Properties

Label 1323.4.a.bf.1.4
Level $1323$
Weight $4$
Character 1323.1
Self dual yes
Analytic conductor $78.060$
Analytic rank $1$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,4,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1323.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.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: \(78.0595269376\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 42x^{4} + 369x^{2} - 112 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.560992\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.560992 q^{2} -7.68529 q^{4} -12.9561 q^{5} -8.79932 q^{8} +O(q^{10})\) \(q+0.560992 q^{2} -7.68529 q^{4} -12.9561 q^{5} -8.79932 q^{8} -7.26827 q^{10} +48.2286 q^{11} -36.3241 q^{13} +56.5460 q^{16} -83.2713 q^{17} +67.4359 q^{19} +99.5714 q^{20} +27.0559 q^{22} -30.6300 q^{23} +42.8607 q^{25} -20.3775 q^{26} +294.244 q^{29} +270.844 q^{31} +102.116 q^{32} -46.7145 q^{34} -204.315 q^{37} +37.8310 q^{38} +114.005 q^{40} -287.529 q^{41} -55.4901 q^{43} -370.651 q^{44} -17.1832 q^{46} +191.301 q^{47} +24.0445 q^{50} +279.161 q^{52} +521.107 q^{53} -624.855 q^{55} +165.069 q^{58} +381.074 q^{59} +155.465 q^{61} +151.941 q^{62} -395.081 q^{64} +470.619 q^{65} -65.1959 q^{67} +639.964 q^{68} -256.370 q^{71} +318.625 q^{73} -114.619 q^{74} -518.264 q^{76} +77.7532 q^{79} -732.615 q^{80} -161.301 q^{82} -836.549 q^{83} +1078.87 q^{85} -31.1295 q^{86} -424.379 q^{88} -1590.06 q^{89} +235.400 q^{92} +107.319 q^{94} -873.706 q^{95} -1189.88 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 36 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 36 q^{4} - 180 q^{10} - 108 q^{13} + 420 q^{16} - 198 q^{19} - 84 q^{22} + 420 q^{25} + 90 q^{31} + 648 q^{34} - 402 q^{37} - 2844 q^{40} - 660 q^{43} - 1332 q^{46} - 1224 q^{52} - 846 q^{55} - 1800 q^{58} - 1152 q^{61} + 2964 q^{64} + 924 q^{67} - 1260 q^{73} - 5868 q^{76} - 1500 q^{79} - 4500 q^{82} - 2232 q^{85} - 2460 q^{88} + 4968 q^{94} - 3312 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.560992 0.198341 0.0991703 0.995070i \(-0.468381\pi\)
0.0991703 + 0.995070i \(0.468381\pi\)
\(3\) 0 0
\(4\) −7.68529 −0.960661
\(5\) −12.9561 −1.15883 −0.579415 0.815033i \(-0.696719\pi\)
−0.579415 + 0.815033i \(0.696719\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −8.79932 −0.388879
\(9\) 0 0
\(10\) −7.26827 −0.229843
\(11\) 48.2286 1.32195 0.660976 0.750407i \(-0.270142\pi\)
0.660976 + 0.750407i \(0.270142\pi\)
\(12\) 0 0
\(13\) −36.3241 −0.774962 −0.387481 0.921878i \(-0.626655\pi\)
−0.387481 + 0.921878i \(0.626655\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 56.5460 0.883531
\(17\) −83.2713 −1.18802 −0.594008 0.804459i \(-0.702455\pi\)
−0.594008 + 0.804459i \(0.702455\pi\)
\(18\) 0 0
\(19\) 67.4359 0.814255 0.407128 0.913371i \(-0.366530\pi\)
0.407128 + 0.913371i \(0.366530\pi\)
\(20\) 99.5714 1.11324
\(21\) 0 0
\(22\) 27.0559 0.262197
\(23\) −30.6300 −0.277687 −0.138843 0.990314i \(-0.544338\pi\)
−0.138843 + 0.990314i \(0.544338\pi\)
\(24\) 0 0
\(25\) 42.8607 0.342885
\(26\) −20.3775 −0.153706
\(27\) 0 0
\(28\) 0 0
\(29\) 294.244 1.88413 0.942065 0.335431i \(-0.108882\pi\)
0.942065 + 0.335431i \(0.108882\pi\)
\(30\) 0 0
\(31\) 270.844 1.56919 0.784597 0.620006i \(-0.212870\pi\)
0.784597 + 0.620006i \(0.212870\pi\)
\(32\) 102.116 0.564119
\(33\) 0 0
\(34\) −46.7145 −0.235632
\(35\) 0 0
\(36\) 0 0
\(37\) −204.315 −0.907817 −0.453909 0.891048i \(-0.649971\pi\)
−0.453909 + 0.891048i \(0.649971\pi\)
\(38\) 37.8310 0.161500
\(39\) 0 0
\(40\) 114.005 0.450644
\(41\) −287.529 −1.09523 −0.547615 0.836730i \(-0.684464\pi\)
−0.547615 + 0.836730i \(0.684464\pi\)
\(42\) 0 0
\(43\) −55.4901 −0.196794 −0.0983972 0.995147i \(-0.531372\pi\)
−0.0983972 + 0.995147i \(0.531372\pi\)
\(44\) −370.651 −1.26995
\(45\) 0 0
\(46\) −17.1832 −0.0550765
\(47\) 191.301 0.593706 0.296853 0.954923i \(-0.404063\pi\)
0.296853 + 0.954923i \(0.404063\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 24.0445 0.0680081
\(51\) 0 0
\(52\) 279.161 0.744475
\(53\) 521.107 1.35056 0.675278 0.737563i \(-0.264024\pi\)
0.675278 + 0.737563i \(0.264024\pi\)
\(54\) 0 0
\(55\) −624.855 −1.53192
\(56\) 0 0
\(57\) 0 0
\(58\) 165.069 0.373699
\(59\) 381.074 0.840874 0.420437 0.907322i \(-0.361877\pi\)
0.420437 + 0.907322i \(0.361877\pi\)
\(60\) 0 0
\(61\) 155.465 0.326315 0.163158 0.986600i \(-0.447832\pi\)
0.163158 + 0.986600i \(0.447832\pi\)
