Properties

Label 2304.4.a.bt.1.3
Level $2304$
Weight $4$
Character 2304.1
Self dual yes
Analytic conductor $135.940$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,4,Mod(1,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, 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("2304.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2304.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: \(135.940400653\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1436.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 11x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.28282\) of defining polynomial
Character \(\chi\) \(=\) 2304.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.15486 q^{5} -27.4175 q^{7} +O(q^{10})\) \(q+9.15486 q^{5} -27.4175 q^{7} +20.5252 q^{11} +32.0471 q^{13} +111.764 q^{17} -129.764 q^{19} +9.16510 q^{23} -41.1885 q^{25} -41.0606 q^{29} -187.606 q^{31} -251.003 q^{35} -114.127 q^{37} +282.915 q^{41} +89.3870 q^{43} +54.6464 q^{47} +408.717 q^{49} -726.878 q^{53} +187.905 q^{55} -216.579 q^{59} +754.222 q^{61} +293.387 q^{65} -379.433 q^{67} +302.080 q^{71} +504.396 q^{73} -562.748 q^{77} +301.780 q^{79} +599.003 q^{83} +1023.18 q^{85} -277.528 q^{89} -878.651 q^{91} -1187.97 q^{95} -765.905 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 10 q^{5} - 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 10 q^{5} - 14 q^{7} + 52 q^{13} - 26 q^{17} - 28 q^{19} + 164 q^{23} + 53 q^{25} - 174 q^{29} - 318 q^{31} - 92 q^{35} + 296 q^{37} + 118 q^{41} + 260 q^{43} + 204 q^{47} + 327 q^{49} - 1086 q^{53} - 512 q^{55} + 196 q^{59} + 1536 q^{61} + 872 q^{65} + 660 q^{67} - 852 q^{71} - 478 q^{73} - 2304 q^{77} - 22 q^{79} + 1136 q^{83} + 2732 q^{85} - 110 q^{89} + 632 q^{91} - 2552 q^{95} - 1222 q^{97}+O(q^{100}) \) 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
\(5\) 9.15486 0.818836 0.409418 0.912347i \(-0.365732\pi\)
0.409418 + 0.912347i \(0.365732\pi\)
\(6\) 0 0
\(7\) −27.4175 −1.48040 −0.740202 0.672385i \(-0.765270\pi\)
−0.740202 + 0.672385i \(0.765270\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 20.5252 0.562598 0.281299 0.959620i \(-0.409235\pi\)
0.281299 + 0.959620i \(0.409235\pi\)
\(12\) 0 0
\(13\) 32.0471 0.683713 0.341857 0.939752i \(-0.388944\pi\)
0.341857 + 0.939752i \(0.388944\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 111.764 1.59452 0.797258 0.603639i \(-0.206283\pi\)
0.797258 + 0.603639i \(0.206283\pi\)
\(18\) 0 0
\(19\) −129.764 −1.56684 −0.783419 0.621494i \(-0.786526\pi\)
−0.783419 + 0.621494i \(0.786526\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 9.16510 0.0830893 0.0415447 0.999137i \(-0.486772\pi\)
0.0415447 + 0.999137i \(0.486772\pi\)
\(24\) 0 0
\(25\) −41.1885 −0.329508
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −41.0606 −0.262923 −0.131461 0.991321i \(-0.541967\pi\)
−0.131461 + 0.991321i \(0.541967\pi\)
\(30\) 0 0
\(31\) −187.606 −1.08694 −0.543468 0.839430i \(-0.682889\pi\)
−0.543468 + 0.839430i \(0.682889\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −251.003 −1.21221
\(36\) 0 0
\(37\) −114.127 −0.507093 −0.253546 0.967323i \(-0.581597\pi\)
−0.253546 + 0.967323i \(0.581597\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 282.915 1.07766 0.538828 0.842416i \(-0.318867\pi\)
0.538828 + 0.842416i \(0.318867\pi\)
\(42\) 0 0
\(43\) 89.3870 0.317009 0.158505 0.987358i \(-0.449333\pi\)
0.158505 + 0.987358i \(0.449333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 54.6464 0.169596 0.0847978 0.996398i \(-0.472976\pi\)
0.0847978 + 0.996398i \(0.472976\pi\)
\(48\) 0 0
\(49\) 408.717 1.19159
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −726.878 −1.88386 −0.941928 0.335815i \(-0.890988\pi\)
−0.941928 + 0.335815i \(0.890988\pi\)
\(54\) 0 0
\(55\) 187.905 0.460675
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −216.579 −0.477900 −0.238950 0.971032i \(-0.576803\pi\)
−0.238950 + 0.971032i \(0.576803\pi\)
\(60\) 0 0
\(61\) 754.222 1.58309 0.791543 0.611114i \(-0.209278\pi\)
0.791543 + 0.611114i \(0.209278\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 293.387 0.559849
\(66\) 0 0
