Properties

Label 768.4.a.u.1.2
Level $768$
Weight $4$
Character 768.1
Self dual yes
Analytic conductor $45.313$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,4,Mod(1,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, 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("768.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.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: \(45.3134668844\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.9792.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 2x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.63019\) of defining polynomial
Character \(\chi\) \(=\) 768.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -5.67763 q^{5} +33.0917 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -5.67763 q^{5} +33.0917 q^{7} +9.00000 q^{9} -34.6274 q^{11} -82.2421 q^{13} +17.0329 q^{15} +97.8823 q^{17} +55.8823 q^{19} -99.2750 q^{21} -130.418 q^{23} -92.7645 q^{25} -27.0000 q^{27} +147.451 q^{29} +101.223 q^{31} +103.882 q^{33} -187.882 q^{35} +184.439 q^{37} +246.726 q^{39} -237.411 q^{41} -199.882 q^{43} -51.0987 q^{45} -334.813 q^{47} +752.058 q^{49} -293.647 q^{51} +102.030 q^{53} +196.602 q^{55} -167.647 q^{57} -105.961 q^{59} +717.803 q^{61} +297.825 q^{63} +466.940 q^{65} -316.471 q^{67} +391.255 q^{69} -800.045 q^{71} -301.058 q^{73} +278.294 q^{75} -1145.88 q^{77} -42.8329 q^{79} +81.0000 q^{81} -1236.67 q^{83} -555.739 q^{85} -442.354 q^{87} -1325.29 q^{89} -2721.53 q^{91} -303.670 q^{93} -317.279 q^{95} -505.765 q^{97} -311.647 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} + 36 q^{9} - 48 q^{11} + 120 q^{17} - 48 q^{19} + 172 q^{25} - 108 q^{27} + 144 q^{33} - 480 q^{35} + 408 q^{41} - 528 q^{43} + 836 q^{49} - 360 q^{51} + 144 q^{57} - 1872 q^{59} - 576 q^{65} - 2352 q^{67} + 968 q^{73} - 516 q^{75} + 324 q^{81} - 3408 q^{83} - 3672 q^{89} - 5184 q^{91} - 1480 q^{97} - 432 q^{99}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −5.67763 −0.507823 −0.253911 0.967227i \(-0.581717\pi\)
−0.253911 + 0.967227i \(0.581717\pi\)
\(6\) 0 0
\(7\) 33.0917 1.78678 0.893391 0.449280i \(-0.148320\pi\)
0.893391 + 0.449280i \(0.148320\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −34.6274 −0.949142 −0.474571 0.880217i \(-0.657397\pi\)
−0.474571 + 0.880217i \(0.657397\pi\)
\(12\) 0 0
\(13\) −82.2421 −1.75460 −0.877302 0.479939i \(-0.840659\pi\)
−0.877302 + 0.479939i \(0.840659\pi\)
\(14\) 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
\(19\) 55.8823 0.674751 0.337375 0.941370i \(-0.390461\pi\)
0.337375 + 0.941370i \(0.390461\pi\)
\(20\) 0 0
\(21\) −99.2750 −1.03160
\(22\) 0 0
\(23\) −130.418 −1.18235 −0.591176 0.806542i \(-0.701336\pi\)
−0.591176 + 0.806542i \(0.701336\pi\)
\(24\) 0 0
\(25\) −92.7645 −0.742116
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 147.451 0.944173 0.472086 0.881552i \(-0.343501\pi\)
0.472086 + 0.881552i \(0.343501\pi\)
\(30\) 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
\(35\) −187.882 −0.907368
\(36\) 0 0
\(37\) 184.439 0.819504 0.409752 0.912197i \(-0.365615\pi\)
0.409752 + 0.912197i \(0.365615\pi\)
\(38\) 0 0
\(39\) 246.726 1.01302
\(40\) 0 0
\(41\) −237.411 −0.904327 −0.452164 0.891935i \(-0.649348\pi\)
−0.452164 + 0.891935i \(0.649348\pi\)
\(42\) 0 0
\(43\) −199.882 −0.708878 −0.354439 0.935079i \(-0.615328\pi\)
−0.354439 + 0.935079i \(0.615328\pi\)
\(44\) 0 0
\(45\) −51.0987 −0.169274
\(46\) 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
\(51\) −293.647 −0.806250
\(52\) 0 0
\(53\) 102.030 0.264433 0.132216 0.991221i \(-0.457791\pi\)
0.132216 + 0.991221i \(0.457791\pi\)
\(54\) 0 0
\(55\) 196.602 0.481996
\(56\) 0 0
\(57\) −167.647 −0.389568
\(58\) 0 0
\(59\) −105.961 −0.233813 −0.116907 0.993143i \(-0.537298\pi\)
−0.116907 + 0.993143i \(0.537298\pi\)
\(60\) 0 0
