Properties

Label 6480.2.a.bc.1.1
Level $6480$
Weight $2$
Character 6480.1
Self dual yes
Analytic conductor $51.743$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6480,2,Mod(1,6480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6480, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("6480.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6480.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: \(51.7430605098\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 6480.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} -3.44949 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -3.44949 q^{7} -2.00000 q^{17} +6.89898 q^{19} +7.44949 q^{23} +1.00000 q^{25} -1.89898 q^{29} -1.10102 q^{31} +3.44949 q^{35} -6.00000 q^{37} -9.89898 q^{41} -11.7980 q^{43} +9.44949 q^{47} +4.89898 q^{49} +7.79796 q^{53} +1.10102 q^{59} -3.00000 q^{61} +13.2474 q^{67} -9.79796 q^{71} +13.7980 q^{73} +6.89898 q^{79} +5.44949 q^{83} +2.00000 q^{85} +2.79796 q^{89} -6.89898 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{7} - 4 q^{17} + 4 q^{19} + 10 q^{23} + 2 q^{25} + 6 q^{29} - 12 q^{31} + 2 q^{35} - 12 q^{37} - 10 q^{41} - 4 q^{43} + 14 q^{47} - 4 q^{53} + 12 q^{59} - 6 q^{61} + 2 q^{67} + 8 q^{73} + 4 q^{79} + 6 q^{83} + 4 q^{85} - 14 q^{89} - 4 q^{95} + 4 q^{97}+O(q^{100}) \) 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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.44949 −1.30378 −0.651892 0.758312i \(-0.726025\pi\)
−0.651892 + 0.758312i \(0.726025\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 6.89898 1.58273 0.791367 0.611341i \(-0.209370\pi\)
0.791367 + 0.611341i \(0.209370\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.44949 1.55333 0.776663 0.629916i \(-0.216911\pi\)
0.776663 + 0.629916i \(0.216911\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.89898 −0.352632 −0.176316 0.984334i \(-0.556418\pi\)
−0.176316 + 0.984334i \(0.556418\pi\)
\(30\) 0 0
\(31\) −1.10102 −0.197749 −0.0988746 0.995100i \(-0.531524\pi\)
−0.0988746 + 0.995100i \(0.531524\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.44949 0.583070
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.89898 −1.54596 −0.772980 0.634430i \(-0.781235\pi\)
−0.772980 + 0.634430i \(0.781235\pi\)
\(42\) 0 0
\(43\) −11.7980 −1.79917 −0.899586 0.436744i \(-0.856132\pi\)
−0.899586 + 0.436744i \(0.856132\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.44949 1.37835 0.689175 0.724595i \(-0.257973\pi\)
0.689175 + 0.724595i \(0.257973\pi\)
\(48\) 0 0
\(49\) 4.89898 0.699854
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.79796 1.07113 0.535566 0.844493i \(-0.320098\pi\)
0.535566 + 0.844493i \(0.320098\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.10102 0.143341 0.0716703 0.997428i \(-0.477167\pi\)
0.0716703 + 0.997428i \(0.477167\pi\)
\(60\) 0 0
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.2474 1.61843 0.809217 0.587510i \(-0.199892\pi\)
0.809217 + 0.587510i \(0.199892\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.79796 −1.16280 −0.581402 0.813617i \(-0.697496\pi\)
−0.581402 + 0.813617i \(0.697496\pi\)
\(72\) 0 0
\(73\) 13.7980 1.61493 0.807464 0.589916i \(-0.200839\pi\)
0.807464 + 0.589916i \(0.200839\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.89898 0.776196 0.388098 0.921618i \(-0.373132\pi\)
0.388098 + 0.921618i \(0.373132\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.44949 0.598159 0.299080 0.954228i \(-0.403320\pi\)
0.299080 + 0.954228i \(0.403320\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.79796 0.296583 0.148292 0.988944i \(-0.452623\pi\)
0.148292 + 0.988944i \(0.452623\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.89898 −0.707820
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.3485 1.19377 0.596886 0.802326i \(-0.296405\pi\)
0.596886 + 0.802326i \(0.296405\pi\)
\(108\) 0 0
