Properties

Label 1575.4.a.br.1.8
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
Dimension $8$
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] = [1575,4,Mod(1,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, 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("1575.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.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: \(92.9280082590\)
Analytic rank: \(0\)
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: \( x^{8} - 50x^{6} + 698x^{4} - 2653x^{2} + 2268 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(5.38335\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.38335 q^{2} +20.9805 q^{4} -7.00000 q^{7} +69.8783 q^{8} +O(q^{10})\) \(q+5.38335 q^{2} +20.9805 q^{4} -7.00000 q^{7} +69.8783 q^{8} +15.8856 q^{11} +40.8573 q^{13} -37.6834 q^{14} +208.336 q^{16} -47.2555 q^{17} +141.113 q^{19} +85.5176 q^{22} +97.3015 q^{23} +219.949 q^{26} -146.863 q^{28} -236.682 q^{29} +95.7947 q^{31} +562.518 q^{32} -254.393 q^{34} -33.8738 q^{37} +759.663 q^{38} -197.295 q^{41} -437.132 q^{43} +333.287 q^{44} +523.808 q^{46} +256.598 q^{47} +49.0000 q^{49} +857.205 q^{52} -64.7048 q^{53} -489.148 q^{56} -1274.14 q^{58} +809.137 q^{59} +318.721 q^{61} +515.696 q^{62} +1361.55 q^{64} +726.284 q^{67} -991.443 q^{68} +68.6547 q^{71} +680.433 q^{73} -182.355 q^{74} +2960.62 q^{76} -111.199 q^{77} +1169.94 q^{79} -1062.11 q^{82} -642.901 q^{83} -2353.24 q^{86} +1110.06 q^{88} +1382.18 q^{89} -286.001 q^{91} +2041.43 q^{92} +1381.36 q^{94} +317.080 q^{97} +263.784 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{4} - 56 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 36 q^{4} - 56 q^{7} + 38 q^{13} + 320 q^{16} + 68 q^{19} - 110 q^{22} - 252 q^{28} + 534 q^{31} + 118 q^{34} - 14 q^{37} - 816 q^{43} + 1588 q^{46} + 392 q^{49} + 266 q^{52} - 716 q^{58} + 874 q^{61} + 2502 q^{64} - 254 q^{67} + 1532 q^{73} + 4304 q^{76} + 5190 q^{79} - 1158 q^{82} + 4300 q^{88} - 266 q^{91} + 6236 q^{94} - 428 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.38335 1.90330 0.951651 0.307182i \(-0.0993860\pi\)
0.951651 + 0.307182i \(0.0993860\pi\)
\(3\) 0 0
\(4\) 20.9805 2.62256
\(5\) 0 0
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 69.8783 3.08822
\(9\) 0 0
\(10\) 0 0
\(11\) 15.8856 0.435426 0.217713 0.976013i \(-0.430140\pi\)
0.217713 + 0.976013i \(0.430140\pi\)
\(12\) 0 0
\(13\) 40.8573 0.871676 0.435838 0.900025i \(-0.356452\pi\)
0.435838 + 0.900025i \(0.356452\pi\)
\(14\) −37.6834 −0.719380
\(15\) 0 0
\(16\) 208.336 3.25525
\(17\) −47.2555 −0.674185 −0.337093 0.941471i \(-0.609444\pi\)
−0.337093 + 0.941471i \(0.609444\pi\)
\(18\) 0 0
\(19\) 141.113 1.70388 0.851938 0.523643i \(-0.175427\pi\)
0.851938 + 0.523643i \(0.175427\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 85.5176 0.828746
\(23\) 97.3015 0.882120 0.441060 0.897478i \(-0.354603\pi\)
0.441060 + 0.897478i \(0.354603\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 219.949 1.65906
\(27\) 0 0
\(28\) −146.863 −0.991233
\(29\) −236.682 −1.51555 −0.757773 0.652519i \(-0.773712\pi\)
−0.757773 + 0.652519i \(0.773712\pi\)
\(30\) 0 0
\(31\) 95.7947 0.555008 0.277504 0.960725i \(-0.410493\pi\)
0.277504 + 0.960725i \(0.410493\pi\)
\(32\) 562.518 3.10750
\(33\) 0 0
\(34\) −254.393 −1.28318
\(35\) 0 0
\(36\) 0 0
\(37\) −33.8738 −0.150509 −0.0752544 0.997164i \(-0.523977\pi\)
−0.0752544 + 0.997164i \(0.523977\pi\)
\(38\) 759.663 3.24299
\(39\) 0 0
\(40\) 0 0
\(41\) −197.295 −0.751520 −0.375760 0.926717i \(-0.622618\pi\)
−0.375760 + 0.926717i \(0.622618\pi\)
\(42\) 0 0
\(43\) −437.132 −1.55028 −0.775140 0.631790i \(-0.782321\pi\)
−0.775140 + 0.631790i \(0.782321\pi\)
\(44\) 333.287 1.14193
\(45\) 0 0
\(46\) 523.808 1.67894
\(47\) 256.598 0.796354 0.398177 0.917309i \(-0.369643\pi\)
0.398177 + 0.917309i \(0.369643\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 857.205 2.28602
\(53\) −64.7048 −0.167696 −0.0838480 0.996479i \(-0.526721\pi\)
−0.0838480 + 0.996479i \(0.526721\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −489.148 −1.16724
\(57\) 0 0
\(58\) −1274.14 −2.88454
\(59\) 809.137 1.78543 0.892717 0.450618i \(-0.148796\pi\)
0.892717 + 0.450618i \(0.148796\pi\)
\(60\) 0 0
\(61\) 318.721 0.668985 0.334492 0.942398i \(-0.391435\pi\)
0.334492 + 0.942398i \(0.391435\pi\)
