Properties

Label 384.4.d.f.193.6
Level $384$
Weight $4$
Character 384.193
Analytic conductor $22.657$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,4,Mod(193,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.193");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1534132224.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 107x^{4} + 210x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.6
Root \(2.21597i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.4.d.f.193.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} -5.67763i q^{5} -33.0917 q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} -5.67763i q^{5} -33.0917 q^{7} -9.00000 q^{9} -34.6274i q^{11} +82.2421i q^{13} +17.0329 q^{15} +97.8823 q^{17} -55.8823i q^{19} -99.2750i q^{21} +130.418 q^{23} +92.7645 q^{25} -27.0000i q^{27} -147.451i q^{29} +101.223 q^{31} +103.882 q^{33} +187.882i q^{35} +184.439i q^{37} -246.726 q^{39} +237.411 q^{41} -199.882i q^{43} +51.0987i q^{45} -334.813 q^{47} +752.058 q^{49} +293.647i q^{51} +102.030i q^{53} -196.602 q^{55} +167.647 q^{57} -105.961i q^{59} -717.803i q^{61} +297.825 q^{63} +466.940 q^{65} +316.471i q^{67} +391.255i q^{69} +800.045 q^{71} +301.058 q^{73} +278.294i q^{75} +1145.88i q^{77} -42.8329 q^{79} +81.0000 q^{81} +1236.67i q^{83} -555.739i q^{85} +442.354 q^{87} +1325.29 q^{89} -2721.53i q^{91} +303.670i q^{93} -317.279 q^{95} -505.765 q^{97} +311.647i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 72 q^{9} + 240 q^{17} - 344 q^{25} + 288 q^{33} - 816 q^{41} + 1672 q^{49} - 288 q^{57} - 1152 q^{65} - 1936 q^{73} + 648 q^{81} + 7344 q^{89} - 2960 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) − 5.67763i − 0.507823i −0.967227 0.253911i \(-0.918283\pi\)
0.967227 0.253911i \(-0.0817172\pi\)
\(6\) 0 0
\(7\) −33.0917 −1.78678 −0.893391 0.449280i \(-0.851680\pi\)
−0.893391 + 0.449280i \(0.851680\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) − 34.6274i − 0.949142i −0.880217 0.474571i \(-0.842603\pi\)
0.880217 0.474571i \(-0.157397\pi\)
\(12\) 0 0
\(13\) 82.2421i 1.75460i 0.479939 + 0.877302i \(0.340659\pi\)
−0.479939 + 0.877302i \(0.659341\pi\)
\(14\) 0 0
\(15\) 17.0329 0.293192
\(16\) 0 0
\(17\) 97.8823 1.39647 0.698233 0.715870i \(-0.253970\pi\)
0.698233 + 0.715870i \(0.253970\pi\)
\(18\) 0 0
\(19\) − 55.8823i − 0.674751i −0.941370 0.337375i \(-0.890461\pi\)
0.941370 0.337375i \(-0.109539\pi\)
\(20\) 0 0
\(21\) − 99.2750i − 1.03160i
\(22\) 0 0
\(23\) 130.418 1.18235 0.591176 0.806542i \(-0.298664\pi\)
0.591176 + 0.806542i \(0.298664\pi\)
\(24\) 0 0
\(25\) 92.7645 0.742116
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) − 147.451i − 0.944173i −0.881552 0.472086i \(-0.843501\pi\)
0.881552 0.472086i \(-0.156499\pi\)
\(30\) 0 0
\(31\) 101.223 0.586459 0.293230 0.956042i \(-0.405270\pi\)
0.293230 + 0.956042i \(0.405270\pi\)
\(32\) 0 0
\(33\) 103.882 0.547987
\(34\) 0 0
\(35\) 187.882i 0.907368i
\(36\) 0 0
\(37\) 184.439i 0.819504i 0.912197 + 0.409752i \(0.134385\pi\)
−0.912197 + 0.409752i \(0.865615\pi\)
\(38\) 0 0
\(39\) −246.726 −1.01302
\(40\) 0 0
\(41\) 237.411 0.904327 0.452164 0.891935i \(-0.350652\pi\)
0.452164 + 0.891935i \(0.350652\pi\)
\(42\) 0 0
\(43\) − 199.882i − 0.708878i −0.935079 0.354439i \(-0.884672\pi\)
0.935079 0.354439i \(-0.115328\pi\)
\(44\) 0 0
\(45\) 51.0987i 0.169274i
\(46\) 0 0
\(47\) −334.813 −1.03910 −0.519548 0.854441i \(-0.673900\pi\)
−0.519548 + 0.854441i \(0.673900\pi\)
\(48\) 0 0
\(49\) 752.058 2.19259
\(50\) 0 0
\(51\) 293.647i 0.806250i
\(52\) 0 0
\(53\) 102.030i 0.264433i 0.991221 + 0.132216i \(0.0422094\pi\)
−0.991221 + 0.132216i \(0.957791\pi\)
\(54\) 0 0
\(55\) −196.602 −0.481996
\(56\) 0 0
\(57\) 167.647 0.389568
\(58\) 0 0
\(59\) − 105.961i − 0.233813i −0.993143 0.116907i \(-0.962702\pi\)
0.993143 0.116907i \(-0.0372978\pi\)
\(60\) 0 0
\(61\) − 717.803i − 1.50664i −0.657653 0.753321i \(-0.728451\pi\)
0.657653 0.753321i \(-0.271549\pi\)
\(62\) 0 0
\(63\) 297.825 0.595594
\(64\) 0 0
\(65\) 466.940 0.891028
\(66\) 0 0
\(67\) 316.471i 0.577061i 0.957471 + 0.288530i \(0.0931667\pi\)
−0.957471 + 0.288530i \(0.906833\pi\)
\(68\) 0 0
\(69\) 391.255i 0.682632i
\(70\) 0 0
\(71\) 800.045 1.33729 0.668647 0.743580i \(-0.266874\pi\)
0.668647 + 0.743580i \(0.266874\pi\)
\(72\) 0 0
\(73\) 301.058 0.482687 0.241344 0.970440i \(-0.422412\pi\)
0.241344 + 0.970440i \(0.422412\pi\)
