Properties

Label 2352.2.k.g.881.2
Level $2352$
Weight $2$
Character 2352.881
Analytic conductor $18.781$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2352,2,Mod(881,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, 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("2352.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.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: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3288334336.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 8x^{6} - 8x^{5} + 14x^{4} + 8x^{3} - 16x^{2} + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 588)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.2
Root \(0.707107 - 0.349249i\) of defining polynomial
Character \(\chi\) \(=\) 2352.881
Dual form 2352.2.k.g.881.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.64533 + 0.541196i) q^{3} +0.963811 q^{5} +(2.41421 - 1.78089i) q^{9} +O(q^{10})\) \(q+(-1.64533 + 0.541196i) q^{3} +0.963811 q^{5} +(2.41421 - 1.78089i) q^{9} -4.29945i q^{11} +3.37849i q^{13} +(-1.58579 + 0.521611i) q^{15} +6.98054 q^{17} +3.06147i q^{19} +6.81801i q^{23} -4.07107 q^{25} +(-3.00836 + 4.23671i) q^{27} -7.86123i q^{29} -8.47343i q^{31} +(2.32685 + 7.07401i) q^{33} -6.24264 q^{37} +(-1.82843 - 5.55873i) q^{39} +3.68988 q^{41} +4.82843 q^{43} +(2.32685 - 1.71644i) q^{45} +3.29066 q^{47} +(-11.4853 + 3.77784i) q^{51} -1.04322i q^{53} -4.14386i q^{55} +(-1.65685 - 5.03712i) q^{57} +2.72607 q^{59} -3.37849i q^{61} +3.25623i q^{65} +2.00000 q^{67} +(-3.68988 - 11.2179i) q^{69} -10.3798i q^{71} +1.39942i q^{73} +(6.69825 - 2.20325i) q^{75} +11.6569 q^{79} +(2.65685 - 8.59890i) q^{81} +1.36303 q^{83} +6.72792 q^{85} +(4.25447 + 12.9343i) q^{87} +11.6342 q^{89} +(4.58579 + 13.9416i) q^{93} +2.95068i q^{95} +16.1815i q^{97} +(-7.65685 - 10.3798i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} - 24 q^{15} + 24 q^{25} - 16 q^{37} + 8 q^{39} + 16 q^{43} - 24 q^{51} + 32 q^{57} + 16 q^{67} + 48 q^{79} - 24 q^{81} - 48 q^{85} + 48 q^{93} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.64533 + 0.541196i −0.949931 + 0.312460i
\(4\) 0 0
\(5\) 0.963811 0.431030 0.215515 0.976501i \(-0.430857\pi\)
0.215515 + 0.976501i \(0.430857\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.41421 1.78089i 0.804738 0.593630i
\(10\) 0 0
\(11\) 4.29945i 1.29633i −0.761499 0.648167i \(-0.775536\pi\)
0.761499 0.648167i \(-0.224464\pi\)
\(12\) 0 0
\(13\) 3.37849i 0.937025i 0.883457 + 0.468513i \(0.155210\pi\)
−0.883457 + 0.468513i \(0.844790\pi\)
\(14\) 0 0
\(15\) −1.58579 + 0.521611i −0.409448 + 0.134679i
\(16\) 0 0
\(17\) 6.98054 1.69303 0.846515 0.532365i \(-0.178697\pi\)
0.846515 + 0.532365i \(0.178697\pi\)
\(18\) 0 0
\(19\) 3.06147i 0.702349i 0.936310 + 0.351174i \(0.114218\pi\)
−0.936310 + 0.351174i \(0.885782\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.81801i 1.42165i 0.703367 + 0.710827i \(0.251679\pi\)
−0.703367 + 0.710827i \(0.748321\pi\)
\(24\) 0 0
\(25\) −4.07107 −0.814214
\(26\) 0 0
\(27\) −3.00836 + 4.23671i −0.578960 + 0.815356i
\(28\) 0 0
\(29\) 7.86123i 1.45979i −0.683557 0.729897i \(-0.739568\pi\)
0.683557 0.729897i \(-0.260432\pi\)
\(30\) 0 0
\(31\) 8.47343i 1.52187i −0.648827 0.760936i \(-0.724740\pi\)
0.648827 0.760936i \(-0.275260\pi\)
\(32\) 0 0
\(33\) 2.32685 + 7.07401i 0.405052 + 1.23143i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.24264 −1.02628 −0.513142 0.858304i \(-0.671519\pi\)
−0.513142 + 0.858304i \(0.671519\pi\)
\(38\) 0 0
\(39\) −1.82843 5.55873i −0.292783 0.890109i
\(40\) 0 0
\(41\) 3.68988 0.576263 0.288131 0.957591i \(-0.406966\pi\)
0.288131 + 0.957591i \(0.406966\pi\)
\(42\) 0 0
\(43\) 4.82843 0.736328 0.368164 0.929761i \(-0.379986\pi\)
0.368164 + 0.929761i \(0.379986\pi\)
\(44\) 0 0
\(45\) 2.32685 1.71644i 0.346866 0.255872i
\(46\) 0 0
\(47\) 3.29066 0.479992 0.239996 0.970774i \(-0.422854\pi\)
0.239996 + 0.970774i \(0.422854\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −11.4853 + 3.77784i −1.60826 + 0.529003i
\(52\) 0 0
\(53\) 1.04322i 0.143298i −0.997430 0.0716488i \(-0.977174\pi\)
0.997430 0.0716488i \(-0.0228261\pi\)
\(54\) 0 0
\(55\) 4.14386i 0.558758i
\(56\) 0 0
\(57\) −1.65685 5.03712i −0.219456 0.667183i
\(58\) 0 0
\(59\) 2.72607 0.354904 0.177452 0.984129i \(-0.443215\pi\)
0.177452 + 0.984129i \(0.443215\pi\)
\(60\) 0 0
\(61\) 3.37849i 0.432572i −0.976330 0.216286i \(-0.930606\pi\)
0.976330 0.216286i \(-0.0693943\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.25623i 0.403886i
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) −3.68988 11.2179i −0.444210 1.35047i
\(70\) 0 0
\(71\) 10.3798i 1.23185i −0.787803 0.615927i \(-0.788781\pi\)
