(715\) −11087.0 −0.579905
\(716\) 0 0
\(717\) −8567.13 −0.446228
\(718\) 0 0
\(719\) −25216.5 −1.30795 −0.653975 0.756517i \(-0.726900\pi\)
−0.653975 + 0.756517i \(0.726900\pi\)
\(720\) 0 0
\(721\) 9410.69 0.486092
\(722\) 0 0
\(723\) −1290.05 −0.0663586
\(724\) 0 0
\(725\) −1781.42 −0.0912558
\(726\) 0 0
\(727\) 36727.3 1.87364 0.936822 0.349806i \(-0.113753\pi\)
0.936822 + 0.349806i \(0.113753\pi\)
\(728\) 0 0
\(729\) −31055.4 −1.57778
\(730\) 0 0
\(731\) −2008.19 −0.101608
\(732\) 0 0
\(733\) 3401.78 0.171416 0.0857079 0.996320i \(-0.472685\pi\)
0.0857079 + 0.996320i \(0.472685\pi\)
\(734\) 0 0
\(735\) −23774.4 −1.19311
\(736\) 0 0
\(737\) −2855.04 −0.142696
\(738\) 0 0
\(739\) 11982.1 0.596442 0.298221 0.954497i \(-0.403607\pi\)
0.298221 + 0.954497i \(0.403607\pi\)
\(740\) 0 0
\(741\) −9305.74 −0.461343
\(742\) 0 0
\(743\) 32449.8 1.60225 0.801123 0.598499i \(-0.204236\pi\)
0.801123 + 0.598499i \(0.204236\pi\)
\(744\) 0 0
\(745\) 39593.0 1.94708
\(746\) 0 0
\(747\) 600.107 0.0293933
\(748\) 0 0
\(749\) 24764.3 1.20810
\(750\) 0 0
\(751\) 4340.97 0.210925 0.105462 0.994423i \(-0.466368\pi\)
0.105462 + 0.994423i \(0.466368\pi\)
\(752\) 0 0
\(753\) −23293.5 −1.12731
\(754\) 0 0
\(755\) −11480.4 −0.553398
\(756\) 0 0
\(757\) −36757.0 −1.76480 −0.882401 0.470498i \(-0.844074\pi\)
−0.882401 + 0.470498i \(0.844074\pi\)
\(758\) 0 0
\(759\) 3410.85 0.163117
\(760\) 0 0
\(761\) 2408.45 0.114726 0.0573629 0.998353i \(-0.481731\pi\)
0.0573629 + 0.998353i \(0.481731\pi\)
\(762\) 0 0
\(763\) 50443.1 2.39340
\(764\) 0 0
\(765\) −4612.30 −0.217984
\(766\) 0 0
\(767\) −10259.1 −0.482964
\(768\) 0 0
\(769\) −20323.2 −0.953022 −0.476511 0.879168i \(-0.658099\pi\)
−0.476511 + 0.879168i \(0.658099\pi\)
\(770\) 0 0
\(771\) −46559.0 −2.17482
\(772\) 0 0
\(773\) 23521.1 1.09443 0.547214 0.836992i \(-0.315688\pi\)
0.547214 + 0.836992i \(0.315688\pi\)
\(774\) 0 0
\(775\) 1873.81 0.0868508
\(776\) 0 0
\(777\) −7055.82 −0.325773
\(778\) 0 0
\(779\) 9215.54 0.423852
\(780\) 0 0
\(781\) 19933.0 0.913262
\(782\) 0 0
\(783\) 14649.8 0.668635
\(784\) 0 0
\(785\) 2350.40 0.106865
\(786\) 0 0
\(787\) 32043.4 1.45136 0.725682 0.688030i \(-0.241524\pi\)
0.725682 + 0.688030i \(0.241524\pi\)
\(788\) 0 0
\(789\) −24598.1 −1.10991
\(790\) 0 0
\(791\) −29725.1 −1.33616
\(792\) 0 0
\(793\) −13682.9 −0.612730
\(794\) 0 0
\(795\) 70380.6 3.13980
\(796\) 0 0
\(797\) 21774.0 0.967720 0.483860 0.875145i \(-0.339234\pi\)
0.483860 + 0.875145i \(0.339234\pi\)
\(798\) 0 0
\(799\) −6190.75 −0.274109
\(800\) 0 0
\(801\) 7580.71 0.334396
\(802\) 0 0
\(803\) 4129.60 0.181482
\(804\) 0 0
\(805\) −6542.07 −0.286432
\(806\) 0 0
\(807\) 33599.9 1.46564
\(808\) 0 0
\(809\) 14693.1 0.638543 0.319271 0.947663i \(-0.396562\pi\)
0.319271 + 0.947663i \(0.396562\pi\)
\(810\) 0 0
\(811\) 41634.1 1.80268 0.901338 0.433117i \(-0.142586\pi\)
0.901338 + 0.433117i \(0.142586\pi\)
\(812\) 0 0
\(813\) 10743.1 0.463442
\(814\) 0 0
