Properties

Label 1575.4.a.bs.1.2
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
Dimension $8$
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 1.2
Root \(-3.95827\) 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)
 
\(f(q)\) \(=\) \(q-3.95827 q^{2} +7.66789 q^{4} +7.00000 q^{7} +1.31458 q^{8} -18.4481 q^{11} +48.8596 q^{13} -27.7079 q^{14} -66.5466 q^{16} -124.724 q^{17} -122.717 q^{19} +73.0227 q^{22} -140.628 q^{23} -193.400 q^{26} +53.6752 q^{28} +196.100 q^{29} +289.591 q^{31} +252.893 q^{32} +493.690 q^{34} -401.649 q^{37} +485.747 q^{38} +303.093 q^{41} +202.027 q^{43} -141.458 q^{44} +556.643 q^{46} -308.944 q^{47} +49.0000 q^{49} +374.650 q^{52} -461.082 q^{53} +9.20203 q^{56} -776.217 q^{58} -111.900 q^{59} +416.658 q^{61} -1146.28 q^{62} -468.644 q^{64} +452.322 q^{67} -956.367 q^{68} -1008.59 q^{71} +669.043 q^{73} +1589.84 q^{74} -940.980 q^{76} -129.137 q^{77} +517.398 q^{79} -1199.72 q^{82} +35.3950 q^{83} -799.677 q^{86} -24.2515 q^{88} +1082.09 q^{89} +342.018 q^{91} -1078.32 q^{92} +1222.88 q^{94} +203.138 q^{97} -193.955 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{4} + 56 q^{7} - 38 q^{13} + 320 q^{16} + 68 q^{19} + 110 q^{22} + 252 q^{28} + 534 q^{31} + 118 q^{34} + 14 q^{37} + 816 q^{43} + 1588 q^{46} + 392 q^{49} - 266 q^{52} + 716 q^{58} + 874 q^{61}+ \cdots + 428 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.95827 −1.39946 −0.699730 0.714408i \(-0.746696\pi\)
−0.699730 + 0.714408i \(0.746696\pi\)
\(3\) 0 0
\(4\) 7.66789 0.958486
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 1.31458 0.0580966
\(9\) 0 0
\(10\) 0 0
\(11\) −18.4481 −0.505666 −0.252833 0.967510i \(-0.581362\pi\)
−0.252833 + 0.967510i \(0.581362\pi\)
\(12\) 0 0
\(13\) 48.8596 1.04240 0.521201 0.853434i \(-0.325484\pi\)
0.521201 + 0.853434i \(0.325484\pi\)
\(14\) −27.7079 −0.528946
\(15\) 0 0
\(16\) −66.5466 −1.03979
\(17\) −124.724 −1.77941 −0.889703 0.456539i \(-0.849089\pi\)
−0.889703 + 0.456539i \(0.849089\pi\)
\(18\) 0 0
\(19\) −122.717 −1.48175 −0.740874 0.671644i \(-0.765588\pi\)
−0.740874 + 0.671644i \(0.765588\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 73.0227 0.707659
\(23\) −140.628 −1.27491 −0.637456 0.770487i \(-0.720013\pi\)
−0.637456 + 0.770487i \(0.720013\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −193.400 −1.45880
\(27\) 0 0
\(28\) 53.6752 0.362274
\(29\) 196.100 1.25569 0.627843 0.778340i \(-0.283938\pi\)
0.627843 + 0.778340i \(0.283938\pi\)
\(30\) 0 0
\(31\) 289.591 1.67781 0.838905 0.544277i \(-0.183196\pi\)
0.838905 + 0.544277i \(0.183196\pi\)
\(32\) 252.893 1.39705
\(33\) 0 0
\(34\) 493.690 2.49021
\(35\) 0 0
\(36\) 0 0
\(37\) −401.649 −1.78461 −0.892307 0.451429i \(-0.850914\pi\)
−0.892307 + 0.451429i \(0.850914\pi\)
\(38\) 485.747 2.07365
\(39\) 0 0
\(40\) 0 0
\(41\) 303.093 1.15452 0.577258 0.816562i \(-0.304123\pi\)
0.577258 + 0.816562i \(0.304123\pi\)
\(42\) 0 0
\(43\) 202.027 0.716485 0.358242 0.933629i \(-0.383376\pi\)
0.358242 + 0.933629i \(0.383376\pi\)
\(44\) −141.458 −0.484674
\(45\) 0 0
\(46\) 556.643 1.78419
\(47\) −308.944 −0.958811 −0.479405 0.877594i \(-0.659148\pi\)
−0.479405 + 0.877594i \(0.659148\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 374.650 0.999128
\(53\) −461.082 −1.19499 −0.597495 0.801873i \(-0.703837\pi\)
−0.597495 + 0.801873i \(0.703837\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 9.20203 0.0219584
\(57\) 0 0
\(58\) −776.217 −1.75728
\(59\) −111.900 −0.246918 −0.123459 0.992350i \(-0.539399\pi\)
−0.123459 + 0.992350i \(0.539399\pi\)
\(60\) 0 0
\(61\) 416.658 0.874551 0.437276 0.899327i \(-0.355943\pi\)
0.437276 + 0.899327i \(0.355943\pi\)
\(62\) −1146.28 −2.34803
\(63\) 0 0
\(64\) −468.644 −0.915321
\(65\) 0 0
\(66\) 0 0
\(67\) 452.322 0.824775 0.412388 0.911009i \(-0.364695\pi\)
0.412388 + 0.911009i \(0.364695\pi\)
\(68\) −956.367 −1.70554
\(69\) 0 0
\(70\) 0 0
\(71\) −1008.59 −1.68589 −0.842944 0.538001i \(-0.819180\pi\)
−0.842944 + 0.538001i \(0.819180\pi\)
\(72\) 0 0
\(73\) 669.043 1.07268 0.536339 0.844002i \(-0.319807\pi\)
0.536339 + 0.844002i \(0.319807\pi\)
\(74\) 1589.84 2.49749
\(75\) 0 0
\(76\) −940.980 −1.42024
\(77\) −129.137 −0.191124
\(78\) 0 0
\(79\) 517.398 0.736859 0.368429 0.929656i \(-0.379896\pi\)
