Properties

Label 1008.2.k.c.881.5
Level $1008$
Weight $2$
Character 1008.881
Analytic conductor $8.049$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(881,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.k (of order \(2\), degree \(1\), not 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.5
Root \(-0.382683 - 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 1008.881
Dual form 1008.2.k.c.881.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.53073 q^{5} +(0.414214 - 2.61313i) q^{7} +O(q^{10})\) \(q+1.53073 q^{5} +(0.414214 - 2.61313i) q^{7} -0.585786i q^{11} +2.16478i q^{13} +5.86030 q^{17} -5.22625i q^{19} +2.24264i q^{23} -2.65685 q^{25} -5.41421i q^{29} -4.32957i q^{31} +(0.634051 - 4.00000i) q^{35} +4.00000 q^{37} +8.92177 q^{41} -10.4853 q^{43} +7.39104 q^{47} +(-6.65685 - 2.16478i) q^{49} +5.41421i q^{53} -0.896683i q^{55} +3.31371i q^{65} +9.65685 q^{67} +4.58579i q^{71} -12.6173i q^{73} +(-1.53073 - 0.242641i) q^{77} +2.34315 q^{79} +13.5140 q^{83} +8.97056 q^{85} +5.86030 q^{89} +(5.65685 + 0.896683i) q^{91} -8.00000i q^{95} -8.28772i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} + 24 q^{25} + 32 q^{37} - 16 q^{43} - 8 q^{49} + 32 q^{67} + 64 q^{79} - 64 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.53073 0.684565 0.342282 0.939597i \(-0.388800\pi\)
0.342282 + 0.939597i \(0.388800\pi\)
\(6\) 0 0
\(7\) 0.414214 2.61313i 0.156558 0.987669i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.585786i 0.176621i −0.996093 0.0883106i \(-0.971853\pi\)
0.996093 0.0883106i \(-0.0281468\pi\)
\(12\) 0 0
\(13\) 2.16478i 0.600403i 0.953876 + 0.300202i \(0.0970540\pi\)
−0.953876 + 0.300202i \(0.902946\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.86030 1.42133 0.710666 0.703529i \(-0.248394\pi\)
0.710666 + 0.703529i \(0.248394\pi\)
\(18\) 0 0
\(19\) 5.22625i 1.19898i −0.800381 0.599492i \(-0.795369\pi\)
0.800381 0.599492i \(-0.204631\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.24264i 0.467623i 0.972282 + 0.233811i \(0.0751198\pi\)
−0.972282 + 0.233811i \(0.924880\pi\)
\(24\) 0 0
\(25\) −2.65685 −0.531371
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.41421i 1.00539i −0.864463 0.502697i \(-0.832341\pi\)
0.864463 0.502697i \(-0.167659\pi\)
\(30\) 0 0
\(31\) 4.32957i 0.777614i −0.921319 0.388807i \(-0.872887\pi\)
0.921319 0.388807i \(-0.127113\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.634051 4.00000i 0.107174 0.676123i
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.92177 1.39335 0.696673 0.717389i \(-0.254663\pi\)
0.696673 + 0.717389i \(0.254663\pi\)
\(42\) 0 0
\(43\) −10.4853 −1.59899 −0.799495 0.600672i \(-0.794900\pi\)
−0.799495 + 0.600672i \(0.794900\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.39104 1.07809 0.539047 0.842276i \(-0.318785\pi\)
0.539047 + 0.842276i \(0.318785\pi\)
\(48\) 0 0
\(49\) −6.65685 2.16478i −0.950979 0.309255i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.41421i 0.743699i 0.928293 + 0.371850i \(0.121276\pi\)
−0.928293 + 0.371850i \(0.878724\pi\)
\(54\) 0 0
\(55\) 0.896683i 0.120909i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.31371i 0.411015i
\(66\) 0 0
\(67\) 9.65685 1.17977 0.589886 0.807486i \(-0.299173\pi\)
0.589886 + 0.807486i \(0.299173\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.58579i 0.544233i 0.962264 + 0.272116i \(0.0877236\pi\)
−0.962264 + 0.272116i \(0.912276\pi\)
\(72\) 0 0
\(73\) 12.6173i 1.47674i −0.674395 0.738371i \(-0.735595\pi\)
0.674395 0.738371i \(-0.264405\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.53073 0.242641i −0.174443 0.0276515i
\(78\) 0 0
\(79\) 2.34315 0.263624 0.131812 0.991275i \(-0.457920\pi\)
0.131812 + 0.991275i \(0.457920\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.5140 1.48335 0.741676 0.670759i \(-0.234031\pi\)
0.741676 + 0.670759i \(0.234031\pi\)
\(84\) 0 0
\(85\) 8.97056 0.972994
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.86030 0.621191 0.310595 0.950542i \(-0.399472\pi\)
0.310595 + 0.950542i \(0.399472\pi\)
\(90\) 0 0
\(91\) 5.65685 + 0.896683i 0.592999 + 0.0939979i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000i 0.820783i
