Properties

Label 1200.3.e.h.751.2
Level $1200$
Weight $3$
Character 1200.751
Analytic conductor $32.698$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,3,Mod(751,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.751");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.e (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 751.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1200.751
Dual form 1200.3.e.h.751.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} +6.92820i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{3} +6.92820i q^{7} -3.00000 q^{9} +20.7846i q^{11} +14.0000 q^{13} +6.00000 q^{17} +6.92820i q^{19} -12.0000 q^{21} -5.19615i q^{27} +30.0000 q^{29} -20.7846i q^{31} -36.0000 q^{33} -26.0000 q^{37} +24.2487i q^{39} -54.0000 q^{41} -20.7846i q^{43} +41.5692i q^{47} +1.00000 q^{49} +10.3923i q^{51} +18.0000 q^{53} -12.0000 q^{57} -20.7846i q^{59} -70.0000 q^{61} -20.7846i q^{63} +117.779i q^{67} +83.1384i q^{71} -82.0000 q^{73} -144.000 q^{77} -76.2102i q^{79} +9.00000 q^{81} +20.7846i q^{83} +51.9615i q^{87} +114.000 q^{89} +96.9948i q^{91} +36.0000 q^{93} -34.0000 q^{97} -62.3538i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{9} + 28 q^{13} + 12 q^{17} - 24 q^{21} + 60 q^{29} - 72 q^{33} - 52 q^{37} - 108 q^{41} + 2 q^{49} + 36 q^{53} - 24 q^{57} - 140 q^{61} - 164 q^{73} - 288 q^{77} + 18 q^{81} + 228 q^{89} + 72 q^{93} - 68 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.73205i 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 6.92820i 0.989743i 0.868966 + 0.494872i \(0.164785\pi\)
−0.868966 + 0.494872i \(0.835215\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) 20.7846i 1.88951i 0.327777 + 0.944755i \(0.393700\pi\)
−0.327777 + 0.944755i \(0.606300\pi\)
\(12\) 0 0
\(13\) 14.0000 1.07692 0.538462 0.842650i \(-0.319006\pi\)
0.538462 + 0.842650i \(0.319006\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 0.352941 0.176471 0.984306i \(-0.443532\pi\)
0.176471 + 0.984306i \(0.443532\pi\)
\(18\) 0 0
\(19\) 6.92820i 0.364642i 0.983239 + 0.182321i \(0.0583610\pi\)
−0.983239 + 0.182321i \(0.941639\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.571429
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.19615i − 0.192450i
\(28\) 0 0
\(29\) 30.0000 1.03448 0.517241 0.855840i \(-0.326959\pi\)
0.517241 + 0.855840i \(0.326959\pi\)
\(30\) 0 0
\(31\) − 20.7846i − 0.670471i −0.942134 0.335236i \(-0.891184\pi\)
0.942134 0.335236i \(-0.108816\pi\)
\(32\) 0 0
\(33\) −36.0000 −1.09091
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −26.0000 −0.702703 −0.351351 0.936244i \(-0.614278\pi\)
−0.351351 + 0.936244i \(0.614278\pi\)
\(38\) 0 0
\(39\) 24.2487i 0.621762i
\(40\) 0 0
\(41\) −54.0000 −1.31707 −0.658537 0.752549i \(-0.728824\pi\)
−0.658537 + 0.752549i \(0.728824\pi\)
\(42\) 0 0
\(43\) − 20.7846i − 0.483363i −0.970356 0.241682i \(-0.922301\pi\)
0.970356 0.241682i \(-0.0776989\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 41.5692i 0.884451i 0.896904 + 0.442226i \(0.145811\pi\)
−0.896904 + 0.442226i \(0.854189\pi\)
\(48\) 0 0
\(49\) 1.00000 0.0204082
\(50\) 0 0
\(51\) 10.3923i 0.203771i
\(52\) 0 0
\(53\) 18.0000 0.339623 0.169811 0.985477i \(-0.445684\pi\)
0.169811 + 0.985477i \(0.445684\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −12.0000 −0.210526
\(58\) 0 0
\(59\) − 20.7846i − 0.352282i −0.984365 0.176141i \(-0.943639\pi\)
0.984365 0.176141i \(-0.0563614\pi\)
\(60\) 0 0
\(61\) −70.0000 −1.14754 −0.573770 0.819016i \(-0.694520\pi\)
−0.573770 + 0.819016i \(0.694520\pi\)
\(62\) 0 0
\(63\) − 20.7846i − 0.329914i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 117.779i 1.75790i 0.476912 + 0.878951i \(0.341756\pi\)
−0.476912 + 0.878951i \(0.658244\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 83.1384i 1.17096i 0.810685 + 0.585482i \(0.199095\pi\)
−0.810685 + 0.585482i \(0.800905\pi\)
\(72\) 0 0
\(73\) −82.0000 −1.12329 −0.561644 0.827379i \(-0.689831\pi\)
−0.561644 + 0.827379i \(0.689831\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −144.000 −1.87013
\(78\) 0 0
\(79\) − 76.2102i − 0.964687i −0.875982 0.482343i \(-0.839786\pi\)
0.875982 0.482343i \(-0.160214\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 20.7846i 0.250417i 0.992130 + 0.125208i \(0.0399600\pi\)
−0.992130 + 0.125208i \(0.960040\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 51.9615i 0.597259i