0.787803 0.615927i \(-0.211219\pi\)
\(72\) 0 0
\(73\) 1.39942i 0.163789i 0.996641 + 0.0818947i \(0.0260971\pi\)
−0.996641 + 0.0818947i \(0.973903\pi\)
\(74\) 0 0
\(75\) 6.69825 2.20325i 0.773447 0.254409i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.6569 1.31150 0.655749 0.754979i \(-0.272353\pi\)
0.655749 + 0.754979i \(0.272353\pi\)
\(80\) 0 0
\(81\) 2.65685 8.59890i 0.295206 0.955434i
\(82\) 0 0
\(83\) 1.36303 0.149613 0.0748063 0.997198i \(-0.476166\pi\)
0.0748063 + 0.997198i \(0.476166\pi\)
\(84\) 0 0
\(85\) 6.72792 0.729746
\(86\) 0 0
\(87\) 4.25447 + 12.9343i 0.456127 + 1.38670i
\(88\) 0 0
\(89\) 11.6342 1.23323 0.616613 0.787266i \(-0.288504\pi\)
0.616613 + 0.787266i \(0.288504\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.58579 + 13.9416i 0.475524 + 1.44567i
\(94\) 0 0
\(95\) 2.95068i 0.302733i
\(96\) 0 0
\(97\) 16.1815i 1.64298i 0.570222 + 0.821491i \(0.306857\pi\)
−0.570222 + 0.821491i \(0.693143\pi\)
\(98\) 0 0
\(99\) −7.65685 10.3798i −0.769543 1.04321i
\(100\) 0 0
\(101\) −8.90816 −0.886395 −0.443198 0.896424i \(-0.646156\pi\)
−0.443198 + 0.896424i \(0.646156\pi\)
\(102\) 0 0
\(103\) 13.0656i 1.28739i 0.765280 + 0.643697i \(0.222600\pi\)
−0.765280 + 0.643697i \(0.777400\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.33657i 0.902600i −0.892372 0.451300i \(-0.850960\pi\)
0.892372 0.451300i \(-0.149040\pi\)
\(108\) 0 0
\(109\) 6.58579 0.630804 0.315402 0.948958i \(-0.397861\pi\)
0.315402 + 0.948958i \(0.397861\pi\)
\(110\) 0 0
\(111\) 10.2712 3.37849i 0.974899 0.320672i
\(112\) 0 0
\(113\) 6.08034i 0.571990i 0.958231 + 0.285995i \(0.0923242\pi\)
−0.958231 + 0.285995i \(0.907676\pi\)
\(114\) 0 0
\(115\) 6.57128i 0.612775i
\(116\) 0 0
\(117\) 6.01673 + 8.15640i 0.556247 + 0.754060i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.48528 −0.680480
\(122\) 0 0
\(123\) −6.07107 + 1.99695i −0.547410 + 0.180059i
\(124\) 0 0
\(125\) −8.74280 −0.781980
\(126\) 0 0
\(127\) 11.1716 0.991317 0.495658 0.868518i \(-0.334927\pi\)
0.495658 + 0.868518i \(0.334927\pi\)
\(128\) 0 0
\(129\) −7.94435 + 2.61313i −0.699461 + 0.230073i
\(130\) 0 0
\(131\) 19.1794 1.67571 0.837854 0.545894i \(-0.183810\pi\)
0.837854 + 0.545894i \(0.183810\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.89949 + 4.08339i −0.249549 + 0.351443i
\(136\) 0 0
\(137\) 20.0219i 1.71059i −0.518143 0.855294i \(-0.673377\pi\)
0.518143 0.855294i \(-0.326623\pi\)
\(138\) 0 0
\(139\) 7.83938i 0.664927i 0.943116 + 0.332464i \(0.107880\pi\)
−0.943116 + 0.332464i \(0.892120\pi\)
\(140\) 0 0
\(141\) −5.41421 + 1.78089i −0.455959 + 0.149978i
\(142\) 0 0
\(143\) 14.5257 1.21470
\(144\) 0 0
\(145\) 7.57675i 0.629214i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16.7657i 1.37350i 0.726894 + 0.686749i \(0.240963\pi\)
−0.726894 + 0.686749i \(0.759037\pi\)
\(150\) 0 0
\(151\) 13.1716 1.07189 0.535944 0.844254i \(-0.319956\pi\)
0.535944 + 0.844254i \(0.319956\pi\)
\(152\) 0 0
\(153\) 16.8525 12.4316i 1.36244 1.00503i
\(154\) 0 0
\(155\) 8.16679i 0.655972i
\(156\) 0 0
\(157\) 0.131316i 0.0104802i 0.999986 + 0.00524009i \(0.00166798\pi\)
−0.999986 + 0.00524009i \(0.998332\pi\)
\(158\) 0 0
\(159\) 0.564588 + 1.71644i 0.0447747 + 0.136123i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.82843 0.221540 0.110770 0.993846i \(-0.464668\pi\)
0.110770 + 0.993846i \(0.464668\pi\)
\(164\) 0 0
\(165\) 2.24264 + 6.81801i 0.174589 + 0.530781i
\(166\) 0 0
\(167\) 7.14590 0.552967 0.276483 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(168\) 0 0
\(169\) 1.58579 0.121984
\(170\) 0 0
\(171\) 5.45214 + 7.39104i 0.416936 + 0.565207i
\(172\) 0 0
\(173\) 5.61750 0.427091 0.213545 0.976933i \(-0.431499\pi\)
0.213545 + 0.976933i \(0.431499\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.48528 + 1.47534i −0.337134 + 0.110893i
\(178\) 0 0
\(179\) 0.305553i 0.0228381i 0.999935 + 0.0114190i \(0.00363487\pi\)
−0.999935 + 0.0114190i \(0.996365\pi\)
\(180\) 0 0
\(181\) 6.62567i 0.492482i −0.969209 0.246241i \(-0.920805\pi\)
0.969209 0.246241i \(-0.0791955\pi\)
\(182\) 0 0
\(183\) 1.82843 + 5.55873i 0.135161 + 0.410913i
\(184\) 0 0
\(185\) −6.01673 −0.442359
\(186\) 0 0
\(187\) 30.0125i 2.19473i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.29335i 0.600086i 0.953926 + 0.300043i \(0.0970010\pi\)
−0.953926 + 0.300043i \(0.902999\pi\)
\(192\) 0 0
\(193\) 19.6569 1.41493 0.707466 0.706748i \(-0.249838\pi\)
0.707466 + 0.706748i \(0.249838\pi\)
\(194\) 0 0
\(195\) −1.76226 5.35757i −0.126198 0.383663i
\(196\) 0 0
\(197\) 22.2349i 1.58417i −0.610409 0.792086i \(-0.708995\pi\)
0.610409 0.792086i \(-0.291005\pi\)
\(198\) 0 0