\(61\) 717.803 1.50664 0.753321 0.657653i \(-0.228451\pi\)
0.753321 + 0.657653i \(0.228451\pi\)
\(62\) 0 0
\(63\) 297.825 0.595594
\(64\) 0 0
\(65\) 466.940 0.891028
\(66\) 0 0
\(67\) −316.471 −0.577061 −0.288530 0.957471i \(-0.593167\pi\)
−0.288530 + 0.957471i \(0.593167\pi\)
\(68\) 0 0
\(69\) 391.255 0.682632
\(70\) 0 0
\(71\) −800.045 −1.33729 −0.668647 0.743580i \(-0.733126\pi\)
−0.668647 + 0.743580i \(0.733126\pi\)
\(72\) 0 0
\(73\) −301.058 −0.482687 −0.241344 0.970440i \(-0.577588\pi\)
−0.241344 + 0.970440i \(0.577588\pi\)
\(74\) 0 0
\(75\) 278.294 0.428461
\(76\) 0 0
\(77\) −1145.88 −1.69591
\(78\) 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
\(83\) −1236.67 −1.63544 −0.817721 0.575614i \(-0.804763\pi\)
−0.817721 + 0.575614i \(0.804763\pi\)
\(84\) 0 0
\(85\) −555.739 −0.709158
\(86\) 0 0
\(87\) −442.354 −0.545119
\(88\) 0 0
\(89\) −1325.29 −1.57844 −0.789218 0.614113i \(-0.789514\pi\)
−0.789218 + 0.614113i \(0.789514\pi\)
\(90\) 0 0
\(91\) −2721.53 −3.13509
\(92\) 0 0
\(93\) −303.670 −0.338592
\(94\) 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
\(99\) −311.647 −0.316381
\(100\) 0 0
\(101\) −1281.97 −1.26298 −0.631491 0.775383i \(-0.717557\pi\)
−0.631491 + 0.775383i \(0.717557\pi\)
\(102\) 0 0
\(103\) −161.562 −0.154555 −0.0772775 0.997010i \(-0.524623\pi\)
−0.0772775 + 0.997010i \(0.524623\pi\)
\(104\) 0 0
\(105\) 563.647 0.523869
\(106\) 0 0
\(107\) −481.726 −0.435235 −0.217618 0.976034i \(-0.569829\pi\)
−0.217618 + 0.976034i \(0.569829\pi\)
\(108\) 0 0
\(109\) −286.637 −0.251879 −0.125940 0.992038i \(-0.540195\pi\)
−0.125940 + 0.992038i \(0.540195\pi\)
\(110\) 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
\(115\) 740.468 0.600426
\(116\) 0 0
\(117\) −740.179 −0.584868
\(118\) 0 0
\(119\) 3239.09 2.49518
\(120\) 0 0
\(121\) −131.942 −0.0991300
\(122\) 0 0
\(123\) 712.234 0.522113
\(124\) 0 0
\(125\) 1236.39 0.884686
\(126\) 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
\(131\) −1365.57 −0.910765 −0.455382 0.890296i \(-0.650497\pi\)
−0.455382 + 0.890296i \(0.650497\pi\)
\(132\) 0 0
\(133\) 1849.24 1.20563
\(134\) 0 0
\(135\) 153.296 0.0977305
\(136\) 0 0
\(137\) 1223.06 0.762721 0.381360 0.924426i \(-0.375456\pi\)
0.381360 + 0.924426i \(0.375456\pi\)
\(138\) 0 0
\(139\) −2176.70 −1.32824 −0.664121 0.747625i \(-0.731194\pi\)
−0.664121 + 0.747625i \(0.731194\pi\)
\(140\) 0 0
\(141\) 1004.44 0.599922
\(142\) 0 0
\(143\) 2847.83 1.66537
\(144\) 0 0
\(145\) −837.174 −0.479473
\(146\) 0 0
\(147\) −2256.17 −1.26589
\(148\) 0 0
\(149\) 2875.88 1.58122 0.790610 0.612320i \(-0.209764\pi\)
0.790610 + 0.612320i \(0.209764\pi\)
\(150\) 0 0
\(151\) −2330.03 −1.25573 −0.627863 0.778323i \(-0.716070\pi\)
−0.627863 + 0.778323i \(0.716070\pi\)
\(152\) 0 0
\(153\) 880.940 0.465489
\(154\) 0 0
\(155\) −574.708 −0.297817
\(156\) 0 0
\(157\) −2447.31 −1.24405 −0.622027 0.782996i \(-0.713690\pi\)
−0.622027 + 0.782996i \(0.713690\pi\)
\(158\) 0 0
\(159\) −306.091 −0.152670
\(160\) 0 0
\(161\) −4315.76 −2.11261
\(162\) 0 0
\(163\) 1436.82 0.690432 0.345216 0.938523i \(-0.387806\pi\)
0.345216 + 0.938523i \(0.387806\pi\)
\(164\) 0 0
\(165\) −589.805 −0.278280
\(166\) 0 0
\(167\) −1769.42 −0.819889 −0.409945 0.912110i \(-0.634452\pi\)
−0.409945 + 0.912110i \(0.634452\pi\)
\(168\) 0 0
\(169\) 4566.76 2.07863
\(170\) 0 0
\(171\) 502.940 0.224917
\(172\) 0 0
\(173\) 3897.86 1.71300 0.856499 0.516148i \(-0.172635\pi\)
0.856499 + 0.516148i \(0.172635\pi\)
\(174\) 0 0
\(175\) −3069.73 −1.32600
\(176\) 0 0
\(177\) 317.884 0.134992
\(178\) 0 0
\(179\) −4146.74 −1.73152 −0.865760 0.500460i \(-0.833164\pi\)
−0.865760 + 0.500460i \(0.833164\pi\)
\(180\) 0 0
\(181\) −1104.22 −0.453457 −0.226728 0.973958i \(-0.572803\pi\)
−0.226728 + 0.973958i \(0.572803\pi\)
\(182\) 0 0
\(183\) −2153.41 −0.869860
\(184\) 0 0
\(185\) −1047.18 −0.416163