\(62\) 151.941 0.311235
\(63\) 0 0
\(64\) −395.081 −0.771643
\(65\) 470.619 0.898048
\(66\) 0 0
\(67\) −65.1959 −0.118880 −0.0594399 0.998232i \(-0.518931\pi\)
−0.0594399 + 0.998232i \(0.518931\pi\)
\(68\) 639.964 1.14128
\(69\) 0 0
\(70\) 0 0
\(71\) −256.370 −0.428529 −0.214265 0.976776i \(-0.568735\pi\)
−0.214265 + 0.976776i \(0.568735\pi\)
\(72\) 0 0
\(73\) 318.625 0.510852 0.255426 0.966829i \(-0.417784\pi\)
0.255426 + 0.966829i \(0.417784\pi\)
\(74\) −114.619 −0.180057
\(75\) 0 0
\(76\) −518.264 −0.782223
\(77\) 0 0
\(78\) 0 0
\(79\) 77.7532 0.110733 0.0553666 0.998466i \(-0.482367\pi\)
0.0553666 + 0.998466i \(0.482367\pi\)
\(80\) −732.615 −1.02386
\(81\) 0 0
\(82\) −161.301 −0.217229
\(83\) −836.549 −1.10630 −0.553152 0.833081i \(-0.686575\pi\)
−0.553152 + 0.833081i \(0.686575\pi\)
\(84\) 0 0
\(85\) 1078.87 1.37671
\(86\) −31.1295 −0.0390323
\(87\) 0 0
\(88\) −424.379 −0.514079
\(89\) −1590.06 −1.89377 −0.946887 0.321565i \(-0.895791\pi\)
−0.946887 + 0.321565i \(0.895791\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 235.400 0.266763
\(93\) 0 0
\(94\) 107.319 0.117756
\(95\) −873.706 −0.943583
\(96\) 0 0
\(97\) −1189.88 −1.24550 −0.622751 0.782420i \(-0.713985\pi\)
−0.622751 + 0.782420i \(0.713985\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −329.397 −0.329397
\(101\) −507.845 −0.500321 −0.250161 0.968204i \(-0.580483\pi\)
−0.250161 + 0.968204i \(0.580483\pi\)
\(102\) 0 0
\(103\) −1852.78 −1.77242 −0.886212 0.463280i \(-0.846672\pi\)
−0.886212 + 0.463280i \(0.846672\pi\)
\(104\) 319.628 0.301366
\(105\) 0 0
\(106\) 292.337 0.267870
\(107\) 614.904 0.555561 0.277781 0.960645i \(-0.410401\pi\)
0.277781 + 0.960645i \(0.410401\pi\)
\(108\) 0 0
\(109\) 1017.92 0.894487 0.447243 0.894412i \(-0.352406\pi\)
0.447243 + 0.894412i \(0.352406\pi\)
\(110\) −350.539 −0.303841
\(111\) 0 0
\(112\) 0 0
\(113\) −1204.42 −1.00267 −0.501337 0.865252i \(-0.667158\pi\)
−0.501337 + 0.865252i \(0.667158\pi\)
\(114\) 0 0
\(115\) 396.845 0.321792
\(116\) −2261.35 −1.81001
\(117\) 0 0
\(118\) 213.779 0.166779
\(119\) 0 0
\(120\) 0 0
\(121\) 995.000 0.747559
\(122\) 87.2145 0.0647215
\(123\) 0 0
\(124\) −2081.52 −1.50746
\(125\) 1064.21 0.761484
\(126\) 0 0
\(127\) −2515.55 −1.75763 −0.878813 0.477165i \(-0.841664\pi\)
−0.878813 + 0.477165i \(0.841664\pi\)
\(128\) −1038.57 −0.717167
\(129\) 0 0
\(130\) 264.014 0.178119
\(131\) 2165.37 1.44419 0.722096 0.691793i \(-0.243179\pi\)
0.722096 + 0.691793i \(0.243179\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −36.5743 −0.0235787
\(135\) 0 0
\(136\) 732.731 0.461994
\(137\) 795.668 0.496194 0.248097 0.968735i \(-0.420195\pi\)
0.248097 + 0.968735i \(0.420195\pi\)
\(138\) 0 0
\(139\) −2400.60 −1.46487 −0.732433 0.680839i \(-0.761615\pi\)
−0.732433 + 0.680839i \(0.761615\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −143.822 −0.0849947
\(143\) −1751.86 −1.02446
\(144\) 0 0
\(145\) −3812.26 −2.18338
\(146\) 178.746 0.101323
\(147\) 0 0
\(148\) 1570.22 0.872105
\(149\) 1404.26 0.772091 0.386046 0.922480i \(-0.373841\pi\)
0.386046 + 0.922480i \(0.373841\pi\)
\(150\) 0 0
\(151\) −1761.48 −0.949322 −0.474661 0.880169i \(-0.657429\pi\)
−0.474661 + 0.880169i \(0.657429\pi\)
\(152\) −593.390 −0.316646
\(153\) 0 0
\(154\) 0 0
\(155\) −3509.09 −1.81843
\(156\) 0 0
\(157\) −1082.03 −0.550034 −0.275017 0.961439i \(-0.588683\pi\)
−0.275017 + 0.961439i \(0.588683\pi\)
\(158\) 43.6189 0.0219629
\(159\) 0 0
\(160\) −1323.03 −0.653717
\(161\) 0 0
\(162\) 0 0
\(163\) −2473.27 −1.18847 −0.594237 0.804290i \(-0.702546\pi\)
−0.594237 + 0.804290i \(0.702546\pi\)
\(164\) 2209.74 1.05215
\(165\) 0 0
\(166\) −469.297 −0.219425
\(167\) −2240.74 −1.03828 −0.519142 0.854688i \(-0.673748\pi\)
−0.519142 + 0.854688i \(0.673748\pi\)
\(168\) 0 0
\(169\) −877.557 −0.399434
\(170\) 605.238 0.273057
\(171\) 0 0
\(172\) 426.457 0.189053
\(173\) 2121.28 0.932242 0.466121 0.884721i \(-0.345651\pi\)
0.466121 + 0.884721i \(0.345651\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2727.13 1.16799
\(177\) 0 0
\(178\) −892.010 −0.375612
\(179\) 1578.84 0.659262 0.329631 0.944110i \(-0.393076\pi\)
0.329631 + 0.944110i \(0.393076\pi\)
\(180\) 0 0
\(181\) −4609.64 −1.89299 −0.946496 0.322716i \(-0.895404\pi\)
−0.946496 + 0.322716i \(0.895404\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 269.523 0.107986
\(185\) 2647.13 1.05201
\(186\) 0 0
\(187\) −4016.06 −1.57050
