Properties

Label 1323.2.a.u.1.1
Level $1323$
Weight $2$
Character 1323.1
Self dual yes
Analytic conductor $10.564$
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] = [1323,2,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1323.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.44949 q^{2} +4.00000 q^{4} +2.44949 q^{5} -4.89898 q^{8} +O(q^{10})\) \(q-2.44949 q^{2} +4.00000 q^{4} +2.44949 q^{5} -4.89898 q^{8} -6.00000 q^{10} -4.89898 q^{11} +4.00000 q^{13} +4.00000 q^{16} -2.44949 q^{17} +1.00000 q^{19} +9.79796 q^{20} +12.0000 q^{22} -2.44949 q^{23} +1.00000 q^{25} -9.79796 q^{26} +7.34847 q^{29} +7.00000 q^{31} +6.00000 q^{34} +8.00000 q^{37} -2.44949 q^{38} -12.0000 q^{40} +7.34847 q^{41} -1.00000 q^{43} -19.5959 q^{44} +6.00000 q^{46} +2.44949 q^{47} -2.44949 q^{50} +16.0000 q^{52} +2.44949 q^{53} -12.0000 q^{55} -18.0000 q^{58} -9.79796 q^{59} -5.00000 q^{61} -17.1464 q^{62} -8.00000 q^{64} +9.79796 q^{65} +2.00000 q^{67} -9.79796 q^{68} +1.00000 q^{73} -19.5959 q^{74} +4.00000 q^{76} -4.00000 q^{79} +9.79796 q^{80} -18.0000 q^{82} +14.6969 q^{83} -6.00000 q^{85} +2.44949 q^{86} +24.0000 q^{88} +2.44949 q^{89} -9.79796 q^{92} -6.00000 q^{94} +2.44949 q^{95} +1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{4} - 12 q^{10} + 8 q^{13} + 8 q^{16} + 2 q^{19} + 24 q^{22} + 2 q^{25} + 14 q^{31} + 12 q^{34} + 16 q^{37} - 24 q^{40} - 2 q^{43} + 12 q^{46} + 32 q^{52} - 24 q^{55} - 36 q^{58} - 10 q^{61} - 16 q^{64} + 4 q^{67} + 2 q^{73} + 8 q^{76} - 8 q^{79} - 36 q^{82} - 12 q^{85} + 48 q^{88} - 12 q^{94} + 2 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.44949 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 0 0
\(4\) 4.00000 2.00000
\(5\) 2.44949 1.09545 0.547723 0.836660i \(-0.315495\pi\)
0.547723 + 0.836660i \(0.315495\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −4.89898 −1.73205
\(9\) 0 0
\(10\) −6.00000 −1.89737
\(11\) −4.89898 −1.47710 −0.738549 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) −2.44949 −0.594089 −0.297044 0.954864i \(-0.596001\pi\)
−0.297044 + 0.954864i \(0.596001\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 9.79796 2.19089
\(21\) 0 0
\(22\) 12.0000 2.55841
\(23\) −2.44949 −0.510754 −0.255377 0.966842i \(-0.582200\pi\)
−0.255377 + 0.966842i \(0.582200\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −9.79796 −1.92154
\(27\) 0 0
\(28\) 0 0
\(29\) 7.34847 1.36458 0.682288 0.731083i \(-0.260985\pi\)
0.682288 + 0.731083i \(0.260985\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −2.44949 −0.397360
\(39\) 0 0
\(40\) −12.0000 −1.89737
\(41\) 7.34847 1.14764 0.573819 0.818982i \(-0.305461\pi\)
0.573819 + 0.818982i \(0.305461\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −19.5959 −2.95420
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 2.44949 0.357295 0.178647 0.983913i \(-0.442828\pi\)
0.178647 + 0.983913i \(0.442828\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2.44949 −0.346410
\(51\) 0 0
\(52\) 16.0000 2.21880
\(53\) 2.44949 0.336463 0.168232 0.985747i \(-0.446194\pi\)
0.168232 + 0.985747i \(0.446194\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) −18.0000 −2.36352
\(59\) −9.79796 −1.27559 −0.637793 0.770208i \(-0.720152\pi\)
−0.637793 + 0.770208i \(0.720152\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) −17.1464 −2.17760
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 9.79796 1.21529
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −9.79796 −1.18818
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) −19.5959 −2.27798
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 9.79796 1.09545
\(81\) 0 0
\(82\) −18.0000 −1.98777
\(83\) 14.6969 1.61320 0.806599 0.591099i \(-0.201306\pi\)
0.806599 + 0.591099i \(0.201306\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 2.44949 0.264135
\(87\) 0 0
\(88\) 24.0000 2.55841
\(89\) 2.44949 0.259645 0.129823 0.991537i \(-0.458559\pi\)
0.129823 + 0.991537i \(0.458559\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −9.79796 −1.02151
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 2.44949 0.251312