\(96\) 0 0
\(97\) 8.28772i 0.841490i −0.907179 0.420745i \(-0.861769\pi\)
0.907179 0.420745i \(-0.138231\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −16.3128 −1.62318 −0.811592 0.584224i \(-0.801399\pi\)
−0.811592 + 0.584224i \(0.801399\pi\)
\(102\) 0 0
\(103\) 14.7821i 1.45652i 0.685300 + 0.728260i \(0.259671\pi\)
−0.685300 + 0.728260i \(0.740329\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.0711i 1.26363i 0.775120 + 0.631814i \(0.217689\pi\)
−0.775120 + 0.631814i \(0.782311\pi\)
\(108\) 0 0
\(109\) −9.65685 −0.924959 −0.462479 0.886630i \(-0.653040\pi\)
−0.462479 + 0.886630i \(0.653040\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 20.2426i 1.90427i −0.305682 0.952134i \(-0.598884\pi\)
0.305682 0.952134i \(-0.401116\pi\)
\(114\) 0 0
\(115\) 3.43289i 0.320118i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.42742 15.3137i 0.222521 1.40381i
\(120\) 0 0
\(121\) 10.6569 0.968805
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.7206 −1.04832
\(126\) 0 0
\(127\) −16.9706 −1.50589 −0.752947 0.658081i \(-0.771368\pi\)
−0.752947 + 0.658081i \(0.771368\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7.39104 −0.645758 −0.322879 0.946440i \(-0.604651\pi\)
−0.322879 + 0.946440i \(0.604651\pi\)
\(132\) 0 0
\(133\) −13.6569 2.16478i −1.18420 0.187711i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.89949i 0.845771i 0.906183 + 0.422885i \(0.138983\pi\)
−0.906183 + 0.422885i \(0.861017\pi\)
\(138\) 0 0
\(139\) 10.4525i 0.886570i 0.896381 + 0.443285i \(0.146187\pi\)
−0.896381 + 0.443285i \(0.853813\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.26810 0.106044
\(144\) 0 0
\(145\) 8.28772i 0.688258i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16.2426i 1.33065i 0.746554 + 0.665324i \(0.231707\pi\)
−0.746554 + 0.665324i \(0.768293\pi\)
\(150\) 0 0
\(151\) 12.1421 0.988113 0.494056 0.869430i \(-0.335514\pi\)
0.494056 + 0.869430i \(0.335514\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.62742i 0.532327i
\(156\) 0 0
\(157\) 16.5754i 1.32286i 0.750005 + 0.661432i \(0.230051\pi\)
−0.750005 + 0.661432i \(0.769949\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.86030 + 0.928932i 0.461857 + 0.0732101i
\(162\) 0 0
\(163\) 6.34315 0.496834 0.248417 0.968653i \(-0.420090\pi\)
0.248417 + 0.968653i \(0.420090\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.5140 −1.04574 −0.522871 0.852412i \(-0.675139\pi\)
−0.522871 + 0.852412i \(0.675139\pi\)
\(168\) 0 0
\(169\) 8.31371 0.639516
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −22.4357 −1.70576 −0.852879 0.522109i \(-0.825145\pi\)
−0.852879 + 0.522109i \(0.825145\pi\)
\(174\) 0 0
\(175\) −1.10051 + 6.94269i −0.0831904 + 0.524818i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.89949i 0.291462i 0.989324 + 0.145731i \(0.0465534\pi\)
−0.989324 + 0.145731i \(0.953447\pi\)
\(180\) 0 0
\(181\) 14.4107i 1.07114i 0.844492 + 0.535568i \(0.179902\pi\)
−0.844492 + 0.535568i \(0.820098\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.12293 0.450167
\(186\) 0 0
\(187\) 3.43289i 0.251037i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 22.7279i 1.64453i 0.569101 + 0.822267i \(0.307291\pi\)
−0.569101 + 0.822267i \(0.692709\pi\)
\(192\) 0 0
\(193\) −5.65685 −0.407189 −0.203595 0.979055i \(-0.565262\pi\)
−0.203595 + 0.979055i \(0.565262\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.3848i 1.02487i −0.858725 0.512436i \(-0.828743\pi\)
0.858725 0.512436i \(-0.171257\pi\)
\(198\) 0 0
\(199\) 3.43289i 0.243351i −0.992570 0.121675i \(-0.961173\pi\)
0.992570 0.121675i \(-0.0388267\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −14.1480 2.24264i −0.992996 0.157403i
\(204\) 0 0
\(205\) 13.6569 0.953836
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.06147 −0.211766
\(210\) 0 0
\(211\) −24.8284 −1.70926 −0.854630 0.519238i \(-0.826216\pi\)
−0.854630 + 0.519238i \(0.826216\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −16.0502 −1.09461
\(216\) 0 0
\(217\) −11.3137 1.79337i −0.768025 0.121742i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12.6863i 0.853372i
\(222\) 0 0
\(223\) 3.43289i 0.229883i −0.993372 0.114942i \(-0.963332\pi\)