\(74\) 0 0
\(75\) 278.294i 0.428461i
\(76\) 0 0
\(77\) 1145.88i 1.69591i
\(78\) 0 0
\(79\) −42.8329 −0.0610010 −0.0305005 0.999535i \(-0.509710\pi\)
−0.0305005 + 0.999535i \(0.509710\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1236.67i 1.63544i 0.575614 + 0.817721i \(0.304763\pi\)
−0.575614 + 0.817721i \(0.695237\pi\)
\(84\) 0 0
\(85\) − 555.739i − 0.709158i
\(86\) 0 0
\(87\) 442.354 0.545119
\(88\) 0 0
\(89\) 1325.29 1.57844 0.789218 0.614113i \(-0.210486\pi\)
0.789218 + 0.614113i \(0.210486\pi\)
\(90\) 0 0
\(91\) − 2721.53i − 3.13509i
\(92\) 0 0
\(93\) 303.670i 0.338592i
\(94\) 0 0
\(95\) −317.279 −0.342654
\(96\) 0 0
\(97\) −505.765 −0.529408 −0.264704 0.964330i \(-0.585274\pi\)
−0.264704 + 0.964330i \(0.585274\pi\)
\(98\) 0 0
\(99\) 311.647i 0.316381i
\(100\) 0 0
\(101\) − 1281.97i − 1.26298i −0.775383 0.631491i \(-0.782443\pi\)
0.775383 0.631491i \(-0.217557\pi\)
\(102\) 0 0
\(103\) 161.562 0.154555 0.0772775 0.997010i \(-0.475377\pi\)
0.0772775 + 0.997010i \(0.475377\pi\)
\(104\) 0 0
\(105\) −563.647 −0.523869
\(106\) 0 0
\(107\) − 481.726i − 0.435235i −0.976034 0.217618i \(-0.930171\pi\)
0.976034 0.217618i \(-0.0698286\pi\)
\(108\) 0 0
\(109\) 286.637i 0.251879i 0.992038 + 0.125940i \(0.0401945\pi\)
−0.992038 + 0.125940i \(0.959805\pi\)
\(110\) 0 0
\(111\) −553.318 −0.473141
\(112\) 0 0
\(113\) −1397.29 −1.16324 −0.581621 0.813460i \(-0.697581\pi\)
−0.581621 + 0.813460i \(0.697581\pi\)
\(114\) 0 0
\(115\) − 740.468i − 0.600426i
\(116\) 0 0
\(117\) − 740.179i − 0.584868i
\(118\) 0 0
\(119\) −3239.09 −2.49518
\(120\) 0 0
\(121\) 131.942 0.0991300
\(122\) 0 0
\(123\) 712.234i 0.522113i
\(124\) 0 0
\(125\) − 1236.39i − 0.884686i
\(126\) 0 0
\(127\) 443.829 0.310106 0.155053 0.987906i \(-0.450445\pi\)
0.155053 + 0.987906i \(0.450445\pi\)
\(128\) 0 0
\(129\) 599.647 0.409271
\(130\) 0 0
\(131\) 1365.57i 0.910765i 0.890296 + 0.455382i \(0.150497\pi\)
−0.890296 + 0.455382i \(0.849503\pi\)
\(132\) 0 0
\(133\) 1849.24i 1.20563i
\(134\) 0 0
\(135\) −153.296 −0.0977305
\(136\) 0 0
\(137\) −1223.06 −0.762721 −0.381360 0.924426i \(-0.624544\pi\)
−0.381360 + 0.924426i \(0.624544\pi\)
\(138\) 0 0
\(139\) − 2176.70i − 1.32824i −0.747625 0.664121i \(-0.768806\pi\)
0.747625 0.664121i \(-0.231194\pi\)
\(140\) 0 0
\(141\) − 1004.44i − 0.599922i
\(142\) 0 0
\(143\) 2847.83 1.66537
\(144\) 0 0
\(145\) −837.174 −0.479473
\(146\) 0 0
\(147\) 2256.17i 1.26589i
\(148\) 0 0
\(149\) 2875.88i 1.58122i 0.612320 + 0.790610i \(0.290236\pi\)
−0.612320 + 0.790610i \(0.709764\pi\)
\(150\) 0 0
\(151\) 2330.03 1.25573 0.627863 0.778323i \(-0.283930\pi\)
0.627863 + 0.778323i \(0.283930\pi\)
\(152\) 0 0
\(153\) −880.940 −0.465489
\(154\) 0 0
\(155\) − 574.708i − 0.297817i
\(156\) 0 0
\(157\) 2447.31i 1.24405i 0.782996 + 0.622027i \(0.213690\pi\)
−0.782996 + 0.622027i \(0.786310\pi\)
\(158\) 0 0
\(159\) −306.091 −0.152670
\(160\) 0 0
\(161\) −4315.76 −2.11261
\(162\) 0 0
\(163\) − 1436.82i − 0.690432i −0.938523 0.345216i \(-0.887806\pi\)
0.938523 0.345216i \(-0.112194\pi\)
\(164\) 0 0
\(165\) − 589.805i − 0.278280i
\(166\) 0 0
\(167\) 1769.42 0.819889 0.409945 0.912110i \(-0.365548\pi\)
0.409945 + 0.912110i \(0.365548\pi\)
\(168\) 0 0
\(169\) −4566.76 −2.07863
\(170\) 0 0
\(171\) 502.940i 0.224917i
\(172\) 0 0
\(173\) − 3897.86i − 1.71300i −0.516148 0.856499i \(-0.672635\pi\)
0.516148 0.856499i \(-0.327365\pi\)
\(174\) 0 0
\(175\) −3069.73 −1.32600
\(176\) 0 0
\(177\) 317.884 0.134992
\(178\) 0 0
\(179\) 4146.74i 1.73152i 0.500460 + 0.865760i \(0.333164\pi\)
−0.500460 + 0.865760i \(0.666836\pi\)
\(180\) 0 0
\(181\) − 1104.22i − 0.453457i −0.973958 0.226728i \(-0.927197\pi\)
0.973958 0.226728i \(-0.0728030\pi\)
\(182\) 0 0
\(183\) 2153.41 0.869860
\(184\) 0 0
\(185\) 1047.18 0.416163
\(186\) 0 0
\(187\) − 3389.41i − 1.32544i
\(188\) 0 0
\(189\) 893.475i 0.343866i
\(190\) 0 0
\(191\) −35.0686 −0.0132852 −0.00664260 0.999978i \(-0.502114\pi\)
−0.00664260 + 0.999978i \(0.502114\pi\)
\(192\) 0 0
\(193\) 615.884 0.229701 0.114851 0.993383i \(-0.463361\pi\)
0.114851 + 0.993383i \(0.463361\pi\)
\(194\) 0 0
\(195\) 1400.82i 0.514435i
\(196\) 0 0
\(197\) − 589.638i − 0.213249i −0.994299 0.106624i \(-0.965996\pi\)
0.994299 0.106624i \(-0.0340042\pi\)
\(198\) 0 0
\(199\) −4813.82 −1.71479 −0.857394 0.514661i \(-0.827918\pi\)
−0.857394 + 0.514661i \(0.827918\pi\)