\(88\) 0 0
\(89\) 114.000 1.28090 0.640449 0.768000i \(-0.278748\pi\)
0.640449 + 0.768000i \(0.278748\pi\)
\(90\) 0 0
\(91\) 96.9948i 1.06588i
\(92\) 0 0
\(93\) 36.0000 0.387097
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −34.0000 −0.350515 −0.175258 0.984523i \(-0.556076\pi\)
−0.175258 + 0.984523i \(0.556076\pi\)
\(98\) 0 0
\(99\) − 62.3538i − 0.629837i
\(100\) 0 0
\(101\) −18.0000 −0.178218 −0.0891089 0.996022i \(-0.528402\pi\)
−0.0891089 + 0.996022i \(0.528402\pi\)
\(102\) 0 0
\(103\) 131.636i 1.27802i 0.769199 + 0.639009i \(0.220655\pi\)
−0.769199 + 0.639009i \(0.779345\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 145.492i − 1.35974i −0.733332 0.679870i \(-0.762036\pi\)
0.733332 0.679870i \(-0.237964\pi\)
\(108\) 0 0
\(109\) 34.0000 0.311927 0.155963 0.987763i \(-0.450152\pi\)
0.155963 + 0.987763i \(0.450152\pi\)
\(110\) 0 0
\(111\) − 45.0333i − 0.405706i
\(112\) 0 0
\(113\) 78.0000 0.690265 0.345133 0.938554i \(-0.387834\pi\)
0.345133 + 0.938554i \(0.387834\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −42.0000 −0.358974
\(118\) 0 0
\(119\) 41.5692i 0.349321i
\(120\) 0 0
\(121\) −311.000 −2.57025
\(122\) 0 0
\(123\) − 93.5307i − 0.760413i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 103.923i − 0.818292i −0.912469 0.409146i \(-0.865827\pi\)
0.912469 0.409146i \(-0.134173\pi\)
\(128\) 0 0
\(129\) 36.0000 0.279070
\(130\) 0 0
\(131\) 103.923i 0.793306i 0.917969 + 0.396653i \(0.129828\pi\)
−0.917969 + 0.396653i \(0.870172\pi\)
\(132\) 0 0
\(133\) −48.0000 −0.360902
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −186.000 −1.35766 −0.678832 0.734294i \(-0.737514\pi\)
−0.678832 + 0.734294i \(0.737514\pi\)
\(138\) 0 0
\(139\) − 48.4974i − 0.348902i −0.984666 0.174451i \(-0.944185\pi\)
0.984666 0.174451i \(-0.0558151\pi\)
\(140\) 0 0
\(141\) −72.0000 −0.510638
\(142\) 0 0
\(143\) 290.985i 2.03486i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.73205i 0.0117827i
\(148\) 0 0
\(149\) −186.000 −1.24832 −0.624161 0.781296i \(-0.714559\pi\)
−0.624161 + 0.781296i \(0.714559\pi\)
\(150\) 0 0
\(151\) − 34.6410i − 0.229411i −0.993400 0.114705i \(-0.963408\pi\)
0.993400 0.114705i \(-0.0365924\pi\)
\(152\) 0 0
\(153\) −18.0000 −0.117647
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −170.000 −1.08280 −0.541401 0.840764i \(-0.682106\pi\)
−0.541401 + 0.840764i \(0.682106\pi\)
\(158\) 0 0
\(159\) 31.1769i 0.196081i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 284.056i − 1.74268i −0.490683 0.871338i \(-0.663253\pi\)
0.490683 0.871338i \(-0.336747\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 207.846i 1.24459i 0.782784 + 0.622294i \(0.213799\pi\)
−0.782784 + 0.622294i \(0.786201\pi\)
\(168\) 0 0
\(169\) 27.0000 0.159763
\(170\) 0 0
\(171\) − 20.7846i − 0.121547i
\(172\) 0 0
\(173\) 42.0000 0.242775 0.121387 0.992605i \(-0.461266\pi\)
0.121387 + 0.992605i \(0.461266\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 36.0000 0.203390
\(178\) 0 0
\(179\) − 145.492i − 0.812806i −0.913694 0.406403i \(-0.866783\pi\)
0.913694 0.406403i \(-0.133217\pi\)
\(180\) 0 0
\(181\) 82.0000 0.453039 0.226519 0.974007i \(-0.427265\pi\)
0.226519 + 0.974007i \(0.427265\pi\)
\(182\) 0 0
\(183\) − 121.244i − 0.662533i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 124.708i 0.666886i
\(188\) 0 0
\(189\) 36.0000 0.190476
\(190\) 0 0
\(191\) 332.554i 1.74112i 0.492063 + 0.870560i \(0.336243\pi\)
−0.492063 + 0.870560i \(0.663757\pi\)
\(192\) 0 0
\(193\) 94.0000 0.487047 0.243523 0.969895i \(-0.421697\pi\)
0.243523 + 0.969895i \(0.421697\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 258.000 1.30964 0.654822 0.755783i \(-0.272743\pi\)
0.654822 + 0.755783i \(0.272743\pi\)
\(198\) 0 0
\(199\) 117.779i 0.591857i 0.955210 + 0.295928i \(0.0956289\pi\)
−0.955210 + 0.295928i \(0.904371\pi\)
\(200\) 0 0
\(201\) −204.000 −1.01493
\(202\) 0 0
\(203\) 207.846i 1.02387i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −144.000 −0.688995
\(210\) 0 0
\(211\) − 90.0666i − 0.426856i −0.976959 0.213428i \(-0.931537\pi\)
0.976959 0.213428i \(-0.0684629\pi\)
\(212\) 0 0
\(213\) −144.000 −0.676056
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 144.000 0.663594
\(218\) 0 0
\(219\) − 142.028i − 0.648530i
\(220\) 0 0
\(221\) 84.0000 0.380090