0.993372 0.114942i \(-0.0366681\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.12293 0.406393 0.203197 0.979138i \(-0.434867\pi\)
0.203197 + 0.979138i \(0.434867\pi\)
\(228\) 0 0
\(229\) 23.0698i 1.52449i 0.647286 + 0.762247i \(0.275904\pi\)
−0.647286 + 0.762247i \(0.724096\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.72792i 0.309736i 0.987935 + 0.154868i \(0.0494953\pi\)
−0.987935 + 0.154868i \(0.950505\pi\)
\(234\) 0 0
\(235\) 11.3137 0.738025
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 17.0711i 1.10424i 0.833766 + 0.552118i \(0.186180\pi\)
−0.833766 + 0.552118i \(0.813820\pi\)
\(240\) 0 0
\(241\) 16.9469i 1.09164i 0.837901 + 0.545822i \(0.183782\pi\)
−0.837901 + 0.545822i \(0.816218\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −10.1899 3.31371i −0.651007 0.211705i
\(246\) 0 0
\(247\) 11.3137 0.719874
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −28.2960 −1.78603 −0.893015 0.450026i \(-0.851415\pi\)
−0.893015 + 0.450026i \(0.851415\pi\)
\(252\) 0 0
\(253\) 1.31371 0.0825921
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.1062 1.12943 0.564716 0.825285i \(-0.308986\pi\)
0.564716 + 0.825285i \(0.308986\pi\)
\(258\) 0 0
\(259\) 1.65685 10.4525i 0.102952 0.649487i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.72792i 0.414861i −0.978250 0.207431i \(-0.933490\pi\)
0.978250 0.207431i \(-0.0665102\pi\)
\(264\) 0 0
\(265\) 8.28772i 0.509111i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.59220 0.279991 0.139996 0.990152i \(-0.455291\pi\)
0.139996 + 0.990152i \(0.455291\pi\)
\(270\) 0 0
\(271\) 16.5754i 1.00689i −0.864028 0.503443i \(-0.832066\pi\)
0.864028 0.503443i \(-0.167934\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.55635i 0.0938514i
\(276\) 0 0
\(277\) 8.34315 0.501291 0.250646 0.968079i \(-0.419357\pi\)
0.250646 + 0.968079i \(0.419357\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.89949i 0.590554i −0.955412 0.295277i \(-0.904588\pi\)
0.955412 0.295277i \(-0.0954120\pi\)
\(282\) 0 0
\(283\) 24.3379i 1.44674i 0.690462 + 0.723369i \(0.257407\pi\)
−0.690462 + 0.723369i \(0.742593\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.69552 23.3137i 0.218140 1.37616i
\(288\) 0 0
\(289\) 17.3431 1.02019
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −10.7151 −0.625985 −0.312992 0.949756i \(-0.601331\pi\)
−0.312992 + 0.949756i \(0.601331\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.85483 −0.280762
\(300\) 0 0
\(301\) −4.34315 + 27.3994i −0.250335 + 1.57927i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 26.1313i 1.49139i −0.666288 0.745695i \(-0.732118\pi\)
0.666288 0.745695i \(-0.267882\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −29.5641 −1.67643 −0.838214 0.545341i \(-0.816400\pi\)
−0.838214 + 0.545341i \(0.816400\pi\)
\(312\) 0 0
\(313\) 8.65914i 0.489443i −0.969593 0.244722i \(-0.921303\pi\)
0.969593 0.244722i \(-0.0786966\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.3848i 0.807930i 0.914775 + 0.403965i \(0.132368\pi\)
−0.914775 + 0.403965i \(0.867632\pi\)
\(318\) 0 0
\(319\) −3.17157 −0.177574
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 30.6274i 1.70416i
\(324\) 0 0
\(325\) 5.75152i 0.319037i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.06147 19.3137i 0.168784 1.06480i
\(330\) 0 0
\(331\) 0.828427 0.0455345 0.0227672 0.999741i \(-0.492752\pi\)
0.0227672 + 0.999741i \(0.492752\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 14.7821 0.807631
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.53620 −0.137343
\(342\) 0 0
\(343\) −8.41421 + 16.4985i −0.454325 + 0.890836i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 17.5563i 0.942474i 0.882007 + 0.471237i \(0.156192\pi\)
−0.882007 + 0.471237i \(0.843808\pi\)
\(348\) 0 0
\(349\) 16.5754i 0.887263i −0.896209 0.443631i \(-0.853690\pi\)
0.896209 0.443631i \(-0.146310\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.262632 −0.0139785 −0.00698926 0.999976i \(-0.502225\pi\)
−0.00698926 + 0.999976i \(0.502225\pi\)
\(354\) 0 0
\(355\) 7.01962i 0.372563i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 28.3848i 1.49809i 0.662518 + 0.749046i \(0.269488\pi\)