\(199\) 17.2095i 1.21995i −0.792421 0.609974i \(-0.791180\pi\)
0.792421 0.609974i \(-0.208820\pi\)
\(200\) 0 0
\(201\) −3.29066 + 1.08239i −0.232105 + 0.0763461i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3.55635 0.248386
\(206\) 0 0
\(207\) 12.1421 + 16.4601i 0.843937 + 1.14406i
\(208\) 0 0
\(209\) 13.1626 0.910478
\(210\) 0 0
\(211\) 13.3137 0.916553 0.458277 0.888810i \(-0.348467\pi\)
0.458277 + 0.888810i \(0.348467\pi\)
\(212\) 0 0
\(213\) 5.61750 + 17.0782i 0.384905 + 1.17018i
\(214\) 0 0
\(215\) 4.65369 0.317379
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −0.757359 2.30250i −0.0511776 0.155589i
\(220\) 0 0
\(221\) 23.5837i 1.58641i
\(222\) 0 0
\(223\) 0.896683i 0.0600463i 0.999549 + 0.0300232i \(0.00955811\pi\)
−0.999549 + 0.0300232i \(0.990442\pi\)
\(224\) 0 0
\(225\) −9.82843 + 7.25013i −0.655228 + 0.483342i
\(226\) 0 0
\(227\) −16.6871 −1.10756 −0.553782 0.832661i \(-0.686816\pi\)
−0.553782 + 0.832661i \(0.686816\pi\)
\(228\) 0 0
\(229\) 11.8519i 0.783197i −0.920136 0.391599i \(-0.871922\pi\)
0.920136 0.391599i \(-0.128078\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 11.4230i 0.748347i −0.927359 0.374173i \(-0.877927\pi\)
0.927359 0.374173i \(-0.122073\pi\)
\(234\) 0 0
\(235\) 3.17157 0.206891
\(236\) 0 0
\(237\) −19.1794 + 6.30864i −1.24583 + 0.409790i
\(238\) 0 0
\(239\) 3.86733i 0.250157i 0.992147 + 0.125079i \(0.0399183\pi\)
−0.992147 + 0.125079i \(0.960082\pi\)
\(240\) 0 0
\(241\) 20.5880i 1.32619i −0.748536 0.663094i \(-0.769243\pi\)
0.748536 0.663094i \(-0.230757\pi\)
\(242\) 0 0
\(243\) 0.282294 + 15.5859i 0.0181092 + 0.999836i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −10.3431 −0.658119
\(248\) 0 0
\(249\) −2.24264 + 0.737669i −0.142122 + 0.0467479i
\(250\) 0 0
\(251\) −9.30739 −0.587477 −0.293738 0.955886i \(-0.594899\pi\)
−0.293738 + 0.955886i \(0.594899\pi\)
\(252\) 0 0
\(253\) 29.3137 1.84294
\(254\) 0 0
\(255\) −11.0696 + 3.64113i −0.693208 + 0.228016i
\(256\) 0 0
\(257\) 12.1988 0.760941 0.380471 0.924793i \(-0.375762\pi\)
0.380471 + 0.924793i \(0.375762\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −14.0000 18.9787i −0.866578 1.17475i
\(262\) 0 0
\(263\) 15.4169i 0.950648i −0.879811 0.475324i \(-0.842331\pi\)
0.879811 0.475324i \(-0.157669\pi\)
\(264\) 0 0
\(265\) 1.00547i 0.0617655i
\(266\) 0 0
\(267\) −19.1421 + 6.29640i −1.17148 + 0.385333i
\(268\) 0 0
\(269\) 16.2879 0.993092 0.496546 0.868010i \(-0.334601\pi\)
0.496546 + 0.868010i \(0.334601\pi\)
\(270\) 0 0
\(271\) 2.87576i 0.174690i 0.996178 + 0.0873449i \(0.0278382\pi\)
−0.996178 + 0.0873449i \(0.972162\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 17.5034i 1.05549i
\(276\) 0 0
\(277\) −29.4558 −1.76983 −0.884915 0.465752i \(-0.845784\pi\)
−0.884915 + 0.465752i \(0.845784\pi\)
\(278\) 0 0
\(279\) −15.0903 20.4567i −0.903430 1.22471i
\(280\) 0 0
\(281\) 19.4108i 1.15795i 0.815345 + 0.578976i \(0.196548\pi\)
−0.815345 + 0.578976i \(0.803452\pi\)
\(282\) 0 0
\(283\) 3.32410i 0.197597i −0.995107 0.0987986i \(-0.968500\pi\)
0.995107 0.0987986i \(-0.0315000\pi\)
\(284\) 0 0
\(285\) −1.59689 4.85483i −0.0945919 0.287576i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 31.7279 1.86635
\(290\) 0 0
\(291\) −8.75736 26.6239i −0.513365 1.56072i
\(292\) 0 0
\(293\) −10.8358 −0.633033 −0.316517 0.948587i \(-0.602513\pi\)
−0.316517 + 0.948587i \(0.602513\pi\)
\(294\) 0 0
\(295\) 2.62742 0.152974
\(296\) 0 0
\(297\) 18.2155 + 12.9343i 1.05697 + 0.750525i
\(298\) 0 0
\(299\) −23.0346 −1.33213
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 14.6569 4.82106i 0.842014 0.276963i
\(304\) 0 0
\(305\) 3.25623i 0.186451i
\(306\) 0 0
\(307\) 11.9832i 0.683919i 0.939715 + 0.341960i \(0.111091\pi\)
−0.939715 + 0.341960i \(0.888909\pi\)
\(308\) 0 0
\(309\) −7.07107 21.4973i −0.402259 1.22294i
\(310\) 0 0
\(311\) −24.9622 −1.41548 −0.707739 0.706474i \(-0.750285\pi\)
−0.707739 + 0.706474i \(0.750285\pi\)
\(312\) 0 0
\(313\) 18.8715i 1.06668i 0.845900 + 0.533341i \(0.179064\pi\)
−0.845900 + 0.533341i \(0.820936\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.08644i 0.117186i −0.998282 0.0585932i \(-0.981339\pi\)
0.998282 0.0585932i \(-0.0186615\pi\)
\(318\) 0 0
\(319\) −33.7990 −1.89238
\(320\) 0 0
\(321\) 5.05292 + 15.3617i 0.282026 + 0.857408i
\(322\) 0 0
\(323\) 21.3707i 1.18910i
\(324\) 0 0
\(325\) 13.7541i 0.762939i
\(326\) 0 0
\(327\) −10.8358 + 3.56420i −0.599220 + 0.197101i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −15.7990 −0.868391 −0.434196 0.900819i \(-0.642967\pi\)
−0.434196 + 0.900819i \(0.642967\pi\)
\(332\) 0 0
\(333\) −15.0711 + 11.1175i −0.825889 + 0.609233i
\(334\) 0 0
\(335\) 1.92762 0.105317