\(222\) 0 0
\(223\) − 353.338i − 1.58448i −0.610212 0.792238i \(-0.708916\pi\)
0.610212 0.792238i \(-0.291084\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 145.492i − 0.640935i −0.947259 0.320468i \(-0.896160\pi\)
0.947259 0.320468i \(-0.103840\pi\)
\(228\) 0 0
\(229\) 226.000 0.986900 0.493450 0.869774i \(-0.335736\pi\)
0.493450 + 0.869774i \(0.335736\pi\)
\(230\) 0 0
\(231\) − 249.415i − 1.07972i
\(232\) 0 0
\(233\) −114.000 −0.489270 −0.244635 0.969615i \(-0.578668\pi\)
−0.244635 + 0.969615i \(0.578668\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 132.000 0.556962
\(238\) 0 0
\(239\) 332.554i 1.39144i 0.718314 + 0.695719i \(0.244914\pi\)
−0.718314 + 0.695719i \(0.755086\pi\)
\(240\) 0 0
\(241\) 178.000 0.738589 0.369295 0.929312i \(-0.379599\pi\)
0.369295 + 0.929312i \(0.379599\pi\)
\(242\) 0 0
\(243\) 15.5885i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 96.9948i 0.392692i
\(248\) 0 0
\(249\) −36.0000 −0.144578
\(250\) 0 0
\(251\) 103.923i 0.414036i 0.978337 + 0.207018i \(0.0663759\pi\)
−0.978337 + 0.207018i \(0.933624\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −258.000 −1.00389 −0.501946 0.864899i \(-0.667382\pi\)
−0.501946 + 0.864899i \(0.667382\pi\)
\(258\) 0 0
\(259\) − 180.133i − 0.695495i
\(260\) 0 0
\(261\) −90.0000 −0.344828
\(262\) 0 0
\(263\) 374.123i 1.42252i 0.702929 + 0.711260i \(0.251875\pi\)
−0.702929 + 0.711260i \(0.748125\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 197.454i 0.739527i
\(268\) 0 0
\(269\) 510.000 1.89591 0.947955 0.318403i \(-0.103147\pi\)
0.947955 + 0.318403i \(0.103147\pi\)
\(270\) 0 0
\(271\) − 450.333i − 1.66175i −0.556462 0.830873i \(-0.687842\pi\)
0.556462 0.830873i \(-0.312158\pi\)
\(272\) 0 0
\(273\) −168.000 −0.615385
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 14.0000 0.0505415 0.0252708 0.999681i \(-0.491955\pi\)
0.0252708 + 0.999681i \(0.491955\pi\)
\(278\) 0 0
\(279\) 62.3538i 0.223490i
\(280\) 0 0
\(281\) 354.000 1.25979 0.629893 0.776682i \(-0.283099\pi\)
0.629893 + 0.776682i \(0.283099\pi\)
\(282\) 0 0
\(283\) − 145.492i − 0.514107i −0.966397 0.257053i \(-0.917248\pi\)
0.966397 0.257053i \(-0.0827517\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 374.123i − 1.30356i
\(288\) 0 0
\(289\) −253.000 −0.875433
\(290\) 0 0
\(291\) − 58.8897i − 0.202370i
\(292\) 0 0
\(293\) 498.000 1.69966 0.849829 0.527058i \(-0.176705\pi\)
0.849829 + 0.527058i \(0.176705\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 108.000 0.363636
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 144.000 0.478405
\(302\) 0 0
\(303\) − 31.1769i − 0.102894i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 187.061i − 0.609321i −0.952461 0.304660i \(-0.901457\pi\)
0.952461 0.304660i \(-0.0985430\pi\)
\(308\) 0 0
\(309\) −228.000 −0.737864
\(310\) 0 0
\(311\) − 41.5692i − 0.133663i −0.997764 0.0668315i \(-0.978711\pi\)
0.997764 0.0668315i \(-0.0212890\pi\)
\(312\) 0 0
\(313\) −290.000 −0.926518 −0.463259 0.886223i \(-0.653320\pi\)
−0.463259 + 0.886223i \(0.653320\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 210.000 0.662461 0.331230 0.943550i \(-0.392536\pi\)
0.331230 + 0.943550i \(0.392536\pi\)
\(318\) 0 0
\(319\) 623.538i 1.95467i
\(320\) 0 0
\(321\) 252.000 0.785047
\(322\) 0 0
\(323\) 41.5692i 0.128697i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 58.8897i 0.180091i
\(328\) 0 0
\(329\) −288.000 −0.875380
\(330\) 0 0
\(331\) 200.918i 0.607003i 0.952831 + 0.303501i \(0.0981557\pi\)
−0.952831 + 0.303501i \(0.901844\pi\)
\(332\) 0 0
\(333\) 78.0000 0.234234
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 302.000 0.896142 0.448071 0.893998i \(-0.352111\pi\)
0.448071 + 0.893998i \(0.352111\pi\)
\(338\) 0 0
\(339\) 135.100i 0.398525i
\(340\) 0 0
\(341\) 432.000 1.26686
\(342\) 0 0
\(343\) 346.410i 1.00994i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 62.3538i 0.179694i 0.995956 + 0.0898470i \(0.0286378\pi\)
−0.995956 + 0.0898470i \(0.971362\pi\)
\(348\) 0 0
\(349\) −358.000 −1.02579 −0.512894 0.858452i \(-0.671427\pi\)
−0.512894 + 0.858452i \(0.671427\pi\)
\(350\) 0 0
\(351\) − 72.7461i − 0.207254i
\(352\) 0 0
\(353\) 558.000 1.58074 0.790368 0.612632i \(-0.209889\pi\)
0.790368 + 0.612632i \(0.209889\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −72.0000 −0.201681
\(358\) 0 0
