Properties

Label 4864.2.a.bn.1.6
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $8$
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] = [4864,2,Mod(1,4864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4864, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4864.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.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: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 13x^{6} + 24x^{5} + 48x^{4} - 68x^{3} - 62x^{2} + 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-2.81542\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.70663 q^{3} +1.66222 q^{5} -1.99556 q^{7} -0.0874066 q^{9} +O(q^{10})\) \(q+1.70663 q^{3} +1.66222 q^{5} -1.99556 q^{7} -0.0874066 q^{9} +2.08741 q^{11} -4.77322 q^{13} +2.83680 q^{15} +2.10657 q^{17} +1.00000 q^{19} -3.40569 q^{21} -4.84437 q^{23} -2.23703 q^{25} -5.26907 q^{27} +0.695946 q^{29} -9.77587 q^{31} +3.56244 q^{33} -3.31706 q^{35} -0.0772780 q^{37} -8.14614 q^{39} -10.7743 q^{41} -1.43840 q^{43} -0.145289 q^{45} +2.88133 q^{47} -3.01774 q^{49} +3.59513 q^{51} -9.00264 q^{53} +3.46973 q^{55} +1.70663 q^{57} +11.5253 q^{59} +8.82910 q^{61} +0.174425 q^{63} -7.93414 q^{65} +1.27859 q^{67} -8.26756 q^{69} -4.66655 q^{71} -4.44578 q^{73} -3.81779 q^{75} -4.16555 q^{77} -1.10044 q^{79} -8.73014 q^{81} +2.47419 q^{83} +3.50157 q^{85} +1.18772 q^{87} +15.8259 q^{89} +9.52526 q^{91} -16.6838 q^{93} +1.66222 q^{95} -13.9046 q^{97} -0.182453 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5} - 4 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} - 4 q^{7} + 12 q^{9} + 4 q^{11} - 8 q^{13} - 4 q^{17} + 8 q^{19} - 16 q^{21} + 12 q^{25} - 28 q^{29} - 8 q^{31} - 12 q^{35} - 4 q^{37} + 4 q^{39} - 8 q^{41} - 4 q^{43} - 24 q^{45} - 12 q^{47} + 12 q^{49} + 12 q^{51} - 32 q^{53} + 8 q^{55} + 12 q^{59} - 8 q^{61} + 16 q^{63} + 8 q^{65} - 4 q^{67} - 28 q^{69} + 24 q^{71} - 24 q^{77} + 24 q^{79} - 8 q^{81} + 40 q^{83} - 24 q^{85} - 24 q^{87} + 8 q^{89} - 4 q^{91} - 32 q^{93} - 8 q^{95} + 16 q^{97} - 76 q^{99}+O(q^{100}) \) 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
\(3\) 1.70663 0.985325 0.492662 0.870221i \(-0.336024\pi\)
0.492662 + 0.870221i \(0.336024\pi\)
\(4\) 0 0
\(5\) 1.66222 0.743367 0.371683 0.928360i \(-0.378781\pi\)
0.371683 + 0.928360i \(0.378781\pi\)
\(6\) 0 0
\(7\) −1.99556 −0.754251 −0.377125 0.926162i \(-0.623087\pi\)
−0.377125 + 0.926162i \(0.623087\pi\)
\(8\) 0 0
\(9\) −0.0874066 −0.0291355
\(10\) 0 0
\(11\) 2.08741 0.629377 0.314688 0.949195i \(-0.398100\pi\)
0.314688 + 0.949195i \(0.398100\pi\)
\(12\) 0 0
\(13\) −4.77322 −1.32385 −0.661927 0.749568i \(-0.730261\pi\)
−0.661927 + 0.749568i \(0.730261\pi\)
\(14\) 0 0
\(15\) 2.83680 0.732457
\(16\) 0 0
\(17\) 2.10657 0.510917 0.255459 0.966820i \(-0.417774\pi\)
0.255459 + 0.966820i \(0.417774\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.40569 −0.743182
\(22\) 0 0
\(23\) −4.84437 −1.01012 −0.505061 0.863084i \(-0.668530\pi\)
−0.505061 + 0.863084i \(0.668530\pi\)
\(24\) 0 0
\(25\) −2.23703 −0.447406
\(26\) 0 0
\(27\) −5.26907 −1.01403
\(28\) 0 0
\(29\) 0.695946 0.129234 0.0646170 0.997910i \(-0.479417\pi\)
0.0646170 + 0.997910i \(0.479417\pi\)
\(30\) 0 0
\(31\) −9.77587 −1.75580 −0.877899 0.478846i \(-0.841055\pi\)
−0.877899 + 0.478846i \(0.841055\pi\)
\(32\) 0 0
\(33\) 3.56244 0.620140
\(34\) 0 0
\(35\) −3.31706 −0.560685
\(36\) 0 0
\(37\) −0.0772780 −0.0127044 −0.00635221 0.999980i \(-0.502022\pi\)
−0.00635221 + 0.999980i \(0.502022\pi\)
\(38\) 0 0
\(39\) −8.14614 −1.30443
\(40\) 0 0
\(41\) −10.7743 −1.68267 −0.841334 0.540516i \(-0.818229\pi\)
−0.841334 + 0.540516i \(0.818229\pi\)
\(42\) 0 0
\(43\) −1.43840 −0.219354 −0.109677 0.993967i \(-0.534982\pi\)
−0.109677 + 0.993967i \(0.534982\pi\)
\(44\) 0 0
\(45\) −0.145289 −0.0216584
\(46\) 0 0
\(47\) 2.88133 0.420285 0.210143 0.977671i \(-0.432607\pi\)
0.210143 + 0.977671i \(0.432607\pi\)
\(48\) 0 0
\(49\) −3.01774 −0.431106
\(50\) 0 0
\(51\) 3.59513 0.503419
\(52\) 0 0
\(53\) −9.00264 −1.23661 −0.618304 0.785939i \(-0.712180\pi\)
−0.618304 + 0.785939i \(0.712180\pi\)
\(54\) 0 0
\(55\) 3.46973 0.467858
\(56\) 0 0
\(57\) 1.70663 0.226049
\(58\) 0 0
\(59\) 11.5253 1.50046 0.750230 0.661177i \(-0.229943\pi\)
0.750230 + 0.661177i \(0.229943\pi\)
\(60\) 0 0