\(200\) 0 0
\(201\) −949.413 −0.333166
\(202\) 0 0
\(203\) 4879.41i 1.68703i
\(204\) 0 0
\(205\) − 1347.93i − 0.459238i
\(206\) 0 0
\(207\) −1173.77 −0.394118
\(208\) 0 0
\(209\) −1935.06 −0.640434
\(210\) 0 0
\(211\) − 3294.35i − 1.07484i −0.843313 0.537422i \(-0.819398\pi\)
0.843313 0.537422i \(-0.180602\pi\)
\(212\) 0 0
\(213\) 2400.13i 0.772087i
\(214\) 0 0
\(215\) −1134.86 −0.359984
\(216\) 0 0
\(217\) −3349.65 −1.04787
\(218\) 0 0
\(219\) 903.174i 0.278680i
\(220\) 0 0
\(221\) 8050.04i 2.45025i
\(222\) 0 0
\(223\) 1572.78 0.472294 0.236147 0.971717i \(-0.424115\pi\)
0.236147 + 0.971717i \(0.424115\pi\)
\(224\) 0 0
\(225\) −834.881 −0.247372
\(226\) 0 0
\(227\) − 1492.51i − 0.436394i −0.975905 0.218197i \(-0.929982\pi\)
0.975905 0.218197i \(-0.0700176\pi\)
\(228\) 0 0
\(229\) − 6163.31i − 1.77853i −0.457393 0.889265i \(-0.651217\pi\)
0.457393 0.889265i \(-0.348783\pi\)
\(230\) 0 0
\(231\) −3437.64 −0.979134
\(232\) 0 0
\(233\) 2540.35 0.714266 0.357133 0.934054i \(-0.383754\pi\)
0.357133 + 0.934054i \(0.383754\pi\)
\(234\) 0 0
\(235\) 1900.95i 0.527677i
\(236\) 0 0
\(237\) − 128.499i − 0.0352190i
\(238\) 0 0
\(239\) −1339.25 −0.362465 −0.181232 0.983440i \(-0.558009\pi\)
−0.181232 + 0.983440i \(0.558009\pi\)
\(240\) 0 0
\(241\) 2542.71 0.679627 0.339814 0.940493i \(-0.389636\pi\)
0.339814 + 0.940493i \(0.389636\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) − 4269.91i − 1.11345i
\(246\) 0 0
\(247\) 4595.87 1.18392
\(248\) 0 0
\(249\) −3710.00 −0.944223
\(250\) 0 0
\(251\) − 4701.45i − 1.18228i −0.806568 0.591141i \(-0.798678\pi\)
0.806568 0.591141i \(-0.201322\pi\)
\(252\) 0 0
\(253\) − 4516.05i − 1.12222i
\(254\) 0 0
\(255\) 1667.22 0.409432
\(256\) 0 0
\(257\) −863.884 −0.209679 −0.104840 0.994489i \(-0.533433\pi\)
−0.104840 + 0.994489i \(0.533433\pi\)
\(258\) 0 0
\(259\) − 6103.41i − 1.46428i
\(260\) 0 0
\(261\) 1327.06i 0.314724i
\(262\) 0 0
\(263\) 7091.36 1.66263 0.831315 0.555801i \(-0.187588\pi\)
0.831315 + 0.555801i \(0.187588\pi\)
\(264\) 0 0
\(265\) 579.290 0.134285
\(266\) 0 0
\(267\) 3975.88i 0.911311i
\(268\) 0 0
\(269\) 395.596i 0.0896650i 0.998995 + 0.0448325i \(0.0142754\pi\)
−0.998995 + 0.0448325i \(0.985725\pi\)
\(270\) 0 0
\(271\) 5532.21 1.24007 0.620033 0.784576i \(-0.287119\pi\)
0.620033 + 0.784576i \(0.287119\pi\)
\(272\) 0 0
\(273\) 8164.58 1.81005
\(274\) 0 0
\(275\) − 3212.20i − 0.704373i
\(276\) 0 0
\(277\) 3830.31i 0.830834i 0.909631 + 0.415417i \(0.136364\pi\)
−0.909631 + 0.415417i \(0.863636\pi\)
\(278\) 0 0
\(279\) −911.009 −0.195486
\(280\) 0 0
\(281\) 5283.17 1.12159 0.560797 0.827954i \(-0.310495\pi\)
0.560797 + 0.827954i \(0.310495\pi\)
\(282\) 0 0
\(283\) − 4639.76i − 0.974577i −0.873241 0.487288i \(-0.837986\pi\)
0.873241 0.487288i \(-0.162014\pi\)
\(284\) 0 0
\(285\) − 951.836i − 0.197831i
\(286\) 0 0
\(287\) −7856.33 −1.61584
\(288\) 0 0
\(289\) 4667.94 0.950119
\(290\) 0 0
\(291\) − 1517.29i − 0.305654i
\(292\) 0 0
\(293\) 6022.12i 1.20074i 0.799723 + 0.600369i \(0.204980\pi\)
−0.799723 + 0.600369i \(0.795020\pi\)
\(294\) 0 0
\(295\) −601.609 −0.118736
\(296\) 0 0
\(297\) −934.940 −0.182662
\(298\) 0 0
\(299\) 10725.9i 2.07456i
\(300\) 0 0
\(301\) 6614.44i 1.26661i
\(302\) 0 0
\(303\) 3845.92 0.729183
\(304\) 0 0
\(305\) −4075.42 −0.765107
\(306\) 0 0
\(307\) 2998.82i 0.557497i 0.960364 + 0.278749i \(0.0899196\pi\)
−0.960364 + 0.278749i \(0.910080\pi\)
\(308\) 0 0
\(309\) 484.685i 0.0892323i
\(310\) 0 0
\(311\) −2403.97 −0.438318 −0.219159 0.975689i \(-0.570331\pi\)
−0.219159 + 0.975689i \(0.570331\pi\)
\(312\) 0 0
\(313\) 5845.75 1.05566 0.527830 0.849350i \(-0.323006\pi\)
0.527830 + 0.849350i \(0.323006\pi\)
\(314\) 0 0
\(315\) − 1690.94i − 0.302456i
\(316\) 0 0
\(317\) − 8917.38i − 1.57997i −0.613127 0.789984i \(-0.710089\pi\)
0.613127 0.789984i \(-0.289911\pi\)
\(318\) 0 0
\(319\) −5105.86 −0.896154
\(320\) 0 0
\(321\) 1445.18 0.251283
\(322\) 0 0
\(323\) − 5469.88i − 0.942267i
\(324\) 0 0
\(325\) 7629.15i 1.30212i
\(326\) 0 0
\(327\) −859.910 −0.145422
\(328\) 0 0
\(329\) 11079.5 1.85664
\(330\) 0 0
\(331\) 9011.29i 1.49639i 0.663478 + 0.748196i \(0.269080\pi\)
−0.663478 + 0.748196i \(0.730920\pi\)
\(332\) 0 0
\(333\) − 1659.96i − 0.273168i
\(334\) 0 0
\(335\) 1796.81 0.293045
\(336\) 0 0
\(337\) 4516.35 0.730033 0.365017 0.931001i \(-0.381063\pi\)
0.365017 + 0.931001i \(0.381063\pi\)
