Properties

Label 2160.4.a.bq.1.2
Level $2160$
Weight $4$
Character 2160.1
Self dual yes
Analytic conductor $127.444$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2160,4,Mod(1,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2160.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.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: \(127.444125612\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1772.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 12x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 135)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.654334\) of defining polynomial
Character \(\chi\) \(=\) 2160.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.00000 q^{5} +11.8065 q^{7} +O(q^{10})\) \(q+5.00000 q^{5} +11.8065 q^{7} +56.2376 q^{11} +34.5961 q^{13} +39.2675 q^{17} +146.561 q^{19} +23.5777 q^{23} +25.0000 q^{25} +161.003 q^{29} +29.5465 q^{31} +59.0326 q^{35} -217.688 q^{37} +142.290 q^{41} +468.030 q^{43} -394.318 q^{47} -203.606 q^{49} -134.780 q^{53} +281.188 q^{55} +131.195 q^{59} +259.801 q^{61} +172.980 q^{65} -445.244 q^{67} -560.841 q^{71} -88.6681 q^{73} +663.970 q^{77} -450.342 q^{79} -284.295 q^{83} +196.337 q^{85} +625.305 q^{89} +408.459 q^{91} +732.804 q^{95} -193.261 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 15 q^{5} + 4 q^{7} - 5 q^{11} + 7 q^{13} + 155 q^{17} + 50 q^{19} - 285 q^{23} + 75 q^{25} + 115 q^{29} + 115 q^{31} + 20 q^{35} - 384 q^{37} + 580 q^{41} + 797 q^{43} + 145 q^{47} + 577 q^{49} - 400 q^{53} - 25 q^{55} - 380 q^{59} - 152 q^{61} + 35 q^{65} - 2 q^{67} - 40 q^{71} - 980 q^{73} + 1950 q^{77} - 1013 q^{79} - 270 q^{83} + 775 q^{85} + 1020 q^{89} + 632 q^{91} + 250 q^{95} + 720 q^{97}+O(q^{100}) \) 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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 11.8065 0.637492 0.318746 0.947840i \(-0.396738\pi\)
0.318746 + 0.947840i \(0.396738\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 56.2376 1.54148 0.770740 0.637150i \(-0.219887\pi\)
0.770740 + 0.637150i \(0.219887\pi\)
\(12\) 0 0
\(13\) 34.5961 0.738094 0.369047 0.929411i \(-0.379684\pi\)
0.369047 + 0.929411i \(0.379684\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 39.2675 0.560221 0.280111 0.959968i \(-0.409629\pi\)
0.280111 + 0.959968i \(0.409629\pi\)
\(18\) 0 0
\(19\) 146.561 1.76965 0.884825 0.465924i \(-0.154278\pi\)
0.884825 + 0.465924i \(0.154278\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 23.5777 0.213752 0.106876 0.994272i \(-0.465915\pi\)
0.106876 + 0.994272i \(0.465915\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 161.003 1.03095 0.515473 0.856906i \(-0.327616\pi\)
0.515473 + 0.856906i \(0.327616\pi\)
\(30\) 0 0
\(31\) 29.5465 0.171184 0.0855921 0.996330i \(-0.472722\pi\)
0.0855921 + 0.996330i \(0.472722\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 59.0326 0.285095
\(36\) 0 0
\(37\) −217.688 −0.967233 −0.483617 0.875280i \(-0.660677\pi\)
−0.483617 + 0.875280i \(0.660677\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 142.290 0.541999 0.270999 0.962580i \(-0.412646\pi\)
0.270999 + 0.962580i \(0.412646\pi\)
\(42\) 0 0
\(43\) 468.030 1.65986 0.829929 0.557869i \(-0.188381\pi\)
0.829929 + 0.557869i \(0.188381\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −394.318 −1.22377 −0.611886 0.790946i \(-0.709589\pi\)
−0.611886 + 0.790946i \(0.709589\pi\)
\(48\) 0 0
\(49\) −203.606 −0.593604
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −134.780 −0.349311 −0.174655 0.984630i \(-0.555881\pi\)
−0.174655 + 0.984630i \(0.555881\pi\)
\(54\) 0 0
\(55\) 281.188 0.689371
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 131.195 0.289495 0.144747 0.989469i \(-0.453763\pi\)
0.144747 + 0.989469i \(0.453763\pi\)
\(60\) 0 0
\(61\) 259.801 0.545313 0.272657 0.962111i \(-0.412098\pi\)
0.272657 + 0.962111i \(0.412098\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 172.980 0.330086
\(66\) 0 0
\(67\) −445.244 −0.811869 −0.405935 0.913902i \(-0.633054\pi\)