\(62\) 515.696 1.05635
\(63\) 0 0
\(64\) 1361.55 2.65927
\(65\) 0 0
\(66\) 0 0
\(67\) 726.284 1.32432 0.662162 0.749361i \(-0.269639\pi\)
0.662162 + 0.749361i \(0.269639\pi\)
\(68\) −991.443 −1.76809
\(69\) 0 0
\(70\) 0 0
\(71\) 68.6547 0.114758 0.0573790 0.998352i \(-0.481726\pi\)
0.0573790 + 0.998352i \(0.481726\pi\)
\(72\) 0 0
\(73\) 680.433 1.09094 0.545471 0.838130i \(-0.316351\pi\)
0.545471 + 0.838130i \(0.316351\pi\)
\(74\) −182.355 −0.286464
\(75\) 0 0
\(76\) 2960.62 4.46851
\(77\) −111.199 −0.164575
\(78\) 0 0
\(79\) 1169.94 1.66619 0.833093 0.553132i \(-0.186568\pi\)
0.833093 + 0.553132i \(0.186568\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1062.11 −1.43037
\(83\) −642.901 −0.850211 −0.425105 0.905144i \(-0.639763\pi\)
−0.425105 + 0.905144i \(0.639763\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2353.24 −2.95065
\(87\) 0 0
\(88\) 1110.06 1.34469
\(89\) 1382.18 1.64619 0.823096 0.567903i \(-0.192245\pi\)
0.823096 + 0.567903i \(0.192245\pi\)
\(90\) 0 0
\(91\) −286.001 −0.329462
\(92\) 2041.43 2.31341
\(93\) 0 0
\(94\) 1381.36 1.51570
\(95\) 0 0
\(96\) 0 0
\(97\) 317.080 0.331903 0.165951 0.986134i \(-0.446930\pi\)
0.165951 + 0.986134i \(0.446930\pi\)
\(98\) 263.784 0.271900
\(99\) 0 0
\(100\) 0 0
\(101\) −1140.37 −1.12347 −0.561736 0.827316i \(-0.689866\pi\)
−0.561736 + 0.827316i \(0.689866\pi\)
\(102\) 0 0
\(103\) −970.893 −0.928786 −0.464393 0.885629i \(-0.653727\pi\)
−0.464393 + 0.885629i \(0.653727\pi\)
\(104\) 2855.04 2.69192
\(105\) 0 0
\(106\) −348.329 −0.319176
\(107\) 707.342 0.639078 0.319539 0.947573i \(-0.396472\pi\)
0.319539 + 0.947573i \(0.396472\pi\)
\(108\) 0 0
\(109\) −1538.49 −1.35193 −0.675967 0.736932i \(-0.736274\pi\)
−0.675967 + 0.736932i \(0.736274\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1458.35 −1.23037
\(113\) −465.734 −0.387722 −0.193861 0.981029i \(-0.562101\pi\)
−0.193861 + 0.981029i \(0.562101\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4965.70 −3.97460
\(117\) 0 0
\(118\) 4355.87 3.39822
\(119\) 330.789 0.254818
\(120\) 0 0
\(121\) −1078.65 −0.810405
\(122\) 1715.79 1.27328
\(123\) 0 0
\(124\) 2009.82 1.45554
\(125\) 0 0
\(126\) 0 0
\(127\) −964.749 −0.674076 −0.337038 0.941491i \(-0.609425\pi\)
−0.337038 + 0.941491i \(0.609425\pi\)
\(128\) 2829.53 1.95389
\(129\) 0 0
\(130\) 0 0
\(131\) 2733.39 1.82303 0.911517 0.411262i \(-0.134912\pi\)
0.911517 + 0.411262i \(0.134912\pi\)
\(132\) 0 0
\(133\) −987.794 −0.644005
\(134\) 3909.84 2.52059
\(135\) 0 0
\(136\) −3302.14 −2.08203
\(137\) −2707.00 −1.68813 −0.844067 0.536238i \(-0.819845\pi\)
−0.844067 + 0.536238i \(0.819845\pi\)
\(138\) 0 0
\(139\) −2459.38 −1.50073 −0.750367 0.661021i \(-0.770123\pi\)
−0.750367 + 0.661021i \(0.770123\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 369.592 0.218419
\(143\) 649.042 0.379550
\(144\) 0 0
\(145\) 0 0
\(146\) 3663.01 2.07639
\(147\) 0 0
\(148\) −710.689 −0.394718
\(149\) −3484.84 −1.91604 −0.958018 0.286707i \(-0.907439\pi\)
−0.958018 + 0.286707i \(0.907439\pi\)
\(150\) 0 0
\(151\) 444.052 0.239314 0.119657 0.992815i \(-0.461820\pi\)
0.119657 + 0.992815i \(0.461820\pi\)
\(152\) 9860.77 5.26194
\(153\) 0 0
\(154\) −598.623 −0.313237
\(155\) 0 0
\(156\) 0 0
\(157\) 831.556 0.422709 0.211355 0.977409i \(-0.432212\pi\)
0.211355 + 0.977409i \(0.432212\pi\)
\(158\) 6298.21 3.17126
\(159\) 0 0
\(160\) 0 0
\(161\) −681.110 −0.333410
\(162\) 0 0
\(163\) 419.137 0.201407 0.100703 0.994916i \(-0.467891\pi\)
0.100703 + 0.994916i \(0.467891\pi\)
\(164\) −4139.34 −1.97090
\(165\) 0 0
\(166\) −3460.96 −1.61821
\(167\) 2441.03 1.13109 0.565547 0.824716i \(-0.308665\pi\)
0.565547 + 0.824716i \(0.308665\pi\)
\(168\) 0 0
\(169\) −527.678 −0.240181
\(170\) 0 0
\(171\) 0 0
\(172\) −9171.23 −4.06570
\(173\) −2630.72 −1.15613 −0.578064 0.815991i \(-0.696192\pi\)
−0.578064 + 0.815991i \(0.696192\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3309.54 1.41742
\(177\) 0 0
\(178\) 7440.77 3.13320
\(179\) 739.987 0.308990 0.154495 0.987994i \(-0.450625\pi\)
0.154495 + 0.987994i \(0.450625\pi\)
\(180\) 0 0
\(181\) −2061.53 −0.846588 −0.423294 0.905992i \(-0.639126\pi\)
−0.423294 + 0.905992i \(0.639126\pi\)
\(182\) −1539.65 −0.627066
\(183\) 0 0
\(184\) 6799.27 2.72418
\(185\) 0 0
\(186\) 0 0
\(187\) −750.681 −0.293558
\(188\) 5383.54 2.08848
\(189\) 0 0
\(190\) 0 0