\(109\) −10.7980 −1.03426 −0.517128 0.855908i \(-0.672999\pi\)
−0.517128 + 0.855908i \(0.672999\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.79796 −0.921714 −0.460857 0.887474i \(-0.652458\pi\)
−0.460857 + 0.887474i \(0.652458\pi\)
\(114\) 0 0
\(115\) −7.44949 −0.694669
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.89898 0.632428
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −0.348469 −0.0309216 −0.0154608 0.999880i \(-0.504922\pi\)
−0.0154608 + 0.999880i \(0.504922\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.89898 −0.253285 −0.126643 0.991948i \(-0.540420\pi\)
−0.126643 + 0.991948i \(0.540420\pi\)
\(132\) 0 0
\(133\) −23.7980 −2.06354
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.5959 1.67419 0.837096 0.547056i \(-0.184251\pi\)
0.837096 + 0.547056i \(0.184251\pi\)
\(138\) 0 0
\(139\) −19.5959 −1.66210 −0.831052 0.556195i \(-0.812261\pi\)
−0.831052 + 0.556195i \(0.812261\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 1.89898 0.157702
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −21.0000 −1.72039 −0.860194 0.509968i \(-0.829657\pi\)
−0.860194 + 0.509968i \(0.829657\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.10102 0.0884361
\(156\) 0 0
\(157\) −23.7980 −1.89928 −0.949642 0.313337i \(-0.898553\pi\)
−0.949642 + 0.313337i \(0.898553\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −25.6969 −2.02520
\(162\) 0 0
\(163\) 7.79796 0.610783 0.305392 0.952227i \(-0.401213\pi\)
0.305392 + 0.952227i \(0.401213\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.34847 −0.646024 −0.323012 0.946395i \(-0.604695\pi\)
−0.323012 + 0.946395i \(0.604695\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 0 0
\(175\) −3.44949 −0.260757
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.7980 1.03131 0.515654 0.856797i \(-0.327549\pi\)
0.515654 + 0.856797i \(0.327549\pi\)
\(180\) 0 0
\(181\) −9.69694 −0.720768 −0.360384 0.932804i \(-0.617354\pi\)
−0.360384 + 0.932804i \(0.617354\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 14.8990 1.07805 0.539026 0.842289i \(-0.318792\pi\)
0.539026 + 0.842289i \(0.318792\pi\)
\(192\) 0 0
\(193\) −2.20204 −0.158506 −0.0792532 0.996855i \(-0.525254\pi\)
−0.0792532 + 0.996855i \(0.525254\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −15.5959 −1.11116 −0.555582 0.831462i \(-0.687504\pi\)
−0.555582 + 0.831462i \(0.687504\pi\)
\(198\) 0 0
\(199\) −22.8990 −1.62327 −0.811633 0.584168i \(-0.801421\pi\)
−0.811633 + 0.584168i \(0.801421\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.55051 0.459756
\(204\) 0 0
\(205\) 9.89898 0.691375
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 11.7980 0.804614
\(216\) 0 0
\(217\) 3.79796 0.257822
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −5.44949 −0.364925 −0.182462 0.983213i \(-0.558407\pi\)
−0.182462 + 0.983213i \(0.558407\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.20204 0.544389 0.272194 0.962242i \(-0.412251\pi\)
0.272194 + 0.962242i \(0.412251\pi\)
\(228\) 0 0
\(229\) −4.10102 −0.271003 −0.135502 0.990777i \(-0.543265\pi\)
−0.135502 + 0.990777i \(0.543265\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −19.7980 −1.29701 −0.648504 0.761211i \(-0.724605\pi\)
−0.648504 + 0.761211i \(0.724605\pi\)
\(234\) 0 0
\(235\) −9.44949 −0.616417
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2.20204 −0.142438 −0.0712191 0.997461i \(-0.522689\pi\)
−0.0712191 + 0.997461i \(0.522689\pi\)
\(240\) 0 0
\(241\) −3.69694 −0.238141 −0.119070 0.992886i \(-0.537991\pi\)
−0.119070 + 0.992886i \(0.537991\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.89898 −0.312984
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.89898 −0.182982 −0.0914910 0.995806i \(-0.529163\pi\)
−0.0914910 + 0.995806i \(0.529163\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 27.7980 1.73399 0.866995 0.498318i \(-0.166049\pi\)
0.866995 + 0.498318i \(0.166049\pi\)