\(61\) 8.82910 1.13045 0.565225 0.824937i \(-0.308789\pi\)
0.565225 + 0.824937i \(0.308789\pi\)
\(62\) 0 0
\(63\) 0.174425 0.0219755
\(64\) 0 0
\(65\) −7.93414 −0.984109
\(66\) 0 0
\(67\) 1.27859 0.156205 0.0781023 0.996945i \(-0.475114\pi\)
0.0781023 + 0.996945i \(0.475114\pi\)
\(68\) 0 0
\(69\) −8.26756 −0.995297
\(70\) 0 0
\(71\) −4.66655 −0.553818 −0.276909 0.960896i \(-0.589310\pi\)
−0.276909 + 0.960896i \(0.589310\pi\)
\(72\) 0 0
\(73\) −4.44578 −0.520339 −0.260170 0.965563i \(-0.583779\pi\)
−0.260170 + 0.965563i \(0.583779\pi\)
\(74\) 0 0
\(75\) −3.81779 −0.440840
\(76\) 0 0
\(77\) −4.16555 −0.474708
\(78\) 0 0
\(79\) −1.10044 −0.123809 −0.0619044 0.998082i \(-0.519717\pi\)
−0.0619044 + 0.998082i \(0.519717\pi\)
\(80\) 0 0
\(81\) −8.73014 −0.970016
\(82\) 0 0
\(83\) 2.47419 0.271578 0.135789 0.990738i \(-0.456643\pi\)
0.135789 + 0.990738i \(0.456643\pi\)
\(84\) 0 0
\(85\) 3.50157 0.379799
\(86\) 0 0
\(87\) 1.18772 0.127337
\(88\) 0 0
\(89\) 15.8259 1.67755 0.838773 0.544481i \(-0.183273\pi\)
0.838773 + 0.544481i \(0.183273\pi\)
\(90\) 0 0
\(91\) 9.52526 0.998518
\(92\) 0 0
\(93\) −16.6838 −1.73003
\(94\) 0 0
\(95\) 1.66222 0.170540
\(96\) 0 0
\(97\) −13.9046 −1.41180 −0.705898 0.708313i \(-0.749456\pi\)
−0.705898 + 0.708313i \(0.749456\pi\)
\(98\) 0 0
\(99\) −0.182453 −0.0183372
\(100\) 0 0
\(101\) −1.00493 −0.0999945 −0.0499973 0.998749i \(-0.515921\pi\)
−0.0499973 + 0.998749i \(0.515921\pi\)
\(102\) 0 0
\(103\) 12.4581 1.22753 0.613767 0.789487i \(-0.289653\pi\)
0.613767 + 0.789487i \(0.289653\pi\)
\(104\) 0 0
\(105\) −5.66100 −0.552457
\(106\) 0 0
\(107\) 15.3004 1.47915 0.739573 0.673076i \(-0.235027\pi\)
0.739573 + 0.673076i \(0.235027\pi\)
\(108\) 0 0
\(109\) −7.65850 −0.733551 −0.366775 0.930309i \(-0.619538\pi\)
−0.366775 + 0.930309i \(0.619538\pi\)
\(110\) 0 0
\(111\) −0.131885 −0.0125180
\(112\) 0 0
\(113\) 17.3938 1.63627 0.818134 0.575028i \(-0.195009\pi\)
0.818134 + 0.575028i \(0.195009\pi\)
\(114\) 0 0
\(115\) −8.05240 −0.750891
\(116\) 0 0
\(117\) 0.417211 0.0385712
\(118\) 0 0
\(119\) −4.20378 −0.385360
\(120\) 0 0
\(121\) −6.64273 −0.603885
\(122\) 0 0
\(123\) −18.3878 −1.65797
\(124\) 0 0
\(125\) −12.0295 −1.07595
\(126\) 0 0
\(127\) −18.1488 −1.61045 −0.805225 0.592970i \(-0.797955\pi\)
−0.805225 + 0.592970i \(0.797955\pi\)
\(128\) 0 0
\(129\) −2.45482 −0.216135
\(130\) 0 0
\(131\) −9.70500 −0.847930 −0.423965 0.905679i \(-0.639362\pi\)
−0.423965 + 0.905679i \(0.639362\pi\)
\(132\) 0 0
\(133\) −1.99556 −0.173037
\(134\) 0 0
\(135\) −8.75834 −0.753798
\(136\) 0 0
\(137\) 20.2253 1.72797 0.863983 0.503521i \(-0.167962\pi\)
0.863983 + 0.503521i \(0.167962\pi\)
\(138\) 0 0
\(139\) −16.7746 −1.42280 −0.711401 0.702786i \(-0.751939\pi\)
−0.711401 + 0.702786i \(0.751939\pi\)
\(140\) 0 0
\(141\) 4.91737 0.414117
\(142\) 0 0
\(143\) −9.96366 −0.833203
\(144\) 0 0
\(145\) 1.15681 0.0960682
\(146\) 0 0
\(147\) −5.15017 −0.424779
\(148\) 0 0
\(149\) −13.6533 −1.11853 −0.559263 0.828991i \(-0.688916\pi\)
−0.559263 + 0.828991i \(0.688916\pi\)
\(150\) 0 0
\(151\) 6.77602 0.551425 0.275712 0.961240i \(-0.411086\pi\)
0.275712 + 0.961240i \(0.411086\pi\)
\(152\) 0 0
\(153\) −0.184128 −0.0148858
\(154\) 0 0
\(155\) −16.2496 −1.30520
\(156\) 0 0
\(157\) −23.4178 −1.86895 −0.934473 0.356035i \(-0.884129\pi\)
−0.934473 + 0.356035i \(0.884129\pi\)
\(158\) 0 0
\(159\) −15.3642 −1.21846
\(160\) 0 0
\(161\) 9.66724 0.761885
\(162\) 0 0
\(163\) −1.06973 −0.0837875 −0.0418938 0.999122i \(-0.513339\pi\)
−0.0418938 + 0.999122i \(0.513339\pi\)
\(164\) 0 0
\(165\) 5.92155 0.460992
\(166\) 0 0
\(167\) 13.6217 1.05408 0.527039 0.849841i \(-0.323302\pi\)
0.527039 + 0.849841i \(0.323302\pi\)
\(168\) 0 0
\(169\) 9.78367 0.752590
\(170\) 0 0
\(171\) −0.0874066 −0.00668415
\(172\) 0 0
\(173\) 11.5684 0.879527 0.439763 0.898114i \(-0.355062\pi\)
0.439763 + 0.898114i \(0.355062\pi\)
\(174\) 0 0
\(175\) 4.46413 0.337456
\(176\) 0 0
\(177\) 19.6694 1.47844
\(178\) 0 0
\(179\) 20.4983 1.53211 0.766057 0.642772i \(-0.222216\pi\)
0.766057 + 0.642772i \(0.222216\pi\)
\(180\) 0 0
\(181\) −18.7745 −1.39550 −0.697750 0.716341i \(-0.745815\pi\)
−0.697750 + 0.716341i \(0.745815\pi\)
\(182\) 0 0
\(183\) 15.0680 1.11386
\(184\) 0 0
\(185\) −0.128453 −0.00944404
\(186\) 0 0
\(187\) 4.39726 0.321560
\(188\) 0 0
\(189\) 10.5147 0.764835