0.368429 + 0.929656i \(0.379896\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1199.72 −1.61570
\(83\) 35.3950 0.0468085 0.0234042 0.999726i \(-0.492550\pi\)
0.0234042 + 0.999726i \(0.492550\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −799.677 −1.00269
\(87\) 0 0
\(88\) −24.2515 −0.0293775
\(89\) 1082.09 1.28878 0.644389 0.764698i \(-0.277112\pi\)
0.644389 + 0.764698i \(0.277112\pi\)
\(90\) 0 0
\(91\) 342.018 0.393991
\(92\) −1078.32 −1.22199
\(93\) 0 0
\(94\) 1222.88 1.34182
\(95\) 0 0
\(96\) 0 0
\(97\) 203.138 0.212634 0.106317 0.994332i \(-0.466094\pi\)
0.106317 + 0.994332i \(0.466094\pi\)
\(98\) −193.955 −0.199923
\(99\) 0 0
\(100\) 0 0
\(101\) −1567.11 −1.54390 −0.771949 0.635684i \(-0.780718\pi\)
−0.771949 + 0.635684i \(0.780718\pi\)
\(102\) 0 0
\(103\) −473.964 −0.453409 −0.226704 0.973964i \(-0.572795\pi\)
−0.226704 + 0.973964i \(0.572795\pi\)
\(104\) 64.2297 0.0605600
\(105\) 0 0
\(106\) 1825.09 1.67234
\(107\) −496.898 −0.448943 −0.224472 0.974481i \(-0.572066\pi\)
−0.224472 + 0.974481i \(0.572066\pi\)
\(108\) 0 0
\(109\) 1125.46 0.988983 0.494492 0.869182i \(-0.335354\pi\)
0.494492 + 0.869182i \(0.335354\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −465.826 −0.393004
\(113\) 184.508 0.153603 0.0768013 0.997046i \(-0.475529\pi\)
0.0768013 + 0.997046i \(0.475529\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1503.67 1.20356
\(117\) 0 0
\(118\) 442.931 0.345552
\(119\) −873.065 −0.672553
\(120\) 0 0
\(121\) −990.666 −0.744302
\(122\) −1649.25 −1.22390
\(123\) 0 0
\(124\) 2220.55 1.60816
\(125\) 0 0
\(126\) 0 0
\(127\) 2835.92 1.98148 0.990738 0.135784i \(-0.0433553\pi\)
0.990738 + 0.135784i \(0.0433553\pi\)
\(128\) −168.121 −0.116093
\(129\) 0 0
\(130\) 0 0
\(131\) −189.998 −0.126719 −0.0633595 0.997991i \(-0.520181\pi\)
−0.0633595 + 0.997991i \(0.520181\pi\)
\(132\) 0 0
\(133\) −859.019 −0.560048
\(134\) −1790.41 −1.15424
\(135\) 0 0
\(136\) −163.959 −0.103377
\(137\) −2230.24 −1.39082 −0.695408 0.718615i \(-0.744777\pi\)
−0.695408 + 0.718615i \(0.744777\pi\)
\(138\) 0 0
\(139\) −398.286 −0.243037 −0.121519 0.992589i \(-0.538776\pi\)
−0.121519 + 0.992589i \(0.538776\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 3992.29 2.35933
\(143\) −901.370 −0.527107
\(144\) 0 0
\(145\) 0 0
\(146\) −2648.25 −1.50117
\(147\) 0 0
\(148\) −3079.80 −1.71053
\(149\) 3556.30 1.95532 0.977662 0.210183i \(-0.0674060\pi\)
0.977662 + 0.210183i \(0.0674060\pi\)
\(150\) 0 0
\(151\) −955.955 −0.515196 −0.257598 0.966252i \(-0.582931\pi\)
−0.257598 + 0.966252i \(0.582931\pi\)
\(152\) −161.321 −0.0860845
\(153\) 0 0
\(154\) 511.159 0.267470
\(155\) 0 0
\(156\) 0 0
\(157\) −3095.78 −1.57370 −0.786848 0.617147i \(-0.788288\pi\)
−0.786848 + 0.617147i \(0.788288\pi\)
\(158\) −2048.00 −1.03120
\(159\) 0 0
\(160\) 0 0
\(161\) −984.396 −0.481871
\(162\) 0 0
\(163\) −2665.44 −1.28082 −0.640409 0.768034i \(-0.721235\pi\)
−0.640409 + 0.768034i \(0.721235\pi\)
\(164\) 2324.08 1.10659
\(165\) 0 0
\(166\) −140.103 −0.0655065
\(167\) 967.998 0.448538 0.224269 0.974527i \(-0.428000\pi\)
0.224269 + 0.974527i \(0.428000\pi\)
\(168\) 0 0
\(169\) 190.265 0.0866023
\(170\) 0 0
\(171\) 0 0
\(172\) 1549.12 0.686741
\(173\) 2793.34 1.22759 0.613796 0.789465i \(-0.289642\pi\)
0.613796 + 0.789465i \(0.289642\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1227.66 0.525786
\(177\) 0 0
\(178\) −4283.20 −1.80359
\(179\) 449.319 0.187618 0.0938091 0.995590i \(-0.470096\pi\)
0.0938091 + 0.995590i \(0.470096\pi\)
\(180\) 0 0
\(181\) 2054.72 0.843790 0.421895 0.906645i \(-0.361365\pi\)
0.421895 + 0.906645i \(0.361365\pi\)
\(182\) −1353.80 −0.551374
\(183\) 0 0
\(184\) −184.866 −0.0740680
\(185\) 0 0
\(186\) 0 0
\(187\) 2300.92 0.899785
\(188\) −2368.95 −0.919007
\(189\) 0 0
\(190\) 0 0
\(191\) −42.1490 −0.0159675 −0.00798376 0.999968i \(-0.502541\pi\)
−0.00798376 + 0.999968i \(0.502541\pi\)
\(192\) 0 0
\(193\) 3613.73 1.34778 0.673890 0.738831i \(-0.264622\pi\)
0.673890 + 0.738831i \(0.264622\pi\)
\(194\) −804.074 −0.297573
\(195\) 0 0
\(196\) 375.727 0.136927
\(197\) −2429.72 −0.878733 −0.439366 0.898308i \(-0.644797\pi\)
−0.439366 + 0.898308i \(0.644797\pi\)
\(198\) 0 0
\(199\) 3540.39 1.26116 0.630582 0.776123i \(-0.282816\pi\)