−0.405935 + 0.913902i \(0.633054\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −560.841 −0.937459 −0.468729 0.883342i \(-0.655288\pi\)
−0.468729 + 0.883342i \(0.655288\pi\)
\(72\) 0 0
\(73\) −88.6681 −0.142162 −0.0710809 0.997471i \(-0.522645\pi\)
−0.0710809 + 0.997471i \(0.522645\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 663.970 0.982681
\(78\) 0 0
\(79\) −450.342 −0.641360 −0.320680 0.947188i \(-0.603911\pi\)
−0.320680 + 0.947188i \(0.603911\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −284.295 −0.375969 −0.187985 0.982172i \(-0.560196\pi\)
−0.187985 + 0.982172i \(0.560196\pi\)
\(84\) 0 0
\(85\) 196.337 0.250539
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 625.305 0.744744 0.372372 0.928083i \(-0.378545\pi\)
0.372372 + 0.928083i \(0.378545\pi\)
\(90\) 0 0
\(91\) 408.459 0.470529
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 732.804 0.791411
\(96\) 0 0
\(97\) −193.261 −0.202296 −0.101148 0.994871i \(-0.532252\pi\)
−0.101148 + 0.994871i \(0.532252\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1374.86 1.35449 0.677245 0.735758i \(-0.263174\pi\)
0.677245 + 0.735758i \(0.263174\pi\)
\(102\) 0 0
\(103\) −2029.60 −1.94158 −0.970789 0.239935i \(-0.922874\pi\)
−0.970789 + 0.239935i \(0.922874\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 823.062 0.743630 0.371815 0.928307i \(-0.378736\pi\)
0.371815 + 0.928307i \(0.378736\pi\)
\(108\) 0 0
\(109\) −829.868 −0.729238 −0.364619 0.931157i \(-0.618801\pi\)
−0.364619 + 0.931157i \(0.618801\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1503.37 −1.25155 −0.625773 0.780005i \(-0.715216\pi\)
−0.625773 + 0.780005i \(0.715216\pi\)
\(114\) 0 0
\(115\) 117.889 0.0955928
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 463.612 0.357137
\(120\) 0 0
\(121\) 1831.67 1.37616
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 576.348 0.402698 0.201349 0.979520i \(-0.435468\pi\)
0.201349 + 0.979520i \(0.435468\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2390.04 −1.59403 −0.797017 0.603957i \(-0.793590\pi\)
−0.797017 + 0.603957i \(0.793590\pi\)
\(132\) 0 0
\(133\) 1730.37 1.12814
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1002.46 0.625152 0.312576 0.949893i \(-0.398808\pi\)
0.312576 + 0.949893i \(0.398808\pi\)
\(138\) 0 0
\(139\) −131.817 −0.0804356 −0.0402178 0.999191i \(-0.512805\pi\)
−0.0402178 + 0.999191i \(0.512805\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1945.60 1.13776
\(144\) 0 0
\(145\) 805.013 0.461053
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1019.49 0.560533 0.280267 0.959922i \(-0.409577\pi\)
0.280267 + 0.959922i \(0.409577\pi\)
\(150\) 0 0
\(151\) −2822.38 −1.52107 −0.760537 0.649295i \(-0.775064\pi\)
−0.760537 + 0.649295i \(0.775064\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 147.733 0.0765559
\(156\) 0 0
\(157\) 476.499 0.242222 0.121111 0.992639i \(-0.461354\pi\)
0.121111 + 0.992639i \(0.461354\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 278.371 0.136265
\(162\) 0 0
\(163\) 2242.26 1.07747 0.538734 0.842476i \(-0.318903\pi\)
0.538734 + 0.842476i \(0.318903\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −95.0390 −0.0440380 −0.0220190 0.999758i \(-0.507009\pi\)
−0.0220190 + 0.999758i \(0.507009\pi\)
\(168\) 0 0
\(169\) −1000.11 −0.455217
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2133.76 −0.937727 −0.468864 0.883271i \(-0.655336\pi\)
−0.468864 + 0.883271i \(0.655336\pi\)
\(174\) 0 0
\(175\) 295.163 0.127498
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1704.68 0.711808 0.355904 0.934523i \(-0.384173\pi\)
0.355904 + 0.934523i \(0.384173\pi\)
\(180\) 0 0
\(181\) −1360.98 −0.558902 −0.279451 0.960160i \(-0.590152\pi\)
−0.279451 + 0.960160i \(0.590152\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1088.44 −0.432560
\(186\) 0 0
\(187\) 2208.31 0.863570
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1096.84 0.415522 0.207761 0.978180i \(-0.433382\pi\)
0.207761 + 0.978180i \(0.433382\pi\)
\(192\) 0 0
\(193\) −2867.27 −1.06938 −0.534691 0.845048i \(-0.679572\pi\)
−0.534691 + 0.845048i \(0.679572\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 724.139 0.261892 0.130946 0.991389i \(-0.458199\pi\)