\(359\) − 83.1384i − 0.231583i −0.993274 0.115792i \(-0.963059\pi\)
0.993274 0.115792i \(-0.0369405\pi\)
\(360\) 0 0
\(361\) 313.000 0.867036
\(362\) 0 0
\(363\) − 538.668i − 1.48393i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 214.774i 0.585216i 0.956232 + 0.292608i \(0.0945231\pi\)
−0.956232 + 0.292608i \(0.905477\pi\)
\(368\) 0 0
\(369\) 162.000 0.439024
\(370\) 0 0
\(371\) 124.708i 0.336139i
\(372\) 0 0
\(373\) −554.000 −1.48525 −0.742627 0.669705i \(-0.766421\pi\)
−0.742627 + 0.669705i \(0.766421\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 420.000 1.11406
\(378\) 0 0
\(379\) − 533.472i − 1.40758i −0.710410 0.703788i \(-0.751490\pi\)
0.710410 0.703788i \(-0.248510\pi\)
\(380\) 0 0
\(381\) 180.000 0.472441
\(382\) 0 0
\(383\) 498.831i 1.30243i 0.758893 + 0.651215i \(0.225740\pi\)
−0.758893 + 0.651215i \(0.774260\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 62.3538i 0.161121i
\(388\) 0 0
\(389\) 198.000 0.508997 0.254499 0.967073i \(-0.418090\pi\)
0.254499 + 0.967073i \(0.418090\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −180.000 −0.458015
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 646.000 1.62720 0.813602 0.581422i \(-0.197504\pi\)
0.813602 + 0.581422i \(0.197504\pi\)
\(398\) 0 0
\(399\) − 83.1384i − 0.208367i
\(400\) 0 0
\(401\) 330.000 0.822943 0.411471 0.911423i \(-0.365015\pi\)
0.411471 + 0.911423i \(0.365015\pi\)
\(402\) 0 0
\(403\) − 290.985i − 0.722046i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 540.400i − 1.32776i
\(408\) 0 0
\(409\) 130.000 0.317848 0.158924 0.987291i \(-0.449197\pi\)
0.158924 + 0.987291i \(0.449197\pi\)
\(410\) 0 0
\(411\) − 322.161i − 0.783848i
\(412\) 0 0
\(413\) 144.000 0.348668
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 84.0000 0.201439
\(418\) 0 0
\(419\) − 353.338i − 0.843290i −0.906761 0.421645i \(-0.861453\pi\)
0.906761 0.421645i \(-0.138547\pi\)
\(420\) 0 0
\(421\) −398.000 −0.945368 −0.472684 0.881232i \(-0.656715\pi\)
−0.472684 + 0.881232i \(0.656715\pi\)
\(422\) 0 0
\(423\) − 124.708i − 0.294817i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 484.974i − 1.13577i
\(428\) 0 0
\(429\) −504.000 −1.17483
\(430\) 0 0
\(431\) − 124.708i − 0.289345i −0.989480 0.144672i \(-0.953787\pi\)
0.989480 0.144672i \(-0.0462128\pi\)
\(432\) 0 0
\(433\) 142.000 0.327945 0.163972 0.986465i \(-0.447569\pi\)
0.163972 + 0.986465i \(0.447569\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 561.184i 1.27832i 0.769072 + 0.639162i \(0.220719\pi\)
−0.769072 + 0.639162i \(0.779281\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.00680272
\(442\) 0 0
\(443\) − 436.477i − 0.985275i −0.870235 0.492637i \(-0.836033\pi\)
0.870235 0.492637i \(-0.163967\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 322.161i − 0.720719i
\(448\) 0 0
\(449\) −198.000 −0.440980 −0.220490 0.975389i \(-0.570766\pi\)
−0.220490 + 0.975389i \(0.570766\pi\)
\(450\) 0 0
\(451\) − 1122.37i − 2.48862i
\(452\) 0 0
\(453\) 60.0000 0.132450
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 446.000 0.975930 0.487965 0.872863i \(-0.337739\pi\)
0.487965 + 0.872863i \(0.337739\pi\)
\(458\) 0 0
\(459\) − 31.1769i − 0.0679236i
\(460\) 0 0
\(461\) 342.000 0.741866 0.370933 0.928660i \(-0.379038\pi\)
0.370933 + 0.928660i \(0.379038\pi\)
\(462\) 0 0
\(463\) − 159.349i − 0.344166i −0.985082 0.172083i \(-0.944950\pi\)
0.985082 0.172083i \(-0.0550497\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 394.908i − 0.845627i −0.906217 0.422813i \(-0.861043\pi\)
0.906217 0.422813i \(-0.138957\pi\)
\(468\) 0 0
\(469\) −816.000 −1.73987
\(470\) 0 0
\(471\) − 294.449i − 0.625156i
\(472\) 0 0
\(473\) 432.000 0.913319
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −54.0000 −0.113208
\(478\) 0 0
\(479\) 789.815i 1.64888i 0.565947 + 0.824442i \(0.308511\pi\)
−0.565947 + 0.824442i \(0.691489\pi\)
\(480\) 0 0
\(481\) −364.000 −0.756757
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 6.92820i 0.0142263i 0.999975 + 0.00711315i \(0.00226420\pi\)
−0.999975 + 0.00711315i \(0.997736\pi\)
\(488\) 0 0
\(489\) 492.000 1.00613
\(490\) 0 0
\(491\) 644.323i 1.31227i 0.754645 + 0.656133i \(0.227809\pi\)
−0.754645 + 0.656133i \(0.772191\pi\)
\(492\) 0 0
\(493\) 180.000 0.365112
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −576.000 −1.15895
\(498\) 0 0
\(499\) − 810.600i − 1.62445i −0.583345 0.812224i \(-0.698257\pi\)