\(67\) −379.433 −0.691868 −0.345934 0.938259i \(-0.612438\pi\)
−0.345934 + 0.938259i \(0.612438\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 302.080 0.504933 0.252467 0.967606i \(-0.418758\pi\)
0.252467 + 0.967606i \(0.418758\pi\)
\(72\) 0 0
\(73\) 504.396 0.808700 0.404350 0.914604i \(-0.367498\pi\)
0.404350 + 0.914604i \(0.367498\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −562.748 −0.832872
\(78\) 0 0
\(79\) 301.780 0.429784 0.214892 0.976638i \(-0.431060\pi\)
0.214892 + 0.976638i \(0.431060\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 599.003 0.792158 0.396079 0.918216i \(-0.370371\pi\)
0.396079 + 0.918216i \(0.370371\pi\)
\(84\) 0 0
\(85\) 1023.18 1.30565
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −277.528 −0.330538 −0.165269 0.986248i \(-0.552849\pi\)
−0.165269 + 0.986248i \(0.552849\pi\)
\(90\) 0 0
\(91\) −878.651 −1.01217
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1187.97 −1.28298
\(96\) 0 0
\(97\) −765.905 −0.801710 −0.400855 0.916141i \(-0.631287\pi\)
−0.400855 + 0.916141i \(0.631287\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 201.253 0.198272 0.0991360 0.995074i \(-0.468392\pi\)
0.0991360 + 0.995074i \(0.468392\pi\)
\(102\) 0 0
\(103\) −682.440 −0.652843 −0.326421 0.945224i \(-0.605843\pi\)
−0.326421 + 0.945224i \(0.605843\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 457.252 0.413123 0.206562 0.978434i \(-0.433773\pi\)
0.206562 + 0.978434i \(0.433773\pi\)
\(108\) 0 0
\(109\) 625.812 0.549926 0.274963 0.961455i \(-0.411334\pi\)
0.274963 + 0.961455i \(0.411334\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −981.151 −0.816805 −0.408402 0.912802i \(-0.633914\pi\)
−0.408402 + 0.912802i \(0.633914\pi\)
\(114\) 0 0
\(115\) 83.9052 0.0680365
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3064.29 −2.36053
\(120\) 0 0
\(121\) −909.717 −0.683484
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1521.43 −1.08865
\(126\) 0 0
\(127\) −808.055 −0.564593 −0.282296 0.959327i \(-0.591096\pi\)
−0.282296 + 0.959327i \(0.591096\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1110.85 −0.740884 −0.370442 0.928856i \(-0.620794\pi\)
−0.370442 + 0.928856i \(0.620794\pi\)
\(132\) 0 0
\(133\) 3557.80 2.31955
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −466.765 −0.291084 −0.145542 0.989352i \(-0.546493\pi\)
−0.145542 + 0.989352i \(0.546493\pi\)
\(138\) 0 0
\(139\) −351.773 −0.214654 −0.107327 0.994224i \(-0.534229\pi\)
−0.107327 + 0.994224i \(0.534229\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 657.773 0.384656
\(144\) 0 0
\(145\) −375.904 −0.215291
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1290.49 −0.709540 −0.354770 0.934954i \(-0.615441\pi\)
−0.354770 + 0.934954i \(0.615441\pi\)
\(150\) 0 0
\(151\) −1175.51 −0.633521 −0.316761 0.948505i \(-0.602595\pi\)
−0.316761 + 0.948505i \(0.602595\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1717.51 −0.890022
\(156\) 0 0
\(157\) −1092.09 −0.555148 −0.277574 0.960704i \(-0.589530\pi\)
−0.277574 + 0.960704i \(0.589530\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −251.284 −0.123006
\(162\) 0 0
\(163\) 3626.97 1.74286 0.871430 0.490519i \(-0.163193\pi\)
0.871430 + 0.490519i \(0.163193\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 45.8012 0.0212228 0.0106114 0.999944i \(-0.496622\pi\)
0.0106114 + 0.999944i \(0.496622\pi\)
\(168\) 0 0
\(169\) −1169.98 −0.532536
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2455.02 −1.07891 −0.539455 0.842014i \(-0.681370\pi\)
−0.539455 + 0.842014i \(0.681370\pi\)
\(174\) 0 0
\(175\) 1129.28 0.487805
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1026.28 0.428533 0.214267 0.976775i \(-0.431264\pi\)
0.214267 + 0.976775i \(0.431264\pi\)
\(180\) 0 0
\(181\) −3699.05 −1.51905 −0.759526 0.650477i \(-0.774569\pi\)
−0.759526 + 0.650477i \(0.774569\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1044.82 −0.415226
\(186\) 0 0
\(187\) 2293.98 0.897071
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5108.93 −1.93544 −0.967721 0.252023i \(-0.918904\pi\)
−0.967721 + 0.252023i \(0.918904\pi\)
\(192\) 0 0