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) 4.89898 0.487467 0.243733 0.969842i \(-0.421628\pi\)
0.243733 + 0.969842i \(0.421628\pi\)
\(102\) 0 0
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) −19.5959 −1.92154
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 19.5959 1.89441 0.947204 0.320630i \(-0.103895\pi\)
0.947204 + 0.320630i \(0.103895\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 29.3939 2.80260
\(111\) 0 0
\(112\) 0 0
\(113\) 14.6969 1.38257 0.691286 0.722581i \(-0.257045\pi\)
0.691286 + 0.722581i \(0.257045\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) 29.3939 2.72915
\(117\) 0 0
\(118\) 24.0000 2.20938
\(119\) 0 0
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 12.2474 1.10883
\(123\) 0 0
\(124\) 28.0000 2.51447
\(125\) −9.79796 −0.876356
\(126\) 0 0
\(127\) 11.0000 0.976092 0.488046 0.872818i \(-0.337710\pi\)
0.488046 + 0.872818i \(0.337710\pi\)
\(128\) 19.5959 1.73205
\(129\) 0 0
\(130\) −24.0000 −2.10494
\(131\) 2.44949 0.214013 0.107006 0.994258i \(-0.465873\pi\)
0.107006 + 0.994258i \(0.465873\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −4.89898 −0.423207
\(135\) 0 0
\(136\) 12.0000 1.02899
\(137\) 17.1464 1.46492 0.732459 0.680811i \(-0.238373\pi\)
0.732459 + 0.680811i \(0.238373\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −19.5959 −1.63869
\(144\) 0 0
\(145\) 18.0000 1.49482
\(146\) −2.44949 −0.202721
\(147\) 0 0
\(148\) 32.0000 2.63038
\(149\) −2.44949 −0.200670 −0.100335 0.994954i \(-0.531991\pi\)
−0.100335 + 0.994954i \(0.531991\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) −4.89898 −0.397360
\(153\) 0 0
\(154\) 0 0
\(155\) 17.1464 1.37723
\(156\) 0 0
\(157\) −20.0000 −1.59617 −0.798087 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) 9.79796 0.779484
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) 29.3939 2.29528
\(165\) 0 0
\(166\) −36.0000 −2.79414
\(167\) −14.6969 −1.13728 −0.568642 0.822585i \(-0.692531\pi\)
−0.568642 + 0.822585i \(0.692531\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 14.6969 1.12720
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 2.44949 0.186231 0.0931156 0.995655i \(-0.470317\pi\)
0.0931156 + 0.995655i \(0.470317\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −19.5959 −1.47710
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 2.44949 0.183083 0.0915417 0.995801i \(-0.470821\pi\)
0.0915417 + 0.995801i \(0.470821\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 12.0000 0.884652
\(185\) 19.5959 1.44072
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 9.79796 0.714590
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) −2.44949 −0.177239 −0.0886194 0.996066i \(-0.528245\pi\)
−0.0886194 + 0.996066i \(0.528245\pi\)
\(192\) 0 0
\(193\) 20.0000 1.43963 0.719816 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) −2.44949 −0.175863
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 19.0000 1.34687 0.673437 0.739244i \(-0.264817\pi\)
0.673437 + 0.739244i \(0.264817\pi\)
\(200\) −4.89898 −0.346410
\(201\) 0 0
\(202\) −12.0000 −0.844317
\(203\) 0 0
\(204\) 0 0
\(205\) 18.0000 1.25717
\(206\) 4.89898 0.341328
\(207\) 0 0
\(208\) 16.0000 1.10940
\(209\) −4.89898 −0.338869
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) 9.79796 0.672927
\(213\) 0 0
\(214\) −48.0000 −3.28121
\(215\) −2.44949 −0.167054
\(216\) 0 0
\(217\) 0 0
\(218\) 2.44949 0.165900
\(219\) 0 0
\(220\) −48.0000 −3.23616
\(221\) −9.79796 −0.659082
\(222\) 0 0
\(223\) −26.0000 −1.74109 −0.870544 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −36.0000 −2.39468
\(227\) −17.1464 −1.13805 −0.569024 0.822321i \(-0.692679\pi\)
−0.569024 + 0.822321i \(0.692679\pi\)
\(228\) 0 0
\(229\) 7.00000 0.462573 0.231287 0.972886i \(-0.425707\pi\)
0.231287 + 0.972886i \(0.425707\pi\)
\(230\) 14.6969 0.969087
\(231\) 0 0
\(232\) −36.0000 −2.36352
\(233\) −2.44949 −0.160471 −0.0802357 0.996776i \(-0.525567\pi\)
−0.0802357 + 0.996776i \(0.525567\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) −39.1918 −2.55117
\(237\) 0 0
\(238\) 0 0
\(239\) −7.34847 −0.475333 −0.237666 0.971347i \(-0.576383\pi\)