\(338\) 0 0
\(339\) − 4191.88i − 0.671598i
\(340\) 0 0
\(341\) − 3505.10i − 0.556633i
\(342\) 0 0
\(343\) −13536.4 −2.13090
\(344\) 0 0
\(345\) 2221.40 0.346656
\(346\) 0 0
\(347\) 3651.48i 0.564904i 0.959281 + 0.282452i \(0.0911478\pi\)
−0.959281 + 0.282452i \(0.908852\pi\)
\(348\) 0 0
\(349\) − 3250.36i − 0.498532i −0.968435 0.249266i \(-0.919811\pi\)
0.968435 0.249266i \(-0.0801894\pi\)
\(350\) 0 0
\(351\) 2220.54 0.337674
\(352\) 0 0
\(353\) 592.345 0.0893125 0.0446563 0.999002i \(-0.485781\pi\)
0.0446563 + 0.999002i \(0.485781\pi\)
\(354\) 0 0
\(355\) − 4542.36i − 0.679108i
\(356\) 0 0
\(357\) − 9717.26i − 1.44059i
\(358\) 0 0
\(359\) −3443.48 −0.506239 −0.253120 0.967435i \(-0.581457\pi\)
−0.253120 + 0.967435i \(0.581457\pi\)
\(360\) 0 0
\(361\) 3736.17 0.544711
\(362\) 0 0
\(363\) 395.826i 0.0572327i
\(364\) 0 0
\(365\) − 1709.30i − 0.245120i
\(366\) 0 0
\(367\) 3297.39 0.468998 0.234499 0.972116i \(-0.424655\pi\)
0.234499 + 0.972116i \(0.424655\pi\)
\(368\) 0 0
\(369\) −2136.70 −0.301442
\(370\) 0 0
\(371\) − 3376.35i − 0.472483i
\(372\) 0 0
\(373\) 50.1819i 0.00696601i 0.999994 + 0.00348300i \(0.00110868\pi\)
−0.999994 + 0.00348300i \(0.998891\pi\)
\(374\) 0 0
\(375\) 3709.16 0.510774
\(376\) 0 0
\(377\) 12126.7 1.65665
\(378\) 0 0
\(379\) 6769.28i 0.917453i 0.888578 + 0.458726i \(0.151694\pi\)
−0.888578 + 0.458726i \(0.848306\pi\)
\(380\) 0 0
\(381\) 1331.49i 0.179040i
\(382\) 0 0
\(383\) 4208.46 0.561468 0.280734 0.959786i \(-0.409422\pi\)
0.280734 + 0.959786i \(0.409422\pi\)
\(384\) 0 0
\(385\) 6505.88 0.861221
\(386\) 0 0
\(387\) 1798.94i 0.236293i
\(388\) 0 0
\(389\) 2490.47i 0.324607i 0.986741 + 0.162303i \(0.0518923\pi\)
−0.986741 + 0.162303i \(0.948108\pi\)
\(390\) 0 0
\(391\) 12765.6 1.65112
\(392\) 0 0
\(393\) −4096.70 −0.525830
\(394\) 0 0
\(395\) 243.190i 0.0309777i
\(396\) 0 0
\(397\) 7905.51i 0.999411i 0.866195 + 0.499706i \(0.166558\pi\)
−0.866195 + 0.499706i \(0.833442\pi\)
\(398\) 0 0
\(399\) −5547.71 −0.696072
\(400\) 0 0
\(401\) −10389.4 −1.29382 −0.646911 0.762566i \(-0.723939\pi\)
−0.646911 + 0.762566i \(0.723939\pi\)
\(402\) 0 0
\(403\) 8324.81i 1.02900i
\(404\) 0 0
\(405\) − 459.888i − 0.0564248i
\(406\) 0 0
\(407\) 6386.66 0.777826
\(408\) 0 0
\(409\) −13448.1 −1.62583 −0.812917 0.582379i \(-0.802122\pi\)
−0.812917 + 0.582379i \(0.802122\pi\)
\(410\) 0 0
\(411\) − 3669.17i − 0.440357i
\(412\) 0 0
\(413\) 3506.44i 0.417773i
\(414\) 0 0
\(415\) 7021.33 0.830515
\(416\) 0 0
\(417\) 6530.11 0.766860
\(418\) 0 0
\(419\) − 5191.34i − 0.605283i −0.953105 0.302641i \(-0.902132\pi\)
0.953105 0.302641i \(-0.0978684\pi\)
\(420\) 0 0
\(421\) − 5061.52i − 0.585946i −0.956121 0.292973i \(-0.905355\pi\)
0.956121 0.292973i \(-0.0946447\pi\)
\(422\) 0 0
\(423\) 3013.32 0.346365
\(424\) 0 0
\(425\) 9080.00 1.03634
\(426\) 0 0
\(427\) 23753.3i 2.69204i
\(428\) 0 0
\(429\) 8543.49i 0.961501i
\(430\) 0 0
\(431\) −3005.64 −0.335909 −0.167954 0.985795i \(-0.553716\pi\)
−0.167954 + 0.985795i \(0.553716\pi\)
\(432\) 0 0
\(433\) 5895.88 0.654360 0.327180 0.944962i \(-0.393902\pi\)
0.327180 + 0.944962i \(0.393902\pi\)
\(434\) 0 0
\(435\) − 2511.52i − 0.276824i
\(436\) 0 0
\(437\) − 7288.07i − 0.797794i
\(438\) 0 0
\(439\) 11556.8 1.25644 0.628218 0.778038i \(-0.283785\pi\)
0.628218 + 0.778038i \(0.283785\pi\)
\(440\) 0 0
\(441\) −6768.52 −0.730863
\(442\) 0 0
\(443\) − 14007.8i − 1.50233i −0.660117 0.751163i \(-0.729493\pi\)
0.660117 0.751163i \(-0.270507\pi\)
\(444\) 0 0
\(445\) − 7524.53i − 0.801566i
\(446\) 0 0
\(447\) −8627.65 −0.912917
\(448\) 0 0
\(449\) 3783.06 0.397625 0.198813 0.980038i \(-0.436291\pi\)
0.198813 + 0.980038i \(0.436291\pi\)
\(450\) 0 0
\(451\) − 8220.94i − 0.858335i
\(452\) 0 0
\(453\) 6990.08i 0.724994i
\(454\) 0 0
\(455\) −15451.8 −1.59207
\(456\) 0 0
\(457\) −1545.53 −0.158198 −0.0790992 0.996867i \(-0.525204\pi\)
−0.0790992 + 0.996867i \(0.525204\pi\)
\(458\) 0 0
\(459\) − 2642.82i − 0.268750i
\(460\) 0 0
\(461\) 12730.2i 1.28613i 0.765811 + 0.643065i \(0.222338\pi\)
−0.765811 + 0.643065i \(0.777662\pi\)
\(462\) 0 0
\(463\) −19656.4 −1.97303 −0.986513 0.163682i \(-0.947663\pi\)
−0.986513 + 0.163682i \(0.947663\pi\)
\(464\) 0 0
\(465\) 1724.12 0.171945
\(466\) 0 0
\(467\) − 6016.26i − 0.596144i −0.954543 0.298072i \(-0.903656\pi\)
0.954543 0.298072i \(-0.0963436\pi\)
\(468\) 0 0
\(469\) − 10472.6i − 1.03108i