\(191\) −4047.47 −1.53332 −0.766662 0.642051i \(-0.778084\pi\)
−0.766662 + 0.642051i \(0.778084\pi\)
\(192\) 0 0
\(193\) 2100.96 0.783578 0.391789 0.920055i \(-0.371856\pi\)
0.391789 + 0.920055i \(0.371856\pi\)
\(194\) 1706.95 0.631711
\(195\) 0 0
\(196\) 1028.04 0.374651
\(197\) 3057.94 1.10594 0.552968 0.833202i \(-0.313495\pi\)
0.552968 + 0.833202i \(0.313495\pi\)
\(198\) 0 0
\(199\) 2915.91 1.03871 0.519356 0.854558i \(-0.326172\pi\)
0.519356 + 0.854558i \(0.326172\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −6138.99 −2.13831
\(203\) 1656.78 0.572822
\(204\) 0 0
\(205\) 0 0
\(206\) −5226.66 −1.76776
\(207\) 0 0
\(208\) 8512.05 2.83752
\(209\) 2241.67 0.741911
\(210\) 0 0
\(211\) 3626.89 1.18334 0.591671 0.806179i \(-0.298468\pi\)
0.591671 + 0.806179i \(0.298468\pi\)
\(212\) −1357.54 −0.439792
\(213\) 0 0
\(214\) 3807.87 1.21636
\(215\) 0 0
\(216\) 0 0
\(217\) −670.563 −0.209773
\(218\) −8282.24 −2.57314
\(219\) 0 0
\(220\) 0 0
\(221\) −1930.74 −0.587671
\(222\) 0 0
\(223\) −5661.60 −1.70013 −0.850064 0.526679i \(-0.823437\pi\)
−0.850064 + 0.526679i \(0.823437\pi\)
\(224\) −3937.63 −1.17453
\(225\) 0 0
\(226\) −2507.21 −0.737951
\(227\) −5746.85 −1.68032 −0.840158 0.542342i \(-0.817538\pi\)
−0.840158 + 0.542342i \(0.817538\pi\)
\(228\) 0 0
\(229\) −1702.78 −0.491367 −0.245684 0.969350i \(-0.579012\pi\)
−0.245684 + 0.969350i \(0.579012\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −16539.0 −4.68033
\(233\) −988.387 −0.277903 −0.138951 0.990299i \(-0.544373\pi\)
−0.138951 + 0.990299i \(0.544373\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 16976.1 4.68240
\(237\) 0 0
\(238\) 1780.75 0.484996
\(239\) −301.963 −0.0817255 −0.0408627 0.999165i \(-0.513011\pi\)
−0.0408627 + 0.999165i \(0.513011\pi\)
\(240\) 0 0
\(241\) −4054.29 −1.08365 −0.541826 0.840491i \(-0.682267\pi\)
−0.541826 + 0.840491i \(0.682267\pi\)
\(242\) −5806.74 −1.54244
\(243\) 0 0
\(244\) 6686.92 1.75445
\(245\) 0 0
\(246\) 0 0
\(247\) 5765.52 1.48523
\(248\) 6693.97 1.71398
\(249\) 0 0
\(250\) 0 0
\(251\) −6426.79 −1.61616 −0.808079 0.589074i \(-0.799493\pi\)
−0.808079 + 0.589074i \(0.799493\pi\)
\(252\) 0 0
\(253\) 1545.69 0.384098
\(254\) −5193.58 −1.28297
\(255\) 0 0
\(256\) 4339.98 1.05957
\(257\) 991.869 0.240744 0.120372 0.992729i \(-0.461591\pi\)
0.120372 + 0.992729i \(0.461591\pi\)
\(258\) 0 0
\(259\) 237.117 0.0568870
\(260\) 0 0
\(261\) 0 0
\(262\) 14714.8 3.46978
\(263\) −5394.91 −1.26488 −0.632442 0.774608i \(-0.717947\pi\)
−0.632442 + 0.774608i \(0.717947\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −5317.64 −1.22573
\(267\) 0 0
\(268\) 15237.8 3.47311
\(269\) 7111.53 1.61189 0.805944 0.591992i \(-0.201658\pi\)
0.805944 + 0.591992i \(0.201658\pi\)
\(270\) 0 0
\(271\) 609.090 0.136530 0.0682649 0.997667i \(-0.478254\pi\)
0.0682649 + 0.997667i \(0.478254\pi\)
\(272\) −9845.02 −2.19464
\(273\) 0 0
\(274\) −14572.7 −3.21303
\(275\) 0 0
\(276\) 0 0
\(277\) −3015.06 −0.653998 −0.326999 0.945025i \(-0.606037\pi\)
−0.326999 + 0.945025i \(0.606037\pi\)
\(278\) −13239.7 −2.85635
\(279\) 0 0
\(280\) 0 0
\(281\) −416.706 −0.0884648 −0.0442324 0.999021i \(-0.514084\pi\)
−0.0442324 + 0.999021i \(0.514084\pi\)
\(282\) 0 0
\(283\) −4483.54 −0.941763 −0.470882 0.882196i \(-0.656064\pi\)
−0.470882 + 0.882196i \(0.656064\pi\)
\(284\) 1440.41 0.300959
\(285\) 0 0
\(286\) 3494.02 0.722398
\(287\) 1381.07 0.284048
\(288\) 0 0
\(289\) −2679.91 −0.545474
\(290\) 0 0
\(291\) 0 0
\(292\) 14275.8 2.86106
\(293\) −8198.32 −1.63464 −0.817322 0.576181i \(-0.804542\pi\)
−0.817322 + 0.576181i \(0.804542\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2367.05 −0.464804
\(297\) 0 0
\(298\) −18760.1 −3.64680
\(299\) 3975.48 0.768923
\(300\) 0 0
\(301\) 3059.93 0.585951
\(302\) 2390.49 0.455487
\(303\) 0 0
\(304\) 29399.0 5.54654
\(305\) 0 0
\(306\) 0 0
\(307\) −5568.07 −1.03514 −0.517568 0.855642i \(-0.673163\pi\)
−0.517568 + 0.855642i \(0.673163\pi\)
\(308\) −2333.01 −0.431608
\(309\) 0 0
\(310\) 0 0
\(311\) 979.150 0.178529 0.0892645 0.996008i \(-0.471548\pi\)
0.0892645 + 0.996008i \(0.471548\pi\)
\(312\) 0 0
\(313\) 1466.51 0.264831 0.132415 0.991194i \(-0.457727\pi\)
0.132415 + 0.991194i \(0.457727\pi\)
\(314\) 4476.56 0.804544
\(315\) 0 0
\(316\) 24545.9 4.36967
\(317\) −9413.22 −1.66782 −0.833910 0.551900i \(-0.813903\pi\)
−0.833910 + 0.551900i \(0.813903\pi\)