−0.237666 + 0.971347i \(0.576383\pi\)
\(240\) 0 0
\(241\) 13.0000 0.837404 0.418702 0.908124i \(-0.362485\pi\)
0.418702 + 0.908124i \(0.362485\pi\)
\(242\) −31.8434 −2.04697
\(243\) 0 0
\(244\) −20.0000 −1.28037
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) −34.2929 −2.17760
\(249\) 0 0
\(250\) 24.0000 1.51789
\(251\) −22.0454 −1.39149 −0.695747 0.718287i \(-0.744926\pi\)
−0.695747 + 0.718287i \(0.744926\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) −26.9444 −1.69064
\(255\) 0 0
\(256\) −32.0000 −2.00000
\(257\) 24.4949 1.52795 0.763975 0.645246i \(-0.223245\pi\)
0.763975 + 0.645246i \(0.223245\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 39.1918 2.43057
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) −4.89898 −0.302084 −0.151042 0.988527i \(-0.548263\pi\)
−0.151042 + 0.988527i \(0.548263\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) −24.4949 −1.49348 −0.746740 0.665116i \(-0.768382\pi\)
−0.746740 + 0.665116i \(0.768382\pi\)
\(270\) 0 0
\(271\) 1.00000 0.0607457 0.0303728 0.999539i \(-0.490331\pi\)
0.0303728 + 0.999539i \(0.490331\pi\)
\(272\) −9.79796 −0.594089
\(273\) 0 0
\(274\) −42.0000 −2.53731
\(275\) −4.89898 −0.295420
\(276\) 0 0
\(277\) 23.0000 1.38194 0.690968 0.722885i \(-0.257185\pi\)
0.690968 + 0.722885i \(0.257185\pi\)
\(278\) 34.2929 2.05675
\(279\) 0 0
\(280\) 0 0
\(281\) −14.6969 −0.876746 −0.438373 0.898793i \(-0.644445\pi\)
−0.438373 + 0.898793i \(0.644445\pi\)
\(282\) 0 0
\(283\) 25.0000 1.48610 0.743048 0.669238i \(-0.233379\pi\)
0.743048 + 0.669238i \(0.233379\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 48.0000 2.83830
\(287\) 0 0
\(288\) 0 0
\(289\) −11.0000 −0.647059
\(290\) −44.0908 −2.58910
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) 29.3939 1.71721 0.858604 0.512639i \(-0.171332\pi\)
0.858604 + 0.512639i \(0.171332\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) −39.1918 −2.27798
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −9.79796 −0.566631
\(300\) 0 0
\(301\) 0 0
\(302\) −12.2474 −0.704761
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) −12.2474 −0.701287
\(306\) 0 0
\(307\) 25.0000 1.42683 0.713413 0.700744i \(-0.247149\pi\)
0.713413 + 0.700744i \(0.247149\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −42.0000 −2.38544
\(311\) 34.2929 1.94457 0.972285 0.233800i \(-0.0751161\pi\)
0.972285 + 0.233800i \(0.0751161\pi\)
\(312\) 0 0
\(313\) −23.0000 −1.30004 −0.650018 0.759918i \(-0.725239\pi\)
−0.650018 + 0.759918i \(0.725239\pi\)
\(314\) 48.9898 2.76465
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) −24.4949 −1.37577 −0.687885 0.725819i \(-0.741461\pi\)
−0.687885 + 0.725819i \(0.741461\pi\)
\(318\) 0 0
\(319\) −36.0000 −2.01561
\(320\) −19.5959 −1.09545
\(321\) 0 0
\(322\) 0 0
\(323\) −2.44949 −0.136293
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 2.44949 0.135665
\(327\) 0 0
\(328\) −36.0000 −1.98777
\(329\) 0 0
\(330\) 0 0
\(331\) −13.0000 −0.714545 −0.357272 0.934000i \(-0.616293\pi\)
−0.357272 + 0.934000i \(0.616293\pi\)
\(332\) 58.7878 3.22640
\(333\) 0 0
\(334\) 36.0000 1.96983
\(335\) 4.89898 0.267660
\(336\) 0 0
\(337\) 5.00000 0.272367 0.136184 0.990684i \(-0.456516\pi\)
0.136184 + 0.990684i \(0.456516\pi\)
\(338\) −7.34847 −0.399704
\(339\) 0 0
\(340\) −24.0000 −1.30158
\(341\) −34.2929 −1.85706
\(342\) 0 0
\(343\) 0 0
\(344\) 4.89898 0.264135
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 9.79796 0.525982 0.262991 0.964798i \(-0.415291\pi\)
0.262991 + 0.964798i \(0.415291\pi\)
\(348\) 0 0
\(349\) −35.0000 −1.87351 −0.936754 0.349990i \(-0.886185\pi\)
−0.936754 + 0.349990i \(0.886185\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.89898 0.260746 0.130373 0.991465i \(-0.458382\pi\)
0.130373 + 0.991465i \(0.458382\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 9.79796 0.519291
\(357\) 0 0
\(358\) −6.00000 −0.317110
\(359\) 19.5959 1.03423 0.517116 0.855915i \(-0.327005\pi\)
0.517116 + 0.855915i \(0.327005\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −17.1464 −0.901196
\(363\) 0 0
\(364\) 0 0
\(365\) 2.44949 0.128212
\(366\) 0 0