\(190\) 0 0
\(191\) 14.0392 1.01584 0.507919 0.861405i \(-0.330415\pi\)
0.507919 + 0.861405i \(0.330415\pi\)
\(192\) 0 0
\(193\) −23.8700 −1.71820 −0.859099 0.511809i \(-0.828976\pi\)
−0.859099 + 0.511809i \(0.828976\pi\)
\(194\) 0 0
\(195\) −13.5407 −0.969667
\(196\) 0 0
\(197\) −1.84890 −0.131729 −0.0658645 0.997829i \(-0.520980\pi\)
−0.0658645 + 0.997829i \(0.520980\pi\)
\(198\) 0 0
\(199\) −18.3608 −1.30156 −0.650781 0.759266i \(-0.725558\pi\)
−0.650781 + 0.759266i \(0.725558\pi\)
\(200\) 0 0
\(201\) 2.18208 0.153912
\(202\) 0 0
\(203\) −1.38880 −0.0974748
\(204\) 0 0
\(205\) −17.9093 −1.25084
\(206\) 0 0
\(207\) 0.423430 0.0294304
\(208\) 0 0
\(209\) 2.08741 0.144389
\(210\) 0 0
\(211\) 15.5147 1.06808 0.534039 0.845460i \(-0.320674\pi\)
0.534039 + 0.845460i \(0.320674\pi\)
\(212\) 0 0
\(213\) −7.96409 −0.545690
\(214\) 0 0
\(215\) −2.39094 −0.163061
\(216\) 0 0
\(217\) 19.5083 1.32431
\(218\) 0 0
\(219\) −7.58731 −0.512703
\(220\) 0 0
\(221\) −10.0551 −0.676380
\(222\) 0 0
\(223\) 13.9260 0.932553 0.466276 0.884639i \(-0.345595\pi\)
0.466276 + 0.884639i \(0.345595\pi\)
\(224\) 0 0
\(225\) 0.195531 0.0130354
\(226\) 0 0
\(227\) −1.49624 −0.0993088 −0.0496544 0.998766i \(-0.515812\pi\)
−0.0496544 + 0.998766i \(0.515812\pi\)
\(228\) 0 0
\(229\) −1.03706 −0.0685310 −0.0342655 0.999413i \(-0.510909\pi\)
−0.0342655 + 0.999413i \(0.510909\pi\)
\(230\) 0 0
\(231\) −7.10905 −0.467741
\(232\) 0 0
\(233\) −4.93372 −0.323219 −0.161609 0.986855i \(-0.551668\pi\)
−0.161609 + 0.986855i \(0.551668\pi\)
\(234\) 0 0
\(235\) 4.78940 0.312426
\(236\) 0 0
\(237\) −1.87804 −0.121992
\(238\) 0 0
\(239\) 3.84533 0.248733 0.124367 0.992236i \(-0.460310\pi\)
0.124367 + 0.992236i \(0.460310\pi\)
\(240\) 0 0
\(241\) −1.41134 −0.0909123 −0.0454562 0.998966i \(-0.514474\pi\)
−0.0454562 + 0.998966i \(0.514474\pi\)
\(242\) 0 0
\(243\) 0.908064 0.0582523
\(244\) 0 0
\(245\) −5.01614 −0.320469
\(246\) 0 0
\(247\) −4.77322 −0.303713
\(248\) 0 0
\(249\) 4.22253 0.267592
\(250\) 0 0
\(251\) −8.24640 −0.520508 −0.260254 0.965540i \(-0.583806\pi\)
−0.260254 + 0.965540i \(0.583806\pi\)
\(252\) 0 0
\(253\) −10.1122 −0.635747
\(254\) 0 0
\(255\) 5.97590 0.374225
\(256\) 0 0
\(257\) 15.0726 0.940200 0.470100 0.882613i \(-0.344218\pi\)
0.470100 + 0.882613i \(0.344218\pi\)
\(258\) 0 0
\(259\) 0.154213 0.00958232
\(260\) 0 0
\(261\) −0.0608303 −0.00376530
\(262\) 0 0
\(263\) −9.49141 −0.585265 −0.292633 0.956225i \(-0.594531\pi\)
−0.292633 + 0.956225i \(0.594531\pi\)
\(264\) 0 0
\(265\) −14.9644 −0.919253
\(266\) 0 0
\(267\) 27.0091 1.65293
\(268\) 0 0
\(269\) −3.98549 −0.243000 −0.121500 0.992591i \(-0.538770\pi\)
−0.121500 + 0.992591i \(0.538770\pi\)
\(270\) 0 0
\(271\) 6.38427 0.387817 0.193908 0.981020i \(-0.437884\pi\)
0.193908 + 0.981020i \(0.437884\pi\)
\(272\) 0 0
\(273\) 16.2561 0.983865
\(274\) 0 0
\(275\) −4.66959 −0.281587
\(276\) 0 0
\(277\) −15.1758 −0.911823 −0.455912 0.890025i \(-0.650687\pi\)
−0.455912 + 0.890025i \(0.650687\pi\)
\(278\) 0 0
\(279\) 0.854475 0.0511561
\(280\) 0 0
\(281\) −14.9058 −0.889207 −0.444603 0.895728i \(-0.646655\pi\)
−0.444603 + 0.895728i \(0.646655\pi\)
\(282\) 0 0
\(283\) 14.9931 0.891250 0.445625 0.895220i \(-0.352981\pi\)
0.445625 + 0.895220i \(0.352981\pi\)
\(284\) 0 0
\(285\) 2.83680 0.168037
\(286\) 0 0
\(287\) 21.5008 1.26915
\(288\) 0 0
\(289\) −12.5624 −0.738963
\(290\) 0 0
\(291\) −23.7300 −1.39108
\(292\) 0 0
\(293\) −21.6983 −1.26763 −0.633814 0.773485i \(-0.718512\pi\)
−0.633814 + 0.773485i \(0.718512\pi\)
\(294\) 0 0
\(295\) 19.1575 1.11539
\(296\) 0 0
\(297\) −10.9987 −0.638208
\(298\) 0 0
\(299\) 23.1233 1.33725
\(300\) 0 0
\(301\) 2.87042 0.165448
\(302\) 0 0
\(303\) −1.71505 −0.0985270
\(304\) 0 0
\(305\) 14.6759 0.840339
\(306\) 0 0
\(307\) −7.81449 −0.445996 −0.222998 0.974819i \(-0.571584\pi\)
−0.222998 + 0.974819i \(0.571584\pi\)
\(308\) 0 0
\(309\) 21.2614 1.20952
\(310\) 0 0
\(311\) 12.8288 0.727455 0.363727 0.931505i \(-0.381504\pi\)
0.363727 + 0.931505i \(0.381504\pi\)
\(312\) 0 0
\(313\) 4.07578 0.230377 0.115188 0.993344i \(-0.463253\pi\)
0.115188 + 0.993344i \(0.463253\pi\)
\(314\) 0 0
\(315\) 0.289933 0.0163358
\(316\) 0 0
\(317\) 2.26448 0.127186 0.0635931 0.997976i \(-0.479744\pi\)
0.0635931 + 0.997976i \(0.479744\pi\)
\(318\) 0 0
\(319\) 1.45272 0.0813368