0.583345 0.812224i \(-0.301743\pi\)
\(500\) 0 0
\(501\) −360.000 −0.718563
\(502\) 0 0
\(503\) 332.554i 0.661141i 0.943781 + 0.330570i \(0.107241\pi\)
−0.943781 + 0.330570i \(0.892759\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 46.7654i 0.0922394i
\(508\) 0 0
\(509\) −306.000 −0.601179 −0.300589 0.953754i \(-0.597183\pi\)
−0.300589 + 0.953754i \(0.597183\pi\)
\(510\) 0 0
\(511\) − 568.113i − 1.11177i
\(512\) 0 0
\(513\) 36.0000 0.0701754
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −864.000 −1.67118
\(518\) 0 0
\(519\) 72.7461i 0.140166i
\(520\) 0 0
\(521\) 522.000 1.00192 0.500960 0.865471i \(-0.332980\pi\)
0.500960 + 0.865471i \(0.332980\pi\)
\(522\) 0 0
\(523\) − 48.4974i − 0.0927293i −0.998925 0.0463646i \(-0.985236\pi\)
0.998925 0.0463646i \(-0.0147636\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 124.708i − 0.236637i
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 62.3538i 0.117427i
\(532\) 0 0
\(533\) −756.000 −1.41839
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 252.000 0.469274
\(538\) 0 0
\(539\) 20.7846i 0.0385614i
\(540\) 0 0
\(541\) 802.000 1.48244 0.741220 0.671262i \(-0.234248\pi\)
0.741220 + 0.671262i \(0.234248\pi\)
\(542\) 0 0
\(543\) 142.028i 0.261562i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 34.6410i − 0.0633291i −0.999499 0.0316645i \(-0.989919\pi\)
0.999499 0.0316645i \(-0.0100808\pi\)
\(548\) 0 0
\(549\) 210.000 0.382514
\(550\) 0 0
\(551\) 207.846i 0.377216i
\(552\) 0 0
\(553\) 528.000 0.954792
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 474.000 0.850987 0.425494 0.904961i \(-0.360100\pi\)
0.425494 + 0.904961i \(0.360100\pi\)
\(558\) 0 0
\(559\) − 290.985i − 0.520545i
\(560\) 0 0
\(561\) −216.000 −0.385027
\(562\) 0 0
\(563\) 685.892i 1.21828i 0.793062 + 0.609140i \(0.208485\pi\)
−0.793062 + 0.609140i \(0.791515\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 62.3538i 0.109971i
\(568\) 0 0
\(569\) −150.000 −0.263620 −0.131810 0.991275i \(-0.542079\pi\)
−0.131810 + 0.991275i \(0.542079\pi\)
\(570\) 0 0
\(571\) 672.036i 1.17695i 0.808517 + 0.588473i \(0.200271\pi\)
−0.808517 + 0.588473i \(0.799729\pi\)
\(572\) 0 0
\(573\) −576.000 −1.00524
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 46.0000 0.0797227 0.0398614 0.999205i \(-0.487308\pi\)
0.0398614 + 0.999205i \(0.487308\pi\)
\(578\) 0 0
\(579\) 162.813i 0.281197i
\(580\) 0 0
\(581\) −144.000 −0.247849
\(582\) 0 0
\(583\) 374.123i 0.641720i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 353.338i 0.601939i 0.953634 + 0.300970i \(0.0973103\pi\)
−0.953634 + 0.300970i \(0.902690\pi\)
\(588\) 0 0
\(589\) 144.000 0.244482
\(590\) 0 0
\(591\) 446.869i 0.756124i
\(592\) 0 0
\(593\) −114.000 −0.192243 −0.0961214 0.995370i \(-0.530644\pi\)
−0.0961214 + 0.995370i \(0.530644\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −204.000 −0.341709
\(598\) 0 0
\(599\) − 249.415i − 0.416386i −0.978088 0.208193i \(-0.933242\pi\)
0.978088 0.208193i \(-0.0667582\pi\)
\(600\) 0 0
\(601\) 626.000 1.04160 0.520799 0.853680i \(-0.325634\pi\)
0.520799 + 0.853680i \(0.325634\pi\)
\(602\) 0 0
\(603\) − 353.338i − 0.585967i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 672.036i 1.10714i 0.832802 + 0.553571i \(0.186735\pi\)
−0.832802 + 0.553571i \(0.813265\pi\)
\(608\) 0 0
\(609\) −360.000 −0.591133
\(610\) 0 0
\(611\) 581.969i 0.952486i
\(612\) 0 0
\(613\) 694.000 1.13214 0.566069 0.824358i \(-0.308464\pi\)
0.566069 + 0.824358i \(0.308464\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 0.0486224 0.0243112 0.999704i \(-0.492261\pi\)
0.0243112 + 0.999704i \(0.492261\pi\)
\(618\) 0 0
\(619\) 339.482i 0.548436i 0.961668 + 0.274218i \(0.0884190\pi\)
−0.961668 + 0.274218i \(0.911581\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 789.815i 1.26776i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 249.415i − 0.397792i
\(628\) 0 0
\(629\) −156.000 −0.248013
\(630\) 0 0
\(631\) 464.190i 0.735641i 0.929897 + 0.367821i \(0.119896\pi\)
−0.929897 + 0.367821i \(0.880104\pi\)
\(632\) 0 0
\(633\) 156.000 0.246445
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 14.0000 0.0219780
\(638\) 0 0
\(639\) − 249.415i − 0.390321i
\(640\) 0 0
\(641\) −390.000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(642\) 0 0
\(643\) − 810.600i − 1.26065i −0.776330 0.630326i \(-0.782921\pi\)