0.130946 + 0.991389i \(0.458199\pi\)
\(198\) 0 0
\(199\) 1693.65 0.603315 0.301658 0.953416i \(-0.402460\pi\)
0.301658 + 0.953416i \(0.402460\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1900.88 0.657220
\(204\) 0 0
\(205\) 711.449 0.242389
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8242.22 2.72788
\(210\) 0 0
\(211\) −947.452 −0.309124 −0.154562 0.987983i \(-0.549397\pi\)
−0.154562 + 0.987983i \(0.549397\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2340.15 0.742311
\(216\) 0 0
\(217\) 348.841 0.109129
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1358.50 0.413496
\(222\) 0 0
\(223\) −111.866 −0.0335923 −0.0167961 0.999859i \(-0.505347\pi\)
−0.0167961 + 0.999859i \(0.505347\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1200.70 −0.351072 −0.175536 0.984473i \(-0.556166\pi\)
−0.175536 + 0.984473i \(0.556166\pi\)
\(228\) 0 0
\(229\) 822.380 0.237312 0.118656 0.992935i \(-0.462141\pi\)
0.118656 + 0.992935i \(0.462141\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5329.21 −1.49840 −0.749202 0.662341i \(-0.769563\pi\)
−0.749202 + 0.662341i \(0.769563\pi\)
\(234\) 0 0
\(235\) −1971.59 −0.547287
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7085.61 1.91770 0.958850 0.283914i \(-0.0916331\pi\)
0.958850 + 0.283914i \(0.0916331\pi\)
\(240\) 0 0
\(241\) −6560.09 −1.75341 −0.876707 0.481025i \(-0.840265\pi\)
−0.876707 + 0.481025i \(0.840265\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1018.03 −0.265468
\(246\) 0 0
\(247\) 5070.42 1.30617
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −714.222 −0.179607 −0.0898033 0.995960i \(-0.528624\pi\)
−0.0898033 + 0.995960i \(0.528624\pi\)
\(252\) 0 0
\(253\) 1325.95 0.329494
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4396.59 1.06713 0.533563 0.845760i \(-0.320853\pi\)
0.533563 + 0.845760i \(0.320853\pi\)
\(258\) 0 0
\(259\) −2570.13 −0.616604
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7550.31 1.77024 0.885118 0.465367i \(-0.154078\pi\)
0.885118 + 0.465367i \(0.154078\pi\)
\(264\) 0 0
\(265\) −673.900 −0.156216
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5536.86 −1.25497 −0.627487 0.778627i \(-0.715917\pi\)
−0.627487 + 0.778627i \(0.715917\pi\)
\(270\) 0 0
\(271\) −3058.25 −0.685518 −0.342759 0.939423i \(-0.611361\pi\)
−0.342759 + 0.939423i \(0.611361\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1405.94 0.308296
\(276\) 0 0
\(277\) −4070.19 −0.882865 −0.441433 0.897294i \(-0.645530\pi\)
−0.441433 + 0.897294i \(0.645530\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7446.19 1.58079 0.790396 0.612597i \(-0.209875\pi\)
0.790396 + 0.612597i \(0.209875\pi\)
\(282\) 0 0
\(283\) 774.651 0.162715 0.0813573 0.996685i \(-0.474075\pi\)
0.0813573 + 0.996685i \(0.474075\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1679.95 0.345520
\(288\) 0 0
\(289\) −3371.06 −0.686152
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6749.23 1.34571 0.672857 0.739772i \(-0.265067\pi\)
0.672857 + 0.739772i \(0.265067\pi\)
\(294\) 0 0
\(295\) 655.977 0.129466
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 815.696 0.157769
\(300\) 0 0
\(301\) 5525.80 1.05815
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1299.01 0.243872
\(306\) 0 0
\(307\) 2204.39 0.409808 0.204904 0.978782i \(-0.434312\pi\)
0.204904 + 0.978782i \(0.434312\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6032.98 1.10000 0.549998 0.835166i \(-0.314628\pi\)
0.549998 + 0.835166i \(0.314628\pi\)
\(312\) 0 0
\(313\) −5772.96 −1.04251 −0.521257 0.853400i \(-0.674537\pi\)
−0.521257 + 0.853400i \(0.674537\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3302.07 −0.585057 −0.292528 0.956257i \(-0.594497\pi\)
−0.292528 + 0.956257i \(0.594497\pi\)
\(318\) 0 0
\(319\) 9054.41 1.58918
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5755.07 0.991395
\(324\) 0 0
\(325\) 864.901 0.147619
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4655.53 −0.780144
\(330\) 0 0
\(331\) −8053.15 −1.33728 −0.668642 0.743584i \(-0.733124\pi\)