\(318\) 0 0
\(319\) −3759.84 −0.659907
\(320\) 0 0
\(321\) 0 0
\(322\) −3666.66 −0.634580
\(323\) −6668.39 −1.14873
\(324\) 0 0
\(325\) 0 0
\(326\) 2256.36 0.383338
\(327\) 0 0
\(328\) −13786.6 −2.32085
\(329\) −1796.18 −0.300993
\(330\) 0 0
\(331\) 6614.38 1.09837 0.549183 0.835702i \(-0.314939\pi\)
0.549183 + 0.835702i \(0.314939\pi\)
\(332\) −13488.3 −2.22973
\(333\) 0 0
\(334\) 13140.9 2.15281
\(335\) 0 0
\(336\) 0 0
\(337\) −717.673 −0.116006 −0.0580032 0.998316i \(-0.518473\pi\)
−0.0580032 + 0.998316i \(0.518473\pi\)
\(338\) −2840.68 −0.457137
\(339\) 0 0
\(340\) 0 0
\(341\) 1521.75 0.241665
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) −30546.1 −4.78760
\(345\) 0 0
\(346\) −14162.1 −2.20046
\(347\) −6994.59 −1.08210 −0.541051 0.840990i \(-0.681973\pi\)
−0.541051 + 0.840990i \(0.681973\pi\)
\(348\) 0 0
\(349\) 9941.33 1.52478 0.762388 0.647120i \(-0.224027\pi\)
0.762388 + 0.647120i \(0.224027\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 8935.93 1.35309
\(353\) −1997.26 −0.301143 −0.150572 0.988599i \(-0.548111\pi\)
−0.150572 + 0.988599i \(0.548111\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 28998.8 4.31723
\(357\) 0 0
\(358\) 3983.61 0.588101
\(359\) 3272.07 0.481040 0.240520 0.970644i \(-0.422682\pi\)
0.240520 + 0.970644i \(0.422682\pi\)
\(360\) 0 0
\(361\) 13054.0 1.90319
\(362\) −11098.0 −1.61131
\(363\) 0 0
\(364\) −6000.44 −0.864034
\(365\) 0 0
\(366\) 0 0
\(367\) 12744.7 1.81272 0.906359 0.422509i \(-0.138850\pi\)
0.906359 + 0.422509i \(0.138850\pi\)
\(368\) 20271.4 2.87152
\(369\) 0 0
\(370\) 0 0
\(371\) 452.934 0.0633831
\(372\) 0 0
\(373\) −1849.96 −0.256803 −0.128401 0.991722i \(-0.540985\pi\)
−0.128401 + 0.991722i \(0.540985\pi\)
\(374\) −4041.18 −0.558729
\(375\) 0 0
\(376\) 17930.6 2.45931
\(377\) −9670.21 −1.32106
\(378\) 0 0
\(379\) 4832.69 0.654983 0.327491 0.944854i \(-0.393797\pi\)
0.327491 + 0.944854i \(0.393797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −21789.0 −2.91838
\(383\) −3561.98 −0.475218 −0.237609 0.971361i \(-0.576364\pi\)
−0.237609 + 0.971361i \(0.576364\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 11310.2 1.49139
\(387\) 0 0
\(388\) 6652.48 0.870434
\(389\) −5095.14 −0.664097 −0.332048 0.943262i \(-0.607740\pi\)
−0.332048 + 0.943262i \(0.607740\pi\)
\(390\) 0 0
\(391\) −4598.03 −0.594712
\(392\) 3424.04 0.441174
\(393\) 0 0
\(394\) 16462.0 2.10493
\(395\) 0 0
\(396\) 0 0
\(397\) 14511.4 1.83453 0.917263 0.398283i \(-0.130394\pi\)
0.917263 + 0.398283i \(0.130394\pi\)
\(398\) 15697.4 1.97698
\(399\) 0 0
\(400\) 0 0
\(401\) −4973.55 −0.619370 −0.309685 0.950839i \(-0.600224\pi\)
−0.309685 + 0.950839i \(0.600224\pi\)
\(402\) 0 0
\(403\) 3913.92 0.483787
\(404\) −23925.4 −2.94637
\(405\) 0 0
\(406\) 8919.01 1.09025
\(407\) −538.106 −0.0655354
\(408\) 0 0
\(409\) 2841.63 0.343545 0.171772 0.985137i \(-0.445051\pi\)
0.171772 + 0.985137i \(0.445051\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −20369.8 −2.43579
\(413\) −5663.96 −0.674831
\(414\) 0 0
\(415\) 0 0
\(416\) 22983.0 2.70874
\(417\) 0 0
\(418\) 12067.7 1.41208
\(419\) 12983.2 1.51377 0.756886 0.653547i \(-0.226720\pi\)
0.756886 + 0.653547i \(0.226720\pi\)
\(420\) 0 0
\(421\) 6323.41 0.732029 0.366015 0.930609i \(-0.380722\pi\)
0.366015 + 0.930609i \(0.380722\pi\)
\(422\) 19524.8 2.25226
\(423\) 0 0
\(424\) −4521.46 −0.517881
\(425\) 0 0
\(426\) 0 0
\(427\) −2231.05 −0.252852
\(428\) 14840.4 1.67602
\(429\) 0 0
\(430\) 0 0
\(431\) 2421.08 0.270579 0.135289 0.990806i \(-0.456804\pi\)
0.135289 + 0.990806i \(0.456804\pi\)
\(432\) 0 0
\(433\) −3187.12 −0.353726 −0.176863 0.984235i \(-0.556595\pi\)
−0.176863 + 0.984235i \(0.556595\pi\)
\(434\) −3609.87 −0.399262
\(435\) 0 0
\(436\) −32278.3 −3.54553
\(437\) 13730.5 1.50302
\(438\) 0 0
\(439\) −2492.32 −0.270961 −0.135480 0.990780i \(-0.543258\pi\)
−0.135480 + 0.990780i \(0.543258\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −10393.8 −1.11852
\(443\) −5862.02 −0.628698 −0.314349 0.949308i \(-0.601786\pi\)
−0.314349 + 0.949308i \(0.601786\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −30478.4 −3.23586
\(447\) 0 0
\(448\) −9530.82 −1.00511
\(449\) 9655.80 1.01489 0.507445 0.861684i \(-0.330590\pi\)
0.507445 + 0.861684i \(0.330590\pi\)
\(450\) 0 0
\(451\) −3134.15 −0.327231
\(452\) −9771.31 −1.01682