0.776330 0.630326i \(-0.217079\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 581.969i 0.899489i 0.893157 + 0.449744i \(0.148485\pi\)
−0.893157 + 0.449744i \(0.851515\pi\)
\(648\) 0 0
\(649\) 432.000 0.665639
\(650\) 0 0
\(651\) 249.415i 0.383126i
\(652\) 0 0
\(653\) −774.000 −1.18530 −0.592649 0.805461i \(-0.701918\pi\)
−0.592649 + 0.805461i \(0.701918\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 246.000 0.374429
\(658\) 0 0
\(659\) − 228.631i − 0.346936i −0.984840 0.173468i \(-0.944503\pi\)
0.984840 0.173468i \(-0.0554973\pi\)
\(660\) 0 0
\(661\) −454.000 −0.686838 −0.343419 0.939182i \(-0.611585\pi\)
−0.343419 + 0.939182i \(0.611585\pi\)
\(662\) 0 0
\(663\) 145.492i 0.219445i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 612.000 0.914798
\(670\) 0 0
\(671\) − 1454.92i − 2.16829i
\(672\) 0 0
\(673\) −434.000 −0.644874 −0.322437 0.946591i \(-0.604502\pi\)
−0.322437 + 0.946591i \(0.604502\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 234.000 0.345643 0.172821 0.984953i \(-0.444712\pi\)
0.172821 + 0.984953i \(0.444712\pi\)
\(678\) 0 0
\(679\) − 235.559i − 0.346920i
\(680\) 0 0
\(681\) 252.000 0.370044
\(682\) 0 0
\(683\) − 270.200i − 0.395608i −0.980242 0.197804i \(-0.936619\pi\)
0.980242 0.197804i \(-0.0633809\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 391.443i 0.569787i
\(688\) 0 0
\(689\) 252.000 0.365747
\(690\) 0 0
\(691\) − 20.7846i − 0.0300790i −0.999887 0.0150395i \(-0.995213\pi\)
0.999887 0.0150395i \(-0.00478741\pi\)
\(692\) 0 0
\(693\) 432.000 0.623377
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −324.000 −0.464849
\(698\) 0 0
\(699\) − 197.454i − 0.282480i
\(700\) 0 0
\(701\) −1074.00 −1.53210 −0.766049 0.642783i \(-0.777780\pi\)
−0.766049 + 0.642783i \(0.777780\pi\)
\(702\) 0 0
\(703\) − 180.133i − 0.256235i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 124.708i − 0.176390i
\(708\) 0 0
\(709\) 898.000 1.26657 0.633286 0.773918i \(-0.281706\pi\)
0.633286 + 0.773918i \(0.281706\pi\)
\(710\) 0 0
\(711\) 228.631i 0.321562i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −576.000 −0.803347
\(718\) 0 0
\(719\) 956.092i 1.32975i 0.746954 + 0.664876i \(0.231516\pi\)
−0.746954 + 0.664876i \(0.768484\pi\)
\(720\) 0 0
\(721\) −912.000 −1.26491
\(722\) 0 0
\(723\) 308.305i 0.426425i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 810.600i 1.11499i 0.830179 + 0.557496i \(0.188238\pi\)
−0.830179 + 0.557496i \(0.811762\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) − 124.708i − 0.170599i
\(732\) 0 0
\(733\) −370.000 −0.504775 −0.252387 0.967626i \(-0.581216\pi\)
−0.252387 + 0.967626i \(0.581216\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2448.00 −3.32157
\(738\) 0 0
\(739\) 852.169i 1.15314i 0.817048 + 0.576569i \(0.195609\pi\)
−0.817048 + 0.576569i \(0.804391\pi\)
\(740\) 0 0
\(741\) −168.000 −0.226721
\(742\) 0 0
\(743\) − 1371.78i − 1.84628i −0.384467 0.923139i \(-0.625615\pi\)
0.384467 0.923139i \(-0.374385\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 62.3538i − 0.0834723i
\(748\) 0 0
\(749\) 1008.00 1.34579
\(750\) 0 0
\(751\) − 76.2102i − 0.101478i −0.998712 0.0507392i \(-0.983842\pi\)
0.998712 0.0507392i \(-0.0161577\pi\)
\(752\) 0 0
\(753\) −180.000 −0.239044
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −514.000 −0.678996 −0.339498 0.940607i \(-0.610257\pi\)
−0.339498 + 0.940607i \(0.610257\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −966.000 −1.26938 −0.634691 0.772766i \(-0.718873\pi\)
−0.634691 + 0.772766i \(0.718873\pi\)
\(762\) 0 0
\(763\) 235.559i 0.308727i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 290.985i − 0.379380i
\(768\) 0 0
\(769\) −958.000 −1.24577 −0.622887 0.782312i \(-0.714040\pi\)
−0.622887 + 0.782312i \(0.714040\pi\)
\(770\) 0 0
\(771\) − 446.869i − 0.579597i
\(772\) 0 0
\(773\) 546.000 0.706339 0.353169 0.935559i \(-0.385104\pi\)
0.353169 + 0.935559i \(0.385104\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 312.000 0.401544
\(778\) 0 0
\(779\) − 374.123i − 0.480261i
\(780\) 0 0
\(781\) −1728.00 −2.21255
\(782\) 0 0
\(783\) − 155.885i − 0.199086i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 242.487i 0.308116i 0.988062 + 0.154058i \(0.0492342\pi\)
−0.988062 + 0.154058i \(0.950766\pi\)
\(788\) 0 0
\(789\) −648.000 −0.821293