−0.662518 + 0.749046i \(0.730512\pi\)
\(360\) 0 0
\(361\) −8.31371 −0.437564
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 19.3137i 1.01093i
\(366\) 0 0
\(367\) 15.6788i 0.818424i −0.912439 0.409212i \(-0.865804\pi\)
0.912439 0.409212i \(-0.134196\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 14.1480 + 2.24264i 0.734529 + 0.116432i
\(372\) 0 0
\(373\) 5.31371 0.275133 0.137567 0.990493i \(-0.456072\pi\)
0.137567 + 0.990493i \(0.456072\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11.7206 0.603642
\(378\) 0 0
\(379\) −16.8284 −0.864418 −0.432209 0.901773i \(-0.642266\pi\)
−0.432209 + 0.901773i \(0.642266\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 28.2960 1.44586 0.722930 0.690921i \(-0.242795\pi\)
0.722930 + 0.690921i \(0.242795\pi\)
\(384\) 0 0
\(385\) −2.34315 0.371418i −0.119418 0.0189292i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0.727922i 0.0369071i 0.999830 + 0.0184536i \(0.00587428\pi\)
−0.999830 + 0.0184536i \(0.994126\pi\)
\(390\) 0 0
\(391\) 13.1426i 0.664647i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.58673 0.180468
\(396\) 0 0
\(397\) 8.65914i 0.434590i −0.976106 0.217295i \(-0.930277\pi\)
0.976106 0.217295i \(-0.0697233\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 31.5563i 1.57585i 0.615772 + 0.787924i \(0.288844\pi\)
−0.615772 + 0.787924i \(0.711156\pi\)
\(402\) 0 0
\(403\) 9.37258 0.466882
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.34315i 0.116145i
\(408\) 0 0
\(409\) 12.6173i 0.623885i −0.950101 0.311942i \(-0.899020\pi\)
0.950101 0.311942i \(-0.100980\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 20.6863 1.01545
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.65914 0.423027 0.211513 0.977375i \(-0.432161\pi\)
0.211513 + 0.977375i \(0.432161\pi\)
\(420\) 0 0
\(421\) −14.9706 −0.729621 −0.364810 0.931082i \(-0.618866\pi\)
−0.364810 + 0.931082i \(0.618866\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −15.5700 −0.755254
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.2426i 0.493371i −0.969096 0.246685i \(-0.920659\pi\)
0.969096 0.246685i \(-0.0793414\pi\)
\(432\) 0 0
\(433\) 33.8937i 1.62883i −0.580284 0.814414i \(-0.697059\pi\)
0.580284 0.814414i \(-0.302941\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 11.7206 0.560673
\(438\) 0 0
\(439\) 3.43289i 0.163843i 0.996639 + 0.0819213i \(0.0261056\pi\)
−0.996639 + 0.0819213i \(0.973894\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8.58579i 0.407923i 0.978979 + 0.203962i \(0.0653817\pi\)
−0.978979 + 0.203962i \(0.934618\pi\)
\(444\) 0 0
\(445\) 8.97056 0.425245
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.41421i 0.444284i 0.975014 + 0.222142i \(0.0713049\pi\)
−0.975014 + 0.222142i \(0.928695\pi\)
\(450\) 0 0
\(451\) 5.22625i 0.246095i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 8.65914 + 1.37258i 0.405947 + 0.0643477i
\(456\) 0 0
\(457\) −11.3137 −0.529233 −0.264616 0.964354i \(-0.585245\pi\)
−0.264616 + 0.964354i \(0.585245\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 10.7151 0.499054 0.249527 0.968368i \(-0.419725\pi\)
0.249527 + 0.968368i \(0.419725\pi\)
\(462\) 0 0
\(463\) 29.6569 1.37827 0.689135 0.724633i \(-0.257990\pi\)
0.689135 + 0.724633i \(0.257990\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 30.8322 1.42675 0.713373 0.700784i \(-0.247167\pi\)
0.713373 + 0.700784i \(0.247167\pi\)
\(468\) 0 0
\(469\) 4.00000 25.2346i 0.184703 1.16522i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.14214i 0.282416i
\(474\) 0 0
\(475\) 13.8854i 0.637105i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 23.4412 1.07106 0.535528 0.844517i \(-0.320113\pi\)
0.535528 + 0.844517i \(0.320113\pi\)
\(480\) 0 0
\(481\) 8.65914i 0.394823i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.6863i 0.576055i
\(486\) 0 0
\(487\) −39.4558 −1.78791 −0.893957 0.448152i \(-0.852082\pi\)
−0.893957 + 0.448152i \(0.852082\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.5269i 1.19714i −0.801069 0.598571i \(-0.795735\pi\)
0.801069 0.598571i \(-0.204265\pi\)
\(492\) 0 0