\(336\) 0 0
\(337\) 14.5858 0.794538 0.397269 0.917702i \(-0.369958\pi\)
0.397269 + 0.917702i \(0.369958\pi\)
\(338\) 0 0
\(339\) −3.29066 10.0042i −0.178724 0.543352i
\(340\) 0 0
\(341\) −36.4311 −1.97285
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −3.55635 10.8119i −0.191467 0.582094i
\(346\) 0 0
\(347\) 10.9909i 0.590022i 0.955494 + 0.295011i \(0.0953234\pi\)
−0.955494 + 0.295011i \(0.904677\pi\)
\(348\) 0 0
\(349\) 7.89377i 0.422544i 0.977427 + 0.211272i \(0.0677606\pi\)
−0.977427 + 0.211272i \(0.932239\pi\)
\(350\) 0 0
\(351\) −14.3137 10.1637i −0.764009 0.542500i
\(352\) 0 0
\(353\) 17.9817 0.957069 0.478534 0.878069i \(-0.341168\pi\)
0.478534 + 0.878069i \(0.341168\pi\)
\(354\) 0 0
\(355\) 10.0042i 0.530966i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.34267i 0.281976i 0.990011 + 0.140988i \(0.0450278\pi\)
−0.990011 + 0.140988i \(0.954972\pi\)
\(360\) 0 0
\(361\) 9.62742 0.506706
\(362\) 0 0
\(363\) 12.3157 4.05101i 0.646409 0.212623i
\(364\) 0 0
\(365\) 1.34877i 0.0705981i
\(366\) 0 0
\(367\) 4.96362i 0.259099i 0.991573 + 0.129549i \(0.0413531\pi\)
−0.991573 + 0.129549i \(0.958647\pi\)
\(368\) 0 0
\(369\) 8.90816 6.57128i 0.463740 0.342087i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 15.7990 0.818041 0.409020 0.912525i \(-0.365871\pi\)
0.409020 + 0.912525i \(0.365871\pi\)
\(374\) 0 0
\(375\) 14.3848 4.73157i 0.742827 0.244337i
\(376\) 0 0
\(377\) 26.5591 1.36786
\(378\) 0 0
\(379\) 1.65685 0.0851069 0.0425534 0.999094i \(-0.486451\pi\)
0.0425534 + 0.999094i \(0.486451\pi\)
\(380\) 0 0
\(381\) −18.3809 + 6.04601i −0.941683 + 0.309747i
\(382\) 0 0
\(383\) −20.5424 −1.04967 −0.524834 0.851205i \(-0.675872\pi\)
−0.524834 + 0.851205i \(0.675872\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 11.6569 8.59890i 0.592551 0.437107i
\(388\) 0 0
\(389\) 16.4601i 0.834562i 0.908778 + 0.417281i \(0.137017\pi\)
−0.908778 + 0.417281i \(0.862983\pi\)
\(390\) 0 0
\(391\) 47.5934i 2.40690i
\(392\) 0 0
\(393\) −31.5563 + 10.3798i −1.59181 + 0.523591i
\(394\) 0 0
\(395\) 11.2350 0.565295
\(396\) 0 0
\(397\) 12.3003i 0.617332i 0.951170 + 0.308666i \(0.0998826\pi\)
−0.951170 + 0.308666i \(0.900117\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 30.0962i 1.50293i −0.659773 0.751465i \(-0.729347\pi\)
0.659773 0.751465i \(-0.270653\pi\)
\(402\) 0 0
\(403\) 28.6274 1.42603
\(404\) 0 0
\(405\) 2.56071 8.28772i 0.127243 0.411820i
\(406\) 0 0
\(407\) 26.8399i 1.33041i
\(408\) 0 0
\(409\) 9.68714i 0.478998i 0.970897 + 0.239499i \(0.0769832\pi\)
−0.970897 + 0.239499i \(0.923017\pi\)
\(410\) 0 0
\(411\) 10.8358 + 32.9426i 0.534490 + 1.62494i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1.31371 0.0644874
\(416\) 0 0
\(417\) −4.24264 12.8984i −0.207763 0.631635i
\(418\) 0 0
\(419\) −12.3642 −0.604030 −0.302015 0.953303i \(-0.597659\pi\)
−0.302015 + 0.953303i \(0.597659\pi\)
\(420\) 0 0
\(421\) −27.7990 −1.35484 −0.677420 0.735597i \(-0.736902\pi\)
−0.677420 + 0.735597i \(0.736902\pi\)
\(422\) 0 0
\(423\) 7.94435 5.86030i 0.386268 0.284938i
\(424\) 0 0
\(425\) −28.4182 −1.37849
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −23.8995 + 7.86123i −1.15388 + 0.379544i
\(430\) 0 0
\(431\) 2.82411i 0.136033i −0.997684 0.0680164i \(-0.978333\pi\)
0.997684 0.0680164i \(-0.0216670\pi\)
\(432\) 0 0
\(433\) 5.91470i 0.284242i −0.989849 0.142121i \(-0.954608\pi\)
0.989849 0.142121i \(-0.0453922\pi\)
\(434\) 0 0
\(435\) 4.10051 + 12.4662i 0.196604 + 0.597710i
\(436\) 0 0
\(437\) −20.8731 −0.998497
\(438\) 0 0
\(439\) 34.4190i 1.64273i −0.570404 0.821364i \(-0.693213\pi\)
0.570404 0.821364i \(-0.306787\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 29.6640i 1.40938i −0.709515 0.704691i \(-0.751086\pi\)
0.709515 0.704691i \(-0.248914\pi\)
\(444\) 0 0
\(445\) 11.2132 0.531557
\(446\) 0 0
\(447\) −9.07353 27.5851i −0.429163 1.30473i
\(448\) 0 0
\(449\) 21.8028i 1.02894i 0.857509 + 0.514469i \(0.172011\pi\)
−0.857509 + 0.514469i \(0.827989\pi\)
\(450\) 0 0
\(451\) 15.8645i 0.747028i
\(452\) 0 0
\(453\) −21.6716 + 7.12840i −1.01822 + 0.334922i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4.00000 −0.187112 −0.0935561 0.995614i \(-0.529823\pi\)
−0.0935561 + 0.995614i \(0.529823\pi\)
\(458\) 0 0
\(459\) −21.0000 + 29.5745i −0.980196 + 1.38042i
\(460\) 0 0
\(461\) −40.4517 −1.88402 −0.942012 0.335580i \(-0.891068\pi\)
−0.942012 + 0.335580i \(0.891068\pi\)
\(462\) 0 0
\(463\) −16.2843 −0.756794 −0.378397 0.925643i \(-0.623525\pi\)
−0.378397 + 0.925643i \(0.623525\pi\)
\(464\) 0 0
\(465\) 4.41983 + 13.4370i 0.204965 + 0.623128i
\(466\) 0 0
\(467\) −36.4311 −1.68583 −0.842915 0.538047i \(-0.819162\pi\)