\(367\) 13.0000 0.678594 0.339297 0.940679i \(-0.389811\pi\)
0.339297 + 0.940679i \(0.389811\pi\)
\(368\) −9.79796 −0.510754
\(369\) 0 0
\(370\) −48.0000 −2.49540
\(371\) 0 0
\(372\) 0 0
\(373\) 11.0000 0.569558 0.284779 0.958593i \(-0.408080\pi\)
0.284779 + 0.958593i \(0.408080\pi\)
\(374\) −29.3939 −1.51992
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 29.3939 1.51386
\(378\) 0 0
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) 9.79796 0.502625
\(381\) 0 0
\(382\) 6.00000 0.306987
\(383\) −19.5959 −1.00130 −0.500652 0.865648i \(-0.666906\pi\)
−0.500652 + 0.865648i \(0.666906\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −48.9898 −2.49351
\(387\) 0 0
\(388\) 4.00000 0.203069
\(389\) 17.1464 0.869358 0.434679 0.900585i \(-0.356862\pi\)
0.434679 + 0.900585i \(0.356862\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.79796 −0.492989
\(396\) 0 0
\(397\) −17.0000 −0.853206 −0.426603 0.904439i \(-0.640290\pi\)
−0.426603 + 0.904439i \(0.640290\pi\)
\(398\) −46.5403 −2.33285
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −24.4949 −1.22322 −0.611608 0.791161i \(-0.709477\pi\)
−0.611608 + 0.791161i \(0.709477\pi\)
\(402\) 0 0
\(403\) 28.0000 1.39478
\(404\) 19.5959 0.974933
\(405\) 0 0
\(406\) 0 0
\(407\) −39.1918 −1.94267
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) −44.0908 −2.17749
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) 36.0000 1.76717
\(416\) 0 0
\(417\) 0 0
\(418\) 12.0000 0.586939
\(419\) 7.34847 0.358996 0.179498 0.983758i \(-0.442553\pi\)
0.179498 + 0.983758i \(0.442553\pi\)
\(420\) 0 0
\(421\) 35.0000 1.70580 0.852898 0.522078i \(-0.174843\pi\)
0.852898 + 0.522078i \(0.174843\pi\)
\(422\) −12.2474 −0.596196
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) −2.44949 −0.118818
\(426\) 0 0
\(427\) 0 0
\(428\) 78.3837 3.78882
\(429\) 0 0
\(430\) 6.00000 0.289346
\(431\) −19.5959 −0.943902 −0.471951 0.881625i \(-0.656450\pi\)
−0.471951 + 0.881625i \(0.656450\pi\)
\(432\) 0 0
\(433\) 7.00000 0.336399 0.168199 0.985753i \(-0.446205\pi\)
0.168199 + 0.985753i \(0.446205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) −2.44949 −0.117175
\(438\) 0 0
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 58.7878 2.80260
\(441\) 0 0
\(442\) 24.0000 1.14156
\(443\) −2.44949 −0.116379 −0.0581894 0.998306i \(-0.518533\pi\)
−0.0581894 + 0.998306i \(0.518533\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 63.6867 3.01565
\(447\) 0 0
\(448\) 0 0
\(449\) −22.0454 −1.04039 −0.520194 0.854048i \(-0.674140\pi\)
−0.520194 + 0.854048i \(0.674140\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) 58.7878 2.76514
\(453\) 0 0
\(454\) 42.0000 1.97116
\(455\) 0 0
\(456\) 0 0
\(457\) 23.0000 1.07589 0.537947 0.842978i \(-0.319200\pi\)
0.537947 + 0.842978i \(0.319200\pi\)
\(458\) −17.1464 −0.801200
\(459\) 0 0
\(460\) −24.0000 −1.11901
\(461\) −29.3939 −1.36901 −0.684505 0.729008i \(-0.739981\pi\)
−0.684505 + 0.729008i \(0.739981\pi\)
\(462\) 0 0
\(463\) 23.0000 1.06890 0.534450 0.845200i \(-0.320519\pi\)
0.534450 + 0.845200i \(0.320519\pi\)
\(464\) 29.3939 1.36458
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 24.4949 1.13349 0.566744 0.823894i \(-0.308203\pi\)
0.566744 + 0.823894i \(0.308203\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −14.6969 −0.677919
\(471\) 0 0
\(472\) 48.0000 2.20938
\(473\) 4.89898 0.225255
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 18.0000 0.823301
\(479\) 26.9444 1.23112 0.615560 0.788090i \(-0.288930\pi\)
0.615560 + 0.788090i \(0.288930\pi\)
\(480\) 0 0
\(481\) 32.0000 1.45907
\(482\) −31.8434 −1.45043
\(483\) 0 0
\(484\) 52.0000 2.36364
\(485\) 2.44949 0.111226
\(486\) 0 0
\(487\) −19.0000 −0.860972 −0.430486 0.902597i \(-0.641658\pi\)
−0.430486 + 0.902597i \(0.641658\pi\)
\(488\) 24.4949 1.10883
\(489\) 0 0
\(490\) 0 0
\(491\) 14.6969 0.663264 0.331632 0.943409i \(-0.392401\pi\)
0.331632 + 0.943409i \(0.392401\pi\)
\(492\) 0 0
\(493\) −18.0000 −0.810679
\(494\) −9.79796 −0.440831
\(495\) 0 0
\(496\) 28.0000 1.25724
\(497\) 0 0
\(498\) 0 0
\(499\) −7.00000 −0.313363 −0.156682 0.987649i \(-0.550080\pi\)