\(493\) 31.7289i 1.42900i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 11.9832 + 1.89949i 0.537522 + 0.0852040i
\(498\) 0 0
\(499\) 7.17157 0.321044 0.160522 0.987032i \(-0.448682\pi\)
0.160522 + 0.987032i \(0.448682\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 19.6369 0.875566 0.437783 0.899081i \(-0.355764\pi\)
0.437783 + 0.899081i \(0.355764\pi\)
\(504\) 0 0
\(505\) −24.9706 −1.11118
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 10.1899 0.451658 0.225829 0.974167i \(-0.427491\pi\)
0.225829 + 0.974167i \(0.427491\pi\)
\(510\) 0 0
\(511\) −32.9706 5.22625i −1.45853 0.231196i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 22.6274i 0.997083i
\(516\) 0 0
\(517\) 4.32957i 0.190414i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5.86030 −0.256745 −0.128372 0.991726i \(-0.540975\pi\)
−0.128372 + 0.991726i \(0.540975\pi\)
\(522\) 0 0
\(523\) 8.65914i 0.378638i 0.981916 + 0.189319i \(0.0606280\pi\)
−0.981916 + 0.189319i \(0.939372\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 25.3726i 1.10525i
\(528\) 0 0
\(529\) 17.9706 0.781329
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 19.3137i 0.836570i
\(534\) 0 0
\(535\) 20.0083i 0.865035i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.26810 + 3.89949i −0.0546210 + 0.167963i
\(540\) 0 0
\(541\) 16.3431 0.702647 0.351323 0.936254i \(-0.385732\pi\)
0.351323 + 0.936254i \(0.385732\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −14.7821 −0.633194
\(546\) 0 0
\(547\) −6.34315 −0.271213 −0.135607 0.990763i \(-0.543298\pi\)
−0.135607 + 0.990763i \(0.543298\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −28.2960 −1.20545
\(552\) 0 0
\(553\) 0.970563 6.12293i 0.0412725 0.260374i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 26.1005i 1.10591i −0.833210 0.552957i \(-0.813499\pi\)
0.833210 0.552957i \(-0.186501\pi\)
\(558\) 0 0
\(559\) 22.6984i 0.960039i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 45.6143 1.92241 0.961207 0.275827i \(-0.0889518\pi\)
0.961207 + 0.275827i \(0.0889518\pi\)
\(564\) 0 0
\(565\) 30.9861i 1.30359i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.24264i 0.177861i −0.996038 0.0889304i \(-0.971655\pi\)
0.996038 0.0889304i \(-0.0283449\pi\)
\(570\) 0 0
\(571\) 20.9706 0.877591 0.438795 0.898587i \(-0.355405\pi\)
0.438795 + 0.898587i \(0.355405\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5.95837i 0.248481i
\(576\) 0 0
\(577\) 4.32957i 0.180242i −0.995931 0.0901212i \(-0.971275\pi\)
0.995931 0.0901212i \(-0.0287254\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.59767 35.3137i 0.232230 1.46506i
\(582\) 0 0
\(583\) 3.17157 0.131353
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.85483 0.200380 0.100190 0.994968i \(-0.468055\pi\)
0.100190 + 0.994968i \(0.468055\pi\)
\(588\) 0 0
\(589\) −22.6274 −0.932346
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.32410 0.136504 0.0682522 0.997668i \(-0.478258\pi\)
0.0682522 + 0.997668i \(0.478258\pi\)
\(594\) 0 0
\(595\) 3.71573 23.4412i 0.152330 0.960996i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 11.2132i 0.458159i −0.973408 0.229080i \(-0.926428\pi\)
0.973408 0.229080i \(-0.0735716\pi\)
\(600\) 0 0
\(601\) 33.8937i 1.38255i 0.722590 + 0.691277i \(0.242951\pi\)
−0.722590 + 0.691277i \(0.757049\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 16.3128 0.663210
\(606\) 0 0
\(607\) 5.22625i 0.212127i −0.994359 0.106064i \(-0.966175\pi\)
0.994359 0.106064i \(-0.0338247\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 16.0000i 0.647291i
\(612\) 0 0
\(613\) 1.31371 0.0530602 0.0265301 0.999648i \(-0.491554\pi\)
0.0265301 + 0.999648i \(0.491554\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.2426i 0.492870i −0.969159 0.246435i \(-0.920741\pi\)
0.969159 0.246435i \(-0.0792592\pi\)
\(618\) 0 0
\(619\) 19.1116i 0.768162i −0.923299 0.384081i \(-0.874518\pi\)
0.923299 0.384081i \(-0.125482\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.42742 15.3137i 0.0972524 0.613531i
\(624\) 0 0
\(625\) −4.65685 −0.186274
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 23.4412 0.934662
\(630\) 0 0
\(631\) 39.5980 1.57637 0.788185 0.615438i \(-0.211021\pi\)