\(186\) 0 0
\(187\) −3389.41 −1.32544
\(188\) 0 0
\(189\) −893.475 −0.343866
\(190\) 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
\(195\) −1400.82 −0.514435
\(196\) 0 0
\(197\) −589.638 −0.213249 −0.106624 0.994299i \(-0.534004\pi\)
−0.106624 + 0.994299i \(0.534004\pi\)
\(198\) 0 0
\(199\) 4813.82 1.71479 0.857394 0.514661i \(-0.172082\pi\)
0.857394 + 0.514661i \(0.172082\pi\)
\(200\) 0 0
\(201\) 949.413 0.333166
\(202\) 0 0
\(203\) 4879.41 1.68703
\(204\) 0 0
\(205\) 1347.93 0.459238
\(206\) 0 0
\(207\) −1173.77 −0.394118
\(208\) 0 0
\(209\) −1935.06 −0.640434
\(210\) 0 0
\(211\) 3294.35 1.07484 0.537422 0.843313i \(-0.319398\pi\)
0.537422 + 0.843313i \(0.319398\pi\)
\(212\) 0 0
\(213\) 2400.13 0.772087
\(214\) 0 0
\(215\) 1134.86 0.359984
\(216\) 0 0
\(217\) 3349.65 1.04787
\(218\) 0 0
\(219\) 903.174 0.278680
\(220\) 0 0
\(221\) −8050.04 −2.45025
\(222\) 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
\(227\) 1492.51 0.436394 0.218197 0.975905i \(-0.429982\pi\)
0.218197 + 0.975905i \(0.429982\pi\)
\(228\) 0 0
\(229\) −6163.31 −1.77853 −0.889265 0.457393i \(-0.848783\pi\)
−0.889265 + 0.457393i \(0.848783\pi\)
\(230\) 0 0
\(231\) 3437.64 0.979134
\(232\) 0 0
\(233\) −2540.35 −0.714266 −0.357133 0.934054i \(-0.616246\pi\)
−0.357133 + 0.934054i \(0.616246\pi\)
\(234\) 0 0
\(235\) 1900.95 0.527677
\(236\) 0 0
\(237\) 128.499 0.0352190
\(238\) 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
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −4269.91 −1.11345
\(246\) 0 0
\(247\) −4595.87 −1.18392
\(248\) 0 0
\(249\) 3710.00 0.944223
\(250\) 0 0
\(251\) −4701.45 −1.18228 −0.591141 0.806568i \(-0.701322\pi\)
−0.591141 + 0.806568i \(0.701322\pi\)
\(252\) 0 0
\(253\) 4516.05 1.12222
\(254\) 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
\(259\) 6103.41 1.46428
\(260\) 0 0
\(261\) 1327.06 0.314724
\(262\) 0 0
\(263\) −7091.36 −1.66263 −0.831315 0.555801i \(-0.812412\pi\)
−0.831315 + 0.555801i \(0.812412\pi\)
\(264\) 0 0
\(265\) −579.290 −0.134285
\(266\) 0 0
\(267\) 3975.88 0.911311
\(268\) 0 0
\(269\) −395.596 −0.0896650 −0.0448325 0.998995i \(-0.514275\pi\)
−0.0448325 + 0.998995i \(0.514275\pi\)
\(270\) 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
\(275\) 3212.20 0.704373
\(276\) 0 0
\(277\) 3830.31 0.830834 0.415417 0.909631i \(-0.363636\pi\)
0.415417 + 0.909631i \(0.363636\pi\)
\(278\) 0 0
\(279\) 911.009 0.195486
\(280\) 0 0
\(281\) −5283.17 −1.12159 −0.560797 0.827954i \(-0.689505\pi\)
−0.560797 + 0.827954i \(0.689505\pi\)
\(282\) 0 0
\(283\) −4639.76 −0.974577 −0.487288 0.873241i \(-0.662014\pi\)
−0.487288 + 0.873241i \(0.662014\pi\)
\(284\) 0 0
\(285\) 951.836 0.197831
\(286\) 0 0
\(287\) −7856.33 −1.61584
\(288\) 0 0
\(289\) 4667.94 0.950119
\(290\) 0 0
\(291\) 1517.29 0.305654
\(292\) 0 0
\(293\) 6022.12 1.20074 0.600369 0.799723i \(-0.295020\pi\)
0.600369 + 0.799723i \(0.295020\pi\)
\(294\) 0 0
\(295\) 601.609 0.118736
\(296\) 0 0
\(297\) 934.940 0.182662
\(298\) 0 0
\(299\) 10725.9 2.07456
\(300\) 0 0
\(301\) −6614.44 −1.26661
\(302\) 0 0
\(303\) 3845.92 0.729183
\(304\) 0 0
\(305\) −4075.42 −0.765107
\(306\) 0 0
\(307\) −2998.82 −0.557497 −0.278749 0.960364i \(-0.589920\pi\)
−0.278749 + 0.960364i \(0.589920\pi\)
\(308\) 0 0
\(309\) 484.685 0.0892323
\(310\) 0 0
\(311\) 2403.97 0.438318 0.219159 0.975689i \(-0.429669\pi\)
0.219159 + 0.975689i \(0.429669\pi\)
\(312\) 0 0
\(313\) −5845.75 −1.05566 −0.527830 0.849350i \(-0.676994\pi\)
−0.527830 + 0.849350i \(0.676994\pi\)
\(314\) 0 0
\(315\) −1690.94 −0.302456
\(316\) 0 0
\(317\) 8917.38 1.57997 0.789984 0.613127i \(-0.210089\pi\)
0.789984 + 0.613127i \(0.210089\pi\)
\(318\) 0 0
\(319\) −5105.86 −0.896154
\(320\) 0 0
\(321\) 1445.18 0.251283
\(322\) 0 0
\(323\) 5469.88 0.942267
\(324\) 0 0
\(325\) 7629.15 1.30212
\(326\) 0 0
\(327\) 859.910 0.145422