\(320\) 0 0
\(321\) 26.1122 1.45744
\(322\) 0 0
\(323\) 2.10657 0.117212
\(324\) 0 0
\(325\) 10.6778 0.592300
\(326\) 0 0
\(327\) −13.0702 −0.722786
\(328\) 0 0
\(329\) −5.74987 −0.317000
\(330\) 0 0
\(331\) 7.31171 0.401888 0.200944 0.979603i \(-0.435599\pi\)
0.200944 + 0.979603i \(0.435599\pi\)
\(332\) 0 0
\(333\) 0.00675460 0.000370150 0
\(334\) 0 0
\(335\) 2.12529 0.116117
\(336\) 0 0
\(337\) −6.00356 −0.327035 −0.163517 0.986540i \(-0.552284\pi\)
−0.163517 + 0.986540i \(0.552284\pi\)
\(338\) 0 0
\(339\) 29.6848 1.61225
\(340\) 0 0
\(341\) −20.4062 −1.10506
\(342\) 0 0
\(343\) 19.9910 1.07941
\(344\) 0 0
\(345\) −13.7425 −0.739871
\(346\) 0 0
\(347\) 6.66747 0.357929 0.178964 0.983856i \(-0.442725\pi\)
0.178964 + 0.983856i \(0.442725\pi\)
\(348\) 0 0
\(349\) 35.5072 1.90066 0.950329 0.311248i \(-0.100747\pi\)
0.950329 + 0.311248i \(0.100747\pi\)
\(350\) 0 0
\(351\) 25.1504 1.34243
\(352\) 0 0
\(353\) −27.8315 −1.48132 −0.740659 0.671881i \(-0.765487\pi\)
−0.740659 + 0.671881i \(0.765487\pi\)
\(354\) 0 0
\(355\) −7.75683 −0.411690
\(356\) 0 0
\(357\) −7.17431 −0.379705
\(358\) 0 0
\(359\) 10.6778 0.563551 0.281775 0.959480i \(-0.409077\pi\)
0.281775 + 0.959480i \(0.409077\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −11.3367 −0.595023
\(364\) 0 0
\(365\) −7.38986 −0.386803
\(366\) 0 0
\(367\) −16.8950 −0.881911 −0.440956 0.897529i \(-0.645360\pi\)
−0.440956 + 0.897529i \(0.645360\pi\)
\(368\) 0 0
\(369\) 0.941747 0.0490254
\(370\) 0 0
\(371\) 17.9653 0.932713
\(372\) 0 0
\(373\) 33.4640 1.73270 0.866350 0.499438i \(-0.166460\pi\)
0.866350 + 0.499438i \(0.166460\pi\)
\(374\) 0 0
\(375\) −20.5300 −1.06016
\(376\) 0 0
\(377\) −3.32191 −0.171087
\(378\) 0 0
\(379\) −37.0251 −1.90185 −0.950925 0.309423i \(-0.899864\pi\)
−0.950925 + 0.309423i \(0.899864\pi\)
\(380\) 0 0
\(381\) −30.9734 −1.58682
\(382\) 0 0
\(383\) 16.2264 0.829131 0.414566 0.910019i \(-0.363934\pi\)
0.414566 + 0.910019i \(0.363934\pi\)
\(384\) 0 0
\(385\) −6.92405 −0.352882
\(386\) 0 0
\(387\) 0.125726 0.00639100
\(388\) 0 0
\(389\) −11.8131 −0.598947 −0.299473 0.954105i \(-0.596811\pi\)
−0.299473 + 0.954105i \(0.596811\pi\)
\(390\) 0 0
\(391\) −10.2050 −0.516089
\(392\) 0 0
\(393\) −16.5629 −0.835486
\(394\) 0 0
\(395\) −1.82917 −0.0920353
\(396\) 0 0
\(397\) −10.9654 −0.550340 −0.275170 0.961396i \(-0.588734\pi\)
−0.275170 + 0.961396i \(0.588734\pi\)
\(398\) 0 0
\(399\) −3.40569 −0.170498
\(400\) 0 0
\(401\) 11.9479 0.596650 0.298325 0.954464i \(-0.403572\pi\)
0.298325 + 0.954464i \(0.403572\pi\)
\(402\) 0 0
\(403\) 46.6624 2.32442
\(404\) 0 0
\(405\) −14.5114 −0.721077
\(406\) 0 0
\(407\) −0.161311 −0.00799587
\(408\) 0 0
\(409\) 1.28997 0.0637848 0.0318924 0.999491i \(-0.489847\pi\)
0.0318924 + 0.999491i \(0.489847\pi\)
\(410\) 0 0
\(411\) 34.5172 1.70261
\(412\) 0 0
\(413\) −22.9993 −1.13172
\(414\) 0 0
\(415\) 4.11265 0.201882
\(416\) 0 0
\(417\) −28.6281 −1.40192
\(418\) 0 0
\(419\) −15.6802 −0.766030 −0.383015 0.923742i \(-0.625114\pi\)
−0.383015 + 0.923742i \(0.625114\pi\)
\(420\) 0 0
\(421\) −26.2544 −1.27956 −0.639782 0.768557i \(-0.720975\pi\)
−0.639782 + 0.768557i \(0.720975\pi\)
\(422\) 0 0
\(423\) −0.251847 −0.0122452
\(424\) 0 0
\(425\) −4.71245 −0.228587
\(426\) 0 0
\(427\) −17.6190 −0.852643
\(428\) 0 0
\(429\) −17.0043 −0.820975
\(430\) 0 0
\(431\) 33.4648 1.61194 0.805972 0.591954i \(-0.201643\pi\)
0.805972 + 0.591954i \(0.201643\pi\)
\(432\) 0 0
\(433\) 5.89119 0.283112 0.141556 0.989930i \(-0.454789\pi\)
0.141556 + 0.989930i \(0.454789\pi\)
\(434\) 0 0
\(435\) 1.97426 0.0946584
\(436\) 0 0
\(437\) −4.84437 −0.231738
\(438\) 0 0
\(439\) 25.9770 1.23982 0.619908 0.784674i \(-0.287170\pi\)
0.619908 + 0.784674i \(0.287170\pi\)
\(440\) 0 0
\(441\) 0.263770 0.0125605
\(442\) 0 0
\(443\) −11.0652 −0.525722 −0.262861 0.964834i \(-0.584666\pi\)
−0.262861 + 0.964834i \(0.584666\pi\)
\(444\) 0 0
\(445\) 26.3062 1.24703
\(446\) 0 0
\(447\) −23.3012 −1.10211
\(448\) 0 0
\(449\) 3.56950 0.168455 0.0842274 0.996447i \(-0.473158\pi\)
0.0842274 + 0.996447i \(0.473158\pi\)
\(450\) 0 0
\(451\) −22.4904 −1.05903
\(452\) 0 0
\(453\) 11.5642 0.543332
\(454\) 0 0
\(455\) 15.8331 0.742265
\(456\) 0 0
\(457\) 23.3258 1.09113 0.545567 0.838067i \(-0.316314\pi\)
0.545567 + 0.838067i \(0.316314\pi\)
\(458\) 0 0
\(459\) −11.0996 −0.518087