0.788185 + 0.615438i \(0.211021\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −25.9774 −1.03088
\(636\) 0 0
\(637\) 4.68629 14.4107i 0.185678 0.570971i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.8701i 0.745322i −0.927967 0.372661i \(-0.878445\pi\)
0.927967 0.372661i \(-0.121555\pi\)
\(642\) 0 0
\(643\) 13.8854i 0.547586i 0.961789 + 0.273793i \(0.0882784\pi\)
−0.961789 + 0.273793i \(0.911722\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −3.58673 −0.141009 −0.0705045 0.997511i \(-0.522461\pi\)
−0.0705045 + 0.997511i \(0.522461\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 41.2132i 1.61280i 0.591372 + 0.806399i \(0.298586\pi\)
−0.591372 + 0.806399i \(0.701414\pi\)
\(654\) 0 0
\(655\) −11.3137 −0.442063
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 47.4142i 1.84700i −0.383604 0.923498i \(-0.625317\pi\)
0.383604 0.923498i \(-0.374683\pi\)
\(660\) 0 0
\(661\) 37.4804i 1.45782i 0.684609 + 0.728910i \(0.259973\pi\)
−0.684609 + 0.728910i \(0.740027\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −20.9050 3.31371i −0.810661 0.128500i
\(666\) 0 0
\(667\) 12.1421 0.470145
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −13.6569 −0.526433 −0.263217 0.964737i \(-0.584783\pi\)
−0.263217 + 0.964737i \(0.584783\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 45.8770 1.76319 0.881597 0.472002i \(-0.156469\pi\)
0.881597 + 0.472002i \(0.156469\pi\)
\(678\) 0 0
\(679\) −21.6569 3.43289i −0.831114 0.131742i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 34.7279i 1.32883i 0.747365 + 0.664414i \(0.231319\pi\)
−0.747365 + 0.664414i \(0.768681\pi\)
\(684\) 0 0
\(685\) 15.1535i 0.578985i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −11.7206 −0.446519
\(690\) 0 0
\(691\) 31.3575i 1.19290i −0.802652 0.596448i \(-0.796578\pi\)
0.802652 0.596448i \(-0.203422\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.0000i 0.606915i
\(696\) 0 0
\(697\) 52.2843 1.98041
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 28.9289i 1.09263i 0.837580 + 0.546315i \(0.183970\pi\)
−0.837580 + 0.546315i \(0.816030\pi\)
\(702\) 0 0
\(703\) 20.9050i 0.788447i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.75699 + 42.6274i −0.254123 + 1.60317i
\(708\) 0 0
\(709\) −24.2843 −0.912015 −0.456007 0.889976i \(-0.650721\pi\)
−0.456007 + 0.889976i \(0.650721\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 9.70967 0.363630
\(714\) 0 0
\(715\) 1.94113 0.0725940
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 20.9050 0.779625 0.389813 0.920894i \(-0.372540\pi\)
0.389813 + 0.920894i \(0.372540\pi\)
\(720\) 0 0
\(721\) 38.6274 + 6.12293i 1.43856 + 0.228030i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 14.3848i 0.534237i
\(726\) 0 0
\(727\) 6.12293i 0.227087i −0.993533 0.113544i \(-0.963780\pi\)
0.993533 0.113544i \(-0.0362201\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −61.4469 −2.27270
\(732\) 0 0
\(733\) 6.49435i 0.239874i −0.992781 0.119937i \(-0.961731\pi\)
0.992781 0.119937i \(-0.0382693\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.65685i 0.208373i
\(738\) 0 0
\(739\) −40.2843 −1.48188 −0.740940 0.671571i \(-0.765620\pi\)
−0.740940 + 0.671571i \(0.765620\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 47.6985i 1.74989i 0.484224 + 0.874944i \(0.339102\pi\)
−0.484224 + 0.874944i \(0.660898\pi\)
\(744\) 0 0
\(745\) 24.8632i 0.910916i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 34.1563 + 5.41421i 1.24805 + 0.197831i
\(750\) 0 0
\(751\) −13.5147 −0.493159 −0.246580 0.969123i \(-0.579307\pi\)
−0.246580 + 0.969123i \(0.579307\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 18.5864 0.676428
\(756\) 0 0
\(757\) −27.5980 −1.00307 −0.501533 0.865139i \(-0.667230\pi\)
−0.501533 + 0.865139i \(0.667230\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −29.3015 −1.06218 −0.531090 0.847316i \(-0.678217\pi\)
−0.531090 + 0.847316i \(0.678217\pi\)
\(762\) 0 0
\(763\) −4.00000 + 25.2346i −0.144810 + 0.913553i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 33.8937i 1.22224i 0.791539 + 0.611119i \(0.209280\pi\)