\(193\) −1414.13 −0.527417 −0.263709 0.964602i \(-0.584946\pi\)
−0.263709 + 0.964602i \(0.584946\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2816.66 −1.01867 −0.509337 0.860567i \(-0.670109\pi\)
−0.509337 + 0.860567i \(0.670109\pi\)
\(198\) 0 0
\(199\) 948.556 0.337896 0.168948 0.985625i \(-0.445963\pi\)
0.168948 + 0.985625i \(0.445963\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1125.78 0.389232
\(204\) 0 0
\(205\) 2590.05 0.882424
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2663.43 −0.881499
\(210\) 0 0
\(211\) −4487.28 −1.46406 −0.732032 0.681271i \(-0.761428\pi\)
−0.732032 + 0.681271i \(0.761428\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 818.326 0.259578
\(216\) 0 0
\(217\) 5143.68 1.60910
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3581.72 1.09019
\(222\) 0 0
\(223\) −4590.98 −1.37863 −0.689315 0.724462i \(-0.742088\pi\)
−0.689315 + 0.724462i \(0.742088\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2897.47 0.847189 0.423594 0.905852i \(-0.360768\pi\)
0.423594 + 0.905852i \(0.360768\pi\)
\(228\) 0 0
\(229\) 34.6293 0.00999288 0.00499644 0.999988i \(-0.498410\pi\)
0.00499644 + 0.999988i \(0.498410\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1054.02 −0.296355 −0.148178 0.988961i \(-0.547341\pi\)
−0.148178 + 0.988961i \(0.547341\pi\)
\(234\) 0 0
\(235\) 500.280 0.138871
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 654.700 0.177192 0.0885962 0.996068i \(-0.471762\pi\)
0.0885962 + 0.996068i \(0.471762\pi\)
\(240\) 0 0
\(241\) 3194.00 0.853707 0.426854 0.904321i \(-0.359622\pi\)
0.426854 + 0.904321i \(0.359622\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3741.74 0.975719
\(246\) 0 0
\(247\) −4158.57 −1.07127
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5042.90 1.26815 0.634074 0.773273i \(-0.281382\pi\)
0.634074 + 0.773273i \(0.281382\pi\)
\(252\) 0 0
\(253\) 188.115 0.0467459
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5166.64 1.25403 0.627016 0.779007i \(-0.284276\pi\)
0.627016 + 0.779007i \(0.284276\pi\)
\(258\) 0 0
\(259\) 3129.08 0.750702
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7366.11 −1.72705 −0.863524 0.504308i \(-0.831748\pi\)
−0.863524 + 0.504308i \(0.831748\pi\)
\(264\) 0 0
\(265\) −6654.47 −1.54257
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7877.80 1.78557 0.892784 0.450484i \(-0.148749\pi\)
0.892784 + 0.450484i \(0.148749\pi\)
\(270\) 0 0
\(271\) −5399.92 −1.21041 −0.605206 0.796069i \(-0.706909\pi\)
−0.605206 + 0.796069i \(0.706909\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −845.402 −0.185381
\(276\) 0 0
\(277\) −4416.07 −0.957892 −0.478946 0.877844i \(-0.658981\pi\)
−0.478946 + 0.877844i \(0.658981\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8068.94 1.71300 0.856499 0.516148i \(-0.172635\pi\)
0.856499 + 0.516148i \(0.172635\pi\)
\(282\) 0 0
\(283\) 5241.13 1.10089 0.550447 0.834870i \(-0.314457\pi\)
0.550447 + 0.834870i \(0.314457\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −7756.81 −1.59537
\(288\) 0 0
\(289\) 7578.21 1.54248
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6372.75 1.27065 0.635324 0.772246i \(-0.280867\pi\)
0.635324 + 0.772246i \(0.280867\pi\)
\(294\) 0 0
\(295\) −1982.75 −0.391322
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 293.715 0.0568093
\(300\) 0 0
\(301\) −2450.76 −0.469301
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6904.79 1.29629
\(306\) 0 0
\(307\) −3810.22 −0.708342 −0.354171 0.935181i \(-0.615237\pi\)
−0.354171 + 0.935181i \(0.615237\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8106.73 −1.47810 −0.739052 0.673648i \(-0.764726\pi\)
−0.739052 + 0.673648i \(0.764726\pi\)
\(312\) 0 0
\(313\) 559.983 0.101125 0.0505625 0.998721i \(-0.483899\pi\)
0.0505625 + 0.998721i \(0.483899\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5828.98 −1.03277 −0.516385 0.856357i \(-0.672723\pi\)
−0.516385 + 0.856357i \(0.672723\pi\)
\(318\) 0 0
\(319\) −842.776 −0.147920
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −14503.0 −2.49835
\(324\) 0 0
\(325\) −1319.97 −0.225289
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1498.26 −0.251070