−0.156682 + 0.987649i \(0.550080\pi\)
\(500\) −39.1918 −1.75271
\(501\) 0 0
\(502\) 54.0000 2.41014
\(503\) −22.0454 −0.982956 −0.491478 0.870890i \(-0.663543\pi\)
−0.491478 + 0.870890i \(0.663543\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) −29.3939 −1.30672
\(507\) 0 0
\(508\) 44.0000 1.95218
\(509\) −19.5959 −0.868574 −0.434287 0.900775i \(-0.643000\pi\)
−0.434287 + 0.900775i \(0.643000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 39.1918 1.73205
\(513\) 0 0
\(514\) −60.0000 −2.64649
\(515\) −4.89898 −0.215875
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) 0 0
\(520\) −48.0000 −2.10494
\(521\) 19.5959 0.858513 0.429256 0.903183i \(-0.358776\pi\)
0.429256 + 0.903183i \(0.358776\pi\)
\(522\) 0 0
\(523\) −8.00000 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 9.79796 0.428026
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −17.1464 −0.746910
\(528\) 0 0
\(529\) −17.0000 −0.739130
\(530\) −14.6969 −0.638394
\(531\) 0 0
\(532\) 0 0
\(533\) 29.3939 1.27319
\(534\) 0 0
\(535\) 48.0000 2.07522
\(536\) −9.79796 −0.423207
\(537\) 0 0
\(538\) 60.0000 2.58678
\(539\) 0 0
\(540\) 0 0
\(541\) 35.0000 1.50477 0.752384 0.658725i \(-0.228904\pi\)
0.752384 + 0.658725i \(0.228904\pi\)
\(542\) −2.44949 −0.105215
\(543\) 0 0
\(544\) 0 0
\(545\) −2.44949 −0.104925
\(546\) 0 0
\(547\) −37.0000 −1.58201 −0.791003 0.611812i \(-0.790441\pi\)
−0.791003 + 0.611812i \(0.790441\pi\)
\(548\) 68.5857 2.92984
\(549\) 0 0
\(550\) 12.0000 0.511682
\(551\) 7.34847 0.313055
\(552\) 0 0
\(553\) 0 0
\(554\) −56.3383 −2.39358
\(555\) 0 0
\(556\) −56.0000 −2.37493
\(557\) −41.6413 −1.76440 −0.882200 0.470875i \(-0.843938\pi\)
−0.882200 + 0.470875i \(0.843938\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 36.0000 1.51857
\(563\) 12.2474 0.516168 0.258084 0.966122i \(-0.416909\pi\)
0.258084 + 0.966122i \(0.416909\pi\)
\(564\) 0 0
\(565\) 36.0000 1.51453
\(566\) −61.2372 −2.57399
\(567\) 0 0
\(568\) 0 0
\(569\) 19.5959 0.821504 0.410752 0.911747i \(-0.365266\pi\)
0.410752 + 0.911747i \(0.365266\pi\)
\(570\) 0 0
\(571\) 29.0000 1.21361 0.606806 0.794850i \(-0.292450\pi\)
0.606806 + 0.794850i \(0.292450\pi\)
\(572\) −78.3837 −3.27739
\(573\) 0 0
\(574\) 0 0
\(575\) −2.44949 −0.102151
\(576\) 0 0
\(577\) −8.00000 −0.333044 −0.166522 0.986038i \(-0.553254\pi\)
−0.166522 + 0.986038i \(0.553254\pi\)
\(578\) 26.9444 1.12074
\(579\) 0 0
\(580\) 72.0000 2.98964
\(581\) 0 0
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) −4.89898 −0.202721
\(585\) 0 0
\(586\) −72.0000 −2.97429
\(587\) −7.34847 −0.303304 −0.151652 0.988434i \(-0.548459\pi\)
−0.151652 + 0.988434i \(0.548459\pi\)
\(588\) 0 0
\(589\) 7.00000 0.288430
\(590\) 58.7878 2.42025
\(591\) 0 0
\(592\) 32.0000 1.31519
\(593\) 24.4949 1.00588 0.502942 0.864320i \(-0.332251\pi\)
0.502942 + 0.864320i \(0.332251\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −9.79796 −0.401340
\(597\) 0 0
\(598\) 24.0000 0.981433
\(599\) 31.8434 1.30108 0.650542 0.759470i \(-0.274542\pi\)
0.650542 + 0.759470i \(0.274542\pi\)
\(600\) 0 0
\(601\) −17.0000 −0.693444 −0.346722 0.937968i \(-0.612705\pi\)
−0.346722 + 0.937968i \(0.612705\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) 31.8434 1.29462
\(606\) 0 0
\(607\) −29.0000 −1.17707 −0.588537 0.808470i \(-0.700296\pi\)
−0.588537 + 0.808470i \(0.700296\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 30.0000 1.21466
\(611\) 9.79796 0.396383
\(612\) 0 0
\(613\) 17.0000 0.686624 0.343312 0.939222i \(-0.388451\pi\)
0.343312 + 0.939222i \(0.388451\pi\)
\(614\) −61.2372 −2.47133
\(615\) 0 0
\(616\) 0 0
\(617\) −29.3939 −1.18335 −0.591676 0.806176i \(-0.701534\pi\)
−0.591676 + 0.806176i \(0.701534\pi\)
\(618\) 0 0
\(619\) −14.0000 −0.562708 −0.281354 0.959604i \(-0.590783\pi\)
−0.281354 + 0.959604i \(0.590783\pi\)
\(620\) 68.5857 2.75447
\(621\) 0 0
\(622\) −84.0000 −3.36809
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 56.3383 2.25173
\(627\) 0 0
\(628\) −80.0000 −3.19235
\(629\) −19.5959 −0.781340
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 19.5959 0.779484
\(633\) 0 0