−0.791539 + 0.611119i \(0.790720\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −16.3128 −0.586731 −0.293365 0.956000i \(-0.594775\pi\)
−0.293365 + 0.956000i \(0.594775\pi\)
\(774\) 0 0
\(775\) 11.5030i 0.413201i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 46.6274i 1.67060i
\(780\) 0 0
\(781\) 2.68629 0.0961231
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 25.3726i 0.905586i
\(786\) 0 0
\(787\) 38.2233i 1.36251i −0.732045 0.681256i \(-0.761434\pi\)
0.732045 0.681256i \(-0.238566\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −52.8966 8.38478i −1.88079 0.298128i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22.4357 0.794715 0.397357 0.917664i \(-0.369927\pi\)
0.397357 + 0.917664i \(0.369927\pi\)
\(798\) 0 0
\(799\) 43.3137 1.53233
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −7.39104 −0.260824
\(804\) 0 0
\(805\) 8.97056 + 1.42195i 0.316171 + 0.0501171i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 7.07107i 0.248606i −0.992244 0.124303i \(-0.960331\pi\)
0.992244 0.124303i \(-0.0396694\pi\)
\(810\) 0 0
\(811\) 12.2459i 0.430011i 0.976613 + 0.215005i \(0.0689769\pi\)
−0.976613 + 0.215005i \(0.931023\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9.70967 0.340115
\(816\) 0 0
\(817\) 54.7987i 1.91716i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 39.3553i 1.37351i −0.726889 0.686755i \(-0.759034\pi\)
0.726889 0.686755i \(-0.240966\pi\)
\(822\) 0 0
\(823\) −21.6569 −0.754910 −0.377455 0.926028i \(-0.623201\pi\)
−0.377455 + 0.926028i \(0.623201\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 55.0122i 1.91296i −0.291796 0.956481i \(-0.594253\pi\)
0.291796 0.956481i \(-0.405747\pi\)
\(828\) 0 0
\(829\) 10.8239i 0.375930i −0.982176 0.187965i \(-0.939811\pi\)
0.982176 0.187965i \(-0.0601892\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −39.0112 12.6863i −1.35166 0.439554i
\(834\) 0 0
\(835\) −20.6863 −0.715879
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −51.7373 −1.78617 −0.893084 0.449890i \(-0.851463\pi\)
−0.893084 + 0.449890i \(0.851463\pi\)
\(840\) 0 0
\(841\) −0.313708 −0.0108175
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.7261 0.437790
\(846\) 0 0
\(847\) 4.41421 27.8477i 0.151674 0.956858i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.97056i 0.307507i
\(852\) 0 0
\(853\) 29.5641i 1.01226i −0.862458 0.506129i \(-0.831076\pi\)
0.862458 0.506129i \(-0.168924\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.86030 −0.200184 −0.100092 0.994978i \(-0.531914\pi\)
−0.100092 + 0.994978i \(0.531914\pi\)
\(858\) 0 0
\(859\) 13.8854i 0.473763i 0.971539 + 0.236882i \(0.0761254\pi\)
−0.971539 + 0.236882i \(0.923875\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 20.5858i 0.700748i 0.936610 + 0.350374i \(0.113946\pi\)
−0.936610 + 0.350374i \(0.886054\pi\)
\(864\) 0 0
\(865\) −34.3431 −1.16770
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.37258i 0.0465617i
\(870\) 0 0
\(871\) 20.9050i 0.708339i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.85483 + 30.6274i −0.164123 + 1.03540i
\(876\) 0 0
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −39.0112 −1.31432 −0.657160 0.753751i \(-0.728242\pi\)
−0.657160 + 0.753751i \(0.728242\pi\)
\(882\) 0 0
\(883\) 4.14214 0.139394 0.0696970 0.997568i \(-0.477797\pi\)
0.0696970 + 0.997568i \(0.477797\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.0502 0.538912 0.269456 0.963013i \(-0.413156\pi\)
0.269456 + 0.963013i \(0.413156\pi\)
\(888\) 0 0
\(889\) −7.02944 + 44.3462i −0.235760 + 1.48732i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 38.6274i 1.29262i
\(894\) 0 0
\(895\) 5.96909i 0.199525i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −23.4412 −0.781808
\(900\) 0 0
\(901\) 31.7289i 1.05704i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 22.0589i 0.733262i
\(906\) 0 0
\(907\) 10.2010 0.338719 0.169359 0.985554i \(-0.445830\pi\)
0.169359 + 0.985554i \(0.445830\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 20.1838i 0.668718i 0.942446 + 0.334359i \(0.108520\pi\)
−0.942446 + 0.334359i \(0.891480\pi\)
\(912\) 0 0
\(913\) 7.91630i 0.261991i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.06147 + 19.3137i −0.101099 + 0.637795i
