Properties

Label 1856.2.a.z.1.1
Level $1856$
Weight $2$
Character 1856.1
Self dual yes
Analytic conductor $14.820$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1856,2,Mod(1,1856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, 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("1856.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1856.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: \(14.8202346151\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 928)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.17557\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.07768 q^{3} +4.23607 q^{5} -2.35114 q^{7} +6.47214 q^{9} +O(q^{10})\) \(q-3.07768 q^{3} +4.23607 q^{5} -2.35114 q^{7} +6.47214 q^{9} +4.53077 q^{11} +1.76393 q^{13} -13.0373 q^{15} -5.23607 q^{17} +6.15537 q^{19} +7.23607 q^{21} +3.80423 q^{23} +12.9443 q^{25} -10.6861 q^{27} -1.00000 q^{29} -0.726543 q^{31} -13.9443 q^{33} -9.95959 q^{35} -2.47214 q^{37} -5.42882 q^{39} -7.23607 q^{41} -5.98385 q^{43} +27.4164 q^{45} +5.42882 q^{47} -1.47214 q^{49} +16.1150 q^{51} +3.76393 q^{53} +19.1926 q^{55} -18.9443 q^{57} +6.71040 q^{59} +2.76393 q^{61} -15.2169 q^{63} +7.47214 q^{65} -12.3107 q^{67} -11.7082 q^{69} +3.24920 q^{71} -4.94427 q^{73} -39.8384 q^{75} -10.6525 q^{77} +2.17963 q^{79} +13.4721 q^{81} +16.1150 q^{83} -22.1803 q^{85} +3.07768 q^{87} +1.70820 q^{89} -4.14725 q^{91} +2.23607 q^{93} +26.0746 q^{95} +9.70820 q^{97} +29.3238 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{5} + 8 q^{9} + 16 q^{13} - 12 q^{17} + 20 q^{21} + 16 q^{25} - 4 q^{29} - 20 q^{33} + 8 q^{37} - 20 q^{41} + 56 q^{45} + 12 q^{49} + 24 q^{53} - 40 q^{57} + 20 q^{61} + 12 q^{65} - 20 q^{69} + 16 q^{73} + 20 q^{77} + 36 q^{81} - 44 q^{85} - 20 q^{89} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.07768 −1.77690 −0.888451 0.458972i \(-0.848218\pi\)
−0.888451 + 0.458972i \(0.848218\pi\)
\(4\) 0 0
\(5\) 4.23607 1.89443 0.947214 0.320603i \(-0.103886\pi\)
0.947214 + 0.320603i \(0.103886\pi\)
\(6\) 0 0
\(7\) −2.35114 −0.888648 −0.444324 0.895866i \(-0.646556\pi\)
−0.444324 + 0.895866i \(0.646556\pi\)
\(8\) 0 0
\(9\) 6.47214 2.15738
\(10\) 0 0
\(11\) 4.53077 1.36608 0.683039 0.730382i \(-0.260658\pi\)
0.683039 + 0.730382i \(0.260658\pi\)
\(12\) 0 0
\(13\) 1.76393 0.489227 0.244613 0.969621i \(-0.421339\pi\)
0.244613 + 0.969621i \(0.421339\pi\)
\(14\) 0 0
\(15\) −13.0373 −3.36621
\(16\) 0 0
\(17\) −5.23607 −1.26993 −0.634967 0.772540i \(-0.718986\pi\)
−0.634967 + 0.772540i \(0.718986\pi\)
\(18\) 0 0
\(19\) 6.15537 1.41214 0.706069 0.708143i \(-0.250467\pi\)
0.706069 + 0.708143i \(0.250467\pi\)
\(20\) 0 0
\(21\) 7.23607 1.57904
\(22\) 0 0