−0.668642 + 0.743584i \(0.733124\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2226.22 −0.363079
\(336\) 0 0
\(337\) 3460.33 0.559335 0.279668 0.960097i \(-0.409776\pi\)
0.279668 + 0.960097i \(0.409776\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1661.62 0.263877
\(342\) 0 0
\(343\) −6453.51 −1.01591
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9328.27 1.44314 0.721568 0.692344i \(-0.243422\pi\)
0.721568 + 0.692344i \(0.243422\pi\)
\(348\) 0 0
\(349\) 8899.42 1.36497 0.682486 0.730899i \(-0.260899\pi\)
0.682486 + 0.730899i \(0.260899\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3722.45 0.561264 0.280632 0.959816i \(-0.409456\pi\)
0.280632 + 0.959816i \(0.409456\pi\)
\(354\) 0 0
\(355\) −2804.21 −0.419244
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −11029.3 −1.62145 −0.810727 0.585425i \(-0.800928\pi\)
−0.810727 + 0.585425i \(0.800928\pi\)
\(360\) 0 0
\(361\) 14621.1 2.13166
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −443.340 −0.0635767
\(366\) 0 0
\(367\) 4853.11 0.690274 0.345137 0.938552i \(-0.387833\pi\)
0.345137 + 0.938552i \(0.387833\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1591.28 −0.222683
\(372\) 0 0
\(373\) −12373.8 −1.71767 −0.858834 0.512254i \(-0.828811\pi\)
−0.858834 + 0.512254i \(0.828811\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5570.06 0.760935
\(378\) 0 0
\(379\) −11150.6 −1.51127 −0.755634 0.654994i \(-0.772671\pi\)
−0.755634 + 0.654994i \(0.772671\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2199.59 −0.293457 −0.146728 0.989177i \(-0.546874\pi\)
−0.146728 + 0.989177i \(0.546874\pi\)
\(384\) 0 0
\(385\) 3319.85 0.439468
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9218.92 1.20159 0.600794 0.799404i \(-0.294851\pi\)
0.600794 + 0.799404i \(0.294851\pi\)
\(390\) 0 0
\(391\) 925.838 0.119748
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2251.71 −0.286825
\(396\) 0 0
\(397\) 1119.36 0.141509 0.0707544 0.997494i \(-0.477459\pi\)
0.0707544 + 0.997494i \(0.477459\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12296.9 1.53137 0.765683 0.643218i \(-0.222401\pi\)
0.765683 + 0.643218i \(0.222401\pi\)
\(402\) 0 0
\(403\) 1022.19 0.126350
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12242.2 −1.49097
\(408\) 0 0
\(409\) 2500.22 0.302269 0.151134 0.988513i \(-0.451707\pi\)
0.151134 + 0.988513i \(0.451707\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1548.96 0.184551
\(414\) 0 0
\(415\) −1421.48 −0.168139
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8332.97 0.971581 0.485790 0.874075i \(-0.338532\pi\)
0.485790 + 0.874075i \(0.338532\pi\)
\(420\) 0 0
\(421\) 11374.2 1.31673 0.658367 0.752697i \(-0.271248\pi\)
0.658367 + 0.752697i \(0.271248\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 981.687 0.112044
\(426\) 0 0
\(427\) 3067.35 0.347633
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10030.2 1.12097 0.560484 0.828165i \(-0.310615\pi\)
0.560484 + 0.828165i \(0.310615\pi\)
\(432\) 0 0
\(433\) 7609.38 0.844535 0.422267 0.906471i \(-0.361234\pi\)
0.422267 + 0.906471i \(0.361234\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3455.57 0.378266
\(438\) 0 0
\(439\) 12370.5 1.34490 0.672450 0.740143i \(-0.265242\pi\)
0.672450 + 0.740143i \(0.265242\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12084.4 1.29605 0.648023 0.761621i \(-0.275596\pi\)
0.648023 + 0.761621i \(0.275596\pi\)
\(444\) 0 0
\(445\) 3126.53 0.333060
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −625.550 −0.0657495 −0.0328747 0.999459i \(-0.510466\pi\)
−0.0328747 + 0.999459i \(0.510466\pi\)
\(450\) 0 0
\(451\) 8002.04 0.835480
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2042.29 0.210427
\(456\) 0 0
\(457\) 1811.22 0.185395 0.0926974 0.995694i \(-0.470451\pi\)
0.0926974 + 0.995694i \(0.470451\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −11625.0 −1.17447 −0.587233 0.809418i \(-0.699783\pi\)
−0.587233 + 0.809418i \(0.699783\pi\)
\(462\) 0 0
\(463\) −7291.88 −0.731928 −0.365964 0.930629i \(-0.619261\pi\)