\(460\) 0 0
\(461\) −8.16887 −0.380462 −0.190231 0.981739i \(-0.560924\pi\)
−0.190231 + 0.981739i \(0.560924\pi\)
\(462\) 0 0
\(463\) 12.8015 0.594936 0.297468 0.954732i \(-0.403858\pi\)
0.297468 + 0.954732i \(0.403858\pi\)
\(464\) 0 0
\(465\) −27.7321 −1.28605
\(466\) 0 0
\(467\) 0.897425 0.0415279 0.0207639 0.999784i \(-0.493390\pi\)
0.0207639 + 0.999784i \(0.493390\pi\)
\(468\) 0 0
\(469\) −2.55150 −0.117817
\(470\) 0 0
\(471\) −39.9656 −1.84152
\(472\) 0 0
\(473\) −3.00253 −0.138056
\(474\) 0 0
\(475\) −2.23703 −0.102642
\(476\) 0 0
\(477\) 0.786890 0.0360292
\(478\) 0 0
\(479\) −12.9960 −0.593803 −0.296901 0.954908i \(-0.595953\pi\)
−0.296901 + 0.954908i \(0.595953\pi\)
\(480\) 0 0
\(481\) 0.368865 0.0168188
\(482\) 0 0
\(483\) 16.4984 0.750704
\(484\) 0 0
\(485\) −23.1125 −1.04948
\(486\) 0 0
\(487\) 12.0621 0.546588 0.273294 0.961931i \(-0.411887\pi\)
0.273294 + 0.961931i \(0.411887\pi\)
\(488\) 0 0
\(489\) −1.82563 −0.0825579
\(490\) 0 0
\(491\) −30.5337 −1.37797 −0.688983 0.724778i \(-0.741942\pi\)
−0.688983 + 0.724778i \(0.741942\pi\)
\(492\) 0 0
\(493\) 1.46606 0.0660279
\(494\) 0 0
\(495\) −0.303277 −0.0136313
\(496\) 0 0
\(497\) 9.31239 0.417718
\(498\) 0 0
\(499\) −40.2124 −1.80015 −0.900077 0.435731i \(-0.856490\pi\)
−0.900077 + 0.435731i \(0.856490\pi\)
\(500\) 0 0
\(501\) 23.2472 1.03861
\(502\) 0 0
\(503\) 35.7181 1.59259 0.796296 0.604907i \(-0.206790\pi\)
0.796296 + 0.604907i \(0.206790\pi\)
\(504\) 0 0
\(505\) −1.67042 −0.0743326
\(506\) 0 0
\(507\) 16.6971 0.741545
\(508\) 0 0
\(509\) −35.3428 −1.56654 −0.783270 0.621681i \(-0.786450\pi\)
−0.783270 + 0.621681i \(0.786450\pi\)
\(510\) 0 0
\(511\) 8.87183 0.392466
\(512\) 0 0
\(513\) −5.26907 −0.232635
\(514\) 0 0
\(515\) 20.7081 0.912508
\(516\) 0 0
\(517\) 6.01450 0.264518
\(518\) 0 0
\(519\) 19.7429 0.866619
\(520\) 0 0
\(521\) −2.50662 −0.109817 −0.0549086 0.998491i \(-0.517487\pi\)
−0.0549086 + 0.998491i \(0.517487\pi\)
\(522\) 0 0
\(523\) 9.95330 0.435227 0.217614 0.976035i \(-0.430173\pi\)
0.217614 + 0.976035i \(0.430173\pi\)
\(524\) 0 0
\(525\) 7.61863 0.332504
\(526\) 0 0
\(527\) −20.5935 −0.897068
\(528\) 0 0
\(529\) 0.467940 0.0203452
\(530\) 0 0
\(531\) −1.00738 −0.0437167
\(532\) 0 0
\(533\) 51.4283 2.22761
\(534\) 0 0
\(535\) 25.4326 1.09955
\(536\) 0 0
\(537\) 34.9831 1.50963
\(538\) 0 0
\(539\) −6.29925 −0.271328
\(540\) 0 0
\(541\) 34.7061 1.49213 0.746065 0.665873i \(-0.231941\pi\)
0.746065 + 0.665873i \(0.231941\pi\)
\(542\) 0 0
\(543\) −32.0412 −1.37502
\(544\) 0 0
\(545\) −12.7301 −0.545297
\(546\) 0 0
\(547\) −7.21429 −0.308461 −0.154230 0.988035i \(-0.549290\pi\)
−0.154230 + 0.988035i \(0.549290\pi\)
\(548\) 0 0
\(549\) −0.771721 −0.0329363
\(550\) 0 0
\(551\) 0.695946 0.0296483
\(552\) 0 0
\(553\) 2.19599 0.0933829
\(554\) 0 0
\(555\) −0.219222 −0.00930545
\(556\) 0 0
\(557\) 4.27230 0.181023 0.0905116 0.995895i \(-0.471150\pi\)
0.0905116 + 0.995895i \(0.471150\pi\)
\(558\) 0 0
\(559\) 6.86582 0.290393
\(560\) 0 0
\(561\) 7.50451 0.316840
\(562\) 0 0
\(563\) 23.9662 1.01005 0.505027 0.863103i \(-0.331482\pi\)
0.505027 + 0.863103i \(0.331482\pi\)
\(564\) 0 0
\(565\) 28.9122 1.21635
\(566\) 0 0
\(567\) 17.4215 0.731635
\(568\) 0 0
\(569\) −6.53074 −0.273783 −0.136891 0.990586i \(-0.543711\pi\)
−0.136891 + 0.990586i \(0.543711\pi\)
\(570\) 0 0
\(571\) −13.8125 −0.578035 −0.289017 0.957324i \(-0.593329\pi\)
−0.289017 + 0.957324i \(0.593329\pi\)
\(572\) 0 0
\(573\) 23.9597 1.00093
\(574\) 0 0
\(575\) 10.8370 0.451934
\(576\) 0 0
\(577\) −1.75601 −0.0731035 −0.0365517 0.999332i \(-0.511637\pi\)
−0.0365517 + 0.999332i \(0.511637\pi\)
\(578\) 0 0
\(579\) −40.7373 −1.69298
\(580\) 0 0
\(581\) −4.93740 −0.204838
\(582\) 0 0
\(583\) −18.7922 −0.778293
\(584\) 0 0
\(585\) 0.693496 0.0286725
\(586\) 0 0
\(587\) 22.1818 0.915540 0.457770 0.889071i \(-0.348648\pi\)
0.457770 + 0.889071i \(0.348648\pi\)
\(588\) 0 0
\(589\) −9.77587 −0.402808
\(590\) 0 0
\(591\) −3.15540 −0.129796
\(592\) 0 0
\(593\) −14.8347 −0.609186 −0.304593 0.952483i \(-0.598520\pi\)
−0.304593 + 0.952483i \(0.598520\pi\)
\(594\) 0 0
\(595\) −6.98760 −0.286464
\(596\) 0 0
\(597\) −31.3351 −1.28246
\(598\) 0 0
\(599\) −19.9337 −0.814468 −0.407234 0.913324i \(-0.633507\pi\)
−0.407234 + 0.913324i \(0.633507\pi\)
\(600\) 0 0