\(634\) 60.0000 2.38290
\(635\) 26.9444 1.06926
\(636\) 0 0
\(637\) 0 0
\(638\) 88.1816 3.49114
\(639\) 0 0
\(640\) 48.0000 1.89737
\(641\) −26.9444 −1.06424 −0.532120 0.846669i \(-0.678604\pi\)
−0.532120 + 0.846669i \(0.678604\pi\)
\(642\) 0 0
\(643\) −5.00000 −0.197181 −0.0985904 0.995128i \(-0.531433\pi\)
−0.0985904 + 0.995128i \(0.531433\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) 19.5959 0.770395 0.385198 0.922834i \(-0.374133\pi\)
0.385198 + 0.922834i \(0.374133\pi\)
\(648\) 0 0
\(649\) 48.0000 1.88416
\(650\) −9.79796 −0.384308
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −2.44949 −0.0958559 −0.0479280 0.998851i \(-0.515262\pi\)
−0.0479280 + 0.998851i \(0.515262\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 29.3939 1.14764
\(657\) 0 0
\(658\) 0 0
\(659\) 7.34847 0.286256 0.143128 0.989704i \(-0.454284\pi\)
0.143128 + 0.989704i \(0.454284\pi\)
\(660\) 0 0
\(661\) −11.0000 −0.427850 −0.213925 0.976850i \(-0.568625\pi\)
−0.213925 + 0.976850i \(0.568625\pi\)
\(662\) 31.8434 1.23763
\(663\) 0 0
\(664\) −72.0000 −2.79414
\(665\) 0 0
\(666\) 0 0
\(667\) −18.0000 −0.696963
\(668\) −58.7878 −2.27457
\(669\) 0 0
\(670\) −12.0000 −0.463600
\(671\) 24.4949 0.945615
\(672\) 0 0
\(673\) 17.0000 0.655302 0.327651 0.944799i \(-0.393743\pi\)
0.327651 + 0.944799i \(0.393743\pi\)
\(674\) −12.2474 −0.471754
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) −19.5959 −0.753132 −0.376566 0.926390i \(-0.622895\pi\)
−0.376566 + 0.926390i \(0.622895\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 29.3939 1.12720
\(681\) 0 0
\(682\) 84.0000 3.21653
\(683\) 24.4949 0.937271 0.468636 0.883392i \(-0.344746\pi\)
0.468636 + 0.883392i \(0.344746\pi\)
\(684\) 0 0
\(685\) 42.0000 1.60474
\(686\) 0 0
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 9.79796 0.373273
\(690\) 0 0
\(691\) −41.0000 −1.55971 −0.779857 0.625958i \(-0.784708\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) 9.79796 0.372463
\(693\) 0 0
\(694\) −24.0000 −0.911028
\(695\) −34.2929 −1.30080
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 85.7321 3.24501
\(699\) 0 0
\(700\) 0 0
\(701\) −22.0454 −0.832644 −0.416322 0.909217i \(-0.636681\pi\)
−0.416322 + 0.909217i \(0.636681\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 39.1918 1.47710
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 0 0
\(708\) 0 0
\(709\) −49.0000 −1.84023 −0.920117 0.391644i \(-0.871906\pi\)
−0.920117 + 0.391644i \(0.871906\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −12.0000 −0.449719
\(713\) −17.1464 −0.642139
\(714\) 0 0
\(715\) −48.0000 −1.79510
\(716\) 9.79796 0.366167
\(717\) 0 0
\(718\) −48.0000 −1.79134
\(719\) −19.5959 −0.730804 −0.365402 0.930850i \(-0.619069\pi\)
−0.365402 + 0.930850i \(0.619069\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 44.0908 1.64089
\(723\) 0 0
\(724\) 28.0000 1.04061
\(725\) 7.34847 0.272915
\(726\) 0 0
\(727\) 19.0000 0.704671 0.352335 0.935874i \(-0.385388\pi\)
0.352335 + 0.935874i \(0.385388\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −6.00000 −0.222070
\(731\) 2.44949 0.0905977
\(732\) 0 0
\(733\) −11.0000 −0.406294 −0.203147 0.979148i \(-0.565117\pi\)
−0.203147 + 0.979148i \(0.565117\pi\)
\(734\) −31.8434 −1.17536
\(735\) 0 0
\(736\) 0 0
\(737\) −9.79796 −0.360912
\(738\) 0 0
\(739\) −19.0000 −0.698926 −0.349463 0.936950i \(-0.613636\pi\)
−0.349463 + 0.936950i \(0.613636\pi\)
\(740\) 78.3837 2.88144
\(741\) 0 0
\(742\) 0 0
\(743\) −7.34847 −0.269589 −0.134795 0.990874i \(-0.543037\pi\)
−0.134795 + 0.990874i \(0.543037\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) −26.9444 −0.986504
\(747\) 0 0
\(748\) 48.0000 1.75505
\(749\) 0 0
\(750\) 0 0
\(751\) −43.0000 −1.56909 −0.784546 0.620070i \(-0.787104\pi\)
−0.784546 + 0.620070i \(0.787104\pi\)
\(752\) 9.79796 0.357295
\(753\) 0 0
\(754\) −72.0000 −2.62209
\(755\) 12.2474 0.445730
\(756\) 0 0
\(757\) 47.0000 1.70824 0.854122 0.520073i \(-0.174095\pi\)
0.854122 + 0.520073i \(0.174095\pi\)
\(758\) 83.2827 3.02496
\(759\) 0 0
\(760\) −12.0000 −0.435286
\(761\) −19.5959 −0.710351 −0.355176 0.934800i \(-0.615579\pi\)