\(453\) 0 0
\(454\) −30937.3 −3.19815
\(455\) 0 0
\(456\) 0 0
\(457\) 12608.4 1.29059 0.645293 0.763935i \(-0.276735\pi\)
0.645293 + 0.763935i \(0.276735\pi\)
\(458\) −9166.68 −0.935220
\(459\) 0 0
\(460\) 0 0
\(461\) 4611.99 0.465947 0.232974 0.972483i \(-0.425154\pi\)
0.232974 + 0.972483i \(0.425154\pi\)
\(462\) 0 0
\(463\) −11150.2 −1.11921 −0.559603 0.828761i \(-0.689046\pi\)
−0.559603 + 0.828761i \(0.689046\pi\)
\(464\) −49309.4 −4.93348
\(465\) 0 0
\(466\) −5320.83 −0.528933
\(467\) −4106.72 −0.406930 −0.203465 0.979082i \(-0.565220\pi\)
−0.203465 + 0.979082i \(0.565220\pi\)
\(468\) 0 0
\(469\) −5083.99 −0.500547
\(470\) 0 0
\(471\) 0 0
\(472\) 56541.1 5.51380
\(473\) −6944.10 −0.675032
\(474\) 0 0
\(475\) 0 0
\(476\) 6940.10 0.668275
\(477\) 0 0
\(478\) −1625.57 −0.155548
\(479\) −4559.73 −0.434947 −0.217473 0.976066i \(-0.569782\pi\)
−0.217473 + 0.976066i \(0.569782\pi\)
\(480\) 0 0
\(481\) −1383.99 −0.131195
\(482\) −21825.7 −2.06252
\(483\) 0 0
\(484\) −22630.5 −2.12533
\(485\) 0 0
\(486\) 0 0
\(487\) −3854.56 −0.358659 −0.179329 0.983789i \(-0.557393\pi\)
−0.179329 + 0.983789i \(0.557393\pi\)
\(488\) 22271.7 2.06597
\(489\) 0 0
\(490\) 0 0
\(491\) −10619.9 −0.976110 −0.488055 0.872813i \(-0.662293\pi\)
−0.488055 + 0.872813i \(0.662293\pi\)
\(492\) 0 0
\(493\) 11184.6 1.02176
\(494\) 31037.8 2.82684
\(495\) 0 0
\(496\) 19957.5 1.80669
\(497\) −480.583 −0.0433744
\(498\) 0 0
\(499\) −4544.53 −0.407698 −0.203849 0.979002i \(-0.565345\pi\)
−0.203849 + 0.979002i \(0.565345\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −34597.7 −3.07604
\(503\) 8793.98 0.779531 0.389766 0.920914i \(-0.372556\pi\)
0.389766 + 0.920914i \(0.372556\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8320.99 0.731054
\(507\) 0 0
\(508\) −20240.9 −1.76780
\(509\) −10573.9 −0.920782 −0.460391 0.887716i \(-0.652291\pi\)
−0.460391 + 0.887716i \(0.652291\pi\)
\(510\) 0 0
\(511\) −4763.03 −0.412337
\(512\) 727.414 0.0627880
\(513\) 0 0
\(514\) 5339.58 0.458208
\(515\) 0 0
\(516\) 0 0
\(517\) 4076.20 0.346753
\(518\) 1276.48 0.108273
\(519\) 0 0
\(520\) 0 0
\(521\) −15708.4 −1.32092 −0.660458 0.750863i \(-0.729638\pi\)
−0.660458 + 0.750863i \(0.729638\pi\)
\(522\) 0 0
\(523\) −2656.84 −0.222133 −0.111067 0.993813i \(-0.535427\pi\)
−0.111067 + 0.993813i \(0.535427\pi\)
\(524\) 57347.8 4.78101
\(525\) 0 0
\(526\) −29042.7 −2.40745
\(527\) −4526.83 −0.374178
\(528\) 0 0
\(529\) −2699.42 −0.221864
\(530\) 0 0
\(531\) 0 0
\(532\) −20724.4 −1.68894
\(533\) −8060.95 −0.655081
\(534\) 0 0
\(535\) 0 0
\(536\) 50751.5 4.08980
\(537\) 0 0
\(538\) 38283.9 3.06791
\(539\) 778.393 0.0622037
\(540\) 0 0
\(541\) −6340.08 −0.503847 −0.251924 0.967747i \(-0.581063\pi\)
−0.251924 + 0.967747i \(0.581063\pi\)
\(542\) 3278.94 0.259857
\(543\) 0 0
\(544\) −26582.1 −2.09503
\(545\) 0 0
\(546\) 0 0
\(547\) −22317.5 −1.74447 −0.872235 0.489087i \(-0.837330\pi\)
−0.872235 + 0.489087i \(0.837330\pi\)
\(548\) −56794.0 −4.42723
\(549\) 0 0
\(550\) 0 0
\(551\) −33399.1 −2.58230
\(552\) 0 0
\(553\) −8189.59 −0.629759
\(554\) −16231.1 −1.24476
\(555\) 0 0
\(556\) −51599.0 −3.93576
\(557\) 9476.86 0.720911 0.360455 0.932776i \(-0.382621\pi\)
0.360455 + 0.932776i \(0.382621\pi\)
\(558\) 0 0
\(559\) −17860.1 −1.35134
\(560\) 0 0
\(561\) 0 0
\(562\) −2243.28 −0.168375
\(563\) −21003.7 −1.57229 −0.786144 0.618043i \(-0.787926\pi\)
−0.786144 + 0.618043i \(0.787926\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −24136.5 −1.79246
\(567\) 0 0
\(568\) 4797.48 0.354397
\(569\) 22905.7 1.68762 0.843811 0.536640i \(-0.180307\pi\)
0.843811 + 0.536640i \(0.180307\pi\)
\(570\) 0 0
\(571\) 5035.60 0.369060 0.184530 0.982827i \(-0.440924\pi\)
0.184530 + 0.982827i \(0.440924\pi\)
\(572\) 13617.2 0.995392
\(573\) 0 0
\(574\) 7434.76 0.540628
\(575\) 0 0
\(576\) 0 0
\(577\) −7692.81 −0.555036 −0.277518 0.960720i \(-0.589512\pi\)
−0.277518 + 0.960720i \(0.589512\pi\)
\(578\) −14426.9 −1.03820
\(579\) 0 0
\(580\) 0 0
\(581\) 4500.30 0.321350
\(582\) 0 0
\(583\) −1027.87 −0.0730191
\(584\) 47547.6 3.36906
\(585\) 0 0
\(586\) −44134.4 −3.11122
\(587\) −4219.62 −0.296699 −0.148350 0.988935i \(-0.547396\pi\)
−0.148350 + 0.988935i \(0.547396\pi\)
\(588\) 0 0
\(589\) 13517.9 0.945664
\(590\) 0 0
\(591\) 0 0
\(592\) −7057.14 −0.489943
\(593\) −10710.9 −0.741730 −0.370865 0.928687i \(-0.620939\pi\)