−0.365964 + 0.930629i \(0.619261\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11637.6 1.15316 0.576579 0.817042i \(-0.304387\pi\)
0.576579 + 0.817042i \(0.304387\pi\)
\(468\) 0 0
\(469\) −5256.78 −0.517560
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 26320.9 2.55864
\(474\) 0 0
\(475\) 3664.02 0.353930
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −12041.4 −1.14862 −0.574309 0.818639i \(-0.694729\pi\)
−0.574309 + 0.818639i \(0.694729\pi\)
\(480\) 0 0
\(481\) −7531.14 −0.713909
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −966.307 −0.0904695
\(486\) 0 0
\(487\) −7037.81 −0.654853 −0.327427 0.944877i \(-0.606181\pi\)
−0.327427 + 0.944877i \(0.606181\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6603.07 −0.606909 −0.303454 0.952846i \(-0.598140\pi\)
−0.303454 + 0.952846i \(0.598140\pi\)
\(492\) 0 0
\(493\) 6322.17 0.577558
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6621.58 −0.597623
\(498\) 0 0
\(499\) 36.7047 0.00329284 0.00164642 0.999999i \(-0.499476\pi\)
0.00164642 + 0.999999i \(0.499476\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −21242.5 −1.88302 −0.941508 0.336990i \(-0.890591\pi\)
−0.941508 + 0.336990i \(0.890591\pi\)
\(504\) 0 0
\(505\) 6874.29 0.605746
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −11280.4 −0.982309 −0.491155 0.871072i \(-0.663425\pi\)
−0.491155 + 0.871072i \(0.663425\pi\)
\(510\) 0 0
\(511\) −1046.86 −0.0906270
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −10148.0 −0.868300
\(516\) 0 0
\(517\) −22175.5 −1.88642
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10239.3 −0.861023 −0.430511 0.902585i \(-0.641667\pi\)
−0.430511 + 0.902585i \(0.641667\pi\)
\(522\) 0 0
\(523\) 2822.00 0.235942 0.117971 0.993017i \(-0.462361\pi\)
0.117971 + 0.993017i \(0.462361\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1160.22 0.0959010
\(528\) 0 0
\(529\) −11611.1 −0.954310
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4922.67 0.400046
\(534\) 0 0
\(535\) 4115.31 0.332561
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −11450.3 −0.915028
\(540\) 0 0
\(541\) −9409.63 −0.747785 −0.373892 0.927472i \(-0.621977\pi\)
−0.373892 + 0.927472i \(0.621977\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4149.34 −0.326125
\(546\) 0 0
\(547\) 3836.43 0.299879 0.149940 0.988695i \(-0.452092\pi\)
0.149940 + 0.988695i \(0.452092\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 23596.7 1.82441
\(552\) 0 0
\(553\) −5316.97 −0.408862
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6145.92 0.467524 0.233762 0.972294i \(-0.424896\pi\)
0.233762 + 0.972294i \(0.424896\pi\)
\(558\) 0 0
\(559\) 16192.0 1.22513
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −13247.9 −0.991707 −0.495854 0.868406i \(-0.665145\pi\)
−0.495854 + 0.868406i \(0.665145\pi\)
\(564\) 0 0
\(565\) −7516.83 −0.559709
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6544.89 0.482208 0.241104 0.970499i \(-0.422491\pi\)
0.241104 + 0.970499i \(0.422491\pi\)
\(570\) 0 0
\(571\) −20362.1 −1.49234 −0.746170 0.665755i \(-0.768110\pi\)
−0.746170 + 0.665755i \(0.768110\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 589.443 0.0427504
\(576\) 0 0
\(577\) −26247.4 −1.89375 −0.946876 0.321600i \(-0.895779\pi\)
−0.946876 + 0.321600i \(0.895779\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3356.54 −0.239677
\(582\) 0 0
\(583\) −7579.71 −0.538455
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14098.2 0.991301 0.495650 0.868522i \(-0.334930\pi\)
0.495650 + 0.868522i \(0.334930\pi\)
\(588\) 0 0
\(589\) 4330.36 0.302936
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3476.71 0.240761 0.120380 0.992728i \(-0.461589\pi\)
0.120380 + 0.992728i \(0.461589\pi\)
\(594\) 0 0
\(595\) 2318.06 0.159716
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 16179.0 1.10360 0.551798 0.833978i \(-0.313942\pi\)
0.551798 + 0.833978i \(0.313942\pi\)
\(600\) 0 0
\(601\) 8112.16 0.550586 0.275293 0.961360i \(-0.411225\pi\)
0.275293 + 0.961360i \(0.411225\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9158.34 0.615437
\(606\) 0 0