\(188\) −1470.21 −0.570350
\(189\) 0 0
\(190\) −490.142 −0.187151
\(191\) 1291.72 0.489350 0.244675 0.969605i \(-0.421319\pi\)
0.244675 + 0.969605i \(0.421319\pi\)
\(192\) 0 0
\(193\) 1176.01 0.438605 0.219302 0.975657i \(-0.429622\pi\)
0.219302 + 0.975657i \(0.429622\pi\)
\(194\) −667.511 −0.247033
\(195\) 0 0
\(196\) 0 0
\(197\) 4560.71 1.64943 0.824713 0.565552i \(-0.191337\pi\)
0.824713 + 0.565552i \(0.191337\pi\)
\(198\) 0 0
\(199\) 986.014 0.351240 0.175620 0.984458i \(-0.443807\pi\)
0.175620 + 0.984458i \(0.443807\pi\)
\(200\) −377.145 −0.133341
\(201\) 0 0
\(202\) −284.897 −0.0992340
\(203\) 0 0
\(204\) 0 0
\(205\) 3725.25 1.26919
\(206\) −1039.39 −0.351543
\(207\) 0 0
\(208\) −2053.98 −0.684702
\(209\) 3252.34 1.07641
\(210\) 0 0
\(211\) 5459.28 1.78120 0.890598 0.454792i \(-0.150286\pi\)
0.890598 + 0.454792i \(0.150286\pi\)
\(212\) −4004.85 −1.29743
\(213\) 0 0
\(214\) 344.956 0.110190
\(215\) 718.936 0.228051
\(216\) 0 0
\(217\) 0 0
\(218\) 571.045 0.177413
\(219\) 0 0
\(220\) 4802.19 1.47165
\(221\) 3024.76 0.920666
\(222\) 0 0
\(223\) −4626.41 −1.38927 −0.694636 0.719362i \(-0.744434\pi\)
−0.694636 + 0.719362i \(0.744434\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −675.669 −0.198871
\(227\) −2437.77 −0.712776 −0.356388 0.934338i \(-0.615992\pi\)
−0.356388 + 0.934338i \(0.615992\pi\)
\(228\) 0 0
\(229\) −2946.50 −0.850264 −0.425132 0.905131i \(-0.639772\pi\)
−0.425132 + 0.905131i \(0.639772\pi\)
\(230\) 222.627 0.0638243
\(231\) 0 0
\(232\) −2589.15 −0.732698
\(233\) −3441.49 −0.967638 −0.483819 0.875168i \(-0.660751\pi\)
−0.483819 + 0.875168i \(0.660751\pi\)
\(234\) 0 0
\(235\) −2478.52 −0.688004
\(236\) −2928.66 −0.807795
\(237\) 0 0
\(238\) 0 0
\(239\) 744.021 0.201367 0.100684 0.994918i \(-0.467897\pi\)
0.100684 + 0.994918i \(0.467897\pi\)
\(240\) 0 0
\(241\) 4714.37 1.26008 0.630040 0.776562i \(-0.283038\pi\)
0.630040 + 0.776562i \(0.283038\pi\)
\(242\) 558.187 0.148271
\(243\) 0 0
\(244\) −1194.79 −0.313478
\(245\) 0 0
\(246\) 0 0
\(247\) −2449.55 −0.631017
\(248\) −2383.24 −0.610226
\(249\) 0 0
\(250\) 597.011 0.151033
\(251\) 2248.41 0.565411 0.282706 0.959207i \(-0.408768\pi\)
0.282706 + 0.959207i \(0.408768\pi\)
\(252\) 0 0
\(253\) −1477.24 −0.367089
\(254\) −1411.20 −0.348609
\(255\) 0 0
\(256\) 2578.02 0.629400
\(257\) −6099.67 −1.48049 −0.740247 0.672335i \(-0.765291\pi\)
−0.740247 + 0.672335i \(0.765291\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −3616.85 −0.862720
\(261\) 0 0
\(262\) 1214.75 0.286442
\(263\) 3493.26 0.819026 0.409513 0.912304i \(-0.365699\pi\)
0.409513 + 0.912304i \(0.365699\pi\)
\(264\) 0 0
\(265\) −6751.51 −1.56506
\(266\) 0 0
\(267\) 0 0
\(268\) 501.049 0.114203
\(269\) 6393.14 1.44906 0.724529 0.689244i \(-0.242057\pi\)
0.724529 + 0.689244i \(0.242057\pi\)
\(270\) 0 0
\(271\) 4222.81 0.946559 0.473279 0.880912i \(-0.343070\pi\)
0.473279 + 0.880912i \(0.343070\pi\)
\(272\) −4708.66 −1.04965
\(273\) 0 0
\(274\) 446.363 0.0984153
\(275\) 2067.11 0.453278
\(276\) 0 0
\(277\) −5240.37 −1.13669 −0.568345 0.822790i \(-0.692416\pi\)
−0.568345 + 0.822790i \(0.692416\pi\)
\(278\) −1346.72 −0.290542
\(279\) 0 0
\(280\) 0 0
\(281\) 5924.79 1.25781 0.628903 0.777484i \(-0.283504\pi\)
0.628903 + 0.777484i \(0.283504\pi\)
\(282\) 0 0
\(283\) 1885.61 0.396070 0.198035 0.980195i \(-0.436544\pi\)
0.198035 + 0.980195i \(0.436544\pi\)
\(284\) 1970.28 0.411671
\(285\) 0 0
\(286\) −982.781 −0.203192
\(287\) 0 0
\(288\) 0 0
\(289\) 2021.11 0.411380
\(290\) −2138.65 −0.433054
\(291\) 0 0
\(292\) −2448.72 −0.490756
\(293\) 3173.12 0.632680 0.316340 0.948646i \(-0.397546\pi\)
0.316340 + 0.948646i \(0.397546\pi\)
\(294\) 0 0
\(295\) −4937.23 −0.974430
\(296\) 1797.84 0.353031
\(297\) 0 0
\(298\) 787.779 0.153137
\(299\) 1112.61 0.215197
\(300\) 0 0
\(301\) 0 0
\(302\) −988.179 −0.188289
\(303\) 0 0
\(304\) 3813.23 0.719419
\(305\) −2014.22 −0.378144
\(306\) 0 0
\(307\) −6188.30 −1.15044 −0.575220 0.817999i \(-0.695084\pi\)
−0.575220 + 0.817999i \(0.695084\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1968.57 −0.360668
\(311\) −3750.93 −0.683909 −0.341954 0.939717i \(-0.611089\pi\)
−0.341954 + 0.939717i \(0.611089\pi\)
\(312\) 0 0
\(313\) 4172.29 0.753456 0.376728 0.926324i \(-0.377049\pi\)
0.376728 + 0.926324i \(0.377049\pi\)
\(314\) −607.009 −0.109094
\(315\) 0 0
\(316\) −597.556 −0.106377
\(317\) 8410.38 1.49014 0.745069 0.666987i \(-0.232416\pi\)