0.630582 + 0.776123i \(0.282816\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6203.06 2.16062
\(203\) 1372.70 0.474604
\(204\) 0 0
\(205\) 0 0
\(206\) 1876.08 0.634527
\(207\) 0 0
\(208\) −3251.44 −1.08388
\(209\) 2263.90 0.749269
\(210\) 0 0
\(211\) −2183.66 −0.712461 −0.356230 0.934398i \(-0.615938\pi\)
−0.356230 + 0.934398i \(0.615938\pi\)
\(212\) −3535.53 −1.14538
\(213\) 0 0
\(214\) 1966.85 0.628278
\(215\) 0 0
\(216\) 0 0
\(217\) 2027.14 0.634153
\(218\) −4454.86 −1.38404
\(219\) 0 0
\(220\) 0 0
\(221\) −6093.95 −1.85486
\(222\) 0 0
\(223\) −5276.91 −1.58461 −0.792305 0.610126i \(-0.791119\pi\)
−0.792305 + 0.610126i \(0.791119\pi\)
\(224\) 1770.25 0.528034
\(225\) 0 0
\(226\) −730.334 −0.214960
\(227\) 4472.50 1.30771 0.653855 0.756620i \(-0.273151\pi\)
0.653855 + 0.756620i \(0.273151\pi\)
\(228\) 0 0
\(229\) 1698.75 0.490202 0.245101 0.969498i \(-0.421179\pi\)
0.245101 + 0.969498i \(0.421179\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 257.788 0.0729510
\(233\) 4857.72 1.36583 0.682917 0.730496i \(-0.260711\pi\)
0.682917 + 0.730496i \(0.260711\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −858.039 −0.236668
\(237\) 0 0
\(238\) 3455.83 0.941210
\(239\) −375.571 −0.101647 −0.0508235 0.998708i \(-0.516185\pi\)
−0.0508235 + 0.998708i \(0.516185\pi\)
\(240\) 0 0
\(241\) −1005.92 −0.268868 −0.134434 0.990923i \(-0.542922\pi\)
−0.134434 + 0.990923i \(0.542922\pi\)
\(242\) 3921.32 1.04162
\(243\) 0 0
\(244\) 3194.89 0.838246
\(245\) 0 0
\(246\) 0 0
\(247\) −5995.91 −1.54458
\(248\) 380.690 0.0974751
\(249\) 0 0
\(250\) 0 0
\(251\) 5462.83 1.37375 0.686874 0.726776i \(-0.258982\pi\)
0.686874 + 0.726776i \(0.258982\pi\)
\(252\) 0 0
\(253\) 2594.33 0.644679
\(254\) −11225.3 −2.77300
\(255\) 0 0
\(256\) 4414.62 1.07779
\(257\) 427.791 0.103832 0.0519161 0.998651i \(-0.483467\pi\)
0.0519161 + 0.998651i \(0.483467\pi\)
\(258\) 0 0
\(259\) −2811.54 −0.674521
\(260\) 0 0
\(261\) 0 0
\(262\) 752.062 0.177338
\(263\) 520.645 0.122070 0.0610349 0.998136i \(-0.480560\pi\)
0.0610349 + 0.998136i \(0.480560\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 3400.23 0.783764
\(267\) 0 0
\(268\) 3468.36 0.790536
\(269\) 1995.64 0.452328 0.226164 0.974089i \(-0.427381\pi\)
0.226164 + 0.974089i \(0.427381\pi\)
\(270\) 0 0
\(271\) 5212.99 1.16851 0.584256 0.811569i \(-0.301386\pi\)
0.584256 + 0.811569i \(0.301386\pi\)
\(272\) 8299.93 1.85021
\(273\) 0 0
\(274\) 8827.87 1.94639
\(275\) 0 0
\(276\) 0 0
\(277\) 6338.81 1.37495 0.687477 0.726206i \(-0.258718\pi\)
0.687477 + 0.726206i \(0.258718\pi\)
\(278\) 1576.52 0.340121
\(279\) 0 0
\(280\) 0 0
\(281\) 4861.28 1.03203 0.516014 0.856580i \(-0.327415\pi\)
0.516014 + 0.856580i \(0.327415\pi\)
\(282\) 0 0
\(283\) −1040.23 −0.218499 −0.109250 0.994014i \(-0.534845\pi\)
−0.109250 + 0.994014i \(0.534845\pi\)
\(284\) −7733.79 −1.61590
\(285\) 0 0
\(286\) 3567.86 0.737665
\(287\) 2121.65 0.436366
\(288\) 0 0
\(289\) 10643.0 2.16629
\(290\) 0 0
\(291\) 0 0
\(292\) 5130.15 1.02815
\(293\) −3807.01 −0.759072 −0.379536 0.925177i \(-0.623916\pi\)
−0.379536 + 0.925177i \(0.623916\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −527.998 −0.103680
\(297\) 0 0
\(298\) −14076.8 −2.73640
\(299\) −6871.03 −1.32897
\(300\) 0 0
\(301\) 1414.19 0.270806
\(302\) 3783.93 0.720995
\(303\) 0 0
\(304\) 8166.39 1.54071
\(305\) 0 0
\(306\) 0 0
\(307\) −1313.26 −0.244142 −0.122071 0.992521i \(-0.538954\pi\)
−0.122071 + 0.992521i \(0.538954\pi\)
\(308\) −990.208 −0.183189
\(309\) 0 0
\(310\) 0 0
\(311\) 566.183 0.103232 0.0516162 0.998667i \(-0.483563\pi\)
0.0516162 + 0.998667i \(0.483563\pi\)
\(312\) 0 0
\(313\) −1146.47 −0.207036 −0.103518 0.994628i \(-0.533010\pi\)
−0.103518 + 0.994628i \(0.533010\pi\)
\(314\) 12253.9 2.20232
\(315\) 0 0
\(316\) 3967.35 0.706269
\(317\) 2562.45 0.454012 0.227006 0.973893i \(-0.427106\pi\)
0.227006 + 0.973893i \(0.427106\pi\)
\(318\) 0 0
\(319\) −3617.68 −0.634957
\(320\) 0 0
\(321\) 0 0
\(322\) 3896.50 0.674359
\(323\) 15305.7 2.63663
\(324\) 0 0
\(325\) 0 0
\(326\) 10550.5 1.79245
\(327\) 0 0
\(328\) 398.438 0.0670734
\(329\) −2162.61 −0.362396
\(330\) 0 0
\(331\) −917.092 −0.152290 −0.0761450 0.997097i \(-0.524261\pi\)
−0.0761450 + 0.997097i \(0.524261\pi\)
\(332\) 271.405 0.0448653
\(333\) 0 0
\(334\) −3831.60 −0.627711