\(23\) 3.80423 0.793236 0.396618 0.917984i \(-0.370184\pi\)
0.396618 + 0.917984i \(0.370184\pi\)
\(24\) 0 0
\(25\) 12.9443 2.58885
\(26\) 0 0
\(27\) −10.6861 −2.05655
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −0.726543 −0.130491 −0.0652454 0.997869i \(-0.520783\pi\)
−0.0652454 + 0.997869i \(0.520783\pi\)
\(32\) 0 0
\(33\) −13.9443 −2.42739
\(34\) 0 0
\(35\) −9.95959 −1.68348
\(36\) 0 0
\(37\) −2.47214 −0.406417 −0.203208 0.979136i \(-0.565137\pi\)
−0.203208 + 0.979136i \(0.565137\pi\)
\(38\) 0 0
\(39\) −5.42882 −0.869308
\(40\) 0 0
\(41\) −7.23607 −1.13008 −0.565042 0.825062i \(-0.691140\pi\)
−0.565042 + 0.825062i \(0.691140\pi\)
\(42\) 0 0
\(43\) −5.98385 −0.912529 −0.456265 0.889844i \(-0.650813\pi\)
−0.456265 + 0.889844i \(0.650813\pi\)
\(44\) 0 0
\(45\) 27.4164 4.08700
\(46\) 0 0
\(47\) 5.42882 0.791875 0.395938 0.918277i \(-0.370420\pi\)
0.395938 + 0.918277i \(0.370420\pi\)
\(48\) 0 0
\(49\) −1.47214 −0.210305
\(50\) 0 0
\(51\) 16.1150 2.25655
\(52\) 0 0
\(53\) 3.76393 0.517016 0.258508 0.966009i \(-0.416769\pi\)
0.258508 + 0.966009i \(0.416769\pi\)
\(54\) 0 0
\(55\) 19.1926 2.58794
\(56\) 0 0
\(57\) −18.9443 −2.50923
\(58\) 0 0
\(59\) 6.71040 0.873619 0.436810 0.899554i \(-0.356108\pi\)
0.436810 + 0.899554i \(0.356108\pi\)
\(60\) 0 0
\(61\) 2.76393 0.353885 0.176943 0.984221i \(-0.443379\pi\)
0.176943 + 0.984221i \(0.443379\pi\)
\(62\) 0 0
\(63\) −15.2169 −1.91715
\(64\) 0 0
\(65\) 7.47214 0.926804
\(66\) 0 0
\(67\) −12.3107 −1.50400 −0.751998 0.659166i \(-0.770910\pi\)
−0.751998 + 0.659166i \(0.770910\pi\)
\(68\) 0 0
\(69\) −11.7082 −1.40950
\(70\) 0 0
\(71\) 3.24920 0.385609 0.192804 0.981237i \(-0.438242\pi\)
0.192804 + 0.981237i \(0.438242\pi\)
\(72\) 0 0
\(73\) −4.94427 −0.578683 −0.289342 0.957226i \(-0.593436\pi\)
−0.289342 + 0.957226i \(0.593436\pi\)
\(74\) 0 0
\(75\) −39.8384 −4.60014
\(76\) 0 0
\(77\) −10.6525 −1.21396
\(78\) 0 0
\(79\) 2.17963 0.245227 0.122614 0.992454i \(-0.460872\pi\)
0.122614 + 0.992454i \(0.460872\pi\)
\(80\) 0 0
\(81\) 13.4721 1.49690
\(82\) 0 0
\(83\) 16.1150 1.76885 0.884423 0.466685i \(-0.154552\pi\)
0.884423 + 0.466685i \(0.154552\pi\)
\(84\) 0 0
\(85\) −22.1803 −2.40580
\(86\) 0 0
\(87\) 3.07768 0.329962
\(88\) 0 0
\(89\) 1.70820 0.181069 0.0905346 0.995893i \(-0.471142\pi\)
0.0905346 + 0.995893i \(0.471142\pi\)
\(90\) 0 0
\(91\) −4.14725 −0.434750
\(92\) 0 0
\(93\) 2.23607 0.231869
\(94\) 0 0
\(95\) 26.0746 2.67519
\(96\) 0 0
\(97\) 9.70820 0.985719 0.492859 0.870109i \(-0.335952\pi\)
0.492859 + 0.870109i \(0.335952\pi\)
\(98\) 0 0
\(99\) 29.3238 2.94715
\(100\) 0 0
\(101\) 15.4164 1.53399 0.766995 0.641653i \(-0.221751\pi\)
0.766995 + 0.641653i \(0.221751\pi\)
\(102\) 0 0
\(103\) −3.24920 −0.320153 −0.160076 0.987105i \(-0.551174\pi\)