0.745069 + 0.666987i \(0.232416\pi\)
\(318\) 0 0
\(319\) 14191.0 2.49073
\(320\) 5118.71 0.894203
\(321\) 0 0
\(322\) 0 0
\(323\) −5615.47 −0.967347
\(324\) 0 0
\(325\) −1556.88 −0.265723
\(326\) −1387.48 −0.235723
\(327\) 0 0
\(328\) 2530.06 0.425912
\(329\) 0 0
\(330\) 0 0
\(331\) −1217.72 −0.202211 −0.101105 0.994876i \(-0.532238\pi\)
−0.101105 + 0.994876i \(0.532238\pi\)
\(332\) 6429.12 1.06278
\(333\) 0 0
\(334\) −1257.03 −0.205934
\(335\) 844.684 0.137761
\(336\) 0 0
\(337\) 3880.01 0.627175 0.313587 0.949559i \(-0.398469\pi\)
0.313587 + 0.949559i \(0.398469\pi\)
\(338\) −492.302 −0.0792240
\(339\) 0 0
\(340\) −8291.44 −1.32255
\(341\) 13062.4 2.07440
\(342\) 0 0
\(343\) 0 0
\(344\) 488.275 0.0765291
\(345\) 0 0
\(346\) 1190.02 0.184901
\(347\) −9672.45 −1.49638 −0.748191 0.663484i \(-0.769077\pi\)
−0.748191 + 0.663484i \(0.769077\pi\)
\(348\) 0 0
\(349\) −3960.64 −0.607473 −0.303737 0.952756i \(-0.598234\pi\)
−0.303737 + 0.952756i \(0.598234\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4924.93 0.745738
\(353\) 7639.67 1.15189 0.575947 0.817487i \(-0.304633\pi\)
0.575947 + 0.817487i \(0.304633\pi\)
\(354\) 0 0
\(355\) 3321.56 0.496592
\(356\) 12220.1 1.81928
\(357\) 0 0
\(358\) 885.715 0.130758
\(359\) −9411.16 −1.38357 −0.691785 0.722103i \(-0.743176\pi\)
−0.691785 + 0.722103i \(0.743176\pi\)
\(360\) 0 0
\(361\) −2311.40 −0.336989
\(362\) −2585.97 −0.375457
\(363\) 0 0
\(364\) 0 0
\(365\) −4128.14 −0.591991
\(366\) 0 0
\(367\) −5707.80 −0.811838 −0.405919 0.913909i \(-0.633048\pi\)
−0.405919 + 0.913909i \(0.633048\pi\)
\(368\) −1732.00 −0.245345
\(369\) 0 0
\(370\) 1485.02 0.208655
\(371\) 0 0
\(372\) 0 0
\(373\) −9579.55 −1.32979 −0.664893 0.746938i \(-0.731523\pi\)
−0.664893 + 0.746938i \(0.731523\pi\)
\(374\) −2252.98 −0.311494
\(375\) 0 0
\(376\) −1683.32 −0.230880
\(377\) −10688.2 −1.46013
\(378\) 0 0
\(379\) 8730.92 1.18332 0.591658 0.806189i \(-0.298473\pi\)
0.591658 + 0.806189i \(0.298473\pi\)
\(380\) 6714.68 0.906463
\(381\) 0 0
\(382\) 724.646 0.0970579
\(383\) −9623.21 −1.28387 −0.641936 0.766758i \(-0.721869\pi\)
−0.641936 + 0.766758i \(0.721869\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 659.730 0.0869931
\(387\) 0 0
\(388\) 9144.54 1.19650
\(389\) −11319.7 −1.47541 −0.737704 0.675125i \(-0.764090\pi\)
−0.737704 + 0.675125i \(0.764090\pi\)
\(390\) 0 0
\(391\) 2550.60 0.329896
\(392\) 0 0
\(393\) 0 0
\(394\) 2558.52 0.327148
\(395\) −1007.38 −0.128321
\(396\) 0 0
\(397\) −2855.87 −0.361038 −0.180519 0.983572i \(-0.557778\pi\)
−0.180519 + 0.983572i \(0.557778\pi\)
\(398\) 553.146 0.0696651
\(399\) 0 0
\(400\) 2423.60 0.302950
\(401\) −1975.97 −0.246073 −0.123036 0.992402i \(-0.539263\pi\)
−0.123036 + 0.992402i \(0.539263\pi\)
\(402\) 0 0
\(403\) −9838.18 −1.21607
\(404\) 3902.94 0.480639
\(405\) 0 0
\(406\) 0 0
\(407\) −9853.85 −1.20009
\(408\) 0 0
\(409\) −8244.62 −0.996749 −0.498375 0.866962i \(-0.666070\pi\)
−0.498375 + 0.866962i \(0.666070\pi\)
\(410\) 2089.84 0.251731
\(411\) 0 0
\(412\) 14239.1 1.70270
\(413\) 0 0
\(414\) 0 0
\(415\) 10838.4 1.28202
\(416\) −3709.29 −0.437170
\(417\) 0 0
\(418\) 1824.54 0.213495
\(419\) −12861.0 −1.49952 −0.749761 0.661709i \(-0.769831\pi\)
−0.749761 + 0.661709i \(0.769831\pi\)
\(420\) 0 0
\(421\) −16467.7 −1.90638 −0.953188 0.302377i \(-0.902220\pi\)
−0.953188 + 0.302377i \(0.902220\pi\)
\(422\) 3062.61 0.353283
\(423\) 0 0
\(424\) −4585.38 −0.525203
\(425\) −3569.06 −0.407353
\(426\) 0 0
\(427\) 0 0
\(428\) −4725.72 −0.533706
\(429\) 0 0
\(430\) 403.317 0.0452318
\(431\) 273.346 0.0305490 0.0152745 0.999883i \(-0.495138\pi\)
0.0152745 + 0.999883i \(0.495138\pi\)
\(432\) 0 0
\(433\) 13765.3 1.52776 0.763878 0.645361i \(-0.223293\pi\)
0.763878 + 0.645361i \(0.223293\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −7823.01 −0.859298
\(437\) −2065.56 −0.226108
\(438\) 0 0
\(439\) −2965.97 −0.322456 −0.161228 0.986917i \(-0.551545\pi\)
−0.161228 + 0.986917i \(0.551545\pi\)
\(440\) 5498.30 0.595730
\(441\) 0 0
\(442\) 1696.86 0.182605
\(443\) −14038.4 −1.50561 −0.752806 0.658242i \(-0.771300\pi\)
−0.752806 + 0.658242i \(0.771300\pi\)
\(444\) 0 0
\(445\) 20601.0 2.19456
\(446\) −2595.38 −0.275549
\(447\) 0 0
\(448\) 0 0
\(449\) 6338.13 0.666180 0.333090 0.942895i \(-0.391909\pi\)
0.333090 + 0.942895i \(0.391909\pi\)
\(450\) 0 0
\(451\) −13867.1 −1.44784
\(452\) 9256.30 0.963230
\(453\) 0 0
\(454\) −1367.57 −0.141372
\(455\) 0 0