−0.355176 + 0.934800i \(0.615579\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −9.79796 −0.354478
\(765\) 0 0
\(766\) 48.0000 1.73431
\(767\) −39.1918 −1.41514
\(768\) 0 0
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 80.0000 2.87926
\(773\) −46.5403 −1.67394 −0.836969 0.547250i \(-0.815675\pi\)
−0.836969 + 0.547250i \(0.815675\pi\)
\(774\) 0 0
\(775\) 7.00000 0.251447
\(776\) −4.89898 −0.175863
\(777\) 0 0
\(778\) −42.0000 −1.50577
\(779\) 7.34847 0.263286
\(780\) 0 0
\(781\) 0 0
\(782\) −14.6969 −0.525561
\(783\) 0 0
\(784\) 0 0
\(785\) −48.9898 −1.74852
\(786\) 0 0
\(787\) 7.00000 0.249523 0.124762 0.992187i \(-0.460183\pi\)
0.124762 + 0.992187i \(0.460183\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 24.0000 0.853882
\(791\) 0 0
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 41.6413 1.47780
\(795\) 0 0
\(796\) 76.0000 2.69375
\(797\) −14.6969 −0.520592 −0.260296 0.965529i \(-0.583820\pi\)
−0.260296 + 0.965529i \(0.583820\pi\)
\(798\) 0 0
\(799\) −6.00000 −0.212265
\(800\) 0 0
\(801\) 0 0
\(802\) 60.0000 2.11867
\(803\) −4.89898 −0.172881
\(804\) 0 0
\(805\) 0 0
\(806\) −68.5857 −2.41583
\(807\) 0 0
\(808\) −24.0000 −0.844317
\(809\) 24.4949 0.861195 0.430597 0.902544i \(-0.358303\pi\)
0.430597 + 0.902544i \(0.358303\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 96.0000 3.36480
\(815\) −2.44949 −0.0858019
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 48.9898 1.71289
\(819\) 0 0
\(820\) 72.0000 2.51435
\(821\) 19.5959 0.683902 0.341951 0.939718i \(-0.388912\pi\)
0.341951 + 0.939718i \(0.388912\pi\)
\(822\) 0 0
\(823\) −25.0000 −0.871445 −0.435723 0.900081i \(-0.643507\pi\)
−0.435723 + 0.900081i \(0.643507\pi\)
\(824\) 9.79796 0.341328
\(825\) 0 0
\(826\) 0 0
\(827\) −44.0908 −1.53319 −0.766594 0.642132i \(-0.778050\pi\)
−0.766594 + 0.642132i \(0.778050\pi\)
\(828\) 0 0
\(829\) 55.0000 1.91023 0.955114 0.296237i \(-0.0957318\pi\)
0.955114 + 0.296237i \(0.0957318\pi\)
\(830\) −88.1816 −3.06083
\(831\) 0 0
\(832\) −32.0000 −1.10940
\(833\) 0 0
\(834\) 0 0
\(835\) −36.0000 −1.24583
\(836\) −19.5959 −0.677739
\(837\) 0 0
\(838\) −18.0000 −0.621800
\(839\) 14.6969 0.507395 0.253697 0.967284i \(-0.418353\pi\)
0.253697 + 0.967284i \(0.418353\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) −85.7321 −2.95452
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) 7.34847 0.252795
\(846\) 0 0
\(847\) 0 0
\(848\) 9.79796 0.336463
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) −19.5959 −0.671739
\(852\) 0 0
\(853\) 19.0000 0.650548 0.325274 0.945620i \(-0.394544\pi\)
0.325274 + 0.945620i \(0.394544\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −96.0000 −3.28121
\(857\) −39.1918 −1.33877 −0.669384 0.742917i \(-0.733442\pi\)
−0.669384 + 0.742917i \(0.733442\pi\)
\(858\) 0 0
\(859\) 25.0000 0.852989 0.426494 0.904490i \(-0.359748\pi\)
0.426494 + 0.904490i \(0.359748\pi\)
\(860\) −9.79796 −0.334108
\(861\) 0 0
\(862\) 48.0000 1.63489
\(863\) −2.44949 −0.0833816 −0.0416908 0.999131i \(-0.513274\pi\)
−0.0416908 + 0.999131i \(0.513274\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) −17.1464 −0.582659
\(867\) 0 0
\(868\) 0 0
\(869\) 19.5959 0.664746
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 4.89898 0.165900
\(873\) 0 0
\(874\) 6.00000 0.202953
\(875\) 0 0
\(876\) 0 0
\(877\) 47.0000 1.58708 0.793539 0.608520i \(-0.208236\pi\)
0.793539 + 0.608520i \(0.208236\pi\)
\(878\) 34.2929 1.15733
\(879\) 0 0
\(880\) −48.0000 −1.61808
\(881\) −22.0454 −0.742729 −0.371364 0.928487i \(-0.621110\pi\)
−0.371364 + 0.928487i \(0.621110\pi\)
\(882\) 0 0
\(883\) −7.00000 −0.235569 −0.117784 0.993039i \(-0.537579\pi\)
−0.117784 + 0.993039i \(0.537579\pi\)
\(884\) −39.1918 −1.31816
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) −19.5959 −0.657967 −0.328983 0.944336i \(-0.606706\pi\)
−0.328983 + 0.944336i \(0.606706\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −14.6969 −0.492642
\(891\) 0 0
\(892\) −104.000 −3.48218
\(893\) 2.44949 0.0819690
\(894\) 0 0
\(895\) 6.00000 0.200558
\(896\) 0 0
\(897\) 0 0
\(898\) 54.0000 1.80200