\(328\) 0 0
\(329\) −11079.5 −1.85664
\(330\) 0 0
\(331\) 9011.29 1.49639 0.748196 0.663478i \(-0.230920\pi\)
0.748196 + 0.663478i \(0.230920\pi\)
\(332\) 0 0
\(333\) 1659.96 0.273168
\(334\) 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
\(339\) 4191.88 0.671598
\(340\) 0 0
\(341\) −3505.10 −0.556633
\(342\) 0 0
\(343\) 13536.4 2.13090
\(344\) 0 0
\(345\) −2221.40 −0.346656
\(346\) 0 0
\(347\) 3651.48 0.564904 0.282452 0.959281i \(-0.408852\pi\)
0.282452 + 0.959281i \(0.408852\pi\)
\(348\) 0 0
\(349\) 3250.36 0.498532 0.249266 0.968435i \(-0.419811\pi\)
0.249266 + 0.968435i \(0.419811\pi\)
\(350\) 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
\(355\) 4542.36 0.679108
\(356\) 0 0
\(357\) −9717.26 −1.44059
\(358\) 0 0
\(359\) 3443.48 0.506239 0.253120 0.967435i \(-0.418543\pi\)
0.253120 + 0.967435i \(0.418543\pi\)
\(360\) 0 0
\(361\) −3736.17 −0.544711
\(362\) 0 0
\(363\) 395.826 0.0572327
\(364\) 0 0
\(365\) 1709.30 0.245120
\(366\) 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
\(371\) 3376.35 0.472483
\(372\) 0 0
\(373\) 50.1819 0.00696601 0.00348300 0.999994i \(-0.498891\pi\)
0.00348300 + 0.999994i \(0.498891\pi\)
\(374\) 0 0
\(375\) −3709.16 −0.510774
\(376\) 0 0
\(377\) −12126.7 −1.65665
\(378\) 0 0
\(379\) 6769.28 0.917453 0.458726 0.888578i \(-0.348306\pi\)
0.458726 + 0.888578i \(0.348306\pi\)
\(380\) 0 0
\(381\) −1331.49 −0.179040
\(382\) 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
\(387\) −1798.94 −0.236293
\(388\) 0 0
\(389\) 2490.47 0.324607 0.162303 0.986741i \(-0.448108\pi\)
0.162303 + 0.986741i \(0.448108\pi\)
\(390\) 0 0
\(391\) −12765.6 −1.65112
\(392\) 0 0
\(393\) 4096.70 0.525830
\(394\) 0 0
\(395\) 243.190 0.0309777
\(396\) 0 0
\(397\) −7905.51 −0.999411 −0.499706 0.866195i \(-0.666558\pi\)
−0.499706 + 0.866195i \(0.666558\pi\)
\(398\) 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
\(403\) −8324.81 −1.02900
\(404\) 0 0
\(405\) −459.888 −0.0564248
\(406\) 0 0
\(407\) −6386.66 −0.777826
\(408\) 0 0
\(409\) 13448.1 1.62583 0.812917 0.582379i \(-0.197878\pi\)
0.812917 + 0.582379i \(0.197878\pi\)
\(410\) 0 0
\(411\) −3669.17 −0.440357
\(412\) 0 0
\(413\) −3506.44 −0.417773
\(414\) 0 0
\(415\) 7021.33 0.830515
\(416\) 0 0
\(417\) 6530.11 0.766860
\(418\) 0 0
\(419\) 5191.34 0.605283 0.302641 0.953105i \(-0.402132\pi\)
0.302641 + 0.953105i \(0.402132\pi\)
\(420\) 0 0
\(421\) −5061.52 −0.585946 −0.292973 0.956121i \(-0.594645\pi\)
−0.292973 + 0.956121i \(0.594645\pi\)
\(422\) 0 0
\(423\) −3013.32 −0.346365
\(424\) 0 0
\(425\) −9080.00 −1.03634
\(426\) 0 0
\(427\) 23753.3 2.69204
\(428\) 0 0
\(429\) −8543.49 −0.961501
\(430\) 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
\(435\) 2511.52 0.276824
\(436\) 0 0
\(437\) −7288.07 −0.797794
\(438\) 0 0
\(439\) −11556.8 −1.25644 −0.628218 0.778038i \(-0.716215\pi\)
−0.628218 + 0.778038i \(0.716215\pi\)
\(440\) 0 0
\(441\) 6768.52 0.730863
\(442\) 0 0
\(443\) −14007.8 −1.50233 −0.751163 0.660117i \(-0.770507\pi\)
−0.751163 + 0.660117i \(0.770507\pi\)
\(444\) 0 0
\(445\) 7524.53 0.801566
\(446\) 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
\(451\) 8220.94 0.858335
\(452\) 0 0
\(453\) 6990.08 0.724994
\(454\) 0 0
\(455\) 15451.8 1.59207
\(456\) 0 0
\(457\) 1545.53 0.158198 0.0790992 0.996867i \(-0.474796\pi\)
0.0790992 + 0.996867i \(0.474796\pi\)
\(458\) 0 0
\(459\) −2642.82 −0.268750
\(460\) 0 0
\(461\) −12730.2 −1.28613 −0.643065 0.765811i \(-0.722338\pi\)
−0.643065 + 0.765811i \(0.722338\pi\)
\(462\) 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
\(467\) 6016.26 0.596144 0.298072 0.954543i \(-0.403656\pi\)
0.298072 + 0.954543i \(0.403656\pi\)
\(468\) 0 0
\(469\) −10472.6 −1.03108
\(470\) 0 0
\(471\) 7341.92 0.718255
\(472\) 0 0
\(473\) 6921.41 0.672826
\(474\) 0 0
\(475\) −5183.89 −0.500743