\(456\) 0 0
\(457\) 9120.26 0.933540 0.466770 0.884379i \(-0.345418\pi\)
0.466770 + 0.884379i \(0.345418\pi\)
\(458\) −1652.97 −0.168642
\(459\) 0 0
\(460\) −3049.87 −0.309133
\(461\) 15109.6 1.52651 0.763256 0.646096i \(-0.223600\pi\)
0.763256 + 0.646096i \(0.223600\pi\)
\(462\) 0 0
\(463\) 6418.72 0.644283 0.322142 0.946692i \(-0.395597\pi\)
0.322142 + 0.946692i \(0.395597\pi\)
\(464\) 16638.3 1.66469
\(465\) 0 0
\(466\) −1930.65 −0.191922
\(467\) −4304.81 −0.426559 −0.213279 0.976991i \(-0.568414\pi\)
−0.213279 + 0.976991i \(0.568414\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −1390.43 −0.136459
\(471\) 0 0
\(472\) −3353.19 −0.326998
\(473\) −2676.21 −0.260153
\(474\) 0 0
\(475\) 2890.35 0.279196
\(476\) 0 0
\(477\) 0 0
\(478\) 417.390 0.0399393
\(479\) −5030.65 −0.479867 −0.239934 0.970789i \(-0.577126\pi\)
−0.239934 + 0.970789i \(0.577126\pi\)
\(480\) 0 0
\(481\) 7421.58 0.703524
\(482\) 2644.72 0.249925
\(483\) 0 0
\(484\) −7646.87 −0.718150
\(485\) 15416.2 1.44332
\(486\) 0 0
\(487\) −13300.6 −1.23759 −0.618794 0.785553i \(-0.712379\pi\)
−0.618794 + 0.785553i \(0.712379\pi\)
\(488\) −1367.98 −0.126897
\(489\) 0 0
\(490\) 0 0
\(491\) 9246.30 0.849857 0.424928 0.905227i \(-0.360299\pi\)
0.424928 + 0.905227i \(0.360299\pi\)
\(492\) 0 0
\(493\) −24502.1 −2.23837
\(494\) −1374.18 −0.125156
\(495\) 0 0
\(496\) 15315.1 1.38643
\(497\) 0 0
\(498\) 0 0
\(499\) −16233.3 −1.45632 −0.728160 0.685407i \(-0.759624\pi\)
−0.728160 + 0.685407i \(0.759624\pi\)
\(500\) −8178.73 −0.731528
\(501\) 0 0
\(502\) 1261.34 0.112144
\(503\) −1664.13 −0.147515 −0.0737575 0.997276i \(-0.523499\pi\)
−0.0737575 + 0.997276i \(0.523499\pi\)
\(504\) 0 0
\(505\) 6579.69 0.579787
\(506\) −828.721 −0.0728086
\(507\) 0 0
\(508\) 19332.7 1.68848
\(509\) −8757.17 −0.762583 −0.381291 0.924455i \(-0.624521\pi\)
−0.381291 + 0.924455i \(0.624521\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 9754.80 0.842002
\(513\) 0 0
\(514\) −3421.86 −0.293642
\(515\) 24004.8 2.05394
\(516\) 0 0
\(517\) 9226.21 0.784851
\(518\) 0 0
\(519\) 0 0
\(520\) −4141.13 −0.349232
\(521\) 1641.01 0.137993 0.0689963 0.997617i \(-0.478020\pi\)
0.0689963 + 0.997617i \(0.478020\pi\)
\(522\) 0 0
\(523\) 16530.6 1.38209 0.691043 0.722813i \(-0.257151\pi\)
0.691043 + 0.722813i \(0.257151\pi\)
\(524\) −16641.5 −1.38738
\(525\) 0 0
\(526\) 1959.69 0.162446
\(527\) −22553.5 −1.86423
\(528\) 0 0
\(529\) −11228.8 −0.922890
\(530\) −3787.54 −0.310416
\(531\) 0 0
\(532\) 0 0
\(533\) 10444.2 0.848762
\(534\) 0 0
\(535\) −7966.77 −0.643801
\(536\) 573.679 0.0462298
\(537\) 0 0
\(538\) 3586.50 0.287407
\(539\) 0 0
\(540\) 0 0
\(541\) −3418.03 −0.271631 −0.135816 0.990734i \(-0.543365\pi\)
−0.135816 + 0.990734i \(0.543365\pi\)
\(542\) 2368.96 0.187741
\(543\) 0 0
\(544\) −8503.36 −0.670181
\(545\) −13188.3 −1.03656
\(546\) 0 0
\(547\) −15983.1 −1.24934 −0.624668 0.780890i \(-0.714766\pi\)
−0.624668 + 0.780890i \(0.714766\pi\)
\(548\) −6114.94 −0.476674
\(549\) 0 0
\(550\) 1159.63 0.0899034
\(551\) 19842.6 1.53416
\(552\) 0 0
\(553\) 0 0
\(554\) −2939.80 −0.225452
\(555\) 0 0
\(556\) 18449.3 1.40724
\(557\) −8736.81 −0.664615 −0.332307 0.943171i \(-0.607827\pi\)
−0.332307 + 0.943171i \(0.607827\pi\)
\(558\) 0 0
\(559\) 2015.63 0.152508
\(560\) 0 0
\(561\) 0 0
\(562\) 3323.76 0.249474
\(563\) 24107.2 1.80461 0.902306 0.431095i \(-0.141873\pi\)
0.902306 + 0.431095i \(0.141873\pi\)
\(564\) 0 0
\(565\) 15604.6 1.16193
\(566\) 1057.81 0.0785567
\(567\) 0 0
\(568\) 2255.89 0.166646
\(569\) −13291.7 −0.979292 −0.489646 0.871921i \(-0.662874\pi\)
−0.489646 + 0.871921i \(0.662874\pi\)
\(570\) 0 0
\(571\) 15208.2 1.11461 0.557305 0.830308i \(-0.311836\pi\)
0.557305 + 0.830308i \(0.311836\pi\)
\(572\) 13463.6 0.984161
\(573\) 0 0
\(574\) 0 0
\(575\) −1312.82 −0.0952147
\(576\) 0 0
\(577\) −10809.5 −0.779904 −0.389952 0.920835i \(-0.627508\pi\)
−0.389952 + 0.920835i \(0.627508\pi\)
\(578\) 1133.83 0.0815933
\(579\) 0 0
\(580\) 29298.3 2.09749
\(581\) 0 0
\(582\) 0 0
\(583\) 25132.3 1.78537
\(584\) −2803.68 −0.198660
\(585\) 0 0
\(586\) 1780.09 0.125486
\(587\) −3843.26 −0.270236 −0.135118 0.990830i \(-0.543141\pi\)
−0.135118 + 0.990830i \(0.543141\pi\)
\(588\) 0 0
\(589\) 18264.6 1.27773
\(590\) −2769.75 −0.193269
\(591\) 0 0
\(592\) −11553.2 −0.802084
\(593\) 1402.87 0.0971484 0.0485742 0.998820i \(-0.484532\pi\)
0.0485742 + 0.998820i \(0.484532\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −10792.2 −0.741718