\(258\) 0 0
\(259\) 20.6969 1.28605
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) −7.79796 −0.479025
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 22.5959 1.37770 0.688849 0.724905i \(-0.258116\pi\)
0.688849 + 0.724905i \(0.258116\pi\)
\(270\) 0 0
\(271\) −30.8990 −1.87698 −0.938490 0.345307i \(-0.887775\pi\)
−0.938490 + 0.345307i \(0.887775\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.10102 −0.483266 −0.241633 0.970368i \(-0.577683\pi\)
−0.241633 + 0.970368i \(0.577683\pi\)
\(282\) 0 0
\(283\) 3.65153 0.217061 0.108530 0.994093i \(-0.465385\pi\)
0.108530 + 0.994093i \(0.465385\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 34.1464 2.01560
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 17.5959 1.02796 0.513982 0.857801i \(-0.328170\pi\)
0.513982 + 0.857801i \(0.328170\pi\)
\(294\) 0 0
\(295\) −1.10102 −0.0641039
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 40.6969 2.34573
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.00000 0.171780
\(306\) 0 0
\(307\) 4.75255 0.271242 0.135621 0.990761i \(-0.456697\pi\)
0.135621 + 0.990761i \(0.456697\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.89898 0.391205 0.195603 0.980683i \(-0.437334\pi\)
0.195603 + 0.980683i \(0.437334\pi\)
\(312\) 0 0
\(313\) −15.5959 −0.881533 −0.440767 0.897622i \(-0.645293\pi\)
−0.440767 + 0.897622i \(0.645293\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −25.7980 −1.44896 −0.724479 0.689297i \(-0.757920\pi\)
−0.724479 + 0.689297i \(0.757920\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −13.7980 −0.767739
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −32.5959 −1.79707
\(330\) 0 0
\(331\) 6.89898 0.379202 0.189601 0.981861i \(-0.439281\pi\)
0.189601 + 0.981861i \(0.439281\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −13.2474 −0.723785
\(336\) 0 0
\(337\) −29.7980 −1.62320 −0.811599 0.584215i \(-0.801403\pi\)
−0.811599 + 0.584215i \(0.801403\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 7.24745 0.391325
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 25.5959 1.37406 0.687030 0.726629i \(-0.258914\pi\)
0.687030 + 0.726629i \(0.258914\pi\)
\(348\) 0 0
\(349\) −2.30306 −0.123280 −0.0616400 0.998098i \(-0.519633\pi\)
−0.0616400 + 0.998098i \(0.519633\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 9.79796 0.520022
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −28.6969 −1.51457 −0.757283 0.653087i \(-0.773474\pi\)
−0.757283 + 0.653087i \(0.773474\pi\)
\(360\) 0 0
\(361\) 28.5959 1.50505
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −13.7980 −0.722218
\(366\) 0 0
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −26.8990 −1.39653
\(372\) 0 0
\(373\) 21.5959 1.11819 0.559097 0.829102i \(-0.311148\pi\)
0.559097 + 0.829102i \(0.311148\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −26.8990 −1.38171 −0.690854 0.722994i \(-0.742765\pi\)
−0.690854 + 0.722994i \(0.742765\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2.00000 −0.102195 −0.0510976 0.998694i \(-0.516272\pi\)
−0.0510976 + 0.998694i \(0.516272\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −16.5959 −0.841446 −0.420723 0.907189i \(-0.638224\pi\)
−0.420723 + 0.907189i \(0.638224\pi\)
\(390\) 0 0
\(391\) −14.8990 −0.753474
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −6.89898 −0.347125
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 25.5959 1.27820 0.639100 0.769124i \(-0.279307\pi\)
0.639100 + 0.769124i \(0.279307\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −9.59592 −0.474488 −0.237244 0.971450i \(-0.576244\pi\)
−0.237244 + 0.971450i \(0.576244\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.79796 −0.186885
\(414\) 0 0
\(415\) −5.44949 −0.267505
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 25.5959 1.24747 0.623734 0.781636i \(-0.285615\pi\)