\(476\) 0 0
\(477\) 918.272 0.0881442
\(478\) 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
\(483\) 12947.3 1.21971
\(484\) 0 0
\(485\) 2871.54 0.268846
\(486\) 0 0
\(487\) 9314.23 0.866669 0.433335 0.901233i \(-0.357337\pi\)
0.433335 + 0.901233i \(0.357337\pi\)
\(488\) 0 0
\(489\) −4310.46 −0.398621
\(490\) 0 0
\(491\) 7253.34 0.666677 0.333339 0.942807i \(-0.391825\pi\)
0.333339 + 0.942807i \(0.391825\pi\)
\(492\) 0 0
\(493\) 14432.9 1.31851
\(494\) 0 0
\(495\) 1769.42 0.160665
\(496\) 0 0
\(497\) −26474.8 −2.38945
\(498\) 0 0
\(499\) −16208.2 −1.45407 −0.727034 0.686602i \(-0.759102\pi\)
−0.727034 + 0.686602i \(0.759102\pi\)
\(500\) 0 0
\(501\) 5308.25 0.473363
\(502\) 0 0
\(503\) −2182.04 −0.193425 −0.0967123 0.995312i \(-0.530833\pi\)
−0.0967123 + 0.995312i \(0.530833\pi\)
\(504\) 0 0
\(505\) 7278.58 0.641371
\(506\) 0 0
\(507\) −13700.3 −1.20010
\(508\) 0 0
\(509\) −1795.97 −0.156395 −0.0781974 0.996938i \(-0.524916\pi\)
−0.0781974 + 0.996938i \(0.524916\pi\)
\(510\) 0 0
\(511\) −9962.51 −0.862457
\(512\) 0 0
\(513\) −1508.82 −0.129856
\(514\) 0 0
\(515\) 917.288 0.0784865
\(516\) 0 0
\(517\) 11593.7 0.986249
\(518\) 0 0
\(519\) −11693.6 −0.989000
\(520\) 0 0
\(521\) 20946.1 1.76135 0.880676 0.473718i \(-0.157088\pi\)
0.880676 + 0.473718i \(0.157088\pi\)
\(522\) 0 0
\(523\) 5363.64 0.448443 0.224221 0.974538i \(-0.428016\pi\)
0.224221 + 0.974538i \(0.428016\pi\)
\(524\) 0 0
\(525\) 9209.19 0.765566
\(526\) 0 0
\(527\) 9907.96 0.818970
\(528\) 0 0
\(529\) 4841.96 0.397958
\(530\) 0 0
\(531\) −953.652 −0.0779378
\(532\) 0 0
\(533\) 19525.2 1.58674
\(534\) 0 0
\(535\) 2735.06 0.221022
\(536\) 0 0
\(537\) 12440.2 0.999693
\(538\) 0 0
\(539\) −26041.8 −2.08108
\(540\) 0 0
\(541\) 3431.20 0.272678 0.136339 0.990662i \(-0.456466\pi\)
0.136339 + 0.990662i \(0.456466\pi\)
\(542\) 0 0
\(543\) 3312.65 0.261803
\(544\) 0 0
\(545\) 1627.42 0.127910
\(546\) 0 0
\(547\) −19449.0 −1.52026 −0.760128 0.649773i \(-0.774864\pi\)
−0.760128 + 0.649773i \(0.774864\pi\)
\(548\) 0 0
\(549\) 6460.22 0.502214
\(550\) 0 0
\(551\) 8239.91 0.637082
\(552\) 0 0
\(553\) −1417.41 −0.108996
\(554\) 0 0
\(555\) 3141.54 0.240272
\(556\) 0 0
\(557\) 9898.44 0.752981 0.376491 0.926420i \(-0.377131\pi\)
0.376491 + 0.926420i \(0.377131\pi\)
\(558\) 0 0
\(559\) 16438.7 1.24380
\(560\) 0 0
\(561\) 10168.2 0.765246
\(562\) 0 0
\(563\) −13809.8 −1.03378 −0.516888 0.856053i \(-0.672909\pi\)
−0.516888 + 0.856053i \(0.672909\pi\)
\(564\) 0 0
\(565\) 7933.32 0.590721
\(566\) 0 0
\(567\) 2680.42 0.198531
\(568\) 0 0
\(569\) 123.053 0.00906615 0.00453308 0.999990i \(-0.498557\pi\)
0.00453308 + 0.999990i \(0.498557\pi\)
\(570\) 0 0
\(571\) −2540.72 −0.186210 −0.0931049 0.995656i \(-0.529679\pi\)
−0.0931049 + 0.995656i \(0.529679\pi\)
\(572\) 0 0
\(573\) 105.206 0.00767022
\(574\) 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
\(579\) −1847.65 −0.132618
\(580\) 0 0
\(581\) −40923.3 −2.92218
\(582\) 0 0
\(583\) −3533.04 −0.250984
\(584\) 0 0
\(585\) 4202.46 0.297009
\(586\) 0 0
\(587\) 1809.56 0.127238 0.0636188 0.997974i \(-0.479736\pi\)
0.0636188 + 0.997974i \(0.479736\pi\)
\(588\) 0 0
\(589\) 5656.58 0.395714
\(590\) 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
\(595\) −18390.3 −1.26711
\(596\) 0 0
\(597\) −14441.5 −0.990033
\(598\) 0 0
\(599\) −22990.9 −1.56825 −0.784125 0.620603i \(-0.786888\pi\)
−0.784125 + 0.620603i \(0.786888\pi\)
\(600\) 0 0
\(601\) 26893.7 1.82532 0.912661 0.408717i \(-0.134024\pi\)
0.912661 + 0.408717i \(0.134024\pi\)
\(602\) 0 0
\(603\) −2848.24 −0.192354
\(604\) 0 0
\(605\) 749.118 0.0503405
\(606\) 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
\(611\) 27535.7 1.82320
\(612\) 0 0
\(613\) −2612.97 −0.172164 −0.0860822 0.996288i \(-0.527435\pi\)
−0.0860822 + 0.996288i \(0.527435\pi\)
\(614\) 0 0
\(615\) −4043.80 −0.265141