\(330\) 0 0
\(331\) −2847.98 −0.472928 −0.236464 0.971640i \(-0.575989\pi\)
−0.236464 + 0.971640i \(0.575989\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3473.66 −0.566526
\(336\) 0 0
\(337\) −10127.8 −1.63707 −0.818537 0.574454i \(-0.805214\pi\)
−0.818537 + 0.574454i \(0.805214\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3850.65 −0.611508
\(342\) 0 0
\(343\) −1801.78 −0.283636
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −10148.2 −1.56999 −0.784993 0.619505i \(-0.787333\pi\)
−0.784993 + 0.619505i \(0.787333\pi\)
\(348\) 0 0
\(349\) −9515.96 −1.45954 −0.729768 0.683695i \(-0.760372\pi\)
−0.729768 + 0.683695i \(0.760372\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2813.56 −0.424223 −0.212111 0.977245i \(-0.568034\pi\)
−0.212111 + 0.977245i \(0.568034\pi\)
\(354\) 0 0
\(355\) 2765.50 0.413457
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2427.25 −0.356839 −0.178419 0.983955i \(-0.557098\pi\)
−0.178419 + 0.983955i \(0.557098\pi\)
\(360\) 0 0
\(361\) 9979.71 1.45498
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4617.67 0.662192
\(366\) 0 0
\(367\) 5021.46 0.714219 0.357109 0.934063i \(-0.383762\pi\)
0.357109 + 0.934063i \(0.383762\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 19929.1 2.78887
\(372\) 0 0
\(373\) −3182.40 −0.441765 −0.220882 0.975300i \(-0.570894\pi\)
−0.220882 + 0.975300i \(0.570894\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1315.87 −0.179764
\(378\) 0 0
\(379\) −5868.93 −0.795426 −0.397713 0.917510i \(-0.630196\pi\)
−0.397713 + 0.917510i \(0.630196\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −7350.18 −0.980618 −0.490309 0.871549i \(-0.663116\pi\)
−0.490309 + 0.871549i \(0.663116\pi\)
\(384\) 0 0
\(385\) −5151.88 −0.681985
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 13009.1 1.69560 0.847800 0.530317i \(-0.177927\pi\)
0.847800 + 0.530317i \(0.177927\pi\)
\(390\) 0 0
\(391\) 1024.33 0.132487
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2762.76 0.351923
\(396\) 0 0
\(397\) −4877.88 −0.616659 −0.308330 0.951280i \(-0.599770\pi\)
−0.308330 + 0.951280i \(0.599770\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5552.33 −0.691446 −0.345723 0.938337i \(-0.612366\pi\)
−0.345723 + 0.938337i \(0.612366\pi\)
\(402\) 0 0
\(403\) −6012.23 −0.743153
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2342.49 −0.285289
\(408\) 0 0
\(409\) −6989.27 −0.844981 −0.422491 0.906367i \(-0.638844\pi\)
−0.422491 + 0.906367i \(0.638844\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5938.03 0.707485
\(414\) 0 0
\(415\) 5483.79 0.648647
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −10461.0 −1.21970 −0.609849 0.792518i \(-0.708770\pi\)
−0.609849 + 0.792518i \(0.708770\pi\)
\(420\) 0 0
\(421\) −4648.55 −0.538139 −0.269070 0.963121i \(-0.586716\pi\)
−0.269070 + 0.963121i \(0.586716\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4603.40 −0.525406
\(426\) 0 0
\(427\) −20678.8 −2.34360
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12490.7 −1.39595 −0.697975 0.716122i \(-0.745915\pi\)
−0.697975 + 0.716122i \(0.745915\pi\)
\(432\) 0 0
\(433\) 9446.37 1.04842 0.524208 0.851590i \(-0.324362\pi\)
0.524208 + 0.851590i \(0.324362\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1189.30 −0.130188
\(438\) 0 0
\(439\) −2793.60 −0.303716 −0.151858 0.988402i \(-0.548526\pi\)
−0.151858 + 0.988402i \(0.548526\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7601.37 −0.815241 −0.407621 0.913151i \(-0.633641\pi\)
−0.407621 + 0.913151i \(0.633641\pi\)
\(444\) 0 0
\(445\) −2540.73 −0.270657
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10708.8 −1.12557 −0.562785 0.826603i \(-0.690270\pi\)
−0.562785 + 0.826603i \(0.690270\pi\)
\(450\) 0 0
\(451\) 5806.89 0.606287
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8043.92 −0.828802
\(456\) 0 0
\(457\) −233.840 −0.0239356 −0.0119678 0.999928i \(-0.503810\pi\)
−0.0119678 + 0.999928i \(0.503810\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 981.307 0.0991410 0.0495705 0.998771i \(-0.484215\pi\)
0.0495705 + 0.998771i \(0.484215\pi\)
\(462\) 0 0