\(607\) −21139.2 −1.41353 −0.706765 0.707449i \(-0.749846\pi\)
−0.706765 + 0.707449i \(0.749846\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −13641.9 −0.903258
\(612\) 0 0
\(613\) −11440.3 −0.753785 −0.376892 0.926257i \(-0.623007\pi\)
−0.376892 + 0.926257i \(0.623007\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21566.2 1.40717 0.703584 0.710612i \(-0.251582\pi\)
0.703584 + 0.710612i \(0.251582\pi\)
\(618\) 0 0
\(619\) 15198.3 0.986866 0.493433 0.869784i \(-0.335742\pi\)
0.493433 + 0.869784i \(0.335742\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7382.68 0.474769
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −8548.05 −0.541865
\(630\) 0 0
\(631\) 4929.66 0.311009 0.155504 0.987835i \(-0.450300\pi\)
0.155504 + 0.987835i \(0.450300\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2881.74 0.180092
\(636\) 0 0
\(637\) −7043.97 −0.438135
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −7535.73 −0.464342 −0.232171 0.972675i \(-0.574583\pi\)
−0.232171 + 0.972675i \(0.574583\pi\)
\(642\) 0 0
\(643\) 15997.2 0.981135 0.490568 0.871403i \(-0.336789\pi\)
0.490568 + 0.871403i \(0.336789\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6020.46 −0.365825 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(648\) 0 0
\(649\) 7378.12 0.446250
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 10948.7 0.656133 0.328067 0.944655i \(-0.393603\pi\)
0.328067 + 0.944655i \(0.393603\pi\)
\(654\) 0 0
\(655\) −11950.2 −0.712874
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12338.3 0.729334 0.364667 0.931138i \(-0.381183\pi\)
0.364667 + 0.931138i \(0.381183\pi\)
\(660\) 0 0
\(661\) 20016.8 1.17786 0.588928 0.808186i \(-0.299550\pi\)
0.588928 + 0.808186i \(0.299550\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8651.86 0.504518
\(666\) 0 0
\(667\) 3796.08 0.220367
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 14610.6 0.840589
\(672\) 0 0
\(673\) −8419.22 −0.482225 −0.241112 0.970497i \(-0.577512\pi\)
−0.241112 + 0.970497i \(0.577512\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −25707.1 −1.45938 −0.729692 0.683776i \(-0.760337\pi\)
−0.729692 + 0.683776i \(0.760337\pi\)
\(678\) 0 0
\(679\) −2281.74 −0.128962
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −19624.3 −1.09942 −0.549708 0.835357i \(-0.685261\pi\)
−0.549708 + 0.835357i \(0.685261\pi\)
\(684\) 0 0
\(685\) 5012.30 0.279577
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4662.86 −0.257824
\(690\) 0 0
\(691\) −7273.23 −0.400415 −0.200207 0.979754i \(-0.564162\pi\)
−0.200207 + 0.979754i \(0.564162\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −659.084 −0.0359719
\(696\) 0 0
\(697\) 5587.37 0.303639
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −17644.3 −0.950664 −0.475332 0.879807i \(-0.657672\pi\)
−0.475332 + 0.879807i \(0.657672\pi\)
\(702\) 0 0
\(703\) −31904.5 −1.71166
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 16232.3 0.863476
\(708\) 0 0
\(709\) 24304.4 1.28741 0.643703 0.765276i \(-0.277397\pi\)
0.643703 + 0.765276i \(0.277397\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 696.639 0.0365909
\(714\) 0 0
\(715\) 9728.00 0.508820
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −15170.2 −0.786863 −0.393431 0.919354i \(-0.628712\pi\)
−0.393431 + 0.919354i \(0.628712\pi\)
\(720\) 0 0
\(721\) −23962.5 −1.23774
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4025.07 0.206189
\(726\) 0 0
\(727\) 17487.0 0.892102 0.446051 0.895008i \(-0.352830\pi\)
0.446051 + 0.895008i \(0.352830\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 18378.4 0.929888
\(732\) 0 0
\(733\) 18698.0 0.942194 0.471097 0.882082i \(-0.343858\pi\)
0.471097 + 0.882082i \(0.343858\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −25039.5 −1.25148
\(738\) 0 0
\(739\) 35250.3 1.75467 0.877336 0.479876i \(-0.159318\pi\)
0.877336 + 0.479876i \(0.159318\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −11133.6 −0.549731 −0.274866 0.961483i \(-0.588633\pi\)
−0.274866 + 0.961483i \(0.588633\pi\)
\(744\) 0 0
\(745\) 5097.43 0.250678