−0.160076 + 0.987105i \(0.551174\pi\)
\(104\) 0 0
\(105\) 30.6525 2.99138
\(106\) 0 0
\(107\) 2.35114 0.227293 0.113647 0.993521i \(-0.463747\pi\)
0.113647 + 0.993521i \(0.463747\pi\)
\(108\) 0 0
\(109\) 1.76393 0.168954 0.0844770 0.996425i \(-0.473078\pi\)
0.0844770 + 0.996425i \(0.473078\pi\)
\(110\) 0 0
\(111\) 7.60845 0.722162
\(112\) 0 0
\(113\) −2.94427 −0.276974 −0.138487 0.990364i \(-0.544224\pi\)
−0.138487 + 0.990364i \(0.544224\pi\)
\(114\) 0 0
\(115\) 16.1150 1.50273
\(116\) 0 0
\(117\) 11.4164 1.05545
\(118\) 0 0
\(119\) 12.3107 1.12852
\(120\) 0 0
\(121\) 9.52786 0.866169
\(122\) 0 0
\(123\) 22.2703 2.00805
\(124\) 0 0
\(125\) 33.6525 3.00997
\(126\) 0 0
\(127\) −6.15537 −0.546201 −0.273100 0.961986i \(-0.588049\pi\)
−0.273100 + 0.961986i \(0.588049\pi\)
\(128\) 0 0
\(129\) 18.4164 1.62147
\(130\) 0 0
\(131\) −4.35926 −0.380870 −0.190435 0.981700i \(-0.560990\pi\)
−0.190435 + 0.981700i \(0.560990\pi\)
\(132\) 0 0
\(133\) −14.4721 −1.25489
\(134\) 0 0
\(135\) −45.2672 −3.89598
\(136\) 0 0
\(137\) 11.4164 0.975370 0.487685 0.873020i \(-0.337842\pi\)
0.487685 + 0.873020i \(0.337842\pi\)
\(138\) 0 0
\(139\) 0.898056 0.0761721 0.0380861 0.999274i \(-0.487874\pi\)
0.0380861 + 0.999274i \(0.487874\pi\)
\(140\) 0 0
\(141\) −16.7082 −1.40708
\(142\) 0 0
\(143\) 7.99197 0.668322
\(144\) 0 0
\(145\) −4.23607 −0.351786
\(146\) 0 0
\(147\) 4.53077 0.373691
\(148\) 0 0
\(149\) 8.23607 0.674725 0.337362 0.941375i \(-0.390465\pi\)
0.337362 + 0.941375i \(0.390465\pi\)
\(150\) 0 0
\(151\) 24.0664 1.95850 0.979250 0.202658i \(-0.0649581\pi\)
0.979250 + 0.202658i \(0.0649581\pi\)
\(152\) 0 0
\(153\) −33.8885 −2.73973
\(154\) 0 0
\(155\) −3.07768 −0.247205
\(156\) 0 0
\(157\) 9.70820 0.774799 0.387400 0.921912i \(-0.373373\pi\)
0.387400 + 0.921912i \(0.373373\pi\)
\(158\) 0 0
\(159\) −11.5842 −0.918686
\(160\) 0 0
\(161\) −8.94427 −0.704907
\(162\) 0 0
\(163\) 1.28157 0.100380 0.0501902 0.998740i \(-0.484017\pi\)
0.0501902 + 0.998740i \(0.484017\pi\)
\(164\) 0 0
\(165\) −59.0689 −4.59851
\(166\) 0 0
\(167\) 10.8576 0.840190 0.420095 0.907480i \(-0.361997\pi\)
0.420095 + 0.907480i \(0.361997\pi\)
\(168\) 0 0
\(169\) −9.88854 −0.760657
\(170\) 0 0
\(171\) 39.8384 3.04652
\(172\) 0 0
\(173\) 20.4721 1.55647 0.778234 0.627975i \(-0.216116\pi\)
0.778234 + 0.627975i \(0.216116\pi\)
\(174\) 0 0
\(175\) −30.4338 −2.30058
\(176\) 0 0
\(177\) −20.6525 −1.55234
\(178\) 0 0
\(179\) −21.3723 −1.59744 −0.798719 0.601704i \(-0.794489\pi\)
−0.798719 + 0.601704i \(0.794489\pi\)
\(180\) 0 0
\(181\) 19.1803 1.42566 0.712832 0.701335i \(-0.247412\pi\)
0.712832 + 0.701335i \(0.247412\pi\)
\(182\) 0 0
\(183\) −8.50651 −0.628819
\(184\) 0 0