\(601\) −31.4765 −1.28395 −0.641976 0.766724i \(-0.721885\pi\)
−0.641976 + 0.766724i \(0.721885\pi\)
\(602\) 0 0
\(603\) −0.111757 −0.00455110
\(604\) 0 0
\(605\) −11.0417 −0.448908
\(606\) 0 0
\(607\) −25.8245 −1.04818 −0.524092 0.851662i \(-0.675595\pi\)
−0.524092 + 0.851662i \(0.675595\pi\)
\(608\) 0 0
\(609\) −2.37018 −0.0960443
\(610\) 0 0
\(611\) −13.7532 −0.556396
\(612\) 0 0
\(613\) −0.844965 −0.0341278 −0.0170639 0.999854i \(-0.505432\pi\)
−0.0170639 + 0.999854i \(0.505432\pi\)
\(614\) 0 0
\(615\) −30.5646 −1.23248
\(616\) 0 0
\(617\) 10.0344 0.403970 0.201985 0.979389i \(-0.435261\pi\)
0.201985 + 0.979389i \(0.435261\pi\)
\(618\) 0 0
\(619\) 29.2896 1.17725 0.588625 0.808406i \(-0.299669\pi\)
0.588625 + 0.808406i \(0.299669\pi\)
\(620\) 0 0
\(621\) 25.5253 1.02430
\(622\) 0 0
\(623\) −31.5816 −1.26529
\(624\) 0 0
\(625\) −8.81055 −0.352422
\(626\) 0 0
\(627\) 3.56244 0.142270
\(628\) 0 0
\(629\) −0.162791 −0.00649091
\(630\) 0 0
\(631\) −1.02437 −0.0407796 −0.0203898 0.999792i \(-0.506491\pi\)
−0.0203898 + 0.999792i \(0.506491\pi\)
\(632\) 0 0
\(633\) 26.4779 1.05240
\(634\) 0 0
\(635\) −30.1673 −1.19715
\(636\) 0 0
\(637\) 14.4043 0.570721
\(638\) 0 0
\(639\) 0.407887 0.0161358
\(640\) 0 0
\(641\) −31.8367 −1.25747 −0.628737 0.777618i \(-0.716428\pi\)
−0.628737 + 0.777618i \(0.716428\pi\)
\(642\) 0 0
\(643\) −34.2996 −1.35264 −0.676322 0.736606i \(-0.736427\pi\)
−0.676322 + 0.736606i \(0.736427\pi\)
\(644\) 0 0
\(645\) −4.08045 −0.160668
\(646\) 0 0
\(647\) 9.45559 0.371738 0.185869 0.982575i \(-0.440490\pi\)
0.185869 + 0.982575i \(0.440490\pi\)
\(648\) 0 0
\(649\) 24.0579 0.944355
\(650\) 0 0
\(651\) 33.2936 1.30488
\(652\) 0 0
\(653\) 6.33451 0.247888 0.123944 0.992289i \(-0.460446\pi\)
0.123944 + 0.992289i \(0.460446\pi\)
\(654\) 0 0
\(655\) −16.1318 −0.630323
\(656\) 0 0
\(657\) 0.388590 0.0151604
\(658\) 0 0
\(659\) 7.22355 0.281390 0.140695 0.990053i \(-0.455066\pi\)
0.140695 + 0.990053i \(0.455066\pi\)
\(660\) 0 0
\(661\) 0.529621 0.0205999 0.0102999 0.999947i \(-0.496721\pi\)
0.0102999 + 0.999947i \(0.496721\pi\)
\(662\) 0 0
\(663\) −17.1604 −0.666454
\(664\) 0 0
\(665\) −3.31706 −0.128630
\(666\) 0 0
\(667\) −3.37142 −0.130542
\(668\) 0 0
\(669\) 23.7665 0.918867
\(670\) 0 0
\(671\) 18.4299 0.711479
\(672\) 0 0
\(673\) 41.7434 1.60909 0.804544 0.593893i \(-0.202410\pi\)
0.804544 + 0.593893i \(0.202410\pi\)
\(674\) 0 0
\(675\) 11.7871 0.453684
\(676\) 0 0
\(677\) 33.4165 1.28430 0.642151 0.766578i \(-0.278042\pi\)
0.642151 + 0.766578i \(0.278042\pi\)
\(678\) 0 0
\(679\) 27.7474 1.06485
\(680\) 0 0
\(681\) −2.55353 −0.0978514
\(682\) 0 0
\(683\) 30.1765 1.15467 0.577336 0.816507i \(-0.304092\pi\)
0.577336 + 0.816507i \(0.304092\pi\)
\(684\) 0 0
\(685\) 33.6189 1.28451
\(686\) 0 0
\(687\) −1.76988 −0.0675252
\(688\) 0 0
\(689\) 42.9716 1.63709
\(690\) 0 0
\(691\) 17.6002 0.669544 0.334772 0.942299i \(-0.391341\pi\)
0.334772 + 0.942299i \(0.391341\pi\)
\(692\) 0 0
\(693\) 0.364096 0.0138309
\(694\) 0 0
\(695\) −27.8830 −1.05766
\(696\) 0 0
\(697\) −22.6968 −0.859704
\(698\) 0 0
\(699\) −8.42005 −0.318475
\(700\) 0 0
\(701\) −30.6993 −1.15950 −0.579748 0.814796i \(-0.696849\pi\)
−0.579748 + 0.814796i \(0.696849\pi\)
\(702\) 0 0
\(703\) −0.0772780 −0.00291459
\(704\) 0 0
\(705\) 8.17374 0.307841
\(706\) 0 0
\(707\) 2.00540 0.0754210
\(708\) 0 0
\(709\) 8.79437 0.330279 0.165140 0.986270i \(-0.447193\pi\)
0.165140 + 0.986270i \(0.447193\pi\)
\(710\) 0 0
\(711\) 0.0961853 0.00360723
\(712\) 0 0
\(713\) 47.3579 1.77357
\(714\) 0 0
\(715\) −16.5618 −0.619375
\(716\) 0 0
\(717\) 6.56256 0.245083
\(718\) 0 0
\(719\) −12.5991 −0.469867 −0.234933 0.972011i \(-0.575487\pi\)
−0.234933 + 0.972011i \(0.575487\pi\)
\(720\) 0 0
\(721\) −24.8609 −0.925869
\(722\) 0 0
\(723\) −2.40864 −0.0895781
\(724\) 0 0
\(725\) −1.55685 −0.0578200
\(726\) 0 0
\(727\) −39.7756 −1.47519 −0.737597 0.675241i \(-0.764040\pi\)
−0.737597 + 0.675241i \(0.764040\pi\)
\(728\) 0 0
\(729\) 27.7402 1.02741
\(730\) 0 0
\(731\) −3.03009 −0.112072
\(732\) 0 0
\(733\) −5.50154 −0.203204 −0.101602 0.994825i \(-0.532397\pi\)
−0.101602 + 0.994825i \(0.532397\pi\)
\(734\) 0 0
\(735\) −8.56071 −0.315766
\(736\) 0 0
\(737\) 2.66894 0.0983115
\(738\) 0 0
\(739\) −52.6937 −1.93837 −0.969185 0.246335i \(-0.920774\pi\)
−0.969185 + 0.246335i \(0.920774\pi\)