−0.370865 + 0.928687i \(0.620939\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −73113.6 −5.02492
\(597\) 0 0
\(598\) 21401.4 1.46349
\(599\) 25086.1 1.71117 0.855584 0.517663i \(-0.173198\pi\)
0.855584 + 0.517663i \(0.173198\pi\)
\(600\) 0 0
\(601\) 25399.6 1.72391 0.861957 0.506982i \(-0.169239\pi\)
0.861957 + 0.506982i \(0.169239\pi\)
\(602\) 16472.6 1.11524
\(603\) 0 0
\(604\) 9316.42 0.627616
\(605\) 0 0
\(606\) 0 0
\(607\) 14501.2 0.969665 0.484833 0.874607i \(-0.338881\pi\)
0.484833 + 0.874607i \(0.338881\pi\)
\(608\) 79378.9 5.29480
\(609\) 0 0
\(610\) 0 0
\(611\) 10483.9 0.694162
\(612\) 0 0
\(613\) 3673.43 0.242036 0.121018 0.992650i \(-0.461384\pi\)
0.121018 + 0.992650i \(0.461384\pi\)
\(614\) −29974.9 −1.97018
\(615\) 0 0
\(616\) −7770.40 −0.508244
\(617\) −9857.06 −0.643161 −0.321581 0.946882i \(-0.604214\pi\)
−0.321581 + 0.946882i \(0.604214\pi\)
\(618\) 0 0
\(619\) 29100.2 1.88956 0.944778 0.327712i \(-0.106277\pi\)
0.944778 + 0.327712i \(0.106277\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 5271.11 0.339794
\(623\) −9675.28 −0.622202
\(624\) 0 0
\(625\) 0 0
\(626\) 7894.74 0.504053
\(627\) 0 0
\(628\) 17446.4 1.10858
\(629\) 1600.73 0.101471
\(630\) 0 0
\(631\) −5707.06 −0.360055 −0.180027 0.983662i \(-0.557619\pi\)
−0.180027 + 0.983662i \(0.557619\pi\)
\(632\) 81753.6 5.14554
\(633\) 0 0
\(634\) −50674.7 −3.17437
\(635\) 0 0
\(636\) 0 0
\(637\) 2002.01 0.124525
\(638\) −20240.5 −1.25600
\(639\) 0 0
\(640\) 0 0
\(641\) 7664.18 0.472257 0.236129 0.971722i \(-0.424121\pi\)
0.236129 + 0.971722i \(0.424121\pi\)
\(642\) 0 0
\(643\) 15108.9 0.926653 0.463326 0.886188i \(-0.346656\pi\)
0.463326 + 0.886188i \(0.346656\pi\)
\(644\) −14290.0 −0.874387
\(645\) 0 0
\(646\) −35898.3 −2.18638
\(647\) −26190.4 −1.59142 −0.795712 0.605675i \(-0.792903\pi\)
−0.795712 + 0.605675i \(0.792903\pi\)
\(648\) 0 0
\(649\) 12853.6 0.777424
\(650\) 0 0
\(651\) 0 0
\(652\) 8793.68 0.528201
\(653\) 609.867 0.0365482 0.0182741 0.999833i \(-0.494183\pi\)
0.0182741 + 0.999833i \(0.494183\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −41103.6 −2.44638
\(657\) 0 0
\(658\) −9669.49 −0.572881
\(659\) −8456.98 −0.499905 −0.249952 0.968258i \(-0.580415\pi\)
−0.249952 + 0.968258i \(0.580415\pi\)
\(660\) 0 0
\(661\) 11260.1 0.662582 0.331291 0.943529i \(-0.392516\pi\)
0.331291 + 0.943529i \(0.392516\pi\)
\(662\) 35607.5 2.09052
\(663\) 0 0
\(664\) −44924.8 −2.62563
\(665\) 0 0
\(666\) 0 0
\(667\) −23029.5 −1.33689
\(668\) 51213.9 2.96636
\(669\) 0 0
\(670\) 0 0
\(671\) 5063.07 0.291293
\(672\) 0 0
\(673\) −21034.3 −1.20477 −0.602386 0.798205i \(-0.705783\pi\)
−0.602386 + 0.798205i \(0.705783\pi\)
\(674\) −3863.48 −0.220795
\(675\) 0 0
\(676\) −11070.9 −0.629889
\(677\) 6576.39 0.373340 0.186670 0.982423i \(-0.440230\pi\)
0.186670 + 0.982423i \(0.440230\pi\)
\(678\) 0 0
\(679\) −2219.56 −0.125448
\(680\) 0 0
\(681\) 0 0
\(682\) 8192.14 0.459961
\(683\) −19573.4 −1.09657 −0.548283 0.836293i \(-0.684718\pi\)
−0.548283 + 0.836293i \(0.684718\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1846.49 −0.102769
\(687\) 0 0
\(688\) −91070.3 −5.04655
\(689\) −2643.67 −0.146177
\(690\) 0 0
\(691\) 10921.3 0.601251 0.300626 0.953742i \(-0.402805\pi\)
0.300626 + 0.953742i \(0.402805\pi\)
\(692\) −55193.8 −3.03201
\(693\) 0 0
\(694\) −37654.3 −2.05957
\(695\) 0 0
\(696\) 0 0
\(697\) 9323.28 0.506663
\(698\) 53517.6 2.90211
\(699\) 0 0
\(700\) 0 0
\(701\) −15067.7 −0.811839 −0.405920 0.913909i \(-0.633049\pi\)
−0.405920 + 0.913909i \(0.633049\pi\)
\(702\) 0 0
\(703\) −4780.05 −0.256448
\(704\) 21628.9 1.15791
\(705\) 0 0
\(706\) −10752.0 −0.573166
\(707\) 7982.57 0.424633
\(708\) 0 0
\(709\) 27173.9 1.43940 0.719702 0.694283i \(-0.244279\pi\)
0.719702 + 0.694283i \(0.244279\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 96584.6 5.08379
\(713\) 9320.97 0.489583
\(714\) 0 0
\(715\) 0 0
\(716\) 15525.3 0.810344
\(717\) 0 0
\(718\) 17614.7 0.915564
\(719\) 11572.1 0.600233 0.300117 0.953902i \(-0.402974\pi\)
0.300117 + 0.953902i \(0.402974\pi\)
\(720\) 0 0
\(721\) 6796.25 0.351048
\(722\) 70274.2 3.62235
\(723\) 0 0
\(724\) −43251.9 −2.22023
\(725\) 0 0
\(726\) 0 0
\(727\) 10390.6 0.530075 0.265038 0.964238i \(-0.414616\pi\)
0.265038 + 0.964238i \(0.414616\pi\)
\(728\) −19985.3 −1.01745
\(729\) 0 0
\(730\) 0 0