0.623734 + 0.781636i \(0.285615\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 10.3485 0.500798
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −14.8990 −0.717659 −0.358829 0.933403i \(-0.616824\pi\)
−0.358829 + 0.933403i \(0.616824\pi\)
\(432\) 0 0
\(433\) −11.7980 −0.566974 −0.283487 0.958976i \(-0.591491\pi\)
−0.283487 + 0.958976i \(0.591491\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 51.3939 2.45850
\(438\) 0 0
\(439\) 17.7980 0.849450 0.424725 0.905322i \(-0.360371\pi\)
0.424725 + 0.905322i \(0.360371\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8.14643 0.387048 0.193524 0.981095i \(-0.438008\pi\)
0.193524 + 0.981095i \(0.438008\pi\)
\(444\) 0 0
\(445\) −2.79796 −0.132636
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 21.5959 1.01917 0.509587 0.860419i \(-0.329798\pi\)
0.509587 + 0.860419i \(0.329798\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 17.5959 0.823102 0.411551 0.911387i \(-0.364987\pi\)
0.411551 + 0.911387i \(0.364987\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.1010 0.563601 0.281800 0.959473i \(-0.409068\pi\)
0.281800 + 0.959473i \(0.409068\pi\)
\(462\) 0 0
\(463\) −12.2020 −0.567077 −0.283538 0.958961i \(-0.591508\pi\)
−0.283538 + 0.958961i \(0.591508\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.79796 −0.360847 −0.180423 0.983589i \(-0.557747\pi\)
−0.180423 + 0.983589i \(0.557747\pi\)
\(468\) 0 0
\(469\) −45.6969 −2.11009
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 6.89898 0.316547
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −30.4949 −1.39335 −0.696674 0.717388i \(-0.745337\pi\)
−0.696674 + 0.717388i \(0.745337\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) 13.5959 0.616090 0.308045 0.951372i \(-0.400325\pi\)
0.308045 + 0.951372i \(0.400325\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 13.7980 0.622693 0.311347 0.950296i \(-0.399220\pi\)
0.311347 + 0.950296i \(0.399220\pi\)
\(492\) 0 0
\(493\) 3.79796 0.171051
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 33.7980 1.51605
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −25.0454 −1.11672 −0.558360 0.829599i \(-0.688569\pi\)
−0.558360 + 0.829599i \(0.688569\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4.10102 −0.181775 −0.0908873 0.995861i \(-0.528970\pi\)
−0.0908873 + 0.995861i \(0.528970\pi\)
\(510\) 0 0
\(511\) −47.5959 −2.10552
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 10.0000 0.440653
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9.20204 0.403149 0.201574 0.979473i \(-0.435394\pi\)
0.201574 + 0.979473i \(0.435394\pi\)
\(522\) 0 0
\(523\) 34.3485 1.50195 0.750977 0.660329i \(-0.229583\pi\)
0.750977 + 0.660329i \(0.229583\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.20204 0.0959224
\(528\) 0 0
\(529\) 32.4949 1.41282
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −12.3485 −0.533871
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 12.1010 0.520264 0.260132 0.965573i \(-0.416234\pi\)
0.260132 + 0.965573i \(0.416234\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.7980 0.462534
\(546\) 0 0
\(547\) 16.7526 0.716287 0.358144 0.933666i \(-0.383410\pi\)
0.358144 + 0.933666i \(0.383410\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −13.1010 −0.558122
\(552\) 0 0
\(553\) −23.7980 −1.01199
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 43.7980 1.85578 0.927890 0.372855i \(-0.121621\pi\)
0.927890 + 0.372855i \(0.121621\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.3485 0.436136 0.218068 0.975934i \(-0.430025\pi\)
0.218068 + 0.975934i \(0.430025\pi\)
\(564\) 0 0
\(565\) 9.79796 0.412203
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −25.7980 −1.07961 −0.539805 0.841790i \(-0.681502\pi\)
−0.539805 + 0.841790i \(0.681502\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 7.44949 0.310665
\(576\) 0 0
\(577\) −26.0000 −1.08239 −0.541197 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −18.7980 −0.779871