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9717.49 0.474058
\(750\) 0 0
\(751\) −17197.6 −0.835619 −0.417809 0.908535i \(-0.637202\pi\)
−0.417809 + 0.908535i \(0.637202\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −14111.9 −0.680245
\(756\) 0 0
\(757\) 804.647 0.0386333 0.0193166 0.999813i \(-0.493851\pi\)
0.0193166 + 0.999813i \(0.493851\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 26208.9 1.24845 0.624225 0.781245i \(-0.285415\pi\)
0.624225 + 0.781245i \(0.285415\pi\)
\(762\) 0 0
\(763\) −9797.85 −0.464883
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4538.85 0.213674
\(768\) 0 0
\(769\) 36544.2 1.71367 0.856837 0.515587i \(-0.172426\pi\)
0.856837 + 0.515587i \(0.172426\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −42387.4 −1.97228 −0.986139 0.165923i \(-0.946940\pi\)
−0.986139 + 0.165923i \(0.946940\pi\)
\(774\) 0 0
\(775\) 738.663 0.0342368
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 20854.1 0.959148
\(780\) 0 0
\(781\) −31540.4 −1.44507
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2382.50 0.108325
\(786\) 0 0
\(787\) −26849.4 −1.21611 −0.608054 0.793896i \(-0.708050\pi\)
−0.608054 + 0.793896i \(0.708050\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −17749.5 −0.797851
\(792\) 0 0
\(793\) 8988.09 0.402492
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −31307.0 −1.39141 −0.695704 0.718328i \(-0.744908\pi\)
−0.695704 + 0.718328i \(0.744908\pi\)
\(798\) 0 0
\(799\) −15483.9 −0.685583
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −4986.48 −0.219140
\(804\) 0 0
\(805\) 1391.85 0.0609396
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −10011.9 −0.435106 −0.217553 0.976049i \(-0.569807\pi\)
−0.217553 + 0.976049i \(0.569807\pi\)
\(810\) 0 0
\(811\) −4603.68 −0.199331 −0.0996653 0.995021i \(-0.531777\pi\)
−0.0996653 + 0.995021i \(0.531777\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 11211.3 0.481858
\(816\) 0 0
\(817\) 68594.8 2.93737
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −35429.0 −1.50607 −0.753033 0.657983i \(-0.771410\pi\)
−0.753033 + 0.657983i \(0.771410\pi\)
\(822\) 0 0
\(823\) 28297.6 1.19853 0.599265 0.800550i \(-0.295459\pi\)
0.599265 + 0.800550i \(0.295459\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −41059.9 −1.72647 −0.863236 0.504800i \(-0.831566\pi\)
−0.863236 + 0.504800i \(0.831566\pi\)
\(828\) 0 0
\(829\) 9030.68 0.378345 0.189173 0.981944i \(-0.439419\pi\)
0.189173 + 0.981944i \(0.439419\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −7995.10 −0.332550
\(834\) 0 0
\(835\) −475.195 −0.0196944
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −5227.81 −0.215118 −0.107559 0.994199i \(-0.534303\pi\)
−0.107559 + 0.994199i \(0.534303\pi\)
\(840\) 0 0
\(841\) 1532.87 0.0628508
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5000.56 −0.203579
\(846\) 0 0
\(847\) 21625.6 0.877290
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −5132.58 −0.206748
\(852\) 0 0
\(853\) −30002.9 −1.20432 −0.602158 0.798377i \(-0.705692\pi\)
−0.602158 + 0.798377i \(0.705692\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15671.7 0.624662 0.312331 0.949973i \(-0.398890\pi\)
0.312331 + 0.949973i \(0.398890\pi\)
\(858\) 0 0
\(859\) 31306.8 1.24351 0.621755 0.783212i \(-0.286420\pi\)
0.621755 + 0.783212i \(0.286420\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −13212.3 −0.521150 −0.260575 0.965454i \(-0.583912\pi\)
−0.260575 + 0.965454i \(0.583912\pi\)
\(864\) 0 0
\(865\) −10668.8 −0.419364
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −25326.2 −0.988643
\(870\) 0 0
\(871\) −15403.7 −0.599236
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1475.81 0.0570190
\(876\) 0 0
\(877\) 15440.6 0.594519 0.297260 0.954797i \(-0.403927\pi\)
0.297260 + 0.954797i \(0.403927\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −27563.1 −1.05406 −0.527029 0.849847i \(-0.676694\pi\)
−0.527029 + 0.849847i \(0.676694\pi\)
\(882\) 0 0
\(883\) −19897.7 −0.758337 −0.379169 0.925328i \(-0.623790\pi\)
−0.379169 + 0.925328i \(0.623790\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −40061.6 −1.51650 −0.758251 0.651963i \(-0.773946\pi\)