\(616\) 0 0
\(617\) 2803.88 0.182949 0.0914747 0.995807i \(-0.470842\pi\)
0.0914747 + 0.995807i \(0.470842\pi\)
\(618\) 0 0
\(619\) 10547.1 0.684849 0.342425 0.939545i \(-0.388752\pi\)
0.342425 + 0.939545i \(0.388752\pi\)
\(620\) 0 0
\(621\) 3521.30 0.227544
\(622\) 0 0
\(623\) −43856.2 −2.82032
\(624\) 0 0
\(625\) 4575.82 0.292852
\(626\) 0 0
\(627\) 5805.17 0.369755
\(628\) 0 0
\(629\) 18053.3 1.14441
\(630\) 0 0
\(631\) 14161.4 0.893431 0.446716 0.894676i \(-0.352594\pi\)
0.446716 + 0.894676i \(0.352594\pi\)
\(632\) 0 0
\(633\) −9883.04 −0.620562
\(634\) 0 0
\(635\) −2519.90 −0.157479
\(636\) 0 0
\(637\) −61850.8 −3.84713
\(638\) 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
\(643\) −29438.7 −1.80552 −0.902759 0.430146i \(-0.858462\pi\)
−0.902759 + 0.430146i \(0.858462\pi\)
\(644\) 0 0
\(645\) −3404.57 −0.207837
\(646\) 0 0
\(647\) 4607.39 0.279962 0.139981 0.990154i \(-0.455296\pi\)
0.139981 + 0.990154i \(0.455296\pi\)
\(648\) 0 0
\(649\) 3669.17 0.221922
\(650\) 0 0
\(651\) −10048.9 −0.604991
\(652\) 0 0
\(653\) 16634.1 0.996850 0.498425 0.866933i \(-0.333912\pi\)
0.498425 + 0.866933i \(0.333912\pi\)
\(654\) 0 0
\(655\) 7753.19 0.462507
\(656\) 0 0
\(657\) −2709.52 −0.160896
\(658\) 0 0
\(659\) −20619.9 −1.21887 −0.609437 0.792835i \(-0.708604\pi\)
−0.609437 + 0.792835i \(0.708604\pi\)
\(660\) 0 0
\(661\) 881.112 0.0518476 0.0259238 0.999664i \(-0.491747\pi\)
0.0259238 + 0.999664i \(0.491747\pi\)
\(662\) 0 0
\(663\) 24150.1 1.41465
\(664\) 0 0
\(665\) −10499.3 −0.612248
\(666\) 0 0
\(667\) −19230.4 −1.11635
\(668\) 0 0
\(669\) −4718.35 −0.272679
\(670\) 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
\(675\) 2504.64 0.142820
\(676\) 0 0
\(677\) 21748.7 1.23467 0.617333 0.786702i \(-0.288213\pi\)
0.617333 + 0.786702i \(0.288213\pi\)
\(678\) 0 0
\(679\) −16736.6 −0.945937
\(680\) 0 0
\(681\) −4477.53 −0.251952
\(682\) 0 0
\(683\) −7922.37 −0.443838 −0.221919 0.975065i \(-0.571232\pi\)
−0.221919 + 0.975065i \(0.571232\pi\)
\(684\) 0 0
\(685\) −6944.06 −0.387327
\(686\) 0 0
\(687\) 18489.9 1.02683
\(688\) 0 0
\(689\) −8391.18 −0.463975
\(690\) 0 0
\(691\) 177.517 0.00977288 0.00488644 0.999988i \(-0.498445\pi\)
0.00488644 + 0.999988i \(0.498445\pi\)
\(692\) 0 0
\(693\) −10312.9 −0.565303
\(694\) 0 0
\(695\) 12358.5 0.674511
\(696\) 0 0
\(697\) −23238.3 −1.26286
\(698\) 0 0
\(699\) 7621.05 0.412382
\(700\) 0 0
\(701\) −767.980 −0.0413783 −0.0206891 0.999786i \(-0.506586\pi\)
−0.0206891 + 0.999786i \(0.506586\pi\)
\(702\) 0 0
\(703\) 10306.9 0.552961
\(704\) 0 0
\(705\) −5702.84 −0.304654
\(706\) 0 0
\(707\) −42422.7 −2.25667
\(708\) 0 0
\(709\) −31634.2 −1.67566 −0.837832 0.545928i \(-0.816177\pi\)
−0.837832 + 0.545928i \(0.816177\pi\)
\(710\) 0 0
\(711\) −385.496 −0.0203337
\(712\) 0 0
\(713\) −13201.4 −0.693401
\(714\) 0 0
\(715\) −16168.9 −0.845712
\(716\) 0 0
\(717\) 4017.76 0.209269
\(718\) 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
\(723\) −7628.12 −0.392383
\(724\) 0 0
\(725\) −13678.2 −0.700686
\(726\) 0 0
\(727\) 16364.6 0.834842 0.417421 0.908713i \(-0.362934\pi\)
0.417421 + 0.908713i \(0.362934\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −19564.9 −0.989925
\(732\) 0 0
\(733\) 23552.7 1.18682 0.593410 0.804900i \(-0.297781\pi\)
0.593410 + 0.804900i \(0.297781\pi\)
\(734\) 0 0
\(735\) 12809.7 0.642849
\(736\) 0 0
\(737\) 10958.6 0.547713
\(738\) 0 0
\(739\) 3519.05 0.175169 0.0875847 0.996157i \(-0.472085\pi\)
0.0875847 + 0.996157i \(0.472085\pi\)
\(740\) 0 0
\(741\) 13787.6 0.683537
\(742\) 0 0
\(743\) −20752.4 −1.02467 −0.512336 0.858785i \(-0.671220\pi\)
−0.512336 + 0.858785i \(0.671220\pi\)
\(744\) 0 0
\(745\) −16328.2 −0.802979
\(746\) 0 0
\(747\) −11130.0 −0.545148
\(748\) 0 0
\(749\) −15941.1 −0.777671
\(750\) 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