\(597\) 0 0
\(598\) 624.164 0.0426822
\(599\) 12738.8 0.868937 0.434469 0.900687i \(-0.356936\pi\)
0.434469 + 0.900687i \(0.356936\pi\)
\(600\) 0 0
\(601\) 17292.7 1.17368 0.586841 0.809702i \(-0.300371\pi\)
0.586841 + 0.809702i \(0.300371\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 13537.5 0.911977
\(605\) −12891.3 −0.866293
\(606\) 0 0
\(607\) 27165.6 1.81650 0.908251 0.418426i \(-0.137418\pi\)
0.908251 + 0.418426i \(0.137418\pi\)
\(608\) 6886.31 0.459336
\(609\) 0 0
\(610\) −1129.96 −0.0750012
\(611\) −6948.86 −0.460100
\(612\) 0 0
\(613\) −10405.5 −0.685599 −0.342800 0.939409i \(-0.611375\pi\)
−0.342800 + 0.939409i \(0.611375\pi\)
\(614\) −3471.59 −0.228179
\(615\) 0 0
\(616\) 0 0
\(617\) 18326.4 1.19578 0.597889 0.801579i \(-0.296006\pi\)
0.597889 + 0.801579i \(0.296006\pi\)
\(618\) 0 0
\(619\) −3575.13 −0.232143 −0.116072 0.993241i \(-0.537030\pi\)
−0.116072 + 0.993241i \(0.537030\pi\)
\(620\) 26968.3 1.74689
\(621\) 0 0
\(622\) −2104.24 −0.135647
\(623\) 0 0
\(624\) 0 0
\(625\) −19145.5 −1.22531
\(626\) 2340.62 0.149441
\(627\) 0 0
\(628\) 8315.70 0.528396
\(629\) 17013.6 1.07850
\(630\) 0 0
\(631\) −11722.1 −0.739538 −0.369769 0.929124i \(-0.620563\pi\)
−0.369769 + 0.929124i \(0.620563\pi\)
\(632\) −684.175 −0.0430617
\(633\) 0 0
\(634\) 4718.15 0.295555
\(635\) 32591.7 2.03679
\(636\) 0 0
\(637\) 0 0
\(638\) 7961.03 0.494013
\(639\) 0 0
\(640\) 13455.8 0.831074
\(641\) −24327.5 −1.49903 −0.749514 0.661988i \(-0.769713\pi\)
−0.749514 + 0.661988i \(0.769713\pi\)
\(642\) 0 0
\(643\) 10684.8 0.655316 0.327658 0.944796i \(-0.393741\pi\)
0.327658 + 0.944796i \(0.393741\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −3150.23 −0.191864
\(647\) −21203.0 −1.28837 −0.644184 0.764870i \(-0.722803\pi\)
−0.644184 + 0.764870i \(0.722803\pi\)
\(648\) 0 0
\(649\) 18378.7 1.11160
\(650\) −873.395 −0.0527037
\(651\) 0 0
\(652\) 19007.8 1.14172
\(653\) 27562.2 1.65175 0.825874 0.563854i \(-0.190682\pi\)
0.825874 + 0.563854i \(0.190682\pi\)
\(654\) 0 0
\(655\) −28054.7 −1.67357
\(656\) −16258.6 −0.967670
\(657\) 0 0
\(658\) 0 0
\(659\) 1389.06 0.0821091 0.0410546 0.999157i \(-0.486928\pi\)
0.0410546 + 0.999157i \(0.486928\pi\)
\(660\) 0 0
\(661\) −12490.6 −0.734989 −0.367494 0.930026i \(-0.619784\pi\)
−0.367494 + 0.930026i \(0.619784\pi\)
\(662\) −683.130 −0.0401066
\(663\) 0 0
\(664\) 7361.06 0.430218
\(665\) 0 0
\(666\) 0 0
\(667\) −9012.70 −0.523198
\(668\) 17220.7 0.997438
\(669\) 0 0
\(670\) 473.861 0.0273237
\(671\) 7497.85 0.431373
\(672\) 0 0
\(673\) −13234.5 −0.758027 −0.379014 0.925391i \(-0.623737\pi\)
−0.379014 + 0.925391i \(0.623737\pi\)
\(674\) 2176.66 0.124394
\(675\) 0 0
\(676\) 6744.28 0.383721
\(677\) −4788.89 −0.271864 −0.135932 0.990718i \(-0.543403\pi\)
−0.135932 + 0.990718i \(0.543403\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −9493.34 −0.535372
\(681\) 0 0
\(682\) 7327.92 0.411438
\(683\) −7543.01 −0.422584 −0.211292 0.977423i \(-0.567767\pi\)
−0.211292 + 0.977423i \(0.567767\pi\)
\(684\) 0 0
\(685\) −10308.8 −0.575004
\(686\) 0 0
\(687\) 0 0
\(688\) −3137.74 −0.173874
\(689\) −18928.7 −1.04663
\(690\) 0 0
\(691\) 27685.3 1.52416 0.762082 0.647481i \(-0.224177\pi\)
0.762082 + 0.647481i \(0.224177\pi\)
\(692\) −16302.6 −0.895568
\(693\) 0 0
\(694\) −5426.16 −0.296793
\(695\) 31102.4 1.69753
\(696\) 0 0
\(697\) 23942.9 1.30115
\(698\) −2221.89 −0.120487
\(699\) 0 0
\(700\) 0 0
\(701\) −19866.5 −1.07040 −0.535198 0.844726i \(-0.679763\pi\)
−0.535198 + 0.844726i \(0.679763\pi\)
\(702\) 0 0
\(703\) −13778.2 −0.739195
\(704\) −19054.2 −1.02008
\(705\) 0 0
\(706\) 4285.79 0.228467
\(707\) 0 0
\(708\) 0 0
\(709\) −3572.12 −0.189216 −0.0946078 0.995515i \(-0.530160\pi\)
−0.0946078 + 0.995515i \(0.530160\pi\)
\(710\) 1863.37 0.0984944
\(711\) 0 0
\(712\) 13991.4 0.736448
\(713\) −8295.95 −0.435745
\(714\) 0 0
\(715\) 22697.3 1.18718
\(716\) −12133.8 −0.633327
\(717\) 0 0
\(718\) −5279.58 −0.274418
\(719\) −32955.5 −1.70936 −0.854681 0.519154i \(-0.826247\pi\)
−0.854681 + 0.519154i \(0.826247\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1296.68 −0.0668385
\(723\) 0 0
\(724\) 35426.4 1.81852
\(725\) 12611.5 0.646041
\(726\) 0 0
\(727\) −6347.31 −0.323808 −0.161904 0.986806i \(-0.551764\pi\)
−0.161904 + 0.986806i \(0.551764\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2315.85 −0.117416
\(731\) 4620.73 0.233795