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7.65153 −0.315812 −0.157906 0.987454i \(-0.550474\pi\)
−0.157906 + 0.987454i \(0.550474\pi\)
\(588\) 0 0
\(589\) −7.59592 −0.312984
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −31.3939 −1.28919 −0.644596 0.764523i \(-0.722974\pi\)
−0.644596 + 0.764523i \(0.722974\pi\)
\(594\) 0 0
\(595\) −6.89898 −0.282831
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 24.6969 1.00909 0.504545 0.863386i \(-0.331660\pi\)
0.504545 + 0.863386i \(0.331660\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 11.0000 0.447214
\(606\) 0 0
\(607\) −2.14643 −0.0871208 −0.0435604 0.999051i \(-0.513870\pi\)
−0.0435604 + 0.999051i \(0.513870\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −29.5959 −1.19537 −0.597684 0.801732i \(-0.703912\pi\)
−0.597684 + 0.801732i \(0.703912\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −35.1918 −1.41677 −0.708385 0.705826i \(-0.750576\pi\)
−0.708385 + 0.705826i \(0.750576\pi\)
\(618\) 0 0
\(619\) 24.6969 0.992654 0.496327 0.868136i \(-0.334682\pi\)
0.496327 + 0.868136i \(0.334682\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −9.65153 −0.386680
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 15.5959 0.620864 0.310432 0.950596i \(-0.399526\pi\)
0.310432 + 0.950596i \(0.399526\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0.348469 0.0138286
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −13.8990 −0.548977 −0.274488 0.961590i \(-0.588508\pi\)
−0.274488 + 0.961590i \(0.588508\pi\)
\(642\) 0 0
\(643\) 20.1464 0.794498 0.397249 0.917711i \(-0.369965\pi\)
0.397249 + 0.917711i \(0.369965\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.55051 −0.257527 −0.128764 0.991675i \(-0.541101\pi\)
−0.128764 + 0.991675i \(0.541101\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27.7980 1.08782 0.543909 0.839144i \(-0.316944\pi\)
0.543909 + 0.839144i \(0.316944\pi\)
\(654\) 0 0
\(655\) 2.89898 0.113273
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −10.2020 −0.397415 −0.198708 0.980059i \(-0.563674\pi\)
−0.198708 + 0.980059i \(0.563674\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 23.7980 0.922845
\(666\) 0 0
\(667\) −14.1464 −0.547752
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −16.0000 −0.616755 −0.308377 0.951264i \(-0.599786\pi\)
−0.308377 + 0.951264i \(0.599786\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.2020 0.545829 0.272914 0.962038i \(-0.412012\pi\)
0.272914 + 0.962038i \(0.412012\pi\)
\(678\) 0 0
\(679\) −6.89898 −0.264759
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −23.3939 −0.895142 −0.447571 0.894248i \(-0.647711\pi\)
−0.447571 + 0.894248i \(0.647711\pi\)
\(684\) 0 0
\(685\) −19.5959 −0.748722
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −38.8990 −1.47979 −0.739893 0.672724i \(-0.765124\pi\)
−0.739893 + 0.672724i \(0.765124\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 19.5959 0.743316
\(696\) 0 0
\(697\) 19.7980 0.749901
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 30.3939 1.14796 0.573980 0.818869i \(-0.305399\pi\)
0.573980 + 0.818869i \(0.305399\pi\)
\(702\) 0 0
\(703\) −41.3939 −1.56120
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.89898 0.259463
\(708\) 0 0
\(709\) 9.69694 0.364176 0.182088 0.983282i \(-0.441714\pi\)
0.182088 + 0.983282i \(0.441714\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −8.20204 −0.307169
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 28.2929 1.05515 0.527573 0.849510i \(-0.323102\pi\)
0.527573 + 0.849510i \(0.323102\pi\)
\(720\) 0 0
\(721\) 34.4949 1.28466
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.89898 −0.0705263
\(726\) 0 0
\(727\) 31.2474 1.15890 0.579452 0.815006i \(-0.303267\pi\)
0.579452 + 0.815006i \(0.303267\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 23.5959 0.872727