\(755\) 13229.0 0.637687
\(756\) 0 0
\(757\) 527.275 0.0253159 0.0126579 0.999920i \(-0.495971\pi\)
0.0126579 + 0.999920i \(0.495971\pi\)
\(758\) 0 0
\(759\) −13548.2 −0.647914
\(760\) 0 0
\(761\) −3231.54 −0.153933 −0.0769666 0.997034i \(-0.524523\pi\)
−0.0769666 + 0.997034i \(0.524523\pi\)
\(762\) 0 0
\(763\) −9485.29 −0.450053
\(764\) 0 0
\(765\) −5001.65 −0.236386
\(766\) 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
\(771\) 2591.65 0.121058
\(772\) 0 0
\(773\) 9216.63 0.428847 0.214424 0.976741i \(-0.431213\pi\)
0.214424 + 0.976741i \(0.431213\pi\)
\(774\) 0 0
\(775\) −9389.92 −0.435221
\(776\) 0 0
\(777\) −18310.2 −0.845400
\(778\) 0 0
\(779\) −13267.1 −0.610196
\(780\) 0 0
\(781\) 27703.5 1.26928
\(782\) 0 0
\(783\) −3981.18 −0.181706
\(784\) 0 0
\(785\) 13894.9 0.631759
\(786\) 0 0
\(787\) 33169.0 1.50235 0.751174 0.660104i \(-0.229488\pi\)
0.751174 + 0.660104i \(0.229488\pi\)
\(788\) 0 0
\(789\) 21274.1 0.959920
\(790\) 0 0
\(791\) −46238.8 −2.07846
\(792\) 0 0
\(793\) −59033.6 −2.64356
\(794\) 0 0
\(795\) 1737.87 0.0775294
\(796\) 0 0
\(797\) 23781.9 1.05696 0.528481 0.848945i \(-0.322762\pi\)
0.528481 + 0.848945i \(0.322762\pi\)
\(798\) 0 0
\(799\) −32772.3 −1.45106
\(800\) 0 0
\(801\) −11927.6 −0.526145
\(802\) 0 0
\(803\) 10424.9 0.458139
\(804\) 0 0
\(805\) 24503.3 1.07283
\(806\) 0 0
\(807\) 1186.79 0.0517681
\(808\) 0 0
\(809\) 29842.4 1.29691 0.648456 0.761252i \(-0.275415\pi\)
0.648456 + 0.761252i \(0.275415\pi\)
\(810\) 0 0
\(811\) 19074.5 0.825888 0.412944 0.910756i \(-0.364500\pi\)
0.412944 + 0.910756i \(0.364500\pi\)
\(812\) 0 0
\(813\) −16596.6 −0.715952
\(814\) 0 0
\(815\) −8157.74 −0.350617
\(816\) 0 0
\(817\) −11169.9 −0.478316
\(818\) 0 0
\(819\) −24493.7 −1.04503
\(820\) 0 0
\(821\) 7493.97 0.318564 0.159282 0.987233i \(-0.449082\pi\)
0.159282 + 0.987233i \(0.449082\pi\)
\(822\) 0 0
\(823\) 1780.90 0.0754294 0.0377147 0.999289i \(-0.487992\pi\)
0.0377147 + 0.999289i \(0.487992\pi\)
\(824\) 0 0
\(825\) −9636.59 −0.406670
\(826\) 0 0
\(827\) −10860.1 −0.456640 −0.228320 0.973586i \(-0.573323\pi\)
−0.228320 + 0.973586i \(0.573323\pi\)
\(828\) 0 0
\(829\) 34105.1 1.42885 0.714426 0.699711i \(-0.246688\pi\)
0.714426 + 0.699711i \(0.246688\pi\)
\(830\) 0 0
\(831\) −11490.9 −0.479682
\(832\) 0 0
\(833\) 73613.1 3.06188
\(834\) 0 0
\(835\) 10046.1 0.416358
\(836\) 0 0
\(837\) −2733.03 −0.112864
\(838\) 0 0
\(839\) 32688.9 1.34511 0.672555 0.740047i \(-0.265197\pi\)
0.672555 + 0.740047i \(0.265197\pi\)
\(840\) 0 0
\(841\) −2647.12 −0.108537
\(842\) 0 0
\(843\) 15849.5 0.647552
\(844\) 0 0
\(845\) −25928.4 −1.05558
\(846\) 0 0
\(847\) −4366.18 −0.177124
\(848\) 0 0
\(849\) 13919.3 0.562672
\(850\) 0 0
\(851\) −24054.3 −0.968943
\(852\) 0 0
\(853\) −17391.2 −0.698080 −0.349040 0.937108i \(-0.613492\pi\)
−0.349040 + 0.937108i \(0.613492\pi\)
\(854\) 0 0
\(855\) −2855.51 −0.114218
\(856\) 0 0
\(857\) 4182.34 0.166705 0.0833525 0.996520i \(-0.473437\pi\)
0.0833525 + 0.996520i \(0.473437\pi\)
\(858\) 0 0
\(859\) −20585.0 −0.817639 −0.408820 0.912615i \(-0.634059\pi\)
−0.408820 + 0.912615i \(0.634059\pi\)
\(860\) 0 0
\(861\) 23569.0 0.932903
\(862\) 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
\(867\) −14003.8 −0.548552
\(868\) 0 0
\(869\) 1483.19 0.0578986
\(870\) 0 0
\(871\) 26027.2 1.01251
\(872\) 0 0
\(873\) −4551.88 −0.176469
\(874\) 0 0
\(875\) 40914.1 1.58074
\(876\) 0 0
\(877\) 5156.38 0.198539 0.0992695 0.995061i \(-0.468349\pi\)
0.0992695 + 0.995061i \(0.468349\pi\)
\(878\) 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
\(883\) −7845.99 −0.299024 −0.149512 0.988760i \(-0.547770\pi\)
−0.149512 + 0.988760i \(0.547770\pi\)
\(884\) 0 0
\(885\) −1804.83 −0.0685521
\(886\) 0 0
\(887\) 38781.9 1.46806 0.734029 0.679118i \(-0.237637\pi\)
0.734029 + 0.679118i \(0.237637\pi\)
\(888\) 0 0