\(790\) 0 0
\(791\) 540.400i 0.683186i
\(792\) 0 0
\(793\) −980.000 −1.23581
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1338.00 1.67880 0.839398 0.543518i \(-0.182908\pi\)
0.839398 + 0.543518i \(0.182908\pi\)
\(798\) 0 0
\(799\) 249.415i 0.312159i
\(800\) 0 0
\(801\) −342.000 −0.426966
\(802\) 0 0
\(803\) − 1704.34i − 2.12246i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 883.346i 1.09460i
\(808\) 0 0
\(809\) −966.000 −1.19407 −0.597033 0.802216i \(-0.703654\pi\)
−0.597033 + 0.802216i \(0.703654\pi\)
\(810\) 0 0
\(811\) 1517.28i 1.87087i 0.353497 + 0.935436i \(0.384992\pi\)
−0.353497 + 0.935436i \(0.615008\pi\)
\(812\) 0 0
\(813\) 780.000 0.959410
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 144.000 0.176255
\(818\) 0 0
\(819\) − 290.985i − 0.355292i
\(820\) 0 0
\(821\) 222.000 0.270402 0.135201 0.990818i \(-0.456832\pi\)
0.135201 + 0.990818i \(0.456832\pi\)
\(822\) 0 0
\(823\) 1281.72i 1.55737i 0.627413 + 0.778686i \(0.284114\pi\)
−0.627413 + 0.778686i \(0.715886\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1434.14i 1.73415i 0.498182 + 0.867073i \(0.334001\pi\)
−0.498182 + 0.867073i \(0.665999\pi\)
\(828\) 0 0
\(829\) 226.000 0.272618 0.136309 0.990666i \(-0.456476\pi\)
0.136309 + 0.990666i \(0.456476\pi\)
\(830\) 0 0
\(831\) 24.2487i 0.0291802i
\(832\) 0 0
\(833\) 6.00000 0.00720288
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −108.000 −0.129032
\(838\) 0 0
\(839\) − 498.831i − 0.594554i −0.954791 0.297277i \(-0.903922\pi\)
0.954791 0.297277i \(-0.0960784\pi\)
\(840\) 0 0
\(841\) 59.0000 0.0701546
\(842\) 0 0
\(843\) 613.146i 0.727338i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 2154.67i − 2.54389i
\(848\) 0 0
\(849\) 252.000 0.296820
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 70.0000 0.0820633 0.0410317 0.999158i \(-0.486936\pi\)
0.0410317 + 0.999158i \(0.486936\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −42.0000 −0.0490082 −0.0245041 0.999700i \(-0.507801\pi\)
−0.0245041 + 0.999700i \(0.507801\pi\)
\(858\) 0 0
\(859\) − 921.451i − 1.07270i −0.843995 0.536351i \(-0.819802\pi\)
0.843995 0.536351i \(-0.180198\pi\)
\(860\) 0 0
\(861\) 648.000 0.752613
\(862\) 0 0
\(863\) − 166.277i − 0.192673i −0.995349 0.0963365i \(-0.969287\pi\)
0.995349 0.0963365i \(-0.0307125\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 438.209i − 0.505431i
\(868\) 0 0
\(869\) 1584.00 1.82278
\(870\) 0 0
\(871\) 1648.91i 1.89313i
\(872\) 0 0
\(873\) 102.000 0.116838
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 166.000 0.189282 0.0946408 0.995511i \(-0.469830\pi\)
0.0946408 + 0.995511i \(0.469830\pi\)
\(878\) 0 0
\(879\) 862.561i 0.981298i
\(880\) 0 0
\(881\) −702.000 −0.796822 −0.398411 0.917207i \(-0.630438\pi\)
−0.398411 + 0.917207i \(0.630438\pi\)
\(882\) 0 0
\(883\) 630.466i 0.714005i 0.934103 + 0.357003i \(0.116201\pi\)
−0.934103 + 0.357003i \(0.883799\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1288.65i 1.45281i 0.687265 + 0.726407i \(0.258811\pi\)
−0.687265 + 0.726407i \(0.741189\pi\)
\(888\) 0 0
\(889\) 720.000 0.809899
\(890\) 0 0
\(891\) 187.061i 0.209946i
\(892\) 0 0
\(893\) −288.000 −0.322508
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 623.538i − 0.693591i
\(900\) 0 0
\(901\) 108.000 0.119867
\(902\) 0 0
\(903\) 249.415i 0.276207i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1447.99i 1.59647i 0.602349 + 0.798233i \(0.294232\pi\)
−0.602349 + 0.798233i \(0.705768\pi\)
\(908\) 0 0
\(909\) 54.0000 0.0594059
\(910\) 0 0
\(911\) 332.554i 0.365043i 0.983202 + 0.182521i \(0.0584258\pi\)
−0.983202 + 0.182521i \(0.941574\pi\)
\(912\) 0 0
\(913\) −432.000 −0.473165
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −720.000 −0.785169
\(918\) 0 0
\(919\) − 561.184i − 0.610647i −0.952249 0.305323i \(-0.901235\pi\)
0.952249 0.305323i \(-0.0987646\pi\)
\(920\) 0 0
\(921\) 324.000 0.351792
\(922\) 0 0
\(923\) 1163.94i 1.26104i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 394.908i − 0.426006i
\(928\) 0 0
\(929\) −438.000 −0.471475 −0.235737 0.971817i \(-0.575751\pi\)
−0.235737 + 0.971817i \(0.575751\pi\)
\(930\) 0 0
\(931\) 6.92820i 0.00744168i
\(932\) 0 0
\(933\) 72.0000 0.0771704
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1826.00 −1.94877 −0.974386 0.224881i \(-0.927801\pi\)
−0.974386 + 0.224881i \(0.927801\pi\)