\(470\) 0 0
\(471\) −7341.92 −0.718255
\(472\) 0 0
\(473\) −6921.41 −0.672826
\(474\) 0 0
\(475\) − 5183.89i − 0.500743i
\(476\) 0 0
\(477\) − 918.272i − 0.0881442i
\(478\) 0 0
\(479\) 13890.6 1.32501 0.662505 0.749057i \(-0.269493\pi\)
0.662505 + 0.749057i \(0.269493\pi\)
\(480\) 0 0
\(481\) −15168.7 −1.43791
\(482\) 0 0
\(483\) − 12947.3i − 1.21971i
\(484\) 0 0
\(485\) 2871.54i 0.268846i
\(486\) 0 0
\(487\) −9314.23 −0.866669 −0.433335 0.901233i \(-0.642663\pi\)
−0.433335 + 0.901233i \(0.642663\pi\)
\(488\) 0 0
\(489\) 4310.46 0.398621
\(490\) 0 0
\(491\) 7253.34i 0.666677i 0.942807 + 0.333339i \(0.108175\pi\)
−0.942807 + 0.333339i \(0.891825\pi\)
\(492\) 0 0
\(493\) − 14432.9i − 1.31851i
\(494\) 0 0
\(495\) 1769.42 0.160665
\(496\) 0 0
\(497\) −26474.8 −2.38945
\(498\) 0 0
\(499\) 16208.2i 1.45407i 0.686602 + 0.727034i \(0.259102\pi\)
−0.686602 + 0.727034i \(0.740898\pi\)
\(500\) 0 0
\(501\) 5308.25i 0.473363i
\(502\) 0 0
\(503\) 2182.04 0.193425 0.0967123 0.995312i \(-0.469167\pi\)
0.0967123 + 0.995312i \(0.469167\pi\)
\(504\) 0 0
\(505\) −7278.58 −0.641371
\(506\) 0 0
\(507\) − 13700.3i − 1.20010i
\(508\) 0 0
\(509\) 1795.97i 0.156395i 0.996938 + 0.0781974i \(0.0249164\pi\)
−0.996938 + 0.0781974i \(0.975084\pi\)
\(510\) 0 0
\(511\) −9962.51 −0.862457
\(512\) 0 0
\(513\) −1508.82 −0.129856
\(514\) 0 0
\(515\) − 917.288i − 0.0784865i
\(516\) 0 0
\(517\) 11593.7i 0.986249i
\(518\) 0 0
\(519\) 11693.6 0.989000
\(520\) 0 0
\(521\) −20946.1 −1.76135 −0.880676 0.473718i \(-0.842912\pi\)
−0.880676 + 0.473718i \(0.842912\pi\)
\(522\) 0 0
\(523\) 5363.64i 0.448443i 0.974538 + 0.224221i \(0.0719838\pi\)
−0.974538 + 0.224221i \(0.928016\pi\)
\(524\) 0 0
\(525\) − 9209.19i − 0.765566i
\(526\) 0 0
\(527\) 9907.96 0.818970
\(528\) 0 0
\(529\) 4841.96 0.397958
\(530\) 0 0
\(531\) 953.652i 0.0779378i
\(532\) 0 0
\(533\) 19525.2i 1.58674i
\(534\) 0 0
\(535\) −2735.06 −0.221022
\(536\) 0 0
\(537\) −12440.2 −0.999693
\(538\) 0 0
\(539\) − 26041.8i − 2.08108i
\(540\) 0 0
\(541\) − 3431.20i − 0.272678i −0.990662 0.136339i \(-0.956466\pi\)
0.990662 0.136339i \(-0.0435337\pi\)
\(542\) 0 0
\(543\) 3312.65 0.261803
\(544\) 0 0
\(545\) 1627.42 0.127910
\(546\) 0 0
\(547\) 19449.0i 1.52026i 0.649773 + 0.760128i \(0.274864\pi\)
−0.649773 + 0.760128i \(0.725136\pi\)
\(548\) 0 0
\(549\) 6460.22i 0.502214i
\(550\) 0 0
\(551\) −8239.91 −0.637082
\(552\) 0 0
\(553\) 1417.41 0.108996
\(554\) 0 0
\(555\) 3141.54i 0.240272i
\(556\) 0 0
\(557\) − 9898.44i − 0.752981i −0.926420 0.376491i \(-0.877131\pi\)
0.926420 0.376491i \(-0.122869\pi\)
\(558\) 0 0
\(559\) 16438.7 1.24380
\(560\) 0 0
\(561\) 10168.2 0.765246
\(562\) 0 0
\(563\) 13809.8i 1.03378i 0.856053 + 0.516888i \(0.172909\pi\)
−0.856053 + 0.516888i \(0.827091\pi\)
\(564\) 0 0
\(565\) 7933.32i 0.590721i
\(566\) 0 0
\(567\) −2680.42 −0.198531
\(568\) 0 0
\(569\) −123.053 −0.00906615 −0.00453308 0.999990i \(-0.501443\pi\)
−0.00453308 + 0.999990i \(0.501443\pi\)
\(570\) 0 0
\(571\) − 2540.72i − 0.186210i −0.995656 0.0931049i \(-0.970321\pi\)
0.995656 0.0931049i \(-0.0296792\pi\)
\(572\) 0 0
\(573\) − 105.206i − 0.00767022i
\(574\) 0 0
\(575\) 12098.2 0.877443
\(576\) 0 0
\(577\) −15618.2 −1.12685 −0.563427 0.826166i \(-0.690517\pi\)
−0.563427 + 0.826166i \(0.690517\pi\)
\(578\) 0 0
\(579\) 1847.65i 0.132618i
\(580\) 0 0
\(581\) − 40923.3i − 2.92218i
\(582\) 0 0
\(583\) 3533.04 0.250984
\(584\) 0 0
\(585\) −4202.46 −0.297009
\(586\) 0 0
\(587\) 1809.56i 0.127238i 0.997974 + 0.0636188i \(0.0202642\pi\)
−0.997974 + 0.0636188i \(0.979736\pi\)
\(588\) 0 0
\(589\) − 5656.58i − 0.395714i
\(590\) 0 0
\(591\) 1768.91 0.123119
\(592\) 0 0
\(593\) 898.229 0.0622021 0.0311010 0.999516i \(-0.490099\pi\)
0.0311010 + 0.999516i \(0.490099\pi\)
\(594\) 0 0
\(595\) 18390.3i 1.26711i
\(596\) 0 0
\(597\) − 14441.5i − 0.990033i
\(598\) 0 0
\(599\) 22990.9 1.56825 0.784125 0.620603i \(-0.213112\pi\)
0.784125 + 0.620603i \(0.213112\pi\)
\(600\) 0 0
\(601\) −26893.7 −1.82532 −0.912661 0.408717i \(-0.865976\pi\)
−0.912661 + 0.408717i \(0.865976\pi\)
\(602\) 0 0
\(603\) − 2848.24i − 0.192354i
\(604\) 0 0
\(605\) − 749.118i − 0.0503405i
\(606\) 0 0
\(607\) 6306.93 0.421730 0.210865 0.977515i \(-0.432372\pi\)
0.210865 + 0.977515i \(0.432372\pi\)
\(608\) 0 0
\(609\) −14638.2 −0.974008
\(610\) 0 0
\(611\) − 27535.7i − 1.82320i
\(612\) 0 0