\(185\) −10.4721 −0.769927
\(186\) 0 0
\(187\) −23.7234 −1.73483
\(188\) 0 0
\(189\) 25.1246 1.82755
\(190\) 0 0
\(191\) −24.2784 −1.75673 −0.878363 0.477994i \(-0.841364\pi\)
−0.878363 + 0.477994i \(0.841364\pi\)
\(192\) 0 0
\(193\) 8.76393 0.630842 0.315421 0.948952i \(-0.397854\pi\)
0.315421 + 0.948952i \(0.397854\pi\)
\(194\) 0 0
\(195\) −22.9969 −1.64684
\(196\) 0 0
\(197\) 10.9443 0.779747 0.389874 0.920868i \(-0.372519\pi\)
0.389874 + 0.920868i \(0.372519\pi\)
\(198\) 0 0
\(199\) 19.3642 1.37269 0.686344 0.727277i \(-0.259214\pi\)
0.686344 + 0.727277i \(0.259214\pi\)
\(200\) 0 0
\(201\) 37.8885 2.67245
\(202\) 0 0
\(203\) 2.35114 0.165018
\(204\) 0 0
\(205\) −30.6525 −2.14086
\(206\) 0 0
\(207\) 24.6215 1.71131
\(208\) 0 0
\(209\) 27.8885 1.92909
\(210\) 0 0
\(211\) −4.53077 −0.311911 −0.155955 0.987764i \(-0.549846\pi\)
−0.155955 + 0.987764i \(0.549846\pi\)
\(212\) 0 0
\(213\) −10.0000 −0.685189
\(214\) 0 0
\(215\) −25.3480 −1.72872
\(216\) 0 0
\(217\) 1.70820 0.115960
\(218\) 0 0
\(219\) 15.2169 1.02826
\(220\) 0 0
\(221\) −9.23607 −0.621285
\(222\) 0 0
\(223\) −27.8707 −1.86636 −0.933179 0.359412i \(-0.882977\pi\)
−0.933179 + 0.359412i \(0.882977\pi\)
\(224\) 0 0
\(225\) 83.7771 5.58514
\(226\) 0 0
\(227\) −10.8576 −0.720647 −0.360324 0.932827i \(-0.617334\pi\)
−0.360324 + 0.932827i \(0.617334\pi\)
\(228\) 0 0
\(229\) −21.7082 −1.43452 −0.717259 0.696806i \(-0.754604\pi\)
−0.717259 + 0.696806i \(0.754604\pi\)
\(230\) 0 0
\(231\) 32.7849 2.15709
\(232\) 0 0
\(233\) −17.9443 −1.17557 −0.587784 0.809018i \(-0.700000\pi\)
−0.587784 + 0.809018i \(0.700000\pi\)
\(234\) 0 0
\(235\) 22.9969 1.50015
\(236\) 0 0
\(237\) −6.70820 −0.435745
\(238\) 0 0
\(239\) −0.898056 −0.0580904 −0.0290452 0.999578i \(-0.509247\pi\)
−0.0290452 + 0.999578i \(0.509247\pi\)
\(240\) 0 0
\(241\) −3.94427 −0.254073 −0.127036 0.991898i \(-0.540547\pi\)
−0.127036 + 0.991898i \(0.540547\pi\)
\(242\) 0 0
\(243\) −9.40456 −0.603303
\(244\) 0 0
\(245\) −6.23607 −0.398408
\(246\) 0 0
\(247\) 10.8576 0.690856
\(248\) 0 0
\(249\) −49.5967 −3.14307
\(250\) 0 0
\(251\) −8.89002 −0.561133 −0.280567 0.959835i \(-0.590522\pi\)
−0.280567 + 0.959835i \(0.590522\pi\)
\(252\) 0 0
\(253\) 17.2361 1.08362
\(254\) 0 0
\(255\) 68.2641 4.27486
\(256\) 0 0
\(257\) 1.47214 0.0918293 0.0459147 0.998945i \(-0.485380\pi\)
0.0459147 + 0.998945i \(0.485380\pi\)
\(258\) 0 0
\(259\) 5.81234 0.361161
\(260\) 0 0
\(261\) −6.47214 −0.400615
\(262\) 0 0
\(263\) −0.726543 −0.0448005 −0.0224003 0.999749i \(-0.507131\pi\)
−0.0224003 + 0.999749i \(0.507131\pi\)
\(264\) 0 0
\(265\) 15.9443 0.979449
\(266\) 0 0
\(267\) −5.25731 −0.321742
\(268\) 0 0
\(269\) 7.70820 0.469977 0.234989 0.971998i \(-0.424495\pi\)