\(463\) −14082.7 −1.41356 −0.706782 0.707431i \(-0.749854\pi\)
−0.706782 + 0.707431i \(0.749854\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9286.49 0.920188 0.460094 0.887870i \(-0.347816\pi\)
0.460094 + 0.887870i \(0.347816\pi\)
\(468\) 0 0
\(469\) 10403.1 1.02424
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1834.68 0.178349
\(474\) 0 0
\(475\) 5344.79 0.516286
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 19409.3 1.85143 0.925715 0.378222i \(-0.123464\pi\)
0.925715 + 0.378222i \(0.123464\pi\)
\(480\) 0 0
\(481\) −3657.46 −0.346706
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7011.76 −0.656469
\(486\) 0 0
\(487\) −12124.8 −1.12818 −0.564091 0.825712i \(-0.690773\pi\)
−0.564091 + 0.825712i \(0.690773\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5100.69 0.468820 0.234410 0.972138i \(-0.424684\pi\)
0.234410 + 0.972138i \(0.424684\pi\)
\(492\) 0 0
\(493\) −4589.10 −0.419235
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8282.26 −0.747505
\(498\) 0 0
\(499\) 85.2797 0.00765058 0.00382529 0.999993i \(-0.498782\pi\)
0.00382529 + 0.999993i \(0.498782\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12287.2 −1.08918 −0.544592 0.838701i \(-0.683316\pi\)
−0.544592 + 0.838701i \(0.683316\pi\)
\(504\) 0 0
\(505\) 1842.45 0.162352
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −450.441 −0.0392248 −0.0196124 0.999808i \(-0.506243\pi\)
−0.0196124 + 0.999808i \(0.506243\pi\)
\(510\) 0 0
\(511\) −13829.2 −1.19720
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6247.64 −0.534571
\(516\) 0 0
\(517\) 1121.63 0.0954141
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −15088.1 −1.26876 −0.634378 0.773023i \(-0.718744\pi\)
−0.634378 + 0.773023i \(0.718744\pi\)
\(522\) 0 0
\(523\) 17719.4 1.48149 0.740743 0.671789i \(-0.234474\pi\)
0.740743 + 0.671789i \(0.234474\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −20967.6 −1.73314
\(528\) 0 0
\(529\) −12083.0 −0.993096
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9066.62 0.736808
\(534\) 0 0
\(535\) 4186.08 0.338280
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 8388.98 0.670388
\(540\) 0 0
\(541\) 12244.5 0.973074 0.486537 0.873660i \(-0.338260\pi\)
0.486537 + 0.873660i \(0.338260\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5729.22 0.450299
\(546\) 0 0
\(547\) 7822.46 0.611452 0.305726 0.952119i \(-0.401101\pi\)
0.305726 + 0.952119i \(0.401101\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5328.19 0.411957
\(552\) 0 0
\(553\) −8274.05 −0.636254
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16555.5 −1.25938 −0.629692 0.776845i \(-0.716819\pi\)
−0.629692 + 0.776845i \(0.716819\pi\)
\(558\) 0 0
\(559\) 2864.60 0.216743
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −12580.7 −0.941766 −0.470883 0.882196i \(-0.656065\pi\)
−0.470883 + 0.882196i \(0.656065\pi\)
\(564\) 0 0
\(565\) −8982.30 −0.668829
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2657.93 0.195828 0.0979141 0.995195i \(-0.468783\pi\)
0.0979141 + 0.995195i \(0.468783\pi\)
\(570\) 0 0
\(571\) 17669.0 1.29496 0.647481 0.762081i \(-0.275822\pi\)
0.647481 + 0.762081i \(0.275822\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −377.497 −0.0273786
\(576\) 0 0
\(577\) 14617.7 1.05467 0.527334 0.849658i \(-0.323192\pi\)
0.527334 + 0.849658i \(0.323192\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −16423.1 −1.17271
\(582\) 0 0
\(583\) −14919.3 −1.05985
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4096.53 0.288044 0.144022 0.989574i \(-0.453996\pi\)
0.144022 + 0.989574i \(0.453996\pi\)
\(588\) 0 0
\(589\) 24344.5 1.70305
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 21988.3 1.52269 0.761343 0.648349i \(-0.224540\pi\)
0.761343 + 0.648349i \(0.224540\pi\)
\(594\) 0 0
\(595\) −28053.1 −1.93288
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 20767.7 1.41660 0.708302 0.705909i \(-0.249461\pi\)
0.708302 + 0.705909i \(0.249461\pi\)
\(600\) 0 0
\(601\) −5382.61 −0.365326 −0.182663 0.983176i \(-0.558472\pi\)
−0.182663 + 0.983176i \(0.558472\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −8328.33 −0.559661
\(606\) 0 0