\(613\) − 2612.97i − 0.172164i −0.996288 0.0860822i \(-0.972565\pi\)
0.996288 0.0860822i \(-0.0274348\pi\)
\(614\) 0 0
\(615\) 4043.80 0.265141
\(616\) 0 0
\(617\) −2803.88 −0.182949 −0.0914747 0.995807i \(-0.529158\pi\)
−0.0914747 + 0.995807i \(0.529158\pi\)
\(618\) 0 0
\(619\) 10547.1i 0.684849i 0.939545 + 0.342425i \(0.111248\pi\)
−0.939545 + 0.342425i \(0.888752\pi\)
\(620\) 0 0
\(621\) − 3521.30i − 0.227544i
\(622\) 0 0
\(623\) −43856.2 −2.82032
\(624\) 0 0
\(625\) 4575.82 0.292852
\(626\) 0 0
\(627\) − 5805.17i − 0.369755i
\(628\) 0 0
\(629\) 18053.3i 1.14441i
\(630\) 0 0
\(631\) −14161.4 −0.893431 −0.446716 0.894676i \(-0.647406\pi\)
−0.446716 + 0.894676i \(0.647406\pi\)
\(632\) 0 0
\(633\) 9883.04 0.620562
\(634\) 0 0
\(635\) − 2519.90i − 0.157479i
\(636\) 0 0
\(637\) 61850.8i 3.84713i
\(638\) 0 0
\(639\) −7200.40 −0.445764
\(640\) 0 0
\(641\) 5622.61 0.346458 0.173229 0.984882i \(-0.444580\pi\)
0.173229 + 0.984882i \(0.444580\pi\)
\(642\) 0 0
\(643\) 29438.7i 1.80552i 0.430146 + 0.902759i \(0.358462\pi\)
−0.430146 + 0.902759i \(0.641538\pi\)
\(644\) 0 0
\(645\) − 3404.57i − 0.207837i
\(646\) 0 0
\(647\) −4607.39 −0.279962 −0.139981 0.990154i \(-0.544704\pi\)
−0.139981 + 0.990154i \(0.544704\pi\)
\(648\) 0 0
\(649\) −3669.17 −0.221922
\(650\) 0 0
\(651\) − 10048.9i − 0.604991i
\(652\) 0 0
\(653\) − 16634.1i − 0.996850i −0.866933 0.498425i \(-0.833912\pi\)
0.866933 0.498425i \(-0.166088\pi\)
\(654\) 0 0
\(655\) 7753.19 0.462507
\(656\) 0 0
\(657\) −2709.52 −0.160896
\(658\) 0 0
\(659\) 20619.9i 1.21887i 0.792835 + 0.609437i \(0.208604\pi\)
−0.792835 + 0.609437i \(0.791396\pi\)
\(660\) 0 0
\(661\) 881.112i 0.0518476i 0.999664 + 0.0259238i \(0.00825273\pi\)
−0.999664 + 0.0259238i \(0.991747\pi\)
\(662\) 0 0
\(663\) −24150.1 −1.41465
\(664\) 0 0
\(665\) 10499.3 0.612248
\(666\) 0 0
\(667\) − 19230.4i − 1.11635i
\(668\) 0 0
\(669\) 4718.35i 0.272679i
\(670\) 0 0
\(671\) −24855.6 −1.43002
\(672\) 0 0
\(673\) 8943.86 0.512274 0.256137 0.966641i \(-0.417550\pi\)
0.256137 + 0.966641i \(0.417550\pi\)
\(674\) 0 0
\(675\) − 2504.64i − 0.142820i
\(676\) 0 0
\(677\) 21748.7i 1.23467i 0.786702 + 0.617333i \(0.211787\pi\)
−0.786702 + 0.617333i \(0.788213\pi\)
\(678\) 0 0
\(679\) 16736.6 0.945937
\(680\) 0 0
\(681\) 4477.53 0.251952
\(682\) 0 0
\(683\) − 7922.37i − 0.443838i −0.975065 0.221919i \(-0.928768\pi\)
0.975065 0.221919i \(-0.0712320\pi\)
\(684\) 0 0
\(685\) 6944.06i 0.387327i
\(686\) 0 0
\(687\) 18489.9 1.02683
\(688\) 0 0
\(689\) −8391.18 −0.463975
\(690\) 0 0
\(691\) − 177.517i − 0.00977288i −0.999988 0.00488644i \(-0.998445\pi\)
0.999988 0.00488644i \(-0.00155541\pi\)
\(692\) 0 0
\(693\) − 10312.9i − 0.565303i
\(694\) 0 0
\(695\) −12358.5 −0.674511
\(696\) 0 0
\(697\) 23238.3 1.26286
\(698\) 0 0
\(699\) 7621.05i 0.412382i
\(700\) 0 0
\(701\) 767.980i 0.0413783i 0.999786 + 0.0206891i \(0.00658603\pi\)
−0.999786 + 0.0206891i \(0.993414\pi\)
\(702\) 0 0
\(703\) 10306.9 0.552961
\(704\) 0 0
\(705\) −5702.84 −0.304654
\(706\) 0 0
\(707\) 42422.7i 2.25667i
\(708\) 0 0
\(709\) − 31634.2i − 1.67566i −0.545928 0.837832i \(-0.683823\pi\)
0.545928 0.837832i \(-0.316177\pi\)
\(710\) 0 0
\(711\) 385.496 0.0203337
\(712\) 0 0
\(713\) 13201.4 0.693401
\(714\) 0 0
\(715\) − 16168.9i − 0.845712i
\(716\) 0 0
\(717\) − 4017.76i − 0.209269i
\(718\) 0 0
\(719\) 17351.7 0.900011 0.450005 0.893026i \(-0.351422\pi\)
0.450005 + 0.893026i \(0.351422\pi\)
\(720\) 0 0
\(721\) −5346.35 −0.276156
\(722\) 0 0
\(723\) 7628.12i 0.392383i
\(724\) 0 0
\(725\) − 13678.2i − 0.700686i
\(726\) 0 0
\(727\) −16364.6 −0.834842 −0.417421 0.908713i \(-0.637066\pi\)
−0.417421 + 0.908713i \(0.637066\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) − 19564.9i − 0.989925i
\(732\) 0 0
\(733\) − 23552.7i − 1.18682i −0.804900 0.593410i \(-0.797781\pi\)
0.804900 0.593410i \(-0.202219\pi\)
\(734\) 0 0
\(735\) 12809.7 0.642849
\(736\) 0 0
\(737\) 10958.6 0.547713
\(738\) 0 0
\(739\) − 3519.05i − 0.175169i −0.996157 0.0875847i \(-0.972085\pi\)
0.996157 0.0875847i \(-0.0279149\pi\)
\(740\) 0 0
\(741\) 13787.6i 0.683537i
\(742\) 0 0
\(743\) 20752.4 1.02467 0.512336 0.858785i \(-0.328780\pi\)
0.512336 + 0.858785i \(0.328780\pi\)
\(744\) 0 0
\(745\) 16328.2 0.802979
\(746\) 0 0
\(747\) − 11130.0i − 0.545148i
\(748\) 0 0
\(749\) 15941.1i 0.777671i
\(750\) 0 0
\(751\) −3679.08 −0.178764 −0.0893818 0.995997i \(-0.528489\pi\)