0.234989 + 0.971998i \(0.424495\pi\)
\(270\) 0 0
\(271\) −13.3803 −0.812796 −0.406398 0.913696i \(-0.633215\pi\)
−0.406398 + 0.913696i \(0.633215\pi\)
\(272\) 0 0
\(273\) 12.7639 0.772508
\(274\) 0 0
\(275\) 58.6475 3.53658
\(276\) 0 0
\(277\) 15.8885 0.954650 0.477325 0.878727i \(-0.341606\pi\)
0.477325 + 0.878727i \(0.341606\pi\)
\(278\) 0 0
\(279\) −4.70228 −0.281518
\(280\) 0 0
\(281\) −10.5279 −0.628040 −0.314020 0.949416i \(-0.601676\pi\)
−0.314020 + 0.949416i \(0.601676\pi\)
\(282\) 0 0
\(283\) 0.898056 0.0533839 0.0266919 0.999644i \(-0.491503\pi\)
0.0266919 + 0.999644i \(0.491503\pi\)
\(284\) 0 0
\(285\) −80.2492 −4.75355
\(286\) 0 0
\(287\) 17.0130 1.00425
\(288\) 0 0
\(289\) 10.4164 0.612730
\(290\) 0 0
\(291\) −29.8788 −1.75153
\(292\) 0 0
\(293\) −9.41641 −0.550112 −0.275056 0.961428i \(-0.588696\pi\)
−0.275056 + 0.961428i \(0.588696\pi\)
\(294\) 0 0
\(295\) 28.4257 1.65501
\(296\) 0 0
\(297\) −48.4164 −2.80940
\(298\) 0 0
\(299\) 6.71040 0.388072
\(300\) 0 0
\(301\) 14.0689 0.810917
\(302\) 0 0
\(303\) −47.4468 −2.72575
\(304\) 0 0
\(305\) 11.7082 0.670410
\(306\) 0 0
\(307\) −27.6992 −1.58087 −0.790437 0.612543i \(-0.790147\pi\)
−0.790437 + 0.612543i \(0.790147\pi\)
\(308\) 0 0
\(309\) 10.0000 0.568880
\(310\) 0 0
\(311\) 15.5599 0.882323 0.441161 0.897428i \(-0.354567\pi\)
0.441161 + 0.897428i \(0.354567\pi\)
\(312\) 0 0
\(313\) 14.4164 0.814864 0.407432 0.913236i \(-0.366424\pi\)
0.407432 + 0.913236i \(0.366424\pi\)
\(314\) 0 0
\(315\) −64.4598 −3.63190
\(316\) 0 0
\(317\) −0.291796 −0.0163889 −0.00819445 0.999966i \(-0.502608\pi\)
−0.00819445 + 0.999966i \(0.502608\pi\)
\(318\) 0 0
\(319\) −4.53077 −0.253674
\(320\) 0 0
\(321\) −7.23607 −0.403878
\(322\) 0 0
\(323\) −32.2299 −1.79332
\(324\) 0 0
\(325\) 22.8328 1.26654
\(326\) 0 0
\(327\) −5.42882 −0.300215
\(328\) 0 0
\(329\) −12.7639 −0.703698
\(330\) 0 0
\(331\) 0.171513 0.00942723 0.00471362 0.999989i \(-0.498500\pi\)
0.00471362 + 0.999989i \(0.498500\pi\)
\(332\) 0 0
\(333\) −16.0000 −0.876795
\(334\) 0 0
\(335\) −52.1491 −2.84921
\(336\) 0 0
\(337\) −27.1246 −1.47757 −0.738786 0.673940i \(-0.764601\pi\)
−0.738786 + 0.673940i \(0.764601\pi\)
\(338\) 0 0
\(339\) 9.06154 0.492155
\(340\) 0 0
\(341\) −3.29180 −0.178261
\(342\) 0 0
\(343\) 19.9192 1.07553
\(344\) 0 0
\(345\) −49.5967 −2.67020
\(346\) 0 0
\(347\) −18.4661 −0.991312 −0.495656 0.868519i \(-0.665072\pi\)
−0.495656 + 0.868519i \(0.665072\pi\)
\(348\) 0 0
\(349\) 4.34752 0.232718 0.116359 0.993207i \(-0.462878\pi\)
0.116359 + 0.993207i \(0.462878\pi\)
\(350\) 0 0
\(351\) −18.8496 −1.00612
\(352\) 0 0
\(353\) 2.94427 0.156708 0.0783539 0.996926i \(-0.475034\pi\)
0.0783539 + 0.996926i \(0.475034\pi\)