−0.842915 + 0.538047i \(0.819162\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −0.0710678 0.216058i −0.00327463 0.00995544i
\(472\) 0 0
\(473\) 20.7596i 0.954527i
\(474\) 0 0
\(475\) 12.4634i 0.571862i
\(476\) 0 0
\(477\) −1.85786 2.51856i −0.0850658 0.115317i
\(478\) 0 0
\(479\) 27.6883 1.26511 0.632555 0.774515i \(-0.282006\pi\)
0.632555 + 0.774515i \(0.282006\pi\)
\(480\) 0 0
\(481\) 21.0907i 0.961654i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 15.5959i 0.708173i
\(486\) 0 0
\(487\) −38.1421 −1.72839 −0.864193 0.503161i \(-0.832170\pi\)
−0.864193 + 0.503161i \(0.832170\pi\)
\(488\) 0 0
\(489\) −4.65369 + 1.53073i −0.210447 + 0.0692222i
\(490\) 0 0
\(491\) 14.9848i 0.676254i −0.941100 0.338127i \(-0.890207\pi\)
0.941100 0.338127i \(-0.109793\pi\)
\(492\) 0 0
\(493\) 54.8756i 2.47147i
\(494\) 0 0
\(495\) −7.37976 10.0042i −0.331696 0.449654i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.14214 0.0958952 0.0479476 0.998850i \(-0.484732\pi\)
0.0479476 + 0.998850i \(0.484732\pi\)
\(500\) 0 0
\(501\) −11.7574 + 3.86733i −0.525280 + 0.172780i
\(502\) 0 0
\(503\) 0.798447 0.0356010 0.0178005 0.999842i \(-0.494334\pi\)
0.0178005 + 0.999842i \(0.494334\pi\)
\(504\) 0 0
\(505\) −8.58579 −0.382062
\(506\) 0 0
\(507\) −2.60914 + 0.858221i −0.115876 + 0.0381150i
\(508\) 0 0
\(509\) −7.77899 −0.344798 −0.172399 0.985027i \(-0.555152\pi\)
−0.172399 + 0.985027i \(0.555152\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −12.9706 9.21001i −0.572664 0.406632i
\(514\) 0 0
\(515\) 12.5928i 0.554905i
\(516\) 0 0
\(517\) 14.1480i 0.622229i
\(518\) 0 0
\(519\) −9.24264 + 3.04017i −0.405707 + 0.133449i
\(520\) 0 0
\(521\) −36.2657 −1.58883 −0.794415 0.607375i \(-0.792223\pi\)
−0.794415 + 0.607375i \(0.792223\pi\)
\(522\) 0 0
\(523\) 14.0711i 0.615286i 0.951502 + 0.307643i \(0.0995403\pi\)
−0.951502 + 0.307643i \(0.900460\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 59.1491i 2.57658i
\(528\) 0 0
\(529\) −23.4853 −1.02110
\(530\) 0 0
\(531\) 6.58132 4.85483i 0.285605 0.210682i
\(532\) 0 0
\(533\) 12.4662i 0.539973i
\(534\) 0 0
\(535\) 8.99869i 0.389047i
\(536\) 0 0
\(537\) −0.165364 0.502734i −0.00713598 0.0216946i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 12.6274 0.542895 0.271448 0.962453i \(-0.412498\pi\)
0.271448 + 0.962453i \(0.412498\pi\)
\(542\) 0 0
\(543\) 3.58579 + 10.9014i 0.153881 + 0.467824i
\(544\) 0 0
\(545\) 6.34746 0.271895
\(546\) 0 0
\(547\) 6.97056 0.298040 0.149020 0.988834i \(-0.452388\pi\)
0.149020 + 0.988834i \(0.452388\pi\)
\(548\) 0 0
\(549\) −6.01673 8.15640i −0.256788 0.348107i
\(550\) 0 0
\(551\) 24.0669 1.02528
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 9.89949 3.25623i 0.420210 0.138219i
\(556\) 0 0
\(557\) 2.08644i 0.0884055i 0.999023 + 0.0442027i \(0.0140748\pi\)
−0.999023 + 0.0442027i \(0.985925\pi\)
\(558\) 0 0
\(559\) 16.3128i 0.689958i
\(560\) 0 0
\(561\) 16.2426 + 49.3804i 0.685765 + 2.08484i
\(562\) 0 0
\(563\) −1.59689 −0.0673011 −0.0336505 0.999434i \(-0.510713\pi\)
−0.0336505 + 0.999434i \(0.510713\pi\)
\(564\) 0 0
\(565\) 5.86030i 0.246545i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.38589i 0.267711i −0.991001 0.133855i \(-0.957264\pi\)
0.991001 0.133855i \(-0.0427357\pi\)
\(570\) 0 0
\(571\) −0.627417 −0.0262566 −0.0131283 0.999914i \(-0.504179\pi\)
−0.0131283 + 0.999914i \(0.504179\pi\)
\(572\) 0 0
\(573\) −4.48833 13.6453i −0.187503 0.570040i
\(574\) 0 0
\(575\) 27.7566i 1.15753i
\(576\) 0 0
\(577\) 12.0376i 0.501133i −0.968099 0.250567i \(-0.919383\pi\)
0.968099 0.250567i \(-0.0806169\pi\)
\(578\) 0 0
\(579\) −32.3420 + 10.6382i −1.34409 + 0.442109i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −4.48528 −0.185761
\(584\) 0 0
\(585\) 5.79899 + 7.86123i 0.239759 + 0.325022i
\(586\) 0 0
\(587\) 17.4856 0.721708 0.360854 0.932622i \(-0.382485\pi\)
0.360854 + 0.932622i \(0.382485\pi\)
\(588\) 0 0
\(589\) 25.9411 1.06889
\(590\) 0 0
\(591\) 12.0335 + 36.5838i 0.494990 + 1.50485i
\(592\) 0 0
\(593\) 0.729951 0.0299755 0.0149878 0.999888i \(-0.495229\pi\)
0.0149878 + 0.999888i \(0.495229\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 9.31371 + 28.3153i 0.381185 + 1.15887i
\(598\) 0 0
\(599\) 31.7505i 1.29729i 0.761091 + 0.648645i \(0.224664\pi\)
−0.761091 + 0.648645i \(0.775336\pi\)
\(600\) 0 0
\(601\) 30.9636i 1.26303i 0.775364 + 0.631515i \(0.217567\pi\)
−0.775364 + 0.631515i \(0.782433\pi\)
\(602\) 0 0
\(603\) 4.82843 3.56178i 0.196629 0.145047i
\(604\) 0 0
\(605\) −7.21440 −0.293307
\(606\) 0 0
\(607\) 14.7821i 0.599986i 0.953941 + 0.299993i \(0.0969843\pi\)
−0.953941 + 0.299993i \(0.903016\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 11.1175i 0.449764i