−0.0893818 + 0.995997i \(0.528489\pi\)
\(752\) 0 0
\(753\) 14104.3 0.682591
\(754\) 0 0
\(755\) − 13229.0i − 0.637687i
\(756\) 0 0
\(757\) 527.275i 0.0253159i 0.999920 + 0.0126579i \(0.00402926\pi\)
−0.999920 + 0.0126579i \(0.995971\pi\)
\(758\) 0 0
\(759\) 13548.2 0.647914
\(760\) 0 0
\(761\) 3231.54 0.153933 0.0769666 0.997034i \(-0.475477\pi\)
0.0769666 + 0.997034i \(0.475477\pi\)
\(762\) 0 0
\(763\) − 9485.29i − 0.450053i
\(764\) 0 0
\(765\) 5001.65i 0.236386i
\(766\) 0 0
\(767\) 8714.48 0.410250
\(768\) 0 0
\(769\) 13681.7 0.641581 0.320791 0.947150i \(-0.396051\pi\)
0.320791 + 0.947150i \(0.396051\pi\)
\(770\) 0 0
\(771\) − 2591.65i − 0.121058i
\(772\) 0 0
\(773\) 9216.63i 0.428847i 0.976741 + 0.214424i \(0.0687873\pi\)
−0.976741 + 0.214424i \(0.931213\pi\)
\(774\) 0 0
\(775\) 9389.92 0.435221
\(776\) 0 0
\(777\) 18310.2 0.845400
\(778\) 0 0
\(779\) − 13267.1i − 0.610196i
\(780\) 0 0
\(781\) − 27703.5i − 1.26928i
\(782\) 0 0
\(783\) −3981.18 −0.181706
\(784\) 0 0
\(785\) 13894.9 0.631759
\(786\) 0 0
\(787\) − 33169.0i − 1.50235i −0.660104 0.751174i \(-0.729488\pi\)
0.660104 0.751174i \(-0.270512\pi\)
\(788\) 0 0
\(789\) 21274.1i 0.959920i
\(790\) 0 0
\(791\) 46238.8 2.07846
\(792\) 0 0
\(793\) 59033.6 2.64356
\(794\) 0 0
\(795\) 1737.87i 0.0775294i
\(796\) 0 0
\(797\) − 23781.9i − 1.05696i −0.848945 0.528481i \(-0.822762\pi\)
0.848945 0.528481i \(-0.177238\pi\)
\(798\) 0 0
\(799\) −32772.3 −1.45106
\(800\) 0 0
\(801\) −11927.6 −0.526145
\(802\) 0 0
\(803\) − 10424.9i − 0.458139i
\(804\) 0 0
\(805\) 24503.3i 1.07283i
\(806\) 0 0
\(807\) −1186.79 −0.0517681
\(808\) 0 0
\(809\) −29842.4 −1.29691 −0.648456 0.761252i \(-0.724585\pi\)
−0.648456 + 0.761252i \(0.724585\pi\)
\(810\) 0 0
\(811\) 19074.5i 0.825888i 0.910756 + 0.412944i \(0.135500\pi\)
−0.910756 + 0.412944i \(0.864500\pi\)
\(812\) 0 0
\(813\) 16596.6i 0.715952i
\(814\) 0 0
\(815\) −8157.74 −0.350617
\(816\) 0 0
\(817\) −11169.9 −0.478316
\(818\) 0 0
\(819\) 24493.7i 1.04503i
\(820\) 0 0
\(821\) 7493.97i 0.318564i 0.987233 + 0.159282i \(0.0509180\pi\)
−0.987233 + 0.159282i \(0.949082\pi\)
\(822\) 0 0
\(823\) −1780.90 −0.0754294 −0.0377147 0.999289i \(-0.512008\pi\)
−0.0377147 + 0.999289i \(0.512008\pi\)
\(824\) 0 0
\(825\) 9636.59 0.406670
\(826\) 0 0
\(827\) − 10860.1i − 0.456640i −0.973586 0.228320i \(-0.926677\pi\)
0.973586 0.228320i \(-0.0733232\pi\)
\(828\) 0 0
\(829\) − 34105.1i − 1.42885i −0.699711 0.714426i \(-0.746688\pi\)
0.699711 0.714426i \(-0.253312\pi\)
\(830\) 0 0
\(831\) −11490.9 −0.479682
\(832\) 0 0
\(833\) 73613.1 3.06188
\(834\) 0 0
\(835\) − 10046.1i − 0.416358i
\(836\) 0 0
\(837\) − 2733.03i − 0.112864i
\(838\) 0 0
\(839\) −32688.9 −1.34511 −0.672555 0.740047i \(-0.734803\pi\)
−0.672555 + 0.740047i \(0.734803\pi\)
\(840\) 0 0
\(841\) 2647.12 0.108537
\(842\) 0 0
\(843\) 15849.5i 0.647552i
\(844\) 0 0
\(845\) 25928.4i 1.05558i
\(846\) 0 0
\(847\) −4366.18 −0.177124
\(848\) 0 0
\(849\) 13919.3 0.562672
\(850\) 0 0
\(851\) 24054.3i 0.968943i
\(852\) 0 0
\(853\) − 17391.2i − 0.698080i −0.937108 0.349040i \(-0.886508\pi\)
0.937108 0.349040i \(-0.113492\pi\)
\(854\) 0 0
\(855\) 2855.51 0.114218
\(856\) 0 0
\(857\) −4182.34 −0.166705 −0.0833525 0.996520i \(-0.526563\pi\)
−0.0833525 + 0.996520i \(0.526563\pi\)
\(858\) 0 0
\(859\) − 20585.0i − 0.817639i −0.912615 0.408820i \(-0.865941\pi\)
0.912615 0.408820i \(-0.134059\pi\)
\(860\) 0 0
\(861\) − 23569.0i − 0.932903i
\(862\) 0 0
\(863\) −14159.7 −0.558518 −0.279259 0.960216i \(-0.590089\pi\)
−0.279259 + 0.960216i \(0.590089\pi\)
\(864\) 0 0
\(865\) −22130.6 −0.869900
\(866\) 0 0
\(867\) 14003.8i 0.548552i
\(868\) 0 0
\(869\) 1483.19i 0.0578986i
\(870\) 0 0
\(871\) −26027.2 −1.01251
\(872\) 0 0
\(873\) 4551.88 0.176469
\(874\) 0 0
\(875\) 40914.1i 1.58074i
\(876\) 0 0
\(877\) − 5156.38i − 0.198539i −0.995061 0.0992695i \(-0.968349\pi\)
0.995061 0.0992695i \(-0.0316506\pi\)
\(878\) 0 0
\(879\) −18066.4 −0.693246
\(880\) 0 0
\(881\) 841.065 0.0321637 0.0160818 0.999871i \(-0.494881\pi\)
0.0160818 + 0.999871i \(0.494881\pi\)
\(882\) 0 0
\(883\) 7845.99i 0.299024i 0.988760 + 0.149512i \(0.0477703\pi\)
−0.988760 + 0.149512i \(0.952230\pi\)
\(884\) 0 0
\(885\) − 1804.83i − 0.0685521i
\(886\) 0 0
\(887\) −38781.9 −1.46806 −0.734029 0.679118i \(-0.762363\pi\)
−0.734029 + 0.679118i \(0.762363\pi\)