\(335\) 0 0
\(336\) 0 0
\(337\) −2446.17 −0.395405 −0.197702 0.980262i \(-0.563348\pi\)
−0.197702 + 0.980262i \(0.563348\pi\)
\(338\) −753.121 −0.121196
\(339\) 0 0
\(340\) 0 0
\(341\) −5342.42 −0.848412
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 265.580 0.0416253
\(345\) 0 0
\(346\) −11056.8 −1.71796
\(347\) 3598.36 0.556687 0.278343 0.960482i \(-0.410215\pi\)
0.278343 + 0.960482i \(0.410215\pi\)
\(348\) 0 0
\(349\) −247.567 −0.0379712 −0.0189856 0.999820i \(-0.506044\pi\)
−0.0189856 + 0.999820i \(0.506044\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4665.40 −0.706439
\(353\) −81.3189 −0.0122611 −0.00613055 0.999981i \(-0.501951\pi\)
−0.00613055 + 0.999981i \(0.501951\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 8297.35 1.23528
\(357\) 0 0
\(358\) −1778.52 −0.262564
\(359\) −4547.08 −0.668483 −0.334242 0.942487i \(-0.608480\pi\)
−0.334242 + 0.942487i \(0.608480\pi\)
\(360\) 0 0
\(361\) 8200.46 1.19558
\(362\) −8133.12 −1.18085
\(363\) 0 0
\(364\) 2622.55 0.377635
\(365\) 0 0
\(366\) 0 0
\(367\) −11297.4 −1.60686 −0.803429 0.595401i \(-0.796993\pi\)
−0.803429 + 0.595401i \(0.796993\pi\)
\(368\) 9358.31 1.32564
\(369\) 0 0
\(370\) 0 0
\(371\) −3227.57 −0.451664
\(372\) 0 0
\(373\) 2880.15 0.399808 0.199904 0.979816i \(-0.435937\pi\)
0.199904 + 0.979816i \(0.435937\pi\)
\(374\) −9107.65 −1.25921
\(375\) 0 0
\(376\) −406.130 −0.0557036
\(377\) 9581.38 1.30893
\(378\) 0 0
\(379\) −10505.6 −1.42384 −0.711922 0.702259i \(-0.752175\pi\)
−0.711922 + 0.702259i \(0.752175\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 166.837 0.0223459
\(383\) −522.066 −0.0696510 −0.0348255 0.999393i \(-0.511088\pi\)
−0.0348255 + 0.999393i \(0.511088\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14304.1 −1.88616
\(387\) 0 0
\(388\) 1557.64 0.203807
\(389\) 6513.24 0.848932 0.424466 0.905444i \(-0.360462\pi\)
0.424466 + 0.905444i \(0.360462\pi\)
\(390\) 0 0
\(391\) 17539.6 2.26859
\(392\) 64.4142 0.00829951
\(393\) 0 0
\(394\) 9617.48 1.22975
\(395\) 0 0
\(396\) 0 0
\(397\) −9106.98 −1.15130 −0.575650 0.817696i \(-0.695251\pi\)
−0.575650 + 0.817696i \(0.695251\pi\)
\(398\) −14013.8 −1.76495
\(399\) 0 0
\(400\) 0 0
\(401\) 3204.43 0.399056 0.199528 0.979892i \(-0.436059\pi\)
0.199528 + 0.979892i \(0.436059\pi\)
\(402\) 0 0
\(403\) 14149.3 1.74895
\(404\) −12016.5 −1.47981
\(405\) 0 0
\(406\) −5433.52 −0.664190
\(407\) 7409.68 0.902418
\(408\) 0 0
\(409\) 6396.33 0.773296 0.386648 0.922227i \(-0.373633\pi\)
0.386648 + 0.922227i \(0.373633\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3634.31 −0.434586
\(413\) −783.302 −0.0933263
\(414\) 0 0
\(415\) 0 0
\(416\) 12356.2 1.45629
\(417\) 0 0
\(418\) −8961.13 −1.04857
\(419\) 11046.1 1.28791 0.643956 0.765062i \(-0.277292\pi\)
0.643956 + 0.765062i \(0.277292\pi\)
\(420\) 0 0
\(421\) 15873.7 1.83762 0.918811 0.394698i \(-0.129151\pi\)
0.918811 + 0.394698i \(0.129151\pi\)
\(422\) 8643.51 0.997060
\(423\) 0 0
\(424\) −606.127 −0.0694248
\(425\) 0 0
\(426\) 0 0
\(427\) 2916.61 0.330549
\(428\) −3810.16 −0.430306
\(429\) 0 0
\(430\) 0 0
\(431\) 4878.52 0.545221 0.272611 0.962124i \(-0.412113\pi\)
0.272611 + 0.962124i \(0.412113\pi\)
\(432\) 0 0
\(433\) 8450.44 0.937880 0.468940 0.883230i \(-0.344636\pi\)
0.468940 + 0.883230i \(0.344636\pi\)
\(434\) −8023.96 −0.887471
\(435\) 0 0
\(436\) 8629.88 0.947927
\(437\) 17257.4 1.88910
\(438\) 0 0
\(439\) 10996.0 1.19546 0.597732 0.801696i \(-0.296069\pi\)
0.597732 + 0.801696i \(0.296069\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 24121.5 2.59580
\(443\) 13494.3 1.44725 0.723626 0.690193i \(-0.242474\pi\)
0.723626 + 0.690193i \(0.242474\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 20887.4 2.21760
\(447\) 0 0
\(448\) −3280.51 −0.345959
\(449\) −15411.4 −1.61984 −0.809920 0.586540i \(-0.800490\pi\)
−0.809920 + 0.586540i \(0.800490\pi\)
\(450\) 0 0
\(451\) −5591.50 −0.583799
\(452\) 1414.79 0.147226
\(453\) 0 0
\(454\) −17703.3 −1.83009
\(455\) 0 0
\(456\) 0 0
\(457\) 11175.7 1.14393 0.571967 0.820277i \(-0.306181\pi\)
0.571967 + 0.820277i \(0.306181\pi\)
\(458\) −6724.10 −0.686018
\(459\) 0 0
\(460\) 0 0
\(461\) 6343.45 0.640876 0.320438 0.947270i \(-0.396170\pi\)
0.320438 + 0.947270i \(0.396170\pi\)
\(462\) 0 0