\(899\) 51.4393 1.71560
\(900\) 0 0
\(901\) −6.00000 −0.199889
\(902\) 88.1816 2.93613
\(903\) 0 0
\(904\) −72.0000 −2.39468
\(905\) 17.1464 0.569967
\(906\) 0 0
\(907\) 32.0000 1.06254 0.531271 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) −68.5857 −2.27610
\(909\) 0 0
\(910\) 0 0
\(911\) 7.34847 0.243466 0.121733 0.992563i \(-0.461155\pi\)
0.121733 + 0.992563i \(0.461155\pi\)
\(912\) 0 0
\(913\) −72.0000 −2.38285
\(914\) −56.3383 −1.86350
\(915\) 0 0
\(916\) 28.0000 0.925146
\(917\) 0 0
\(918\) 0 0
\(919\) −1.00000 −0.0329870 −0.0164935 0.999864i \(-0.505250\pi\)
−0.0164935 + 0.999864i \(0.505250\pi\)
\(920\) 29.3939 0.969087
\(921\) 0 0
\(922\) 72.0000 2.37119
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) −56.3383 −1.85139
\(927\) 0 0
\(928\) 0 0
\(929\) 46.5403 1.52694 0.763469 0.645845i \(-0.223495\pi\)
0.763469 + 0.645845i \(0.223495\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −9.79796 −0.320943
\(933\) 0 0
\(934\) −60.0000 −1.96326
\(935\) 29.3939 0.961283
\(936\) 0 0
\(937\) 16.0000 0.522697 0.261349 0.965244i \(-0.415833\pi\)
0.261349 + 0.965244i \(0.415833\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 24.0000 0.782794
\(941\) −9.79796 −0.319404 −0.159702 0.987165i \(-0.551053\pi\)
−0.159702 + 0.987165i \(0.551053\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) −39.1918 −1.27559
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) −46.5403 −1.51236 −0.756178 0.654366i \(-0.772936\pi\)
−0.756178 + 0.654366i \(0.772936\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) −2.44949 −0.0794719
\(951\) 0 0
\(952\) 0 0
\(953\) 22.0454 0.714121 0.357060 0.934081i \(-0.383779\pi\)
0.357060 + 0.934081i \(0.383779\pi\)
\(954\) 0 0
\(955\) −6.00000 −0.194155
\(956\) −29.3939 −0.950666
\(957\) 0 0
\(958\) −66.0000 −2.13236
\(959\) 0 0
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) −78.3837 −2.52719
\(963\) 0 0
\(964\) 52.0000 1.67481
\(965\) 48.9898 1.57704
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) −63.6867 −2.04697
\(969\) 0 0
\(970\) −6.00000 −0.192648
\(971\) −41.6413 −1.33633 −0.668167 0.744011i \(-0.732921\pi\)
−0.668167 + 0.744011i \(0.732921\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 46.5403 1.49125
\(975\) 0 0
\(976\) −20.0000 −0.640184
\(977\) 9.79796 0.313464 0.156732 0.987641i \(-0.449904\pi\)
0.156732 + 0.987641i \(0.449904\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) 0 0
\(982\) −36.0000 −1.14881
\(983\) −39.1918 −1.25003 −0.625013 0.780615i \(-0.714906\pi\)
−0.625013 + 0.780615i \(0.714906\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 44.0908 1.40414
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) 2.44949 0.0778892
\(990\) 0 0
\(991\) −10.0000 −0.317660 −0.158830 0.987306i \(-0.550772\pi\)
−0.158830 + 0.987306i \(0.550772\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 46.5403 1.47543
\(996\) 0 0
\(997\) −41.0000 −1.29848 −0.649242 0.760582i \(-0.724914\pi\)
−0.649242 + 0.760582i \(0.724914\pi\)
\(998\) 17.1464 0.542761
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1323.2.a.u.1.1 2
3.2 odd 2 inner 1323.2.a.u.1.2 2
7.3 odd 6 189.2.e.d.163.2 yes 4
7.5 odd 6 189.2.e.d.109.2 yes 4
7.6 odd 2 1323.2.a.v.1.1 2
21.5 even 6 189.2.e.d.109.1 4
21.17 even 6 189.2.e.d.163.1 yes 4
21.20 even 2 1323.2.a.v.1.2 2
63.5 even 6 567.2.h.g.298.2 4
63.31 odd 6 567.2.g.g.541.2 4
63.38 even 6 567.2.h.g.352.2 4
63.40 odd 6 567.2.h.g.298.1 4
63.47 even 6 567.2.g.g.109.1 4
63.52 odd 6 567.2.h.g.352.1 4
63.59 even 6 567.2.g.g.541.1 4
63.61 odd 6 567.2.g.g.109.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.e.d.109.1 4 21.5 even 6
189.2.e.d.109.2 yes 4 7.5 odd 6
189.2.e.d.163.1 yes 4 21.17 even 6
189.2.e.d.163.2 yes 4 7.3 odd 6
567.2.g.g.109.1 4 63.47 even 6
567.2.g.g.109.2 4 63.61 odd 6
567.2.g.g.541.1 4 63.59 even 6
567.2.g.g.541.2 4 63.31 odd 6
567.2.h.g.298.1 4 63.40 odd 6
567.2.h.g.298.2 4 63.5 even 6
567.2.h.g.352.1 4 63.52 odd 6
567.2.h.g.352.2 4 63.38 even 6
1323.2.a.u.1.1 2 1.1 even 1 trivial
1323.2.a.u.1.2 2 3.2 odd 2 inner
1323.2.a.v.1.1 2 7.6 odd 2
1323.2.a.v.1.2 2 21.20 even 2