\(732\) 0 0
\(733\) 22488.7 1.13321 0.566603 0.823991i \(-0.308257\pi\)
0.566603 + 0.823991i \(0.308257\pi\)
\(734\) −3202.03 −0.161020
\(735\) 0 0
\(736\) −3127.82 −0.156648
\(737\) −3144.31 −0.157153
\(738\) 0 0
\(739\) 19859.5 0.988559 0.494279 0.869303i \(-0.335432\pi\)
0.494279 + 0.869303i \(0.335432\pi\)
\(740\) −20344.0 −1.01062
\(741\) 0 0
\(742\) 0 0
\(743\) −16900.0 −0.834455 −0.417228 0.908802i \(-0.636998\pi\)
−0.417228 + 0.908802i \(0.636998\pi\)
\(744\) 0 0
\(745\) −18193.8 −0.894722
\(746\) −5374.05 −0.263751
\(747\) 0 0
\(748\) 30864.6 1.50872
\(749\) 0 0
\(750\) 0 0
\(751\) 19441.1 0.944627 0.472314 0.881431i \(-0.343419\pi\)
0.472314 + 0.881431i \(0.343419\pi\)
\(752\) 10817.3 0.524558
\(753\) 0 0
\(754\) −5995.97 −0.289603
\(755\) 22822.0 1.10010
\(756\) 0 0
\(757\) −17481.7 −0.839343 −0.419671 0.907676i \(-0.637855\pi\)
−0.419671 + 0.907676i \(0.637855\pi\)
\(758\) 4897.97 0.234700
\(759\) 0 0
\(760\) 7688.02 0.366939
\(761\) 4160.98 0.198207 0.0991035 0.995077i \(-0.468403\pi\)
0.0991035 + 0.995077i \(0.468403\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −9927.26 −0.470099
\(765\) 0 0
\(766\) −5398.54 −0.254644
\(767\) −13842.2 −0.651645
\(768\) 0 0
\(769\) −33004.0 −1.54766 −0.773832 0.633391i \(-0.781662\pi\)
−0.773832 + 0.633391i \(0.781662\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −9037.94 −0.421351
\(773\) −29291.1 −1.36291 −0.681455 0.731860i \(-0.738652\pi\)
−0.681455 + 0.731860i \(0.738652\pi\)
\(774\) 0 0
\(775\) 11608.6 0.538054
\(776\) 10470.1 0.484349
\(777\) 0 0
\(778\) −6350.28 −0.292633
\(779\) −19389.8 −0.891797
\(780\) 0 0
\(781\) −12364.4 −0.566495
\(782\) 1430.87 0.0654318
\(783\) 0 0
\(784\) 0 0
\(785\) 14018.9 0.637395
\(786\) 0 0
\(787\) −7935.32 −0.359420 −0.179710 0.983720i \(-0.557516\pi\)
−0.179710 + 0.983720i \(0.557516\pi\)
\(788\) −35050.3 −1.58454
\(789\) 0 0
\(790\) −565.131 −0.0254512
\(791\) 0 0
\(792\) 0 0
\(793\) −5647.12 −0.252882
\(794\) −1602.12 −0.0716084
\(795\) 0 0
\(796\) −7577.80 −0.337422
\(797\) −20316.5 −0.902947 −0.451473 0.892285i \(-0.649101\pi\)
−0.451473 + 0.892285i \(0.649101\pi\)
\(798\) 0 0
\(799\) −15929.9 −0.705332
\(800\) 4376.78 0.193428
\(801\) 0 0
\(802\) −1108.50 −0.0488062
\(803\) 15366.8 0.675323
\(804\) 0 0
\(805\) 0 0
\(806\) −5519.14 −0.241195
\(807\) 0 0
\(808\) 4468.69 0.194564
\(809\) −12447.0 −0.540931 −0.270465 0.962730i \(-0.587178\pi\)
−0.270465 + 0.962730i \(0.587178\pi\)
\(810\) 0 0
\(811\) 9478.03 0.410380 0.205190 0.978722i \(-0.434219\pi\)
0.205190 + 0.978722i \(0.434219\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −5527.93 −0.238027
\(815\) 32043.9 1.37724
\(816\) 0 0
\(817\) −3742.02 −0.160241
\(818\) −4625.17 −0.197696
\(819\) 0 0
\(820\) −28629.7 −1.21926
\(821\) −3763.09 −0.159967 −0.0799835 0.996796i \(-0.525487\pi\)
−0.0799835 + 0.996796i \(0.525487\pi\)
\(822\) 0 0
\(823\) 21474.8 0.909554 0.454777 0.890605i \(-0.349719\pi\)
0.454777 + 0.890605i \(0.349719\pi\)
\(824\) 16303.2 0.689258
\(825\) 0 0
\(826\) 0 0
\(827\) −45304.9 −1.90496 −0.952481 0.304597i \(-0.901478\pi\)
−0.952481 + 0.304597i \(0.901478\pi\)
\(828\) 0 0
\(829\) 4614.78 0.193339 0.0966694 0.995317i \(-0.469181\pi\)
0.0966694 + 0.995317i \(0.469181\pi\)
\(830\) 6080.26 0.254276
\(831\) 0 0
\(832\) 14351.0 0.597994
\(833\) 0 0
\(834\) 0 0
\(835\) 29031.2 1.20319
\(836\) −24995.2 −1.03406
\(837\) 0 0
\(838\) −7214.90 −0.297416
\(839\) 18308.4 0.753368 0.376684 0.926342i \(-0.377064\pi\)
0.376684 + 0.926342i \(0.377064\pi\)
\(840\) 0 0
\(841\) 62190.6 2.54995
\(842\) −9238.22 −0.378112
\(843\) 0 0
\(844\) −41956.1 −1.71112
\(845\) 11369.7 0.462876
\(846\) 0 0
\(847\) 0 0
\(848\) 29466.5 1.19326
\(849\) 0 0
\(850\) −2002.22 −0.0807946
\(851\) 6258.18 0.252089
\(852\) 0 0
\(853\) 13373.0 0.536790 0.268395 0.963309i \(-0.413507\pi\)
0.268395 + 0.963309i \(0.413507\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −5410.74 −0.216046
\(857\) 17658.0 0.703835 0.351917 0.936031i \(-0.385530\pi\)
0.351917 + 0.936031i \(0.385530\pi\)
\(858\) 0 0
\(859\) −12413.0 −0.493044 −0.246522 0.969137i \(-0.579288\pi\)
−0.246522 + 0.969137i \(0.579288\pi\)
\(860\) −5525.23 −0.219080
\(861\) 0 0
\(862\) 153.345 0.00605910
\(863\) 10219.2 0.403087 0.201544 0.979480i \(-0.435404\pi\)
0.201544 + 0.979480i \(0.435404\pi\)
\(864\) 0 0
\(865\) −27483.5 −1.08031
\(866\) 7722.22 0.303016
\(867\) 0 0
\(868\) 0 0