\(612\) 0 0
\(613\) 24.7279 0.998751 0.499376 0.866386i \(-0.333563\pi\)
0.499376 + 0.866386i \(0.333563\pi\)
\(614\) 0 0
\(615\) −5.85136 + 1.92468i −0.235950 + 0.0776107i
\(616\) 0 0
\(617\) 6.38589i 0.257086i −0.991704 0.128543i \(-0.958970\pi\)
0.991704 0.128543i \(-0.0410301\pi\)
\(618\) 0 0
\(619\) 42.2584i 1.69851i 0.527986 + 0.849253i \(0.322948\pi\)
−0.527986 + 0.849253i \(0.677052\pi\)
\(620\) 0 0
\(621\) −28.8860 20.5111i −1.15915 0.823080i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.9289 0.477157
\(626\) 0 0
\(627\) −21.6569 + 7.12356i −0.864891 + 0.284488i
\(628\) 0 0
\(629\) −43.5770 −1.73753
\(630\) 0 0
\(631\) 7.51472 0.299156 0.149578 0.988750i \(-0.452208\pi\)
0.149578 + 0.988750i \(0.452208\pi\)
\(632\) 0 0
\(633\) −21.9054 + 7.20533i −0.870663 + 0.286386i
\(634\) 0 0
\(635\) 10.7673 0.427287
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −18.4853 25.0590i −0.731266 0.991320i
\(640\) 0 0
\(641\) 0.126564i 0.00499898i −0.999997 0.00249949i \(-0.999204\pi\)
0.999997 0.00249949i \(-0.000795613\pi\)
\(642\) 0 0
\(643\) 30.4608i 1.20126i −0.799528 0.600629i \(-0.794917\pi\)
0.799528 0.600629i \(-0.205083\pi\)
\(644\) 0 0
\(645\) −7.65685 + 2.51856i −0.301488 + 0.0991682i
\(646\) 0 0
\(647\) −36.1972 −1.42306 −0.711530 0.702656i \(-0.751997\pi\)
−0.711530 + 0.702656i \(0.751997\pi\)
\(648\) 0 0
\(649\) 11.7206i 0.460074i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.33657i 0.365368i 0.983172 + 0.182684i \(0.0584786\pi\)
−0.983172 + 0.182684i \(0.941521\pi\)
\(654\) 0 0
\(655\) 18.4853 0.722280
\(656\) 0 0
\(657\) 2.49221 + 3.37849i 0.0972304 + 0.131808i
\(658\) 0 0
\(659\) 21.4973i 0.837414i 0.908121 + 0.418707i \(0.137517\pi\)
−0.908121 + 0.418707i \(0.862483\pi\)
\(660\) 0 0
\(661\) 47.2764i 1.83884i 0.393280 + 0.919419i \(0.371340\pi\)
−0.393280 + 0.919419i \(0.628660\pi\)
\(662\) 0 0
\(663\) −12.7634 38.8029i −0.495690 1.50698i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 53.5980 2.07532
\(668\) 0 0
\(669\) −0.485281 1.47534i −0.0187621 0.0570399i
\(670\) 0 0
\(671\) −14.5257 −0.560757
\(672\) 0 0
\(673\) 6.92893 0.267091 0.133545 0.991043i \(-0.457364\pi\)
0.133545 + 0.991043i \(0.457364\pi\)
\(674\) 0 0
\(675\) 12.2473 17.2480i 0.471397 0.663874i
\(676\) 0 0
\(677\) 28.0875 1.07949 0.539746 0.841828i \(-0.318520\pi\)
0.539746 + 0.841828i \(0.318520\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 27.4558 9.03102i 1.05211 0.346069i
\(682\) 0 0
\(683\) 9.76869i 0.373788i 0.982380 + 0.186894i \(0.0598422\pi\)
−0.982380 + 0.186894i \(0.940158\pi\)
\(684\) 0 0
\(685\) 19.2974i 0.737314i
\(686\) 0 0
\(687\) 6.41421 + 19.5003i 0.244718 + 0.743983i
\(688\) 0 0
\(689\) 3.52452 0.134273
\(690\) 0 0
\(691\) 50.3922i 1.91701i −0.285074 0.958505i \(-0.592018\pi\)
0.285074 0.958505i \(-0.407982\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.55568i 0.286603i
\(696\) 0 0
\(697\) 25.7574 0.975630
\(698\) 0 0
\(699\) 6.18209 + 18.7946i 0.233828 + 0.710878i
\(700\) 0 0
\(701\) 25.9233i 0.979108i 0.871973 + 0.489554i \(0.162840\pi\)
−0.871973 + 0.489554i \(0.837160\pi\)
\(702\) 0 0
\(703\) 19.1116i 0.720809i
\(704\) 0 0
\(705\) −5.21828 + 1.71644i −0.196532 + 0.0646450i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −47.0711 −1.76779 −0.883896 0.467684i \(-0.845088\pi\)
−0.883896 + 0.467684i \(0.845088\pi\)
\(710\) 0 0
\(711\) 28.1421 20.7596i 1.05541 0.778545i
\(712\) 0 0
\(713\) 57.7719 2.16358
\(714\) 0 0
\(715\) 14.0000 0.523570
\(716\) 0 0
\(717\) −2.09299 6.36304i −0.0781640 0.237632i
\(718\) 0 0
\(719\) 6.91204 0.257776 0.128888 0.991659i \(-0.458859\pi\)
0.128888 + 0.991659i \(0.458859\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 11.1421 + 33.8740i 0.414380 + 1.25979i
\(724\) 0 0
\(725\) 32.0036i 1.18858i
\(726\) 0 0
\(727\) 1.97908i 0.0733998i 0.999326 + 0.0366999i \(0.0116846\pi\)
−0.999326 + 0.0366999i \(0.988315\pi\)
\(728\) 0 0
\(729\) −8.89949 25.4912i −0.329611 0.944117i
\(730\) 0 0
\(731\) 33.7050 1.24663
\(732\) 0 0
\(733\) 46.0083i 1.69935i −0.527303 0.849677i \(-0.676797\pi\)
0.527303 0.849677i \(-0.323203\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.59890i 0.316745i
\(738\) 0 0
\(739\) 38.4264 1.41354 0.706769 0.707444i \(-0.250152\pi\)
0.706769 + 0.707444i \(0.250152\pi\)
\(740\) 0 0
\(741\) 17.0179 5.59767i 0.625167 0.205636i
\(742\) 0 0
\(743\) 28.6208i 1.05000i 0.851103 + 0.524998i \(0.175934\pi\)
−0.851103 + 0.524998i \(0.824066\pi\)
\(744\) 0 0
\(745\) 16.1590i 0.592018i
\(746\) 0 0
\(747\) 3.29066 2.42742i 0.120399 0.0888145i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −16.3431 −0.596370 −0.298185 0.954508i \(-0.596381\pi\)