\(732\) 0 0
\(733\) −29.5959 −1.09315 −0.546575 0.837410i \(-0.684069\pi\)
−0.546575 + 0.837410i \(0.684069\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 33.1010 1.21764 0.608820 0.793308i \(-0.291643\pi\)
0.608820 + 0.793308i \(0.291643\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.04541 0.258471 0.129235 0.991614i \(-0.458748\pi\)
0.129235 + 0.991614i \(0.458748\pi\)
\(744\) 0 0
\(745\) 21.0000 0.769380
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −42.5959 −1.55642
\(750\) 0 0
\(751\) −26.2020 −0.956126 −0.478063 0.878326i \(-0.658661\pi\)
−0.478063 + 0.878326i \(0.658661\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 31.5959 1.14837 0.574187 0.818724i \(-0.305318\pi\)
0.574187 + 0.818724i \(0.305318\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 35.0000 1.26875 0.634375 0.773026i \(-0.281258\pi\)
0.634375 + 0.773026i \(0.281258\pi\)
\(762\) 0 0
\(763\) 37.2474 1.34845
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −5.20204 −0.187590 −0.0937952 0.995592i \(-0.529900\pi\)
−0.0937952 + 0.995592i \(0.529900\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6.20204 0.223072 0.111536 0.993760i \(-0.464423\pi\)
0.111536 + 0.993760i \(0.464423\pi\)
\(774\) 0 0
\(775\) −1.10102 −0.0395498
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −68.2929 −2.44685
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 23.7980 0.849386
\(786\) 0 0
\(787\) 49.5959 1.76790 0.883952 0.467578i \(-0.154873\pi\)
0.883952 + 0.467578i \(0.154873\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 33.7980 1.20172
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −8.00000 −0.283375 −0.141687 0.989911i \(-0.545253\pi\)
−0.141687 + 0.989911i \(0.545253\pi\)
\(798\) 0 0
\(799\) −18.8990 −0.668598
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 25.6969 0.905698
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −13.1010 −0.460039 −0.230020 0.973186i \(-0.573879\pi\)
−0.230020 + 0.973186i \(0.573879\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −7.79796 −0.273151
\(816\) 0 0
\(817\) −81.3939 −2.84761
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.79796 0.0976494 0.0488247 0.998807i \(-0.484452\pi\)
0.0488247 + 0.998807i \(0.484452\pi\)
\(822\) 0 0
\(823\) 8.34847 0.291009 0.145505 0.989358i \(-0.453519\pi\)
0.145505 + 0.989358i \(0.453519\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −53.5403 −1.86178 −0.930889 0.365301i \(-0.880966\pi\)
−0.930889 + 0.365301i \(0.880966\pi\)
\(828\) 0 0
\(829\) 50.1918 1.74323 0.871617 0.490187i \(-0.163072\pi\)
0.871617 + 0.490187i \(0.163072\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −9.79796 −0.339479
\(834\) 0 0
\(835\) 8.34847 0.288911
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −4.69694 −0.162156 −0.0810782 0.996708i \(-0.525836\pi\)
−0.0810782 + 0.996708i \(0.525836\pi\)
\(840\) 0 0
\(841\) −25.3939 −0.875651
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 13.0000 0.447214
\(846\) 0 0
\(847\) 37.9444 1.30378
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −44.6969 −1.53219
\(852\) 0 0
\(853\) 18.2020 0.623226 0.311613 0.950209i \(-0.399131\pi\)
0.311613 + 0.950209i \(0.399131\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.404082 0.0138032 0.00690159 0.999976i \(-0.497803\pi\)
0.00690159 + 0.999976i \(0.497803\pi\)
\(858\) 0 0
\(859\) −39.1918 −1.33721 −0.668604 0.743619i \(-0.733108\pi\)
−0.668604 + 0.743619i \(0.733108\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −57.4495 −1.95560 −0.977802 0.209532i \(-0.932806\pi\)
−0.977802 + 0.209532i \(0.932806\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.44949 0.116614
\(876\) 0 0
\(877\) −0.404082 −0.0136449 −0.00682244 0.999977i \(-0.502172\pi\)
−0.00682244 + 0.999977i \(0.502172\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 6.10102 0.205549 0.102774 0.994705i \(-0.467228\pi\)