\(607\) 11165.4 0.746607 0.373304 0.927709i \(-0.378225\pi\)
0.373304 + 0.927709i \(0.378225\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1751.26 0.115955
\(612\) 0 0
\(613\) 16413.5 1.08146 0.540731 0.841195i \(-0.318148\pi\)
0.540731 + 0.841195i \(0.318148\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −51.5882 −0.00336607 −0.00168303 0.999999i \(-0.500536\pi\)
−0.00168303 + 0.999999i \(0.500536\pi\)
\(618\) 0 0
\(619\) −6349.55 −0.412294 −0.206147 0.978521i \(-0.566093\pi\)
−0.206147 + 0.978521i \(0.566093\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7609.11 0.489330
\(624\) 0 0
\(625\) −8779.94 −0.561916
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12755.3 −0.808567
\(630\) 0 0
\(631\) 13379.1 0.844078 0.422039 0.906578i \(-0.361315\pi\)
0.422039 + 0.906578i \(0.361315\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −7397.63 −0.462309
\(636\) 0 0
\(637\) 13098.2 0.814709
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20406.3 1.25741 0.628705 0.777644i \(-0.283585\pi\)
0.628705 + 0.777644i \(0.283585\pi\)
\(642\) 0 0
\(643\) 19415.1 1.19076 0.595378 0.803446i \(-0.297002\pi\)
0.595378 + 0.803446i \(0.297002\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −8167.12 −0.496264 −0.248132 0.968726i \(-0.579817\pi\)
−0.248132 + 0.968726i \(0.579817\pi\)
\(648\) 0 0
\(649\) −4445.31 −0.268866
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7444.93 −0.446160 −0.223080 0.974800i \(-0.571611\pi\)
−0.223080 + 0.974800i \(0.571611\pi\)
\(654\) 0 0
\(655\) −10169.7 −0.606662
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 23780.4 1.40569 0.702846 0.711342i \(-0.251912\pi\)
0.702846 + 0.711342i \(0.251912\pi\)
\(660\) 0 0
\(661\) −2528.90 −0.148809 −0.0744046 0.997228i \(-0.523706\pi\)
−0.0744046 + 0.997228i \(0.523706\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 32571.2 1.89933
\(666\) 0 0
\(667\) −376.324 −0.0218461
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 15480.5 0.890640
\(672\) 0 0
\(673\) 16733.7 0.958447 0.479224 0.877693i \(-0.340918\pi\)
0.479224 + 0.877693i \(0.340918\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −24191.5 −1.37335 −0.686673 0.726966i \(-0.740930\pi\)
−0.686673 + 0.726966i \(0.740930\pi\)
\(678\) 0 0
\(679\) 20999.2 1.18685
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 13965.2 0.782376 0.391188 0.920311i \(-0.372064\pi\)
0.391188 + 0.920311i \(0.372064\pi\)
\(684\) 0 0
\(685\) −4273.17 −0.238350
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −23294.4 −1.28802
\(690\) 0 0
\(691\) −8685.63 −0.478172 −0.239086 0.970998i \(-0.576848\pi\)
−0.239086 + 0.970998i \(0.576848\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3220.43 −0.175767
\(696\) 0 0
\(697\) 31619.7 1.71834
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 25942.2 1.39775 0.698876 0.715243i \(-0.253684\pi\)
0.698876 + 0.715243i \(0.253684\pi\)
\(702\) 0 0
\(703\) 14809.6 0.794532
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5517.86 −0.293522
\(708\) 0 0
\(709\) 5487.75 0.290687 0.145343 0.989381i \(-0.453571\pi\)
0.145343 + 0.989381i \(0.453571\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1719.43 −0.0903128
\(714\) 0 0
\(715\) 6021.82 0.314970
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −17141.2 −0.889094 −0.444547 0.895756i \(-0.646635\pi\)
−0.444547 + 0.895756i \(0.646635\pi\)
\(720\) 0 0
\(721\) 18710.8 0.966470
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1691.23 0.0866352
\(726\) 0 0
\(727\) 15946.4 0.813508 0.406754 0.913538i \(-0.366661\pi\)
0.406754 + 0.913538i \(0.366661\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 9990.26 0.505476
\(732\) 0 0
\(733\) 15914.2 0.801917 0.400958 0.916096i \(-0.368677\pi\)
0.400958 + 0.916096i \(0.368677\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7787.94 −0.389243
\(738\) 0 0
\(739\) 13555.4 0.674755 0.337377 0.941369i \(-0.390460\pi\)
0.337377 + 0.941369i \(0.390460\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1772.73 −0.0875303 −0.0437652 0.999042i \(-0.513935\pi\)
−0.0437652 + 0.999042i \(0.513935\pi\)
\(744\) 0 0
\(745\) −11814.3 −0.580997