−0.298185 + 0.954508i \(0.596381\pi\)
\(752\) 0 0
\(753\) 15.3137 5.03712i 0.558063 0.183563i
\(754\) 0 0
\(755\) 12.6949 0.462015
\(756\) 0 0
\(757\) −6.58579 −0.239364 −0.119682 0.992812i \(-0.538188\pi\)
−0.119682 + 0.992812i \(0.538188\pi\)
\(758\) 0 0
\(759\) −48.2307 + 15.8645i −1.75066 + 0.575844i
\(760\) 0 0
\(761\) −4.25447 −0.154224 −0.0771122 0.997022i \(-0.524570\pi\)
−0.0771122 + 0.997022i \(0.524570\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 16.2426 11.9817i 0.587254 0.433199i
\(766\) 0 0
\(767\) 9.21001i 0.332554i
\(768\) 0 0
\(769\) 2.48181i 0.0894963i −0.998998 0.0447482i \(-0.985751\pi\)
0.998998 0.0447482i \(-0.0142485\pi\)
\(770\) 0 0
\(771\) −20.0711 + 6.60195i −0.722842 + 0.237764i
\(772\) 0 0
\(773\) 7.54513 0.271379 0.135690 0.990751i \(-0.456675\pi\)
0.135690 + 0.990751i \(0.456675\pi\)
\(774\) 0 0
\(775\) 34.4959i 1.23913i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 11.2965i 0.404737i
\(780\) 0 0
\(781\) −44.6274 −1.59689
\(782\) 0 0
\(783\) 33.3058 + 23.6494i 1.19025 + 0.845162i
\(784\) 0 0
\(785\) 0.126564i 0.00451726i
\(786\) 0 0
\(787\) 35.5014i 1.26549i −0.774361 0.632744i \(-0.781929\pi\)
0.774361 0.632744i \(-0.218071\pi\)
\(788\) 0 0
\(789\) 8.34357 + 25.3659i 0.297039 + 0.903050i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 11.4142 0.405331
\(794\) 0 0
\(795\) 0.544156 + 1.65433i 0.0192992 + 0.0586729i
\(796\) 0 0
\(797\) 18.5463 0.656943 0.328471 0.944514i \(-0.393467\pi\)
0.328471 + 0.944514i \(0.393467\pi\)
\(798\) 0 0
\(799\) 22.9706 0.812640
\(800\) 0 0
\(801\) 28.0875 20.7193i 0.992424 0.732080i
\(802\) 0 0
\(803\) 6.01673 0.212326
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −26.7990 + 8.81496i −0.943369 + 0.310301i
\(808\) 0 0
\(809\) 15.1114i 0.531287i 0.964071 + 0.265644i \(0.0855844\pi\)
−0.964071 + 0.265644i \(0.914416\pi\)
\(810\) 0 0
\(811\) 33.9706i 1.19287i −0.802661 0.596435i \(-0.796583\pi\)
0.802661 0.596435i \(-0.203417\pi\)
\(812\) 0 0
\(813\) −1.55635 4.73157i −0.0545835 0.165943i
\(814\) 0 0
\(815\) 2.72607 0.0954901
\(816\) 0 0
\(817\) 14.7821i 0.517159i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 48.0316i 1.67632i −0.545428 0.838158i \(-0.683633\pi\)
0.545428 0.838158i \(-0.316367\pi\)
\(822\) 0 0
\(823\) −24.3431 −0.848549 −0.424274 0.905534i \(-0.639471\pi\)
−0.424274 + 0.905534i \(0.639471\pi\)
\(824\) 0 0
\(825\) −9.47275 28.7988i −0.329799 1.00264i
\(826\) 0 0
\(827\) 14.9848i 0.521072i 0.965464 + 0.260536i \(0.0838993\pi\)
−0.965464 + 0.260536i \(0.916101\pi\)
\(828\) 0 0
\(829\) 21.1132i 0.733293i 0.930360 + 0.366647i \(0.119494\pi\)
−0.930360 + 0.366647i \(0.880506\pi\)
\(830\) 0 0
\(831\) 48.4645 15.9414i 1.68122 0.553001i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 6.88730 0.238345
\(836\) 0 0
\(837\) 35.8995 + 25.4912i 1.24087 + 0.881103i
\(838\) 0 0
\(839\) −35.8665 −1.23825 −0.619125 0.785293i \(-0.712512\pi\)
−0.619125 + 0.785293i \(0.712512\pi\)
\(840\) 0 0
\(841\) −32.7990 −1.13100
\(842\) 0 0
\(843\) −10.5051 31.9372i −0.361813 1.09997i
\(844\) 0 0
\(845\) 1.52840 0.0525785
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 1.79899 + 5.46924i 0.0617412 + 0.187704i
\(850\) 0 0
\(851\) 42.5624i 1.45902i
\(852\) 0 0
\(853\) 41.6787i 1.42705i 0.700629 + 0.713526i \(0.252903\pi\)
−0.700629 + 0.713526i \(0.747097\pi\)
\(854\) 0 0
\(855\) 5.25483 + 7.12356i 0.179712 + 0.243621i
\(856\) 0 0
\(857\) 37.9595 1.29667 0.648336 0.761355i \(-0.275465\pi\)
0.648336 + 0.761355i \(0.275465\pi\)
\(858\) 0 0
\(859\) 4.59220i 0.156684i 0.996927 + 0.0783419i \(0.0249626\pi\)
−0.996927 + 0.0783419i \(0.975037\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 11.4230i 0.388844i 0.980918 + 0.194422i \(0.0622831\pi\)
−0.980918 + 0.194422i \(0.937717\pi\)
\(864\) 0 0
\(865\) 5.41421 0.184089
\(866\) 0 0
\(867\) −52.2029 + 17.1710i −1.77290 + 0.583159i
\(868\) 0 0
\(869\) 50.1181i 1.70014i
\(870\) 0 0
\(871\) 6.75699i 0.228952i
\(872\) 0 0
\(873\) 28.8175 + 39.0656i 0.975324 + 1.32217i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −15.3553 −0.518513 −0.259256 0.965809i \(-0.583477\pi\)
−0.259256 + 0.965809i \(0.583477\pi\)
\(878\) 0 0
\(879\) 17.8284 5.86428i 0.601338 0.197797i
\(880\) 0 0
\(881\) −11.9650 −0.403110 −0.201555 0.979477i \(-0.564599\pi\)
−0.201555 + 0.979477i \(0.564599\pi\)
\(882\) 0 0
\(883\) −35.3137 −1.18840 −0.594200 0.804317i \(-0.702531\pi\)
−0.594200 + 0.804317i \(0.702531\pi\)
\(884\) 0 0
\(885\) −4.32296 + 1.42195i −0.145315 + 0.0477983i
\(886\) 0 0
\(887\) −51.2875 −1.72207 −0.861033 0.508550i \(-0.830182\pi\)