\(731\) 20656.9 1.04518
\(732\) 0 0
\(733\) −20807.9 −1.04851 −0.524255 0.851562i \(-0.675656\pi\)
−0.524255 + 0.851562i \(0.675656\pi\)
\(734\) 68609.1 3.45015
\(735\) 0 0
\(736\) 54733.9 2.74119
\(737\) 11537.4 0.576644
\(738\) 0 0
\(739\) 8201.89 0.408270 0.204135 0.978943i \(-0.434562\pi\)
0.204135 + 0.978943i \(0.434562\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2438.30 0.120637
\(743\) 6711.14 0.331370 0.165685 0.986179i \(-0.447017\pi\)
0.165685 + 0.986179i \(0.447017\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −9958.99 −0.488773
\(747\) 0 0
\(748\) −15749.6 −0.769871
\(749\) −4951.39 −0.241549
\(750\) 0 0
\(751\) 28845.5 1.40158 0.700789 0.713368i \(-0.252831\pi\)
0.700789 + 0.713368i \(0.252831\pi\)
\(752\) 53458.5 2.59233
\(753\) 0 0
\(754\) −52058.1 −2.51438
\(755\) 0 0
\(756\) 0 0
\(757\) 18678.9 0.896822 0.448411 0.893827i \(-0.351990\pi\)
0.448411 + 0.893827i \(0.351990\pi\)
\(758\) 26016.0 1.24663
\(759\) 0 0
\(760\) 0 0
\(761\) −22486.7 −1.07115 −0.535573 0.844489i \(-0.679904\pi\)
−0.535573 + 0.844489i \(0.679904\pi\)
\(762\) 0 0
\(763\) 10769.4 0.510983
\(764\) −84917.8 −4.02123
\(765\) 0 0
\(766\) −19175.4 −0.904483
\(767\) 33059.2 1.55632
\(768\) 0 0
\(769\) 3982.98 0.186775 0.0933874 0.995630i \(-0.470230\pi\)
0.0933874 + 0.995630i \(0.470230\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 44079.1 2.05498
\(773\) 8594.11 0.399882 0.199941 0.979808i \(-0.435925\pi\)
0.199941 + 0.979808i \(0.435925\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 22157.0 1.02499
\(777\) 0 0
\(778\) −27428.9 −1.26398
\(779\) −27841.0 −1.28050
\(780\) 0 0
\(781\) 1090.62 0.0499685
\(782\) −24752.8 −1.13192
\(783\) 0 0
\(784\) 10208.5 0.465035
\(785\) 0 0
\(786\) 0 0
\(787\) 21490.9 0.973403 0.486702 0.873568i \(-0.338200\pi\)
0.486702 + 0.873568i \(0.338200\pi\)
\(788\) 64157.0 2.90038
\(789\) 0 0
\(790\) 0 0
\(791\) 3260.14 0.146545
\(792\) 0 0
\(793\) 13022.1 0.583138
\(794\) 78120.0 3.49165
\(795\) 0 0
\(796\) 61177.2 2.72408
\(797\) −2703.13 −0.120138 −0.0600688 0.998194i \(-0.519132\pi\)
−0.0600688 + 0.998194i \(0.519132\pi\)
\(798\) 0 0
\(799\) −12125.7 −0.536890
\(800\) 0 0
\(801\) 0 0
\(802\) −26774.4 −1.17885
\(803\) 10809.1 0.475024
\(804\) 0 0
\(805\) 0 0
\(806\) 21070.0 0.920792
\(807\) 0 0
\(808\) −79686.9 −3.46952
\(809\) 16331.3 0.709738 0.354869 0.934916i \(-0.384526\pi\)
0.354869 + 0.934916i \(0.384526\pi\)
\(810\) 0 0
\(811\) 14503.0 0.627953 0.313977 0.949431i \(-0.398339\pi\)
0.313977 + 0.949431i \(0.398339\pi\)
\(812\) 34759.9 1.50226
\(813\) 0 0
\(814\) −2896.81 −0.124734
\(815\) 0 0
\(816\) 0 0
\(817\) −61685.2 −2.64148
\(818\) 15297.5 0.653869
\(819\) 0 0
\(820\) 0 0
\(821\) −27641.5 −1.17503 −0.587513 0.809215i \(-0.699893\pi\)
−0.587513 + 0.809215i \(0.699893\pi\)
\(822\) 0 0
\(823\) −20193.2 −0.855273 −0.427636 0.903951i \(-0.640654\pi\)
−0.427636 + 0.903951i \(0.640654\pi\)
\(824\) −67844.4 −2.86829
\(825\) 0 0
\(826\) −30491.1 −1.28441
\(827\) 14333.6 0.602696 0.301348 0.953514i \(-0.402563\pi\)
0.301348 + 0.953514i \(0.402563\pi\)
\(828\) 0 0
\(829\) 3214.10 0.134657 0.0673283 0.997731i \(-0.478553\pi\)
0.0673283 + 0.997731i \(0.478553\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 55629.1 2.31802
\(833\) −2315.52 −0.0963122
\(834\) 0 0
\(835\) 0 0
\(836\) 47031.2 1.94570
\(837\) 0 0
\(838\) 69893.1 2.88117
\(839\) −729.209 −0.0300061 −0.0150030 0.999887i \(-0.504776\pi\)
−0.0150030 + 0.999887i \(0.504776\pi\)
\(840\) 0 0
\(841\) 31629.5 1.29688
\(842\) 34041.1 1.39327
\(843\) 0 0
\(844\) 76093.8 3.10338
\(845\) 0 0
\(846\) 0 0
\(847\) 7550.54 0.306304
\(848\) −13480.3 −0.545892
\(849\) 0 0
\(850\) 0 0
\(851\) −3295.98 −0.132767
\(852\) 0 0
\(853\) −41938.2 −1.68340 −0.841698 0.539949i \(-0.818443\pi\)
−0.841698 + 0.539949i \(0.818443\pi\)
\(854\) −12010.5 −0.481254
\(855\) 0 0
\(856\) 49427.9 1.97361
\(857\) −19103.8 −0.761464 −0.380732 0.924685i \(-0.624328\pi\)
−0.380732 + 0.924685i \(0.624328\pi\)
\(858\) 0 0
\(859\) 13341.8 0.529936 0.264968 0.964257i \(-0.414639\pi\)
0.264968 + 0.964257i \(0.414639\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 13033.5 0.514993
\(863\) −38561.8 −1.52104 −0.760522 0.649313i \(-0.775057\pi\)
−0.760522 + 0.649313i \(0.775057\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −17157.4 −0.673247
\(867\) 0 0