0.102774 + 0.994705i \(0.467228\pi\)
\(882\) 0 0
\(883\) 0.954592 0.0321246 0.0160623 0.999871i \(-0.494887\pi\)
0.0160623 + 0.999871i \(0.494887\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.59592 −0.0535857 −0.0267928 0.999641i \(-0.508529\pi\)
−0.0267928 + 0.999641i \(0.508529\pi\)
\(888\) 0 0
\(889\) 1.20204 0.0403152
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 65.1918 2.18156
\(894\) 0 0
\(895\) −13.7980 −0.461215
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.09082 0.0697326
\(900\) 0 0
\(901\) −15.5959 −0.519575
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 9.69694 0.322337
\(906\) 0 0
\(907\) 21.6515 0.718927 0.359464 0.933159i \(-0.382960\pi\)
0.359464 + 0.933159i \(0.382960\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −10.4949 −0.347711 −0.173856 0.984771i \(-0.555623\pi\)
−0.173856 + 0.984771i \(0.555623\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 10.0000 0.330229
\(918\) 0 0
\(919\) −24.6969 −0.814677 −0.407338 0.913277i \(-0.633543\pi\)
−0.407338 + 0.913277i \(0.633543\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 37.5959 1.23348 0.616741 0.787166i \(-0.288453\pi\)
0.616741 + 0.787166i \(0.288453\pi\)
\(930\) 0 0
\(931\) 33.7980 1.10768
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −40.9898 −1.33908 −0.669539 0.742777i \(-0.733508\pi\)
−0.669539 + 0.742777i \(0.733508\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.69694 −0.0553186 −0.0276593 0.999617i \(-0.508805\pi\)
−0.0276593 + 0.999617i \(0.508805\pi\)
\(942\) 0 0
\(943\) −73.7423 −2.40138
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11.8536 0.385189 0.192595 0.981278i \(-0.438310\pi\)
0.192595 + 0.981278i \(0.438310\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −14.2020 −0.460049 −0.230025 0.973185i \(-0.573881\pi\)
−0.230025 + 0.973185i \(0.573881\pi\)
\(954\) 0 0
\(955\) −14.8990 −0.482120
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −67.5959 −2.18279
\(960\) 0 0
\(961\) −29.7878 −0.960895
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.20204 0.0708862
\(966\) 0 0
\(967\) −48.8434 −1.57070 −0.785348 0.619054i \(-0.787516\pi\)
−0.785348 + 0.619054i \(0.787516\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −31.3031 −1.00456 −0.502282 0.864704i \(-0.667506\pi\)
−0.502282 + 0.864704i \(0.667506\pi\)
\(972\) 0 0
\(973\) 67.5959 2.16703
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −13.3939 −0.428508 −0.214254 0.976778i \(-0.568732\pi\)
−0.214254 + 0.976778i \(0.568732\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 11.4495 0.365182 0.182591 0.983189i \(-0.441552\pi\)
0.182591 + 0.983189i \(0.441552\pi\)
\(984\) 0 0
\(985\) 15.5959 0.496927
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −87.8888 −2.79470
\(990\) 0 0
\(991\) −28.6969 −0.911588 −0.455794 0.890085i \(-0.650645\pi\)
−0.455794 + 0.890085i \(0.650645\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 22.8990 0.725946
\(996\) 0 0
\(997\) −23.3939 −0.740892 −0.370446 0.928854i \(-0.620795\pi\)
−0.370446 + 0.928854i \(0.620795\pi\)
\(998\) 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 6480.2.a.bc.1.1 2
3.2 odd 2 6480.2.a.bl.1.1 2
4.3 odd 2 3240.2.a.j.1.2 2
9.2 odd 6 2160.2.q.g.1441.2 4
9.4 even 3 720.2.q.g.241.2 4
9.5 odd 6 2160.2.q.g.721.2 4
9.7 even 3 720.2.q.g.481.2 4
12.11 even 2 3240.2.a.o.1.2 2
36.7 odd 6 360.2.q.c.121.1 4
36.11 even 6 1080.2.q.c.361.1 4
36.23 even 6 1080.2.q.c.721.1 4
36.31 odd 6 360.2.q.c.241.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.q.c.121.1 4 36.7 odd 6
360.2.q.c.241.1 yes 4 36.31 odd 6
720.2.q.g.241.2 4 9.4 even 3
720.2.q.g.481.2 4 9.7 even 3
1080.2.q.c.361.1 4 36.11 even 6
1080.2.q.c.721.1 4 36.23 even 6
2160.2.q.g.721.2 4 9.5 odd 6
2160.2.q.g.1441.2 4 9.2 odd 6
3240.2.a.j.1.2 2 4.3 odd 2
3240.2.a.o.1.2 2 12.11 even 2
6480.2.a.bc.1.1 2 1.1 even 1 trivial
6480.2.a.bl.1.1 2 3.2 odd 2