\(869\) 3749.93 0.146384
\(870\) 0 0
\(871\) 2368.18 0.0921272
\(872\) −8957.00 −0.347847
\(873\) 0 0
\(874\) −1158.76 −0.0448464
\(875\) 0 0
\(876\) 0 0
\(877\) 952.207 0.0366633 0.0183317 0.999832i \(-0.494165\pi\)
0.0183317 + 0.999832i \(0.494165\pi\)
\(878\) −1663.89 −0.0639561
\(879\) 0 0
\(880\) −35333.0 −1.35350
\(881\) −25182.5 −0.963021 −0.481510 0.876440i \(-0.659912\pi\)
−0.481510 + 0.876440i \(0.659912\pi\)
\(882\) 0 0
\(883\) −3941.95 −0.150234 −0.0751172 0.997175i \(-0.523933\pi\)
−0.0751172 + 0.997175i \(0.523933\pi\)
\(884\) −23246.1 −0.884448
\(885\) 0 0
\(886\) −7875.45 −0.298624
\(887\) −41630.5 −1.57589 −0.787946 0.615744i \(-0.788856\pi\)
−0.787946 + 0.615744i \(0.788856\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 11557.0 0.435271
\(891\) 0 0
\(892\) 35555.3 1.33462
\(893\) 12900.6 0.483428
\(894\) 0 0
\(895\) −20455.6 −0.763972
\(896\) 0 0
\(897\) 0 0
\(898\) 3555.64 0.132130
\(899\) 79694.3 2.95657
\(900\) 0 0
\(901\) −43393.2 −1.60448
\(902\) −7779.34 −0.287166
\(903\) 0 0
\(904\) 10598.1 0.389918
\(905\) 59722.9 2.19365
\(906\) 0 0
\(907\) 34713.1 1.27081 0.635407 0.772177i \(-0.280832\pi\)
0.635407 + 0.772177i \(0.280832\pi\)
\(908\) 18734.9 0.684737
\(909\) 0 0
\(910\) 0 0
\(911\) −22126.6 −0.804705 −0.402352 0.915485i \(-0.631807\pi\)
−0.402352 + 0.915485i \(0.631807\pi\)
\(912\) 0 0
\(913\) −40345.6 −1.46248
\(914\) 5116.39 0.185159
\(915\) 0 0
\(916\) 22644.7 0.816816
\(917\) 0 0
\(918\) 0 0
\(919\) −38198.8 −1.37112 −0.685561 0.728015i \(-0.740443\pi\)
−0.685561 + 0.728015i \(0.740443\pi\)
\(920\) −3491.97 −0.125138
\(921\) 0 0
\(922\) 8476.33 0.302769
\(923\) 9312.43 0.332094
\(924\) 0 0
\(925\) −8757.09 −0.311277
\(926\) 3600.85 0.127787
\(927\) 0 0
\(928\) 30047.1 1.06287
\(929\) 13387.9 0.472813 0.236407 0.971654i \(-0.424030\pi\)
0.236407 + 0.971654i \(0.424030\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 26448.8 0.929572
\(933\) 0 0
\(934\) −2414.96 −0.0846039
\(935\) 52032.5 1.81994
\(936\) 0 0
\(937\) −34574.1 −1.20543 −0.602715 0.797957i \(-0.705914\pi\)
−0.602715 + 0.797957i \(0.705914\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 19048.2 0.660939
\(941\) 2922.81 0.101255 0.0506274 0.998718i \(-0.483878\pi\)
0.0506274 + 0.998718i \(0.483878\pi\)
\(942\) 0 0
\(943\) 8807.01 0.304131
\(944\) 21548.2 0.742938
\(945\) 0 0
\(946\) −1501.33 −0.0515989
\(947\) −9913.84 −0.340186 −0.170093 0.985428i \(-0.554407\pi\)
−0.170093 + 0.985428i \(0.554407\pi\)
\(948\) 0 0
\(949\) −11573.8 −0.395891
\(950\) 1621.46 0.0553759
\(951\) 0 0
\(952\) 0 0
\(953\) 51173.4 1.73942 0.869710 0.493562i \(-0.164306\pi\)
0.869710 + 0.493562i \(0.164306\pi\)
\(954\) 0 0
\(955\) −16735.7 −0.567073
\(956\) −5718.02 −0.193446
\(957\) 0 0
\(958\) −2822.16 −0.0951772
\(959\) 0 0
\(960\) 0 0
\(961\) 43565.5 1.46237
\(962\) 4163.45 0.139537
\(963\) 0 0
\(964\) −36231.3 −1.21051
\(965\) −15236.5 −0.508268
\(966\) 0 0
\(967\) 8758.72 0.291273 0.145637 0.989338i \(-0.453477\pi\)
0.145637 + 0.989338i \(0.453477\pi\)
\(968\) −8755.33 −0.290710
\(969\) 0 0
\(970\) 8648.34 0.286270
\(971\) −8293.18 −0.274089 −0.137045 0.990565i \(-0.543760\pi\)
−0.137045 + 0.990565i \(0.543760\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −7461.50 −0.245464
\(975\) 0 0
\(976\) 8790.91 0.288309
\(977\) 18969.7 0.621181 0.310590 0.950544i \(-0.399473\pi\)
0.310590 + 0.950544i \(0.399473\pi\)
\(978\) 0 0
\(979\) −76686.4 −2.50348
\(980\) 0 0
\(981\) 0 0
\(982\) 5187.10 0.168561
\(983\) −56313.2 −1.82718 −0.913588 0.406642i \(-0.866700\pi\)
−0.913588 + 0.406642i \(0.866700\pi\)
\(984\) 0 0
\(985\) −59089.0 −1.91140
\(986\) −13745.5 −0.443960
\(987\) 0 0
\(988\) 18825.5 0.606193
\(989\) 1699.66 0.0546472
\(990\) 0 0
\(991\) −8558.30 −0.274332 −0.137166 0.990548i \(-0.543799\pi\)
−0.137166 + 0.990548i \(0.543799\pi\)
\(992\) 27657.6 0.885212
\(993\) 0 0
\(994\) 0 0
\(995\) −12774.9 −0.407027
\(996\) 0 0
\(997\) −36163.8 −1.14876 −0.574382 0.818587i \(-0.694758\pi\)
−0.574382 + 0.818587i \(0.694758\pi\)
\(998\) −9106.77 −0.288847
\(999\) 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 1323.4.a.bf.1.4 yes 6
3.2 odd 2 inner 1323.4.a.bf.1.3 6
7.6 odd 2 1323.4.a.bg.1.4 yes 6
21.20 even 2 1323.4.a.bg.1.3 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1323.4.a.bf.1.3 6 3.2 odd 2 inner
1323.4.a.bf.1.4 yes 6 1.1 even 1 trivial
1323.4.a.bg.1.3 yes 6 21.20 even 2
1323.4.a.bg.1.4 yes 6 7.6 odd 2