\(463\) 5947.99 0.597034 0.298517 0.954404i \(-0.403508\pi\)
0.298517 + 0.954404i \(0.403508\pi\)
\(464\) −13049.8 −1.30565
\(465\) 0 0
\(466\) −19228.1 −1.91143
\(467\) 11775.8 1.16685 0.583424 0.812168i \(-0.301713\pi\)
0.583424 + 0.812168i \(0.301713\pi\)
\(468\) 0 0
\(469\) 3166.25 0.311736
\(470\) 0 0
\(471\) 0 0
\(472\) −147.101 −0.0143451
\(473\) −3727.02 −0.362302
\(474\) 0 0
\(475\) 0 0
\(476\) −6694.57 −0.644632
\(477\) 0 0
\(478\) 1486.61 0.142251
\(479\) 18005.6 1.71753 0.858765 0.512370i \(-0.171232\pi\)
0.858765 + 0.512370i \(0.171232\pi\)
\(480\) 0 0
\(481\) −19624.4 −1.86029
\(482\) 3981.71 0.376269
\(483\) 0 0
\(484\) −7596.32 −0.713403
\(485\) 0 0
\(486\) 0 0
\(487\) 11098.4 1.03268 0.516340 0.856383i \(-0.327294\pi\)
0.516340 + 0.856383i \(0.327294\pi\)
\(488\) 547.729 0.0508085
\(489\) 0 0
\(490\) 0 0
\(491\) −11253.9 −1.03438 −0.517191 0.855870i \(-0.673022\pi\)
−0.517191 + 0.855870i \(0.673022\pi\)
\(492\) 0 0
\(493\) −24458.3 −2.23437
\(494\) 23733.4 2.16157
\(495\) 0 0
\(496\) −19271.3 −1.74457
\(497\) −7060.16 −0.637206
\(498\) 0 0
\(499\) −6099.94 −0.547236 −0.273618 0.961838i \(-0.588220\pi\)
−0.273618 + 0.961838i \(0.588220\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −21623.3 −1.92250
\(503\) 19384.4 1.71830 0.859152 0.511721i \(-0.170992\pi\)
0.859152 + 0.511721i \(0.170992\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −10269.0 −0.902202
\(507\) 0 0
\(508\) 21745.6 1.89922
\(509\) 5257.46 0.457825 0.228912 0.973447i \(-0.426483\pi\)
0.228912 + 0.973447i \(0.426483\pi\)
\(510\) 0 0
\(511\) 4683.30 0.405434
\(512\) −16129.3 −1.39223
\(513\) 0 0
\(514\) −1693.31 −0.145309
\(515\) 0 0
\(516\) 0 0
\(517\) 5699.44 0.484838
\(518\) 11128.8 0.943964
\(519\) 0 0
\(520\) 0 0
\(521\) −7762.65 −0.652760 −0.326380 0.945239i \(-0.605829\pi\)
−0.326380 + 0.945239i \(0.605829\pi\)
\(522\) 0 0
\(523\) −9640.89 −0.806055 −0.403027 0.915188i \(-0.632042\pi\)
−0.403027 + 0.915188i \(0.632042\pi\)
\(524\) −1456.88 −0.121458
\(525\) 0 0
\(526\) −2060.85 −0.170832
\(527\) −36118.9 −2.98551
\(528\) 0 0
\(529\) 7609.24 0.625399
\(530\) 0 0
\(531\) 0 0
\(532\) −6586.86 −0.536798
\(533\) 14809.0 1.20347
\(534\) 0 0
\(535\) 0 0
\(536\) 594.612 0.0479166
\(537\) 0 0
\(538\) −7899.28 −0.633015
\(539\) −903.959 −0.0722380
\(540\) 0 0
\(541\) −2039.89 −0.162110 −0.0810551 0.996710i \(-0.525829\pi\)
−0.0810551 + 0.996710i \(0.525829\pi\)
\(542\) −20634.4 −1.63528
\(543\) 0 0
\(544\) −31541.7 −2.48592
\(545\) 0 0
\(546\) 0 0
\(547\) 1274.24 0.0996025 0.0498013 0.998759i \(-0.484141\pi\)
0.0498013 + 0.998759i \(0.484141\pi\)
\(548\) −17101.2 −1.33308
\(549\) 0 0
\(550\) 0 0
\(551\) −24064.8 −1.86061
\(552\) 0 0
\(553\) 3621.79 0.278506
\(554\) −25090.7 −1.92419
\(555\) 0 0
\(556\) −3054.01 −0.232948
\(557\) 19965.7 1.51880 0.759402 0.650621i \(-0.225492\pi\)
0.759402 + 0.650621i \(0.225492\pi\)
\(558\) 0 0
\(559\) 9870.97 0.746865
\(560\) 0 0
\(561\) 0 0
\(562\) −19242.3 −1.44428
\(563\) −13467.9 −1.00818 −0.504090 0.863651i \(-0.668172\pi\)
−0.504090 + 0.863651i \(0.668172\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 4117.51 0.305781
\(567\) 0 0
\(568\) −1325.87 −0.0979444
\(569\) −8579.29 −0.632096 −0.316048 0.948743i \(-0.602356\pi\)
−0.316048 + 0.948743i \(0.602356\pi\)
\(570\) 0 0
\(571\) −2146.06 −0.157285 −0.0786426 0.996903i \(-0.525059\pi\)
−0.0786426 + 0.996903i \(0.525059\pi\)
\(572\) −6911.61 −0.505225
\(573\) 0 0
\(574\) −8398.06 −0.610676
\(575\) 0 0
\(576\) 0 0
\(577\) −22451.1 −1.61985 −0.809925 0.586534i \(-0.800492\pi\)
−0.809925 + 0.586534i \(0.800492\pi\)
\(578\) −42127.8 −3.03163
\(579\) 0 0
\(580\) 0 0
\(581\) 247.765 0.0176919
\(582\) 0 0
\(583\) 8506.10 0.604266
\(584\) 879.508 0.0623190
\(585\) 0 0
\(586\) 15069.2 1.06229
\(587\) 6015.40 0.422968 0.211484 0.977381i \(-0.432170\pi\)
0.211484 + 0.977381i \(0.432170\pi\)
\(588\) 0 0
\(589\) −35537.8 −2.48609
\(590\) 0 0
\(591\) 0 0
\(592\) 26728.4 1.85562
\(593\) 15737.3 1.08981 0.544903 0.838499i \(-0.316566\pi\)
0.544903 + 0.838499i \(0.316566\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 27269.3 1.87415
\(597\) 0 0
\(598\) 27197.4 1.85984
\(599\) 25634.5 1.74858 0.874288 0.485407i \(-0.161328\pi\)
0.874288 + 0.485407i \(0.161328\pi\)
\(600\) 0 0