\(868\) −14068.7 −0.550142
\(869\) 18585.2 0.725500
\(870\) 0 0
\(871\) 29674.0 1.15438
\(872\) −107507. −4.17507
\(873\) 0 0
\(874\) 73916.3 2.86071
\(875\) 0 0
\(876\) 0 0
\(877\) 28097.0 1.08183 0.540917 0.841076i \(-0.318077\pi\)
0.540917 + 0.841076i \(0.318077\pi\)
\(878\) −13417.0 −0.515720
\(879\) 0 0
\(880\) 0 0
\(881\) 12677.3 0.484801 0.242400 0.970176i \(-0.422065\pi\)
0.242400 + 0.970176i \(0.422065\pi\)
\(882\) 0 0
\(883\) 26590.0 1.01339 0.506695 0.862125i \(-0.330867\pi\)
0.506695 + 0.862125i \(0.330867\pi\)
\(884\) −40507.7 −1.54120
\(885\) 0 0
\(886\) −31557.3 −1.19660
\(887\) −10754.1 −0.407090 −0.203545 0.979066i \(-0.565246\pi\)
−0.203545 + 0.979066i \(0.565246\pi\)
\(888\) 0 0
\(889\) 6753.24 0.254777
\(890\) 0 0
\(891\) 0 0
\(892\) −118783. −4.45868
\(893\) 36209.4 1.35689
\(894\) 0 0
\(895\) 0 0
\(896\) −19806.7 −0.738500
\(897\) 0 0
\(898\) 51980.5 1.93164
\(899\) −22672.9 −0.841139
\(900\) 0 0
\(901\) 3057.66 0.113058
\(902\) −16872.2 −0.622819
\(903\) 0 0
\(904\) −32544.7 −1.19737
\(905\) 0 0
\(906\) 0 0
\(907\) 40275.4 1.47445 0.737224 0.675649i \(-0.236136\pi\)
0.737224 + 0.675649i \(0.236136\pi\)
\(908\) −120572. −4.40672
\(909\) 0 0
\(910\) 0 0
\(911\) 19797.9 0.720015 0.360007 0.932949i \(-0.382774\pi\)
0.360007 + 0.932949i \(0.382774\pi\)
\(912\) 0 0
\(913\) −10212.8 −0.370204
\(914\) 67875.7 2.45638
\(915\) 0 0
\(916\) −35725.2 −1.28864
\(917\) −19133.7 −0.689042
\(918\) 0 0
\(919\) −40237.3 −1.44429 −0.722146 0.691740i \(-0.756844\pi\)
−0.722146 + 0.691740i \(0.756844\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 24827.9 0.886838
\(923\) 2805.05 0.100032
\(924\) 0 0
\(925\) 0 0
\(926\) −60025.3 −2.13019
\(927\) 0 0
\(928\) −133138. −4.70956
\(929\) 13545.9 0.478391 0.239195 0.970971i \(-0.423116\pi\)
0.239195 + 0.970971i \(0.423116\pi\)
\(930\) 0 0
\(931\) 6914.56 0.243411
\(932\) −20736.8 −0.728816
\(933\) 0 0
\(934\) −22107.9 −0.774511
\(935\) 0 0
\(936\) 0 0
\(937\) 32849.2 1.14529 0.572645 0.819803i \(-0.305917\pi\)
0.572645 + 0.819803i \(0.305917\pi\)
\(938\) −27368.9 −0.952692
\(939\) 0 0
\(940\) 0 0
\(941\) −44538.0 −1.54293 −0.771464 0.636272i \(-0.780475\pi\)
−0.771464 + 0.636272i \(0.780475\pi\)
\(942\) 0 0
\(943\) −19197.1 −0.662931
\(944\) 168572. 5.81203
\(945\) 0 0
\(946\) −37382.5 −1.28479
\(947\) 1052.01 0.0360991 0.0180496 0.999837i \(-0.494254\pi\)
0.0180496 + 0.999837i \(0.494254\pi\)
\(948\) 0 0
\(949\) 27800.7 0.950947
\(950\) 0 0
\(951\) 0 0
\(952\) 23115.0 0.786933
\(953\) −35657.1 −1.21201 −0.606005 0.795461i \(-0.707229\pi\)
−0.606005 + 0.795461i \(0.707229\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −6335.33 −0.214330
\(957\) 0 0
\(958\) −24546.6 −0.827835
\(959\) 18949.0 0.638055
\(960\) 0 0
\(961\) −20614.4 −0.691967
\(962\) −7450.53 −0.249703
\(963\) 0 0
\(964\) −85060.9 −2.84194
\(965\) 0 0
\(966\) 0 0
\(967\) 3425.21 0.113906 0.0569531 0.998377i \(-0.481861\pi\)
0.0569531 + 0.998377i \(0.481861\pi\)
\(968\) −75374.2 −2.50270
\(969\) 0 0
\(970\) 0 0
\(971\) 15544.8 0.513754 0.256877 0.966444i \(-0.417306\pi\)
0.256877 + 0.966444i \(0.417306\pi\)
\(972\) 0 0
\(973\) 17215.7 0.567224
\(974\) −20750.5 −0.682636
\(975\) 0 0
\(976\) 66401.1 2.17771
\(977\) −48799.1 −1.59798 −0.798988 0.601347i \(-0.794631\pi\)
−0.798988 + 0.601347i \(0.794631\pi\)
\(978\) 0 0
\(979\) 21956.8 0.716794
\(980\) 0 0
\(981\) 0 0
\(982\) −57170.7 −1.85783
\(983\) 39839.1 1.29264 0.646322 0.763065i \(-0.276306\pi\)
0.646322 + 0.763065i \(0.276306\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 60210.4 1.94471
\(987\) 0 0
\(988\) 120963. 3.89509
\(989\) −42533.6 −1.36753
\(990\) 0 0
\(991\) 4833.44 0.154934 0.0774668 0.996995i \(-0.475317\pi\)
0.0774668 + 0.996995i \(0.475317\pi\)
\(992\) 53886.3 1.72469
\(993\) 0 0
\(994\) −2587.15 −0.0825546
\(995\) 0 0
\(996\) 0 0
\(997\) −2299.66 −0.0730502 −0.0365251 0.999333i \(-0.511629\pi\)
−0.0365251 + 0.999333i \(0.511629\pi\)
\(998\) −24464.8 −0.775971
\(999\) 0 0
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 1575.4.a.br.1.8 yes 8
3.2 odd 2 inner 1575.4.a.br.1.1 8
5.4 even 2 1575.4.a.bs.1.1 yes 8
15.14 odd 2 1575.4.a.bs.1.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1575.4.a.br.1.1 8 3.2 odd 2 inner
1575.4.a.br.1.8 yes 8 1.1 even 1 trivial
1575.4.a.bs.1.1 yes 8 5.4 even 2
1575.4.a.bs.1.8 yes 8 15.14 odd 2