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −12536.7 −0.611589
\(750\) 0 0
\(751\) −1006.65 −0.0489124 −0.0244562 0.999701i \(-0.507785\pi\)
−0.0244562 + 0.999701i \(0.507785\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −10761.6 −0.518750
\(756\) 0 0
\(757\) 28774.0 1.38152 0.690758 0.723086i \(-0.257277\pi\)
0.690758 + 0.723086i \(0.257277\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18393.5 0.876166 0.438083 0.898934i \(-0.355658\pi\)
0.438083 + 0.898934i \(0.355658\pi\)
\(762\) 0 0
\(763\) −17158.2 −0.814112
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −6940.72 −0.326747
\(768\) 0 0
\(769\) −14672.2 −0.688027 −0.344014 0.938965i \(-0.611787\pi\)
−0.344014 + 0.938965i \(0.611787\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 16256.4 0.756405 0.378202 0.925723i \(-0.376542\pi\)
0.378202 + 0.925723i \(0.376542\pi\)
\(774\) 0 0
\(775\) 7727.21 0.358154
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −36712.2 −1.68851
\(780\) 0 0
\(781\) 6200.24 0.284074
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9997.92 −0.454575
\(786\) 0 0
\(787\) 23988.3 1.08652 0.543259 0.839565i \(-0.317190\pi\)
0.543259 + 0.839565i \(0.317190\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 26900.7 1.20920
\(792\) 0 0
\(793\) 24170.6 1.08238
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −32966.9 −1.46518 −0.732589 0.680672i \(-0.761688\pi\)
−0.732589 + 0.680672i \(0.761688\pi\)
\(798\) 0 0
\(799\) 6107.50 0.270423
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10352.8 0.454973
\(804\) 0 0
\(805\) −2300.47 −0.100721
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −41700.3 −1.81224 −0.906122 0.423017i \(-0.860971\pi\)
−0.906122 + 0.423017i \(0.860971\pi\)
\(810\) 0 0
\(811\) 5981.80 0.259000 0.129500 0.991579i \(-0.458663\pi\)
0.129500 + 0.991579i \(0.458663\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 33204.4 1.42712
\(816\) 0 0
\(817\) −11599.2 −0.496702
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9846.06 −0.418550 −0.209275 0.977857i \(-0.567110\pi\)
−0.209275 + 0.977857i \(0.567110\pi\)
\(822\) 0 0
\(823\) −47001.9 −1.99074 −0.995372 0.0960935i \(-0.969365\pi\)
−0.995372 + 0.0960935i \(0.969365\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −21727.4 −0.913587 −0.456794 0.889573i \(-0.651002\pi\)
−0.456794 + 0.889573i \(0.651002\pi\)
\(828\) 0 0
\(829\) −22772.3 −0.954058 −0.477029 0.878888i \(-0.658286\pi\)
−0.477029 + 0.878888i \(0.658286\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 45679.8 1.90002
\(834\) 0 0
\(835\) 419.304 0.0173780
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 11010.4 0.453064 0.226532 0.974004i \(-0.427261\pi\)
0.226532 + 0.974004i \(0.427261\pi\)
\(840\) 0 0
\(841\) −22703.0 −0.930872
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −10711.0 −0.436059
\(846\) 0 0
\(847\) 24942.1 1.01183
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1045.99 −0.0421340
\(852\) 0 0
\(853\) 38177.4 1.53244 0.766219 0.642579i \(-0.222136\pi\)
0.766219 + 0.642579i \(0.222136\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 8848.01 0.352675 0.176337 0.984330i \(-0.443575\pi\)
0.176337 + 0.984330i \(0.443575\pi\)
\(858\) 0 0
\(859\) 4347.66 0.172690 0.0863448 0.996265i \(-0.472481\pi\)
0.0863448 + 0.996265i \(0.472481\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 33669.9 1.32808 0.664042 0.747695i \(-0.268839\pi\)
0.664042 + 0.747695i \(0.268839\pi\)
\(864\) 0 0
\(865\) −22475.3 −0.883450
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6194.10 0.241796
\(870\) 0 0
\(871\) −12159.7 −0.473039
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 41713.8 1.61164
\(876\) 0 0
\(877\) −50102.0 −1.92910 −0.964552 0.263892i \(-0.914994\pi\)
−0.964552 + 0.263892i \(0.914994\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −18716.9 −0.715766 −0.357883 0.933766i \(-0.616501\pi\)
−0.357883 + 0.933766i \(0.616501\pi\)
\(882\) 0 0
\(883\) 7514.19 0.286379 0.143189 0.989695i \(-0.454264\pi\)
0.143189 + 0.989695i \(0.454264\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −15544.6 −0.588429 −0.294215 0.955739i \(-0.595058\pi\)