\(601\) −6955.91 −0.472109 −0.236054 0.971740i \(-0.575854\pi\)
−0.236054 + 0.971740i \(0.575854\pi\)
\(602\) −5597.74 −0.378982
\(603\) 0 0
\(604\) −7330.16 −0.493808
\(605\) 0 0
\(606\) 0 0
\(607\) −952.222 −0.0636730 −0.0318365 0.999493i \(-0.510136\pi\)
−0.0318365 + 0.999493i \(0.510136\pi\)
\(608\) −31034.2 −2.07007
\(609\) 0 0
\(610\) 0 0
\(611\) −15094.9 −0.999466
\(612\) 0 0
\(613\) 2962.98 0.195226 0.0976132 0.995224i \(-0.468879\pi\)
0.0976132 + 0.995224i \(0.468879\pi\)
\(614\) 5198.23 0.341667
\(615\) 0 0
\(616\) −169.760 −0.0111036
\(617\) −28257.7 −1.84378 −0.921891 0.387449i \(-0.873356\pi\)
−0.921891 + 0.387449i \(0.873356\pi\)
\(618\) 0 0
\(619\) −10708.5 −0.695336 −0.347668 0.937618i \(-0.613026\pi\)
−0.347668 + 0.937618i \(0.613026\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −2241.10 −0.144470
\(623\) 7574.63 0.487113
\(624\) 0 0
\(625\) 0 0
\(626\) 4538.03 0.289738
\(627\) 0 0
\(628\) −23738.1 −1.50837
\(629\) 50095.1 3.17555
\(630\) 0 0
\(631\) 17383.2 1.09669 0.548346 0.836251i \(-0.315258\pi\)
0.548346 + 0.836251i \(0.315258\pi\)
\(632\) 680.159 0.0428090
\(633\) 0 0
\(634\) −10142.9 −0.635371
\(635\) 0 0
\(636\) 0 0
\(637\) 2394.12 0.148915
\(638\) 14319.8 0.888597
\(639\) 0 0
\(640\) 0 0
\(641\) −8268.85 −0.509516 −0.254758 0.967005i \(-0.581996\pi\)
−0.254758 + 0.967005i \(0.581996\pi\)
\(642\) 0 0
\(643\) 1676.14 0.102800 0.0514000 0.998678i \(-0.483632\pi\)
0.0514000 + 0.998678i \(0.483632\pi\)
\(644\) −7548.24 −0.461867
\(645\) 0 0
\(646\) −60584.1 −3.68986
\(647\) −17319.8 −1.05242 −0.526208 0.850356i \(-0.676387\pi\)
−0.526208 + 0.850356i \(0.676387\pi\)
\(648\) 0 0
\(649\) 2064.35 0.124858
\(650\) 0 0
\(651\) 0 0
\(652\) −20438.3 −1.22765
\(653\) 17191.7 1.03026 0.515132 0.857111i \(-0.327743\pi\)
0.515132 + 0.857111i \(0.327743\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −20169.8 −1.20045
\(657\) 0 0
\(658\) 8560.18 0.507159
\(659\) −1559.35 −0.0921753 −0.0460877 0.998937i \(-0.514675\pi\)
−0.0460877 + 0.998937i \(0.514675\pi\)
\(660\) 0 0
\(661\) −17710.2 −1.04213 −0.521065 0.853517i \(-0.674465\pi\)
−0.521065 + 0.853517i \(0.674465\pi\)
\(662\) 3630.10 0.213124
\(663\) 0 0
\(664\) 46.5294 0.00271941
\(665\) 0 0
\(666\) 0 0
\(667\) −27577.2 −1.60089
\(668\) 7422.50 0.429918
\(669\) 0 0
\(670\) 0 0
\(671\) −7686.57 −0.442231
\(672\) 0 0
\(673\) −15614.1 −0.894322 −0.447161 0.894453i \(-0.647565\pi\)
−0.447161 + 0.894453i \(0.647565\pi\)
\(674\) 9682.60 0.553353
\(675\) 0 0
\(676\) 1458.93 0.0830071
\(677\) −1964.69 −0.111535 −0.0557675 0.998444i \(-0.517761\pi\)
−0.0557675 + 0.998444i \(0.517761\pi\)
\(678\) 0 0
\(679\) 1421.96 0.0803682
\(680\) 0 0
\(681\) 0 0
\(682\) 21146.7 1.18732
\(683\) −5230.81 −0.293047 −0.146524 0.989207i \(-0.546808\pi\)
−0.146524 + 0.989207i \(0.546808\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1357.69 −0.0755637
\(687\) 0 0
\(688\) −13444.2 −0.744994
\(689\) −22528.3 −1.24566
\(690\) 0 0
\(691\) 28743.3 1.58241 0.791205 0.611551i \(-0.209454\pi\)
0.791205 + 0.611551i \(0.209454\pi\)
\(692\) 21419.0 1.17663
\(693\) 0 0
\(694\) −14243.3 −0.779061
\(695\) 0 0
\(696\) 0 0
\(697\) −37802.8 −2.05435
\(698\) 979.937 0.0531392
\(699\) 0 0
\(700\) 0 0
\(701\) −15018.5 −0.809188 −0.404594 0.914497i \(-0.632587\pi\)
−0.404594 + 0.914497i \(0.632587\pi\)
\(702\) 0 0
\(703\) 49289.2 2.64435
\(704\) 8645.62 0.462847
\(705\) 0 0
\(706\) 321.882 0.0171589
\(707\) −10969.8 −0.583539
\(708\) 0 0
\(709\) 2164.88 0.114674 0.0573368 0.998355i \(-0.481739\pi\)
0.0573368 + 0.998355i \(0.481739\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1422.49 0.0748737
\(713\) −40724.6 −2.13906
\(714\) 0 0
\(715\) 0 0
\(716\) 3445.33 0.179830
\(717\) 0 0
\(718\) 17998.5 0.935515
\(719\) −16570.9 −0.859516 −0.429758 0.902944i \(-0.641401\pi\)
−0.429758 + 0.902944i \(0.641401\pi\)
\(720\) 0 0
\(721\) −3317.75 −0.171372
\(722\) −32459.6 −1.67316
\(723\) 0 0
\(724\) 15755.3 0.808761
\(725\) 0 0
\(726\) 0 0
\(727\) 4179.50 0.213217 0.106609 0.994301i \(-0.466001\pi\)
0.106609 + 0.994301i \(0.466001\pi\)
\(728\) 449.608 0.0228895
\(729\) 0 0
\(730\) 0 0
\(731\) −25197.5 −1.27492
\(732\) 0 0
\(733\) 24312.7 1.22511 0.612557 0.790426i \(-0.290141\pi\)
0.612557 + 0.790426i \(0.290141\pi\)
\(734\) 44717.9 2.24873
\(735\) 0 0