\(740\) 0 0
\(741\) −8.14614 −0.299256
\(742\) 0 0
\(743\) −6.61647 −0.242735 −0.121367 0.992608i \(-0.538728\pi\)
−0.121367 + 0.992608i \(0.538728\pi\)
\(744\) 0 0
\(745\) −22.6948 −0.831474
\(746\) 0 0
\(747\) −0.216261 −0.00791256
\(748\) 0 0
\(749\) −30.5329 −1.11565
\(750\) 0 0
\(751\) 39.9062 1.45620 0.728099 0.685472i \(-0.240404\pi\)
0.728099 + 0.685472i \(0.240404\pi\)
\(752\) 0 0
\(753\) −14.0736 −0.512869
\(754\) 0 0
\(755\) 11.2632 0.409911
\(756\) 0 0
\(757\) −34.2645 −1.24537 −0.622683 0.782475i \(-0.713957\pi\)
−0.622683 + 0.782475i \(0.713957\pi\)
\(758\) 0 0
\(759\) −17.2578 −0.626417
\(760\) 0 0
\(761\) 12.4757 0.452243 0.226122 0.974099i \(-0.427395\pi\)
0.226122 + 0.974099i \(0.427395\pi\)
\(762\) 0 0
\(763\) 15.2830 0.553281
\(764\) 0 0
\(765\) −0.306060 −0.0110656
\(766\) 0 0
\(767\) −55.0126 −1.98639
\(768\) 0 0
\(769\) 3.32384 0.119861 0.0599304 0.998203i \(-0.480912\pi\)
0.0599304 + 0.998203i \(0.480912\pi\)
\(770\) 0 0
\(771\) 25.7233 0.926402
\(772\) 0 0
\(773\) 14.4248 0.518824 0.259412 0.965767i \(-0.416471\pi\)
0.259412 + 0.965767i \(0.416471\pi\)
\(774\) 0 0
\(775\) 21.8689 0.785555
\(776\) 0 0
\(777\) 0.263185 0.00944170
\(778\) 0 0
\(779\) −10.7743 −0.386030
\(780\) 0 0
\(781\) −9.74099 −0.348560
\(782\) 0 0
\(783\) −3.66699 −0.131047
\(784\) 0 0
\(785\) −38.9255 −1.38931
\(786\) 0 0
\(787\) 38.6403 1.37738 0.688689 0.725057i \(-0.258187\pi\)
0.688689 + 0.725057i \(0.258187\pi\)
\(788\) 0 0
\(789\) −16.1983 −0.576676
\(790\) 0 0
\(791\) −34.7103 −1.23416
\(792\) 0 0
\(793\) −42.1433 −1.49655
\(794\) 0 0
\(795\) −25.5387 −0.905763
\(796\) 0 0
\(797\) 30.9316 1.09565 0.547826 0.836592i \(-0.315456\pi\)
0.547826 + 0.836592i \(0.315456\pi\)
\(798\) 0 0
\(799\) 6.06971 0.214731
\(800\) 0 0
\(801\) −1.38329 −0.0488762
\(802\) 0 0
\(803\) −9.28015 −0.327490
\(804\) 0 0
\(805\) 16.0691 0.566360
\(806\) 0 0
\(807\) −6.80177 −0.239434
\(808\) 0 0
\(809\) −56.7269 −1.99441 −0.997206 0.0747056i \(-0.976198\pi\)
−0.997206 + 0.0747056i \(0.976198\pi\)
\(810\) 0 0
\(811\) 4.91800 0.172694 0.0863472 0.996265i \(-0.472481\pi\)
0.0863472 + 0.996265i \(0.472481\pi\)
\(812\) 0 0
\(813\) 10.8956 0.382125
\(814\) 0 0
\(815\) −1.77812 −0.0622849
\(816\) 0 0
\(817\) −1.43840 −0.0503233
\(818\) 0 0
\(819\) −0.832570 −0.0290924
\(820\) 0 0
\(821\) −19.0797 −0.665886 −0.332943 0.942947i \(-0.608042\pi\)
−0.332943 + 0.942947i \(0.608042\pi\)
\(822\) 0 0
\(823\) −19.6263 −0.684131 −0.342066 0.939676i \(-0.611127\pi\)
−0.342066 + 0.939676i \(0.611127\pi\)
\(824\) 0 0
\(825\) −7.96928 −0.277455
\(826\) 0 0
\(827\) −28.0107 −0.974029 −0.487014 0.873394i \(-0.661914\pi\)
−0.487014 + 0.873394i \(0.661914\pi\)
\(828\) 0 0
\(829\) 23.1932 0.805533 0.402766 0.915303i \(-0.368049\pi\)
0.402766 + 0.915303i \(0.368049\pi\)
\(830\) 0 0
\(831\) −25.8995 −0.898442
\(832\) 0 0
\(833\) −6.35707 −0.220259
\(834\) 0 0
\(835\) 22.6422 0.783567
\(836\) 0 0
\(837\) 51.5097 1.78044
\(838\) 0 0
\(839\) 27.6363 0.954110 0.477055 0.878873i \(-0.341704\pi\)
0.477055 + 0.878873i \(0.341704\pi\)
\(840\) 0 0
\(841\) −28.5157 −0.983299
\(842\) 0 0
\(843\) −25.4388 −0.876157
\(844\) 0 0
\(845\) 16.2626 0.559450
\(846\) 0 0
\(847\) 13.2560 0.455481
\(848\) 0 0
\(849\) 25.5878 0.878171
\(850\) 0 0
\(851\) 0.374363 0.0128330
\(852\) 0 0
\(853\) 53.0502 1.81640 0.908202 0.418532i \(-0.137455\pi\)
0.908202 + 0.418532i \(0.137455\pi\)
\(854\) 0 0
\(855\) −0.145289 −0.00496877
\(856\) 0 0
\(857\) 8.06832 0.275609 0.137804 0.990459i \(-0.455996\pi\)
0.137804 + 0.990459i \(0.455996\pi\)
\(858\) 0 0
\(859\) −10.1806 −0.347357 −0.173679 0.984802i \(-0.555565\pi\)
−0.173679 + 0.984802i \(0.555565\pi\)
\(860\) 0 0
\(861\) 36.6940 1.25053
\(862\) 0 0
\(863\) 45.7074 1.55590 0.777949 0.628327i \(-0.216260\pi\)
0.777949 + 0.628327i \(0.216260\pi\)
\(864\) 0 0
\(865\) 19.2292 0.653811
\(866\) 0 0
\(867\) −21.4394 −0.728119
\(868\) 0 0
\(869\) −2.29706 −0.0779223
\(870\) 0 0
\(871\) −6.10299 −0.206792
\(872\) 0 0
\(873\) 1.21535 0.0411334
\(874\) 0 0
\(875\) 24.0056 0.811539
\(876\) 0 0
\(877\) 8.41970 0.284313 0.142157 0.989844i \(-0.454596\pi\)
0.142157 + 0.989844i \(0.454596\pi\)
\(878\) 0 0
\(879\) −37.0310 −1.24903
\(880\) 0 0
\(881\) 8.33625 0.280855 0.140428 0.990091i \(-0.455152\pi\)
0.140428 + 0.990091i \(0.455152\pi\)