\(888\) 0 0
\(889\) −14687.1 −0.554092
\(890\) 0 0
\(891\) − 2804.82i − 0.105460i
\(892\) 0 0
\(893\) 18710.1i 0.701131i
\(894\) 0 0
\(895\) 23543.7 0.879305
\(896\) 0 0
\(897\) −32177.6 −1.19775
\(898\) 0 0
\(899\) − 14925.5i − 0.553719i
\(900\) 0 0
\(901\) 9986.95i 0.369271i
\(902\) 0 0
\(903\) −19843.3 −0.731278
\(904\) 0 0
\(905\) −6269.33 −0.230276
\(906\) 0 0
\(907\) − 36431.2i − 1.33371i −0.745187 0.666856i \(-0.767640\pi\)
0.745187 0.666856i \(-0.232360\pi\)
\(908\) 0 0
\(909\) 11537.8i 0.420994i
\(910\) 0 0
\(911\) 51077.9 1.85762 0.928808 0.370562i \(-0.120835\pi\)
0.928808 + 0.370562i \(0.120835\pi\)
\(912\) 0 0
\(913\) 42822.6 1.55227
\(914\) 0 0
\(915\) − 12226.3i − 0.441735i
\(916\) 0 0
\(917\) − 45188.9i − 1.62734i
\(918\) 0 0
\(919\) 37080.1 1.33097 0.665484 0.746412i \(-0.268225\pi\)
0.665484 + 0.746412i \(0.268225\pi\)
\(920\) 0 0
\(921\) −8996.46 −0.321871
\(922\) 0 0
\(923\) 65797.3i 2.34642i
\(924\) 0 0
\(925\) 17109.4i 0.608167i
\(926\) 0 0
\(927\) −1454.06 −0.0515183
\(928\) 0 0
\(929\) 38259.5 1.35119 0.675595 0.737273i \(-0.263887\pi\)
0.675595 + 0.737273i \(0.263887\pi\)
\(930\) 0 0
\(931\) − 42026.7i − 1.47945i
\(932\) 0 0
\(933\) − 7211.92i − 0.253063i
\(934\) 0 0
\(935\) −19243.8 −0.673091
\(936\) 0 0
\(937\) 4413.55 0.153879 0.0769394 0.997036i \(-0.475485\pi\)
0.0769394 + 0.997036i \(0.475485\pi\)
\(938\) 0 0
\(939\) 17537.3i 0.609486i
\(940\) 0 0
\(941\) 31511.2i 1.09164i 0.837902 + 0.545821i \(0.183782\pi\)
−0.837902 + 0.545821i \(0.816218\pi\)
\(942\) 0 0
\(943\) 30962.8 1.06923
\(944\) 0 0
\(945\) 5072.82 0.174623
\(946\) 0 0
\(947\) 23561.3i 0.808490i 0.914651 + 0.404245i \(0.132466\pi\)
−0.914651 + 0.404245i \(0.867534\pi\)
\(948\) 0 0
\(949\) 24759.6i 0.846925i
\(950\) 0 0
\(951\) 26752.1 0.912195
\(952\) 0 0
\(953\) 35039.7 1.19103 0.595513 0.803346i \(-0.296949\pi\)
0.595513 + 0.803346i \(0.296949\pi\)
\(954\) 0 0
\(955\) 199.107i 0.00674653i
\(956\) 0 0
\(957\) − 15317.6i − 0.517395i
\(958\) 0 0
\(959\) 40473.0 1.36282
\(960\) 0 0
\(961\) −19544.9 −0.656066
\(962\) 0 0
\(963\) 4335.53i 0.145078i
\(964\) 0 0
\(965\) − 3496.76i − 0.116647i
\(966\) 0 0
\(967\) −7975.38 −0.265223 −0.132612 0.991168i \(-0.542336\pi\)
−0.132612 + 0.991168i \(0.542336\pi\)
\(968\) 0 0
\(969\) 16409.6 0.544018
\(970\) 0 0
\(971\) − 21386.8i − 0.706834i −0.935466 0.353417i \(-0.885020\pi\)
0.935466 0.353417i \(-0.114980\pi\)
\(972\) 0 0
\(973\) 72030.7i 2.37328i
\(974\) 0 0
\(975\) −22887.4 −0.751779
\(976\) 0 0
\(977\) −40142.3 −1.31450 −0.657250 0.753673i \(-0.728280\pi\)
−0.657250 + 0.753673i \(0.728280\pi\)
\(978\) 0 0
\(979\) − 45891.5i − 1.49816i
\(980\) 0 0
\(981\) − 2579.73i − 0.0839597i
\(982\) 0 0
\(983\) 9205.44 0.298686 0.149343 0.988785i \(-0.452284\pi\)
0.149343 + 0.988785i \(0.452284\pi\)
\(984\) 0 0
\(985\) −3347.75 −0.108292
\(986\) 0 0
\(987\) 33238.6i 1.07193i
\(988\) 0 0
\(989\) − 26068.3i − 0.838144i
\(990\) 0 0
\(991\) −22082.2 −0.707834 −0.353917 0.935277i \(-0.615150\pi\)
−0.353917 + 0.935277i \(0.615150\pi\)
\(992\) 0 0
\(993\) −27033.9 −0.863942
\(994\) 0 0
\(995\) 27331.1i 0.870808i
\(996\) 0 0
\(997\) 7207.66i 0.228956i 0.993426 + 0.114478i \(0.0365195\pi\)
−0.993426 + 0.114478i \(0.963480\pi\)
\(998\) 0 0
\(999\) 4979.87 0.157714
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 384.4.d.f.193.6 yes 8
3.2 odd 2 1152.4.d.p.577.5 8
4.3 odd 2 inner 384.4.d.f.193.2 8
8.3 odd 2 inner 384.4.d.f.193.7 yes 8
8.5 even 2 inner 384.4.d.f.193.3 yes 8
12.11 even 2 1152.4.d.p.577.6 8
16.3 odd 4 768.4.a.v.1.2 4
16.5 even 4 768.4.a.v.1.3 4
16.11 odd 4 768.4.a.u.1.3 4
16.13 even 4 768.4.a.u.1.2 4
24.5 odd 2 1152.4.d.p.577.3 8
24.11 even 2 1152.4.d.p.577.4 8
48.5 odd 4 2304.4.a.by.1.2 4
48.11 even 4 2304.4.a.cb.1.2 4
48.29 odd 4 2304.4.a.cb.1.3 4
48.35 even 4 2304.4.a.by.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.f.193.2 8 4.3 odd 2 inner
384.4.d.f.193.3 yes 8 8.5 even 2 inner
384.4.d.f.193.6 yes 8 1.1 even 1 trivial
384.4.d.f.193.7 yes 8 8.3 odd 2 inner
768.4.a.u.1.2 4 16.13 even 4
768.4.a.u.1.3 4 16.11 odd 4
768.4.a.v.1.2 4 16.3 odd 4
768.4.a.v.1.3 4 16.5 even 4
1152.4.d.p.577.3 8 24.5 odd 2
1152.4.d.p.577.4 8 24.11 even 2
1152.4.d.p.577.5 8 3.2 odd 2
1152.4.d.p.577.6 8 12.11 even 2
2304.4.a.by.1.2 4 48.5 odd 4
2304.4.a.by.1.3 4 48.35 even 4
2304.4.a.cb.1.2 4 48.11 even 4
2304.4.a.cb.1.3 4 48.29 odd 4