\(736\) −35563.8 −1.78111
\(737\) −8344.50 −0.417061
\(738\) 0 0
\(739\) −8812.62 −0.438671 −0.219335 0.975650i \(-0.570389\pi\)
−0.219335 + 0.975650i \(0.570389\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 12775.6 0.632085
\(743\) −30132.1 −1.48780 −0.743902 0.668288i \(-0.767027\pi\)
−0.743902 + 0.668288i \(0.767027\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −11400.4 −0.559515
\(747\) 0 0
\(748\) 17643.2 0.862432
\(749\) −3478.28 −0.169685
\(750\) 0 0
\(751\) 1000.93 0.0486343 0.0243172 0.999704i \(-0.492259\pi\)
0.0243172 + 0.999704i \(0.492259\pi\)
\(752\) 20559.2 0.996962
\(753\) 0 0
\(754\) −37925.7 −1.83179
\(755\) 0 0
\(756\) 0 0
\(757\) −21708.6 −1.04229 −0.521144 0.853469i \(-0.674495\pi\)
−0.521144 + 0.853469i \(0.674495\pi\)
\(758\) 41584.0 1.99261
\(759\) 0 0
\(760\) 0 0
\(761\) 33904.7 1.61504 0.807520 0.589840i \(-0.200809\pi\)
0.807520 + 0.589840i \(0.200809\pi\)
\(762\) 0 0
\(763\) 7878.20 0.373801
\(764\) −323.194 −0.0153047
\(765\) 0 0
\(766\) 2066.48 0.0974738
\(767\) −5467.41 −0.257388
\(768\) 0 0
\(769\) −6114.14 −0.286712 −0.143356 0.989671i \(-0.545789\pi\)
−0.143356 + 0.989671i \(0.545789\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 27709.7 1.29183
\(773\) 12350.0 0.574644 0.287322 0.957834i \(-0.407235\pi\)
0.287322 + 0.957834i \(0.407235\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 267.040 0.0123533
\(777\) 0 0
\(778\) −25781.2 −1.18805
\(779\) −37194.6 −1.71070
\(780\) 0 0
\(781\) 18606.7 0.852496
\(782\) −69426.6 −3.17479
\(783\) 0 0
\(784\) −3260.78 −0.148541
\(785\) 0 0
\(786\) 0 0
\(787\) −13597.6 −0.615888 −0.307944 0.951405i \(-0.599641\pi\)
−0.307944 + 0.951405i \(0.599641\pi\)
\(788\) −18630.8 −0.842254
\(789\) 0 0
\(790\) 0 0
\(791\) 1291.56 0.0580563
\(792\) 0 0
\(793\) 20357.8 0.911634
\(794\) 36047.9 1.61120
\(795\) 0 0
\(796\) 27147.3 1.20881
\(797\) 18951.1 0.842262 0.421131 0.907000i \(-0.361633\pi\)
0.421131 + 0.907000i \(0.361633\pi\)
\(798\) 0 0
\(799\) 38532.6 1.70611
\(800\) 0 0
\(801\) 0 0
\(802\) −12684.0 −0.558463
\(803\) −12342.6 −0.542417
\(804\) 0 0
\(805\) 0 0
\(806\) −56006.9 −2.44759
\(807\) 0 0
\(808\) −2060.09 −0.0896952
\(809\) 27695.1 1.20359 0.601797 0.798649i \(-0.294451\pi\)
0.601797 + 0.798649i \(0.294451\pi\)
\(810\) 0 0
\(811\) 1439.61 0.0623325 0.0311663 0.999514i \(-0.490078\pi\)
0.0311663 + 0.999514i \(0.490078\pi\)
\(812\) 10525.7 0.454902
\(813\) 0 0
\(814\) −29329.5 −1.26290
\(815\) 0 0
\(816\) 0 0
\(817\) −24792.2 −1.06165
\(818\) −25318.4 −1.08220
\(819\) 0 0
\(820\) 0 0
\(821\) −11212.2 −0.476626 −0.238313 0.971188i \(-0.576594\pi\)
−0.238313 + 0.971188i \(0.576594\pi\)
\(822\) 0 0
\(823\) 14736.8 0.624170 0.312085 0.950054i \(-0.398973\pi\)
0.312085 + 0.950054i \(0.398973\pi\)
\(824\) −623.062 −0.0263415
\(825\) 0 0
\(826\) 3100.52 0.130606
\(827\) 9129.69 0.383882 0.191941 0.981406i \(-0.438522\pi\)
0.191941 + 0.981406i \(0.438522\pi\)
\(828\) 0 0
\(829\) 13177.4 0.552074 0.276037 0.961147i \(-0.410979\pi\)
0.276037 + 0.961147i \(0.410979\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −22897.8 −0.954133
\(833\) −6111.46 −0.254201
\(834\) 0 0
\(835\) 0 0
\(836\) 17359.3 0.718164
\(837\) 0 0
\(838\) −43723.3 −1.80238
\(839\) 40633.0 1.67200 0.835999 0.548731i \(-0.184889\pi\)
0.835999 + 0.548731i \(0.184889\pi\)
\(840\) 0 0
\(841\) 14066.2 0.576746
\(842\) −62832.5 −2.57168
\(843\) 0 0
\(844\) −16744.1 −0.682884
\(845\) 0 0
\(846\) 0 0
\(847\) −6934.66 −0.281320
\(848\) 30683.4 1.24254
\(849\) 0 0
\(850\) 0 0
\(851\) 56483.1 2.27522
\(852\) 0 0
\(853\) 16268.8 0.653027 0.326514 0.945192i \(-0.394126\pi\)
0.326514 + 0.945192i \(0.394126\pi\)
\(854\) −11544.7 −0.462590
\(855\) 0 0
\(856\) −653.210 −0.0260821
\(857\) 2376.68 0.0947325 0.0473663 0.998878i \(-0.484917\pi\)
0.0473663 + 0.998878i \(0.484917\pi\)
\(858\) 0 0
\(859\) −18821.4 −0.747589 −0.373794 0.927512i \(-0.621943\pi\)
−0.373794 + 0.927512i \(0.621943\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −19310.5 −0.763015
\(863\) 20478.1 0.807743 0.403871 0.914816i \(-0.367664\pi\)
0.403871 + 0.914816i \(0.367664\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −33449.1 −1.31252
\(867\) 0 0
\(868\) 15543.9 0.607827
\(869\) −9545.03 −0.372604
\(870\) 0 0