\(815\) −40138.7 −1.72515
\(816\) 0 0
\(817\) 4291.34 0.183764
\(818\) 0 0
\(819\) −47813.0 −2.03995
\(820\) 0 0
\(821\) 311.819 0.0132553 0.00662763 0.999978i \(-0.497890\pi\)
0.00662763 + 0.999978i \(0.497890\pi\)
\(822\) 0 0
\(823\) −18804.4 −0.796451 −0.398225 0.917288i \(-0.630374\pi\)
−0.398225 + 0.917288i \(0.630374\pi\)
\(824\) 0 0
\(825\) 1578.33 0.0666067
\(826\) 0 0
\(827\) −40839.7 −1.71721 −0.858607 0.512635i \(-0.828670\pi\)
−0.858607 + 0.512635i \(0.828670\pi\)
\(828\) 0 0
\(829\) 21272.1 0.891206 0.445603 0.895231i \(-0.352989\pi\)
0.445603 + 0.895231i \(0.352989\pi\)
\(830\) 0 0
\(831\) 33597.7 1.40252
\(832\) 0 0
\(833\) −2650.66 −0.110252
\(834\) 0 0
\(835\) 11316.7 0.469017
\(836\) 0 0
\(837\) −15409.6 −0.636360
\(838\) 0 0
\(839\) −17124.4 −0.704648 −0.352324 0.935878i \(-0.614609\pi\)
−0.352324 + 0.935878i \(0.614609\pi\)
\(840\) 0 0
\(841\) 3627.01 0.148715
\(842\) 0 0
\(843\) −37514.9 −1.53272
\(844\) 0 0
\(845\) 5543.14 0.225668
\(846\) 0 0
\(847\) 24226.2 0.982787
\(848\) 0 0
\(849\) −33683.8 −1.36163
\(850\) 0 0
\(851\) −825.039 −0.0332338
\(852\) 0 0
\(853\) −41037.7 −1.64725 −0.823624 0.567136i \(-0.808052\pi\)
−0.823624 + 0.567136i \(0.808052\pi\)
\(854\) 0 0
\(855\) 9856.10 0.394236
\(856\) 0 0
\(857\) −43891.1 −1.74946 −0.874732 0.484607i \(-0.838963\pi\)
−0.874732 + 0.484607i \(0.838963\pi\)
\(858\) 0 0
\(859\) −27562.0 −1.09477 −0.547383 0.836882i \(-0.684376\pi\)
−0.547383 + 0.836882i \(0.684376\pi\)
\(860\) 0 0
\(861\) 81110.7 3.21050
\(862\) 0 0
\(863\) −24931.7 −0.983412 −0.491706 0.870761i \(-0.663627\pi\)
−0.491706 + 0.870761i \(0.663627\pi\)
\(864\) 0 0
\(865\) 18516.2 0.727826
\(866\) 0 0
\(867\) 38688.5 1.51549
\(868\) 0 0
\(869\) 4822.54 0.188255
\(870\) 0 0
\(871\) 8016.52 0.311859
\(872\) 0 0
\(873\) −4284.88 −0.166118
\(874\) 0 0
\(875\) 32527.4 1.25672
\(876\) 0 0
\(877\) −27169.3 −1.04612 −0.523058 0.852297i \(-0.675209\pi\)
−0.523058 + 0.852297i \(0.675209\pi\)
\(878\) 0 0
\(879\) −5463.80 −0.209658
\(880\) 0 0
\(881\) 41469.9 1.58587 0.792937 0.609303i \(-0.208551\pi\)
0.792937 + 0.609303i \(0.208551\pi\)
\(882\) 0 0
\(883\) −20433.7 −0.778763 −0.389382 0.921076i \(-0.627311\pi\)
−0.389382 + 0.921076i \(0.627311\pi\)
\(884\) 0 0
\(885\) 18613.3 0.706984
\(886\) 0 0
\(887\) 47298.0 1.79043 0.895215 0.445634i \(-0.147022\pi\)
0.895215 + 0.445634i \(0.147022\pi\)
\(888\) 0 0
\(889\) −3389.30 −0.127867
\(890\) 0 0
\(891\) 5845.93 0.219805
\(892\) 0 0
\(893\) 13229.1 0.495740
\(894\) 0 0
\(895\) −27875.0 −1.04107
\(896\) 0 0
\(897\) −9577.13 −0.356489
\(898\) 0 0
\(899\) −29469.0 −1.09327
\(900\) 0 0
\(901\) 7846.89 0.290142
\(902\) 0 0
\(903\) 37770.3 1.39193
\(904\) 0 0
\(905\) 47798.3 1.75566
\(906\) 0 0
\(907\) −38147.2 −1.39653 −0.698267 0.715838i \(-0.746045\pi\)
−0.698267 + 0.715838i \(0.746045\pi\)
\(908\) 0 0
\(909\) −35976.5 −1.31272
\(910\) 0 0
\(911\) −2307.84 −0.0839320 −0.0419660 0.999119i \(-0.513362\pi\)