\(889\) 14687.1 0.554092
\(890\) 0 0
\(891\) −2804.82 −0.105460
\(892\) 0 0
\(893\) −18710.1 −0.701131
\(894\) 0 0
\(895\) 23543.7 0.879305
\(896\) 0 0
\(897\) −32177.6 −1.19775
\(898\) 0 0
\(899\) 14925.5 0.553719
\(900\) 0 0
\(901\) 9986.95 0.369271
\(902\) 0 0
\(903\) 19843.3 0.731278
\(904\) 0 0
\(905\) 6269.33 0.230276
\(906\) 0 0
\(907\) −36431.2 −1.33371 −0.666856 0.745187i \(-0.732360\pi\)
−0.666856 + 0.745187i \(0.732360\pi\)
\(908\) 0 0
\(909\) −11537.8 −0.420994
\(910\) 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
\(915\) 12226.3 0.441735
\(916\) 0 0
\(917\) −45188.9 −1.62734
\(918\) 0 0
\(919\) −37080.1 −1.33097 −0.665484 0.746412i \(-0.731775\pi\)
−0.665484 + 0.746412i \(0.731775\pi\)
\(920\) 0 0
\(921\) 8996.46 0.321871
\(922\) 0 0
\(923\) 65797.3 2.34642
\(924\) 0 0
\(925\) −17109.4 −0.608167
\(926\) 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
\(931\) 42026.7 1.47945
\(932\) 0 0
\(933\) −7211.92 −0.253063
\(934\) 0 0
\(935\) 19243.8 0.673091
\(936\) 0 0
\(937\) −4413.55 −0.153879 −0.0769394 0.997036i \(-0.524515\pi\)
−0.0769394 + 0.997036i \(0.524515\pi\)
\(938\) 0 0
\(939\) 17537.3 0.609486
\(940\) 0 0
\(941\) −31511.2 −1.09164 −0.545821 0.837902i \(-0.683782\pi\)
−0.545821 + 0.837902i \(0.683782\pi\)
\(942\) 0 0
\(943\) 30962.8 1.06923
\(944\) 0 0
\(945\) 5072.82 0.174623
\(946\) 0 0
\(947\) −23561.3 −0.808490 −0.404245 0.914651i \(-0.632466\pi\)
−0.404245 + 0.914651i \(0.632466\pi\)
\(948\) 0 0
\(949\) 24759.6 0.846925
\(950\) 0 0
\(951\) −26752.1 −0.912195
\(952\) 0 0
\(953\) −35039.7 −1.19103 −0.595513 0.803346i \(-0.703051\pi\)
−0.595513 + 0.803346i \(0.703051\pi\)
\(954\) 0 0
\(955\) 199.107 0.00674653
\(956\) 0 0
\(957\) 15317.6 0.517395
\(958\) 0 0
\(959\) 40473.0 1.36282
\(960\) 0 0
\(961\) −19544.9 −0.656066
\(962\) 0 0
\(963\) −4335.53 −0.145078
\(964\) 0 0
\(965\) −3496.76 −0.116647
\(966\) 0 0
\(967\) 7975.38 0.265223 0.132612 0.991168i \(-0.457664\pi\)
0.132612 + 0.991168i \(0.457664\pi\)
\(968\) 0 0
\(969\) −16409.6 −0.544018
\(970\) 0 0
\(971\) −21386.8 −0.706834 −0.353417 0.935466i \(-0.614980\pi\)
−0.353417 + 0.935466i \(0.614980\pi\)
\(972\) 0 0
\(973\) −72030.7 −2.37328
\(974\) 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
\(979\) 45891.5 1.49816
\(980\) 0 0
\(981\) −2579.73 −0.0839597
\(982\) 0 0
\(983\) −9205.44 −0.298686 −0.149343 0.988785i \(-0.547716\pi\)
−0.149343 + 0.988785i \(0.547716\pi\)
\(984\) 0 0
\(985\) 3347.75 0.108292
\(986\) 0 0
\(987\) 33238.6 1.07193
\(988\) 0 0
\(989\) 26068.3 0.838144
\(990\) 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
\(995\) −27331.1 −0.870808
\(996\) 0 0
\(997\) 7207.66 0.228956 0.114478 0.993426i \(-0.463480\pi\)
0.114478 + 0.993426i \(0.463480\pi\)
\(998\) 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 768.4.a.u.1.2 4
3.2 odd 2 2304.4.a.cb.1.3 4
4.3 odd 2 768.4.a.v.1.2 4
8.3 odd 2 inner 768.4.a.u.1.3 4
8.5 even 2 768.4.a.v.1.3 4
12.11 even 2 2304.4.a.by.1.3 4
16.3 odd 4 384.4.d.f.193.7 yes 8
16.5 even 4 384.4.d.f.193.6 yes 8
16.11 odd 4 384.4.d.f.193.2 8
16.13 even 4 384.4.d.f.193.3 yes 8
24.5 odd 2 2304.4.a.by.1.2 4
24.11 even 2 2304.4.a.cb.1.2 4
48.5 odd 4 1152.4.d.p.577.5 8
48.11 even 4 1152.4.d.p.577.6 8
48.29 odd 4 1152.4.d.p.577.3 8
48.35 even 4 1152.4.d.p.577.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.f.193.2 8 16.11 odd 4
384.4.d.f.193.3 yes 8 16.13 even 4
384.4.d.f.193.6 yes 8 16.5 even 4
384.4.d.f.193.7 yes 8 16.3 odd 4
768.4.a.u.1.2 4 1.1 even 1 trivial
768.4.a.u.1.3 4 8.3 odd 2 inner
768.4.a.v.1.2 4 4.3 odd 2
768.4.a.v.1.3 4 8.5 even 2
1152.4.d.p.577.3 8 48.29 odd 4
1152.4.d.p.577.4 8 48.35 even 4
1152.4.d.p.577.5 8 48.5 odd 4
1152.4.d.p.577.6 8 48.11 even 4
2304.4.a.by.1.2 4 24.5 odd 2
2304.4.a.by.1.3 4 12.11 even 2
2304.4.a.cb.1.2 4 24.11 even 2
2304.4.a.cb.1.3 4 3.2 odd 2