−0.758251 + 0.651963i \(0.773946\pi\)
\(888\) 0 0
\(889\) 6804.66 0.256716
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −57791.6 −2.16565
\(894\) 0 0
\(895\) 8523.39 0.318330
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4757.07 0.176482
\(900\) 0 0
\(901\) −5292.47 −0.195691
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6804.92 −0.249948
\(906\) 0 0
\(907\) −27839.6 −1.01918 −0.509591 0.860417i \(-0.670203\pi\)
−0.509591 + 0.860417i \(0.670203\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −22251.0 −0.809232 −0.404616 0.914487i \(-0.632595\pi\)
−0.404616 + 0.914487i \(0.632595\pi\)
\(912\) 0 0
\(913\) −15988.1 −0.579549
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −28218.0 −1.01618
\(918\) 0 0
\(919\) −29480.5 −1.05819 −0.529093 0.848564i \(-0.677468\pi\)
−0.529093 + 0.848564i \(0.677468\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −19402.9 −0.691933
\(924\) 0 0
\(925\) −5442.19 −0.193447
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 42351.8 1.49571 0.747857 0.663860i \(-0.231083\pi\)
0.747857 + 0.663860i \(0.231083\pi\)
\(930\) 0 0
\(931\) −29840.7 −1.05047
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 11041.5 0.386200
\(936\) 0 0
\(937\) −35930.6 −1.25272 −0.626362 0.779532i \(-0.715457\pi\)
−0.626362 + 0.779532i \(0.715457\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −21564.0 −0.747041 −0.373521 0.927622i \(-0.621849\pi\)
−0.373521 + 0.927622i \(0.621849\pi\)
\(942\) 0 0
\(943\) 3354.87 0.115853
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 16632.4 0.570729 0.285364 0.958419i \(-0.407885\pi\)
0.285364 + 0.958419i \(0.407885\pi\)
\(948\) 0 0
\(949\) −3067.57 −0.104929
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 39557.8 1.34460 0.672299 0.740279i \(-0.265307\pi\)
0.672299 + 0.740279i \(0.265307\pi\)
\(954\) 0 0
\(955\) 5484.21 0.185827
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 11835.5 0.398529
\(960\) 0 0
\(961\) −28918.0 −0.970696
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −14336.3 −0.478242
\(966\) 0 0
\(967\) 10666.2 0.354707 0.177354 0.984147i \(-0.443246\pi\)
0.177354 + 0.984147i \(0.443246\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −31821.8 −1.05171 −0.525855 0.850574i \(-0.676254\pi\)
−0.525855 + 0.850574i \(0.676254\pi\)
\(972\) 0 0
\(973\) −1556.30 −0.0512771
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 11126.5 0.364347 0.182173 0.983266i \(-0.441687\pi\)
0.182173 + 0.983266i \(0.441687\pi\)
\(978\) 0 0
\(979\) 35165.7 1.14801
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 991.225 0.0321619 0.0160810 0.999871i \(-0.494881\pi\)
0.0160810 + 0.999871i \(0.494881\pi\)
\(984\) 0 0
\(985\) 3620.69 0.117122
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 11035.1 0.354798
\(990\) 0 0
\(991\) 48714.9 1.56153 0.780767 0.624822i \(-0.214829\pi\)
0.780767 + 0.624822i \(0.214829\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8468.25 0.269811
\(996\) 0 0
\(997\) −42207.2 −1.34074 −0.670369 0.742028i \(-0.733864\pi\)
−0.670369 + 0.742028i \(0.733864\pi\)
\(998\) 0 0
\(999\) 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 2160.4.a.bq.1.2 3
3.2 odd 2 2160.4.a.bi.1.2 3
4.3 odd 2 135.4.a.h.1.3 yes 3
12.11 even 2 135.4.a.e.1.1 3
20.3 even 4 675.4.b.n.649.1 6
20.7 even 4 675.4.b.n.649.6 6
20.19 odd 2 675.4.a.p.1.1 3
36.7 odd 6 405.4.e.q.271.1 6
36.11 even 6 405.4.e.v.271.3 6
36.23 even 6 405.4.e.v.136.3 6
36.31 odd 6 405.4.e.q.136.1 6
60.23 odd 4 675.4.b.m.649.6 6
60.47 odd 4 675.4.b.m.649.1 6
60.59 even 2 675.4.a.s.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.e.1.1 3 12.11 even 2
135.4.a.h.1.3 yes 3 4.3 odd 2
405.4.e.q.136.1 6 36.31 odd 6
405.4.e.q.271.1 6 36.7 odd 6
405.4.e.v.136.3 6 36.23 even 6
405.4.e.v.271.3 6 36.11 even 6
675.4.a.p.1.1 3 20.19 odd 2
675.4.a.s.1.3 3 60.59 even 2
675.4.b.m.649.1 6 60.47 odd 4
675.4.b.m.649.6 6 60.23 odd 4
675.4.b.n.649.1 6 20.3 even 4
675.4.b.n.649.6 6 20.7 even 4
2160.4.a.bi.1.2 3 3.2 odd 2
2160.4.a.bq.1.2 3 1.1 even 1 trivial