−0.0419660 + 0.999119i \(0.513362\pi\)
\(912\) 0 0
\(913\) −291.802 −0.0105775
\(914\) 0 0
\(915\) 24825.3 0.896940
\(916\) 0 0
\(917\) 33477.5 1.20559
\(918\) 0 0
\(919\) −47460.3 −1.70356 −0.851780 0.523900i \(-0.824477\pi\)
−0.851780 + 0.523900i \(0.824477\pi\)
\(920\) 0 0
\(921\) 61828.5 2.21207
\(922\) 0 0
\(923\) −55968.7 −1.99592
\(924\) 0 0
\(925\) −381.778 −0.0135706
\(926\) 0 0
\(927\) −14591.4 −0.516983
\(928\) 0 0
\(929\) −5371.11 −0.189688 −0.0948440 0.995492i \(-0.530235\pi\)
−0.0948440 + 0.995492i \(0.530235\pi\)
\(930\) 0 0
\(931\) 5664.24 0.199396
\(932\) 0 0
\(933\) 38005.1 1.33358
\(934\) 0 0
\(935\) 2242.73 0.0784439
\(936\) 0 0
\(937\) 37291.4 1.30017 0.650084 0.759863i \(-0.274734\pi\)
0.650084 + 0.759863i \(0.274734\pi\)
\(938\) 0 0
\(939\) 61540.4 2.13876
\(940\) 0 0
\(941\) 10354.8 0.358721 0.179361 0.983783i \(-0.442597\pi\)
0.179361 + 0.983783i \(0.442597\pi\)
\(942\) 0 0
\(943\) 9484.30 0.327520
\(944\) 0 0
\(945\) 24895.2 0.856975
\(946\) 0 0
\(947\) −393.905 −0.0135166 −0.00675829 0.999977i \(-0.502151\pi\)
−0.00675829 + 0.999977i \(0.502151\pi\)
\(948\) 0 0
\(949\) −11595.3 −0.396626
\(950\) 0 0
\(951\) 72335.7 2.46651
\(952\) 0 0
\(953\) 20824.5 0.707840 0.353920 0.935276i \(-0.384849\pi\)
0.353920 + 0.935276i \(0.384849\pi\)
\(954\) 0 0
\(955\) −3738.22 −0.126666
\(956\) 0 0
\(957\) −24822.1 −0.838436
\(958\) 0 0
\(959\) 41215.1 1.38781
\(960\) 0 0
\(961\) 1206.32 0.0404929
\(962\) 0 0
\(963\) −38397.2 −1.28487
\(964\) 0 0
\(965\) −8020.97 −0.267569
\(966\) 0 0
\(967\) 36967.5 1.22936 0.614681 0.788776i \(-0.289285\pi\)
0.614681 + 0.788776i \(0.289285\pi\)
\(968\) 0 0
\(969\) 1882.40 0.0624060
\(970\) 0 0
\(971\) −5339.18 −0.176460 −0.0882299 0.996100i \(-0.528121\pi\)
−0.0882299 + 0.996100i \(0.528121\pi\)
\(972\) 0 0
\(973\) 61231.1 2.01745
\(974\) 0 0
\(975\) −4431.72 −0.145568
\(976\) 0 0
\(977\) 8161.14 0.267245 0.133622 0.991032i \(-0.457339\pi\)
0.133622 + 0.991032i \(0.457339\pi\)
\(978\) 0 0
\(979\) −3686.12 −0.120336
\(980\) 0 0
\(981\) −78212.4 −2.54550
\(982\) 0 0
\(983\) −33898.0 −1.09988 −0.549938 0.835206i \(-0.685349\pi\)
−0.549938 + 0.835206i \(0.685349\pi\)
\(984\) 0 0
\(985\) −16182.1 −0.523456
\(986\) 0 0
\(987\) 116436. 3.75502
\(988\) 0 0
\(989\) 4416.49 0.141998
\(990\) 0 0
\(991\) −24087.4 −0.772111 −0.386055 0.922476i \(-0.626163\pi\)
−0.386055 + 0.922476i \(0.626163\pi\)
\(992\) 0 0
\(993\) 27448.1 0.877181
\(994\) 0 0
\(995\) −25084.6 −0.799232
\(996\) 0 0
\(997\) −29288.3 −0.930362 −0.465181 0.885216i \(-0.654011\pi\)
−0.465181 + 0.885216i \(0.654011\pi\)
\(998\) 0 0
\(999\) 3139.61 0.0994322
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 736.4.a.e.1.2 8
4.3 odd 2 736.4.a.f.1.7 yes 8
8.3 odd 2 1472.4.a.bg.1.2 8
8.5 even 2 1472.4.a.bh.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.4.a.e.1.2 8 1.1 even 1 trivial
736.4.a.f.1.7 yes 8 4.3 odd 2
1472.4.a.bg.1.2 8 8.3 odd 2
1472.4.a.bh.1.7 8 8.5 even 2