−0.861033 + 0.508550i \(0.830182\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −36.9706 11.4230i −1.23856 0.382685i
\(892\) 0 0
\(893\) 10.0742i 0.337122i
\(894\) 0 0
\(895\) 0.294495i 0.00984388i
\(896\) 0 0
\(897\) 37.8995 12.4662i 1.26543 0.416236i
\(898\) 0 0
\(899\) −66.6116 −2.22162
\(900\) 0 0
\(901\) 7.28225i 0.242607i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.38589i 0.212274i
\(906\) 0 0
\(907\) 33.9411 1.12700 0.563498 0.826117i \(-0.309455\pi\)
0.563498 + 0.826117i \(0.309455\pi\)
\(908\) 0 0
\(909\) −21.5062 + 15.8645i −0.713316 + 0.526191i
\(910\) 0 0
\(911\) 30.7073i 1.01738i −0.860951 0.508689i \(-0.830130\pi\)
0.860951 0.508689i \(-0.169870\pi\)
\(912\) 0 0
\(913\) 5.86030i 0.193948i
\(914\) 0 0
\(915\) 1.76226 + 5.35757i 0.0582585 + 0.177116i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −1.02944 −0.0339580 −0.0169790 0.999856i \(-0.505405\pi\)
−0.0169790 + 0.999856i \(0.505405\pi\)
\(920\) 0 0
\(921\) −6.48528 19.7164i −0.213697 0.649676i
\(922\) 0 0
\(923\) 35.0681 1.15428
\(924\) 0 0
\(925\) 25.4142 0.835614
\(926\) 0 0
\(927\) 23.2685 + 31.5432i 0.764237 + 1.03602i
\(928\) 0 0
\(929\) 1.52840 0.0501451 0.0250726 0.999686i \(-0.492018\pi\)
0.0250726 + 0.999686i \(0.492018\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 41.0711 13.5095i 1.34461 0.442280i
\(934\) 0 0
\(935\) 28.9264i 0.945994i
\(936\) 0 0
\(937\) 8.04762i 0.262904i −0.991323 0.131452i \(-0.958036\pi\)
0.991323 0.131452i \(-0.0419640\pi\)
\(938\) 0 0
\(939\) −10.2132 31.0499i −0.333295 1.01327i
\(940\) 0 0
\(941\) 45.3393 1.47802 0.739009 0.673696i \(-0.235294\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(942\) 0 0
\(943\) 25.1577i 0.819246i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 23.4047i 0.760551i 0.924873 + 0.380275i \(0.124171\pi\)
−0.924873 + 0.380275i \(0.875829\pi\)
\(948\) 0 0
\(949\) −4.72792 −0.153475
\(950\) 0 0
\(951\) 1.12918 + 3.43289i 0.0366160 + 0.111319i
\(952\) 0 0
\(953\) 5.03712i 0.163168i −0.996666 0.0815842i \(-0.974002\pi\)
0.996666 0.0815842i \(-0.0259979\pi\)
\(954\) 0 0
\(955\) 7.99322i 0.258655i
\(956\) 0 0
\(957\) 55.6105 18.2919i 1.79763 0.591293i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −40.7990 −1.31610
\(962\) 0 0
\(963\) −16.6274 22.5405i −0.535811 0.726357i
\(964\) 0 0
\(965\) 18.9455 0.609877
\(966\) 0 0
\(967\) −51.9411 −1.67031 −0.835157 0.550012i \(-0.814623\pi\)
−0.835157 + 0.550012i \(0.814623\pi\)
\(968\) 0 0
\(969\) −11.5657 35.1618i −0.371545 1.12956i
\(970\) 0 0
\(971\) 28.1560 0.903570 0.451785 0.892127i \(-0.350787\pi\)
0.451785 + 0.892127i \(0.350787\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 7.44365 + 22.6300i 0.238388 + 0.724739i
\(976\) 0 0
\(977\) 17.0712i 0.546157i −0.961992 0.273079i \(-0.911958\pi\)
0.961992 0.273079i \(-0.0880419\pi\)
\(978\) 0 0
\(979\) 50.0208i 1.59867i
\(980\) 0 0
\(981\) 15.8995 11.7286i 0.507632 0.374464i
\(982\) 0 0
\(983\) −24.8654 −0.793082 −0.396541 0.918017i \(-0.629789\pi\)
−0.396541 + 0.918017i \(0.629789\pi\)
\(984\) 0 0
\(985\) 21.4303i 0.682825i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 32.9203i 1.04680i
\(990\) 0 0
\(991\) −22.2843 −0.707883 −0.353942 0.935268i \(-0.615159\pi\)
−0.353942 + 0.935268i \(0.615159\pi\)
\(992\) 0 0
\(993\) 25.9945 8.55035i 0.824912 0.271337i
\(994\) 0 0
\(995\) 16.5867i 0.525834i
\(996\) 0 0
\(997\) 36.5612i 1.15791i 0.815361 + 0.578953i \(0.196539\pi\)
−0.815361 + 0.578953i \(0.803461\pi\)
\(998\) 0 0
\(999\) 18.7801 26.4483i 0.594177 0.836787i
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 2352.2.k.g.881.2 8
3.2 odd 2 inner 2352.2.k.g.881.8 8
4.3 odd 2 588.2.f.d.293.7 yes 8
7.6 odd 2 inner 2352.2.k.g.881.7 8
12.11 even 2 588.2.f.d.293.1 8
21.20 even 2 inner 2352.2.k.g.881.1 8
28.3 even 6 588.2.k.f.509.7 16
28.11 odd 6 588.2.k.f.509.2 16
28.19 even 6 588.2.k.f.521.5 16
28.23 odd 6 588.2.k.f.521.4 16
28.27 even 2 588.2.f.d.293.2 yes 8
84.11 even 6 588.2.k.f.509.5 16
84.23 even 6 588.2.k.f.521.7 16
84.47 odd 6 588.2.k.f.521.2 16
84.59 odd 6 588.2.k.f.509.4 16
84.83 odd 2 588.2.f.d.293.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.2.f.d.293.1 8 12.11 even 2
588.2.f.d.293.2 yes 8 28.27 even 2
588.2.f.d.293.7 yes 8 4.3 odd 2
588.2.f.d.293.8 yes 8 84.83 odd 2
588.2.k.f.509.2 16 28.11 odd 6
588.2.k.f.509.4 16 84.59 odd 6
588.2.k.f.509.5 16 84.11 even 6
588.2.k.f.509.7 16 28.3 even 6
588.2.k.f.521.2 16 84.47 odd 6
588.2.k.f.521.4 16 28.23 odd 6
588.2.k.f.521.5 16 28.19 even 6
588.2.k.f.521.7 16 84.23 even 6
2352.2.k.g.881.1 8 21.20 even 2 inner
2352.2.k.g.881.2 8 1.1 even 1 trivial
2352.2.k.g.881.7 8 7.6 odd 2 inner
2352.2.k.g.881.8 8 3.2 odd 2 inner