−0.294215 + 0.955739i \(0.595058\pi\)
\(888\) 0 0
\(889\) 22154.8 0.835825
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −7091.14 −0.265729
\(894\) 0 0
\(895\) 9395.42 0.350898
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 7703.21 0.285780
\(900\) 0 0
\(901\) −81238.8 −3.00384
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −33864.3 −1.24385
\(906\) 0 0
\(907\) −8713.10 −0.318979 −0.159489 0.987200i \(-0.550985\pi\)
−0.159489 + 0.987200i \(0.550985\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1975.97 −0.0718627 −0.0359313 0.999354i \(-0.511440\pi\)
−0.0359313 + 0.999354i \(0.511440\pi\)
\(912\) 0 0
\(913\) 12294.6 0.445666
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 30456.8 1.09681
\(918\) 0 0
\(919\) 18430.5 0.661552 0.330776 0.943709i \(-0.392689\pi\)
0.330776 + 0.943709i \(0.392689\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 9680.79 0.345230
\(924\) 0 0
\(925\) 4700.74 0.167091
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −12506.8 −0.441697 −0.220848 0.975308i \(-0.570883\pi\)
−0.220848 + 0.975308i \(0.570883\pi\)
\(930\) 0 0
\(931\) −53036.7 −1.86703
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 21001.0 0.734554
\(936\) 0 0
\(937\) −39267.5 −1.36906 −0.684532 0.728982i \(-0.739994\pi\)
−0.684532 + 0.728982i \(0.739994\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 23727.2 0.821981 0.410991 0.911640i \(-0.365183\pi\)
0.410991 + 0.911640i \(0.365183\pi\)
\(942\) 0 0
\(943\) 2592.94 0.0895418
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23399.8 −0.802948 −0.401474 0.915870i \(-0.631502\pi\)
−0.401474 + 0.915870i \(0.631502\pi\)
\(948\) 0 0
\(949\) 16164.4 0.552919
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 41497.1 1.41052 0.705258 0.708950i \(-0.250831\pi\)
0.705258 + 0.708950i \(0.250831\pi\)
\(954\) 0 0
\(955\) −46771.6 −1.58481
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 12797.5 0.430921
\(960\) 0 0
\(961\) 5405.00 0.181431
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −12946.2 −0.431868
\(966\) 0 0
\(967\) 49123.6 1.63362 0.816808 0.576909i \(-0.195741\pi\)
0.816808 + 0.576909i \(0.195741\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −20346.8 −0.672462 −0.336231 0.941780i \(-0.609152\pi\)
−0.336231 + 0.941780i \(0.609152\pi\)
\(972\) 0 0
\(973\) 9644.71 0.317775
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −40602.1 −1.32955 −0.664777 0.747042i \(-0.731474\pi\)
−0.664777 + 0.747042i \(0.731474\pi\)
\(978\) 0 0
\(979\) −5696.32 −0.185960
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 50425.9 1.63615 0.818075 0.575112i \(-0.195041\pi\)
0.818075 + 0.575112i \(0.195041\pi\)
\(984\) 0 0
\(985\) −25786.2 −0.834127
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 819.241 0.0263401
\(990\) 0 0
\(991\) −8511.62 −0.272836 −0.136418 0.990651i \(-0.543559\pi\)
−0.136418 + 0.990651i \(0.543559\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8683.90 0.276681
\(996\) 0 0
\(997\) 25302.1 0.803738 0.401869 0.915697i \(-0.368361\pi\)
0.401869 + 0.915697i \(0.368361\pi\)
\(998\) 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 2304.4.a.bt.1.3 3
3.2 odd 2 768.4.a.r.1.1 3
4.3 odd 2 2304.4.a.bu.1.3 3
8.3 odd 2 2304.4.a.bw.1.1 3
8.5 even 2 2304.4.a.bv.1.1 3
12.11 even 2 768.4.a.t.1.1 3
16.3 odd 4 288.4.d.d.145.2 6
16.5 even 4 72.4.d.d.37.3 6
16.11 odd 4 288.4.d.d.145.5 6
16.13 even 4 72.4.d.d.37.4 6
24.5 odd 2 768.4.a.s.1.3 3
24.11 even 2 768.4.a.q.1.3 3
48.5 odd 4 24.4.d.a.13.4 yes 6
48.11 even 4 96.4.d.a.49.1 6
48.29 odd 4 24.4.d.a.13.3 6
48.35 even 4 96.4.d.a.49.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.4.d.a.13.3 6 48.29 odd 4
24.4.d.a.13.4 yes 6 48.5 odd 4
72.4.d.d.37.3 6 16.5 even 4
72.4.d.d.37.4 6 16.13 even 4
96.4.d.a.49.1 6 48.11 even 4
96.4.d.a.49.6 6 48.35 even 4
288.4.d.d.145.2 6 16.3 odd 4
288.4.d.d.145.5 6 16.11 odd 4
768.4.a.q.1.3 3 24.11 even 2
768.4.a.r.1.1 3 3.2 odd 2
768.4.a.s.1.3 3 24.5 odd 2
768.4.a.t.1.1 3 12.11 even 2
2304.4.a.bt.1.3 3 1.1 even 1 trivial
2304.4.a.bu.1.3 3 4.3 odd 2
2304.4.a.bv.1.1 3 8.5 even 2
2304.4.a.bw.1.1 3 8.3 odd 2