\(882\) 0 0
\(883\) 26.9797 0.907941 0.453970 0.891017i \(-0.350007\pi\)
0.453970 + 0.891017i \(0.350007\pi\)
\(884\) 0 0
\(885\) 32.6948 1.09902
\(886\) 0 0
\(887\) −25.8570 −0.868193 −0.434097 0.900866i \(-0.642932\pi\)
−0.434097 + 0.900866i \(0.642932\pi\)
\(888\) 0 0
\(889\) 36.2171 1.21468
\(890\) 0 0
\(891\) −18.2234 −0.610505
\(892\) 0 0
\(893\) 2.88133 0.0964200
\(894\) 0 0
\(895\) 34.0726 1.13892
\(896\) 0 0
\(897\) 39.4629 1.31763
\(898\) 0 0
\(899\) −6.80348 −0.226909
\(900\) 0 0
\(901\) −18.9647 −0.631805
\(902\) 0 0
\(903\) 4.89875 0.163020
\(904\) 0 0
\(905\) −31.2074 −1.03737
\(906\) 0 0
\(907\) −21.7649 −0.722690 −0.361345 0.932432i \(-0.617682\pi\)
−0.361345 + 0.932432i \(0.617682\pi\)
\(908\) 0 0
\(909\) 0.0878377 0.00291339
\(910\) 0 0
\(911\) 13.3098 0.440974 0.220487 0.975390i \(-0.429235\pi\)
0.220487 + 0.975390i \(0.429235\pi\)
\(912\) 0 0
\(913\) 5.16464 0.170925
\(914\) 0 0
\(915\) 25.0464 0.828007
\(916\) 0 0
\(917\) 19.3669 0.639552
\(918\) 0 0
\(919\) −29.7421 −0.981103 −0.490551 0.871412i \(-0.663205\pi\)
−0.490551 + 0.871412i \(0.663205\pi\)
\(920\) 0 0
\(921\) −13.3365 −0.439451
\(922\) 0 0
\(923\) 22.2745 0.733174
\(924\) 0 0
\(925\) 0.172873 0.00568403
\(926\) 0 0
\(927\) −1.08892 −0.0357648
\(928\) 0 0
\(929\) −30.3181 −0.994705 −0.497352 0.867549i \(-0.665694\pi\)
−0.497352 + 0.867549i \(0.665694\pi\)
\(930\) 0 0
\(931\) −3.01774 −0.0989024
\(932\) 0 0
\(933\) 21.8941 0.716779
\(934\) 0 0
\(935\) 7.30921 0.239037
\(936\) 0 0
\(937\) 8.81465 0.287962 0.143981 0.989580i \(-0.454010\pi\)
0.143981 + 0.989580i \(0.454010\pi\)
\(938\) 0 0
\(939\) 6.95586 0.226996
\(940\) 0 0
\(941\) −42.2936 −1.37873 −0.689366 0.724413i \(-0.742111\pi\)
−0.689366 + 0.724413i \(0.742111\pi\)
\(942\) 0 0
\(943\) 52.1949 1.69970
\(944\) 0 0
\(945\) 17.4778 0.568553
\(946\) 0 0
\(947\) −30.2387 −0.982626 −0.491313 0.870983i \(-0.663483\pi\)
−0.491313 + 0.870983i \(0.663483\pi\)
\(948\) 0 0
\(949\) 21.2207 0.688853
\(950\) 0 0
\(951\) 3.86464 0.125320
\(952\) 0 0
\(953\) 1.52937 0.0495411 0.0247706 0.999693i \(-0.492114\pi\)
0.0247706 + 0.999693i \(0.492114\pi\)
\(954\) 0 0
\(955\) 23.3362 0.755141
\(956\) 0 0
\(957\) 2.47926 0.0801432
\(958\) 0 0
\(959\) −40.3609 −1.30332
\(960\) 0 0
\(961\) 64.5676 2.08283
\(962\) 0 0
\(963\) −1.33736 −0.0430957
\(964\) 0 0
\(965\) −39.6771 −1.27725
\(966\) 0 0
\(967\) −3.67723 −0.118252 −0.0591259 0.998251i \(-0.518831\pi\)
−0.0591259 + 0.998251i \(0.518831\pi\)
\(968\) 0 0
\(969\) 3.59513 0.115492
\(970\) 0 0
\(971\) −5.89852 −0.189293 −0.0946463 0.995511i \(-0.530172\pi\)
−0.0946463 + 0.995511i \(0.530172\pi\)
\(972\) 0 0
\(973\) 33.4747 1.07315
\(974\) 0 0
\(975\) 18.2232 0.583608
\(976\) 0 0
\(977\) −19.5003 −0.623871 −0.311935 0.950103i \(-0.600977\pi\)
−0.311935 + 0.950103i \(0.600977\pi\)
\(978\) 0 0
\(979\) 33.0352 1.05581
\(980\) 0 0
\(981\) 0.669403 0.0213724
\(982\) 0 0
\(983\) −2.43384 −0.0776274 −0.0388137 0.999246i \(-0.512358\pi\)
−0.0388137 + 0.999246i \(0.512358\pi\)
\(984\) 0 0
\(985\) −3.07328 −0.0979229
\(986\) 0 0
\(987\) −9.81291 −0.312348
\(988\) 0 0
\(989\) 6.96815 0.221574
\(990\) 0 0
\(991\) 15.1966 0.482736 0.241368 0.970434i \(-0.422404\pi\)
0.241368 + 0.970434i \(0.422404\pi\)
\(992\) 0 0
\(993\) 12.4784 0.395990
\(994\) 0 0
\(995\) −30.5196 −0.967537
\(996\) 0 0
\(997\) 42.5056 1.34617 0.673083 0.739567i \(-0.264970\pi\)
0.673083 + 0.739567i \(0.264970\pi\)
\(998\) 0 0
\(999\) 0.407183 0.0128827
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 4864.2.a.bn.1.6 8
4.3 odd 2 4864.2.a.bo.1.3 8
8.3 odd 2 4864.2.a.bq.1.6 8
8.5 even 2 4864.2.a.bp.1.3 8
16.3 odd 4 152.2.c.b.77.7 16
16.5 even 4 608.2.c.b.305.6 16
16.11 odd 4 152.2.c.b.77.8 yes 16
16.13 even 4 608.2.c.b.305.11 16
48.5 odd 4 5472.2.g.b.2737.6 16
48.11 even 4 1368.2.g.b.685.9 16
48.29 odd 4 5472.2.g.b.2737.11 16
48.35 even 4 1368.2.g.b.685.10 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.c.b.77.7 16 16.3 odd 4
152.2.c.b.77.8 yes 16 16.11 odd 4
608.2.c.b.305.6 16 16.5 even 4
608.2.c.b.305.11 16 16.13 even 4
1368.2.g.b.685.9 16 48.11 even 4
1368.2.g.b.685.10 16 48.35 even 4
4864.2.a.bn.1.6 8 1.1 even 1 trivial
4864.2.a.bo.1.3 8 4.3 odd 2
4864.2.a.bp.1.3 8 8.5 even 2
4864.2.a.bq.1.6 8 8.3 odd 2
5472.2.g.b.2737.6 16 48.5 odd 4
5472.2.g.b.2737.11 16 48.29 odd 4