\(871\) 22100.3 0.859747
\(872\) 1479.50 0.0574566
\(873\) 0 0
\(874\) −68309.6 −2.64371
\(875\) 0 0
\(876\) 0 0
\(877\) 46886.5 1.80530 0.902648 0.430379i \(-0.141620\pi\)
0.902648 + 0.430379i \(0.141620\pi\)
\(878\) −43525.0 −1.67300
\(879\) 0 0
\(880\) 0 0
\(881\) 16858.4 0.644691 0.322346 0.946622i \(-0.395529\pi\)
0.322346 + 0.946622i \(0.395529\pi\)
\(882\) 0 0
\(883\) 38016.2 1.44886 0.724432 0.689346i \(-0.242102\pi\)
0.724432 + 0.689346i \(0.242102\pi\)
\(884\) −46727.8 −1.77786
\(885\) 0 0
\(886\) −53414.0 −2.02537
\(887\) −25246.2 −0.955675 −0.477838 0.878448i \(-0.658579\pi\)
−0.477838 + 0.878448i \(0.658579\pi\)
\(888\) 0 0
\(889\) 19851.5 0.748928
\(890\) 0 0
\(891\) 0 0
\(892\) −40462.7 −1.51883
\(893\) 37912.7 1.42072
\(894\) 0 0
\(895\) 0 0
\(896\) −1176.85 −0.0438791
\(897\) 0 0
\(898\) 61002.4 2.26690
\(899\) 56788.9 2.10680
\(900\) 0 0
\(901\) 57507.8 2.12637
\(902\) 22132.7 0.817003
\(903\) 0 0
\(904\) 242.550 0.00892378
\(905\) 0 0
\(906\) 0 0
\(907\) −11679.5 −0.427576 −0.213788 0.976880i \(-0.568580\pi\)
−0.213788 + 0.976880i \(0.568580\pi\)
\(908\) 34294.6 1.25342
\(909\) 0 0
\(910\) 0 0
\(911\) 23655.6 0.860312 0.430156 0.902755i \(-0.358459\pi\)
0.430156 + 0.902755i \(0.358459\pi\)
\(912\) 0 0
\(913\) −652.971 −0.0236694
\(914\) −44236.4 −1.60089
\(915\) 0 0
\(916\) 13025.8 0.469852
\(917\) −1329.98 −0.0478953
\(918\) 0 0
\(919\) 19902.0 0.714371 0.357185 0.934033i \(-0.383736\pi\)
0.357185 + 0.934033i \(0.383736\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −25109.1 −0.896880
\(923\) −49279.5 −1.75737
\(924\) 0 0
\(925\) 0 0
\(926\) −23543.8 −0.835525
\(927\) 0 0
\(928\) 49592.3 1.75425
\(929\) −48903.6 −1.72710 −0.863550 0.504262i \(-0.831764\pi\)
−0.863550 + 0.504262i \(0.831764\pi\)
\(930\) 0 0
\(931\) −6013.13 −0.211678
\(932\) 37248.4 1.30913
\(933\) 0 0
\(934\) −46611.7 −1.63296
\(935\) 0 0
\(936\) 0 0
\(937\) −1179.03 −0.0411068 −0.0205534 0.999789i \(-0.506543\pi\)
−0.0205534 + 0.999789i \(0.506543\pi\)
\(938\) −12532.9 −0.436261
\(939\) 0 0
\(940\) 0 0
\(941\) −6253.60 −0.216643 −0.108322 0.994116i \(-0.534548\pi\)
−0.108322 + 0.994116i \(0.534548\pi\)
\(942\) 0 0
\(943\) −42623.3 −1.47191
\(944\) 7446.58 0.256743
\(945\) 0 0
\(946\) 14752.6 0.507027
\(947\) 29027.3 0.996050 0.498025 0.867163i \(-0.334059\pi\)
0.498025 + 0.867163i \(0.334059\pi\)
\(948\) 0 0
\(949\) 32689.2 1.11816
\(950\) 0 0
\(951\) 0 0
\(952\) −1147.71 −0.0390730
\(953\) 47968.6 1.63049 0.815245 0.579116i \(-0.196602\pi\)
0.815245 + 0.579116i \(0.196602\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −2879.83 −0.0974273
\(957\) 0 0
\(958\) −71271.0 −2.40361
\(959\) −15611.7 −0.525679
\(960\) 0 0
\(961\) 54072.1 1.81505
\(962\) 77678.8 2.60339
\(963\) 0 0
\(964\) −7713.29 −0.257706
\(965\) 0 0
\(966\) 0 0
\(967\) −6940.06 −0.230793 −0.115397 0.993319i \(-0.536814\pi\)
−0.115397 + 0.993319i \(0.536814\pi\)
\(968\) −1302.31 −0.0432414
\(969\) 0 0
\(970\) 0 0
\(971\) −25930.9 −0.857015 −0.428507 0.903538i \(-0.640960\pi\)
−0.428507 + 0.903538i \(0.640960\pi\)
\(972\) 0 0
\(973\) −2788.00 −0.0918594
\(974\) −43930.4 −1.44519
\(975\) 0 0
\(976\) −27727.2 −0.909350
\(977\) −9361.03 −0.306536 −0.153268 0.988185i \(-0.548980\pi\)
−0.153268 + 0.988185i \(0.548980\pi\)
\(978\) 0 0
\(979\) −19962.6 −0.651691
\(980\) 0 0
\(981\) 0 0
\(982\) 44546.0 1.44757
\(983\) 5317.34 0.172530 0.0862649 0.996272i \(-0.472507\pi\)
0.0862649 + 0.996272i \(0.472507\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 96812.6 3.12692
\(987\) 0 0
\(988\) −45976.0 −1.48046
\(989\) −28410.7 −0.913455
\(990\) 0 0
\(991\) −38425.3 −1.23170 −0.615852 0.787862i \(-0.711188\pi\)
−0.615852 + 0.787862i \(0.711188\pi\)
\(992\) 73235.5 2.34398
\(993\) 0 0
\(994\) 27946.0 0.891744
\(995\) 0 0
\(996\) 0 0
\(997\) 29596.3 0.940145 0.470073 0.882628i \(-0.344228\pi\)
0.470073 + 0.882628i \(0.344228\pi\)
\(998\) 24145.2 0.765835
\(999\) 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.bs.1.2 yes 8
3.2 odd 2 inner 1575.4.a.bs.1.7 yes 8
5.4 even 2 1575.4.a.br.1.7 yes 8
15.14 odd 2 1575.4.a.br.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1575.4.a.br.1.2 8 15.14 odd 2
1575.4.a.br.1.7 yes 8 5.4 even 2
1575.4.a.bs.1.2 yes 8 1.1 even 1 trivial
1575.4.a.bs.1.7 yes 8 3.2 odd 2 inner