\(918\) 0 0
\(919\) 37.7990 1.24687 0.623437 0.781874i \(-0.285736\pi\)
0.623437 + 0.781874i \(0.285736\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −9.92724 −0.326759
\(924\) 0 0
\(925\) −10.6274 −0.349427
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −20.1171 −0.660021 −0.330010 0.943977i \(-0.607052\pi\)
−0.330010 + 0.943977i \(0.607052\pi\)
\(930\) 0 0
\(931\) −11.3137 + 34.7904i −0.370792 + 1.14021i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.25483i 0.171851i
\(936\) 0 0
\(937\) 3.58673i 0.117173i −0.998282 0.0585867i \(-0.981341\pi\)
0.998282 0.0585867i \(-0.0186594\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −22.4357 −0.731384 −0.365692 0.930736i \(-0.619168\pi\)
−0.365692 + 0.930736i \(0.619168\pi\)
\(942\) 0 0
\(943\) 20.0083i 0.651561i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24.5858i 0.798931i −0.916748 0.399465i \(-0.869196\pi\)
0.916748 0.399465i \(-0.130804\pi\)
\(948\) 0 0
\(949\) 27.3137 0.886640
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 53.2132i 1.72374i 0.507125 + 0.861872i \(0.330708\pi\)
−0.507125 + 0.861872i \(0.669292\pi\)
\(954\) 0 0
\(955\) 34.7904i 1.12579i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 25.8686 + 4.10051i 0.835342 + 0.132412i
\(960\) 0 0
\(961\) 12.2548 0.395317
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −8.65914 −0.278748
\(966\) 0 0
\(967\) 29.6569 0.953700 0.476850 0.878985i \(-0.341778\pi\)
0.476850 + 0.878985i \(0.341778\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −23.4412 −0.752264 −0.376132 0.926566i \(-0.622746\pi\)
−0.376132 + 0.926566i \(0.622746\pi\)
\(972\) 0 0
\(973\) 27.3137 + 4.32957i 0.875637 + 0.138800i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 17.8995i 0.572656i −0.958132 0.286328i \(-0.907565\pi\)
0.958132 0.286328i \(-0.0924346\pi\)
\(978\) 0 0
\(979\) 3.43289i 0.109716i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −18.5864 −0.592813 −0.296407 0.955062i \(-0.595788\pi\)
−0.296407 + 0.955062i \(0.595788\pi\)
\(984\) 0 0
\(985\) 22.0193i 0.701592i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 23.5147i 0.747725i
\(990\) 0 0
\(991\) −27.8579 −0.884934 −0.442467 0.896785i \(-0.645897\pi\)
−0.442467 + 0.896785i \(0.645897\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 5.25483i 0.166589i
\(996\) 0 0
\(997\) 54.0559i 1.71197i −0.517003 0.855984i \(-0.672952\pi\)
0.517003 0.855984i \(-0.327048\pi\)
\(998\) 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 1008.2.k.c.881.5 8
3.2 odd 2 inner 1008.2.k.c.881.3 8
4.3 odd 2 504.2.k.a.377.6 yes 8
7.6 odd 2 inner 1008.2.k.c.881.4 8
8.3 odd 2 4032.2.k.f.3905.4 8
8.5 even 2 4032.2.k.e.3905.3 8
12.11 even 2 504.2.k.a.377.4 yes 8
21.20 even 2 inner 1008.2.k.c.881.6 8
24.5 odd 2 4032.2.k.e.3905.5 8
24.11 even 2 4032.2.k.f.3905.6 8
28.3 even 6 3528.2.bl.b.1097.6 16
28.11 odd 6 3528.2.bl.b.1097.4 16
28.19 even 6 3528.2.bl.b.521.5 16
28.23 odd 6 3528.2.bl.b.521.3 16
28.27 even 2 504.2.k.a.377.3 8
56.13 odd 2 4032.2.k.e.3905.6 8
56.27 even 2 4032.2.k.f.3905.5 8
84.11 even 6 3528.2.bl.b.1097.5 16
84.23 even 6 3528.2.bl.b.521.6 16
84.47 odd 6 3528.2.bl.b.521.4 16
84.59 odd 6 3528.2.bl.b.1097.3 16
84.83 odd 2 504.2.k.a.377.5 yes 8
168.83 odd 2 4032.2.k.f.3905.3 8
168.125 even 2 4032.2.k.e.3905.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.k.a.377.3 8 28.27 even 2
504.2.k.a.377.4 yes 8 12.11 even 2
504.2.k.a.377.5 yes 8 84.83 odd 2
504.2.k.a.377.6 yes 8 4.3 odd 2
1008.2.k.c.881.3 8 3.2 odd 2 inner
1008.2.k.c.881.4 8 7.6 odd 2 inner
1008.2.k.c.881.5 8 1.1 even 1 trivial
1008.2.k.c.881.6 8 21.20 even 2 inner
3528.2.bl.b.521.3 16 28.23 odd 6
3528.2.bl.b.521.4 16 84.47 odd 6
3528.2.bl.b.521.5 16 28.19 even 6
3528.2.bl.b.521.6 16 84.23 even 6
3528.2.bl.b.1097.3 16 84.59 odd 6
3528.2.bl.b.1097.4 16 28.11 odd 6
3528.2.bl.b.1097.5 16 84.11 even 6
3528.2.bl.b.1097.6 16 28.3 even 6
4032.2.k.e.3905.3 8 8.5 even 2
4032.2.k.e.3905.4 8 168.125 even 2
4032.2.k.e.3905.5 8 24.5 odd 2
4032.2.k.e.3905.6 8 56.13 odd 2
4032.2.k.f.3905.3 8 168.83 odd 2
4032.2.k.f.3905.4 8 8.3 odd 2
4032.2.k.f.3905.5 8 56.27 even 2
4032.2.k.f.3905.6 8 24.11 even 2