\(938\) 0 0
\(939\) − 502.295i − 0.534925i
\(940\) 0 0
\(941\) −330.000 −0.350691 −0.175345 0.984507i \(-0.556104\pi\)
−0.175345 + 0.984507i \(0.556104\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 187.061i − 0.197531i −0.995111 0.0987653i \(-0.968511\pi\)
0.995111 0.0987653i \(-0.0314893\pi\)
\(948\) 0 0
\(949\) −1148.00 −1.20969
\(950\) 0 0
\(951\) 363.731i 0.382472i
\(952\) 0 0
\(953\) 1110.00 1.16474 0.582371 0.812923i \(-0.302125\pi\)
0.582371 + 0.812923i \(0.302125\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −1080.00 −1.12853
\(958\) 0 0
\(959\) − 1288.65i − 1.34374i
\(960\) 0 0
\(961\) 529.000 0.550468
\(962\) 0 0
\(963\) 436.477i 0.453247i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 755.174i − 0.780945i −0.920615 0.390473i \(-0.872312\pi\)
0.920615 0.390473i \(-0.127688\pi\)
\(968\) 0 0
\(969\) −72.0000 −0.0743034
\(970\) 0 0
\(971\) − 394.908i − 0.406702i −0.979106 0.203351i \(-0.934817\pi\)
0.979106 0.203351i \(-0.0651832\pi\)
\(972\) 0 0
\(973\) 336.000 0.345324
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 918.000 0.939611 0.469806 0.882770i \(-0.344324\pi\)
0.469806 + 0.882770i \(0.344324\pi\)
\(978\) 0 0
\(979\) 2369.45i 2.42027i
\(980\) 0 0
\(981\) −102.000 −0.103976
\(982\) 0 0
\(983\) − 41.5692i − 0.0422881i −0.999776 0.0211441i \(-0.993269\pi\)
0.999776 0.0211441i \(-0.00673086\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 498.831i − 0.505401i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) − 48.4974i − 0.0489379i −0.999701 0.0244689i \(-0.992211\pi\)
0.999701 0.0244689i \(-0.00778948\pi\)
\(992\) 0 0
\(993\) −348.000 −0.350453
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −554.000 −0.555667 −0.277834 0.960629i \(-0.589616\pi\)
−0.277834 + 0.960629i \(0.589616\pi\)
\(998\) 0 0
\(999\) 135.100i 0.135235i
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 1200.3.e.h.751.2 2
3.2 odd 2 3600.3.e.t.3151.2 2
4.3 odd 2 inner 1200.3.e.h.751.1 2
5.2 odd 4 1200.3.j.a.799.4 4
5.3 odd 4 1200.3.j.a.799.2 4
5.4 even 2 48.3.g.a.31.1 2
12.11 even 2 3600.3.e.t.3151.1 2
15.2 even 4 3600.3.j.i.1999.1 4
15.8 even 4 3600.3.j.i.1999.3 4
15.14 odd 2 144.3.g.b.127.1 2
20.3 even 4 1200.3.j.a.799.3 4
20.7 even 4 1200.3.j.a.799.1 4
20.19 odd 2 48.3.g.a.31.2 yes 2
35.34 odd 2 2352.3.m.a.1471.2 2
40.19 odd 2 192.3.g.a.127.1 2
40.29 even 2 192.3.g.a.127.2 2
45.4 even 6 1296.3.o.c.703.1 2
45.14 odd 6 1296.3.o.p.703.1 2
45.29 odd 6 1296.3.o.n.271.1 2
45.34 even 6 1296.3.o.a.271.1 2
60.23 odd 4 3600.3.j.i.1999.2 4
60.47 odd 4 3600.3.j.i.1999.4 4
60.59 even 2 144.3.g.b.127.2 2
80.19 odd 4 768.3.b.b.127.1 4
80.29 even 4 768.3.b.b.127.3 4
80.59 odd 4 768.3.b.b.127.4 4
80.69 even 4 768.3.b.b.127.2 4
120.29 odd 2 576.3.g.i.127.1 2
120.59 even 2 576.3.g.i.127.2 2
140.139 even 2 2352.3.m.a.1471.1 2
180.59 even 6 1296.3.o.n.703.1 2
180.79 odd 6 1296.3.o.c.271.1 2
180.119 even 6 1296.3.o.p.271.1 2
180.139 odd 6 1296.3.o.a.703.1 2
240.29 odd 4 2304.3.b.n.127.4 4
240.59 even 4 2304.3.b.n.127.1 4
240.149 odd 4 2304.3.b.n.127.2 4
240.179 even 4 2304.3.b.n.127.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.3.g.a.31.1 2 5.4 even 2
48.3.g.a.31.2 yes 2 20.19 odd 2
144.3.g.b.127.1 2 15.14 odd 2
144.3.g.b.127.2 2 60.59 even 2
192.3.g.a.127.1 2 40.19 odd 2
192.3.g.a.127.2 2 40.29 even 2
576.3.g.i.127.1 2 120.29 odd 2
576.3.g.i.127.2 2 120.59 even 2
768.3.b.b.127.1 4 80.19 odd 4
768.3.b.b.127.2 4 80.69 even 4
768.3.b.b.127.3 4 80.29 even 4
768.3.b.b.127.4 4 80.59 odd 4
1200.3.e.h.751.1 2 4.3 odd 2 inner
1200.3.e.h.751.2 2 1.1 even 1 trivial
1200.3.j.a.799.1 4 20.7 even 4
1200.3.j.a.799.2 4 5.3 odd 4
1200.3.j.a.799.3 4 20.3 even 4
1200.3.j.a.799.4 4 5.2 odd 4
1296.3.o.a.271.1 2 45.34 even 6
1296.3.o.a.703.1 2 180.139 odd 6
1296.3.o.c.271.1 2 180.79 odd 6
1296.3.o.c.703.1 2 45.4 even 6
1296.3.o.n.271.1 2 45.29 odd 6
1296.3.o.n.703.1 2 180.59 even 6
1296.3.o.p.271.1 2 180.119 even 6
1296.3.o.p.703.1 2 45.14 odd 6
2304.3.b.n.127.1 4 240.59 even 4
2304.3.b.n.127.2 4 240.149 odd 4
2304.3.b.n.127.3 4 240.179 even 4
2304.3.b.n.127.4 4 240.29 odd 4
2352.3.m.a.1471.1 2 140.139 even 2
2352.3.m.a.1471.2 2 35.34 odd 2
3600.3.e.t.3151.1 2 12.11 even 2
3600.3.e.t.3151.2 2 3.2 odd 2
3600.3.j.i.1999.1 4 15.2 even 4
3600.3.j.i.1999.2 4 60.23 odd 4
3600.3.j.i.1999.3 4 15.8 even 4
3600.3.j.i.1999.4 4 60.47 odd 4