\(354\) 0 0
\(355\) 13.7638 0.730508
\(356\) 0 0
\(357\) −37.8885 −2.00527
\(358\) 0 0
\(359\) 0.383516 0.0202412 0.0101206 0.999949i \(-0.496778\pi\)
0.0101206 + 0.999949i \(0.496778\pi\)
\(360\) 0 0
\(361\) 18.8885 0.994134
\(362\) 0 0
\(363\) −29.3238 −1.53910
\(364\) 0 0
\(365\) −20.9443 −1.09627
\(366\) 0 0
\(367\) −10.8576 −0.566765 −0.283382 0.959007i \(-0.591457\pi\)
−0.283382 + 0.959007i \(0.591457\pi\)
\(368\) 0 0
\(369\) −46.8328 −2.43802
\(370\) 0 0
\(371\) −8.84953 −0.459445
\(372\) 0 0
\(373\) −1.76393 −0.0913329 −0.0456665 0.998957i \(-0.514541\pi\)
−0.0456665 + 0.998957i \(0.514541\pi\)
\(374\) 0 0
\(375\) −103.572 −5.34842
\(376\) 0 0
\(377\) −1.76393 −0.0908471
\(378\) 0 0
\(379\) 27.8707 1.43162 0.715810 0.698295i \(-0.246058\pi\)
0.715810 + 0.698295i \(0.246058\pi\)
\(380\) 0 0
\(381\) 18.9443 0.970544
\(382\) 0 0
\(383\) −23.3804 −1.19468 −0.597341 0.801987i \(-0.703776\pi\)
−0.597341 + 0.801987i \(0.703776\pi\)
\(384\) 0 0
\(385\) −45.1246 −2.29976
\(386\) 0 0
\(387\) −38.7283 −1.96867
\(388\) 0 0
\(389\) −16.0000 −0.811232 −0.405616 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) −19.9192 −1.00736
\(392\) 0 0
\(393\) 13.4164 0.676768
\(394\) 0 0
\(395\) 9.23305 0.464565
\(396\) 0 0
\(397\) 19.7639 0.991923 0.495962 0.868344i \(-0.334816\pi\)
0.495962 + 0.868344i \(0.334816\pi\)
\(398\) 0 0
\(399\) 44.5407 2.22982
\(400\) 0 0
\(401\) 27.0000 1.34832 0.674158 0.738587i \(-0.264507\pi\)
0.674158 + 0.738587i \(0.264507\pi\)
\(402\) 0 0
\(403\) −1.28157 −0.0638396
\(404\) 0 0
\(405\) 57.0689 2.83578
\(406\) 0 0
\(407\) −11.2007 −0.555197
\(408\) 0 0
\(409\) 11.8885 0.587851 0.293925 0.955828i \(-0.405038\pi\)
0.293925 + 0.955828i \(0.405038\pi\)
\(410\) 0 0
\(411\) −35.1361 −1.73314
\(412\) 0 0
\(413\) −15.7771 −0.776340
\(414\) 0 0
\(415\) 68.2641 3.35095
\(416\) 0 0
\(417\) −2.76393 −0.135350
\(418\) 0 0
\(419\) 35.4791 1.73327 0.866634 0.498944i \(-0.166279\pi\)
0.866634 + 0.498944i \(0.166279\pi\)
\(420\) 0 0
\(421\) 8.29180 0.404117 0.202059 0.979373i \(-0.435237\pi\)
0.202059 + 0.979373i \(0.435237\pi\)
\(422\) 0 0
\(423\) 35.1361 1.70838
\(424\) 0 0
\(425\) −67.7771 −3.28767
\(426\) 0 0
\(427\) −6.49839 −0.314479
\(428\) 0 0
\(429\) −24.5967 −1.18754
\(430\) 0 0
\(431\) −5.60034 −0.269759 −0.134879 0.990862i \(-0.543065\pi\)
−0.134879 + 0.990862i \(0.543065\pi\)
\(432\) 0 0
\(433\) 21.8885 1.05190 0.525948 0.850517i \(-0.323711\pi\)
0.525948 + 0.850517i \(0.323711\pi\)
\(434\) 0 0
\(435\) 13.0373 0.625090
\(436\) 0 0
\(437\) 23.4164 1.12016
\(438\) 0 0
\(439\) −13.2088 −0.630421 −0.315211 0.949022i \(-0.602075\pi\)
−0.315211 + 0.949022i \(0.602075\pi\)
\(440\) 0 0
\(441\) −9.52786 −0.453708
\(442\) 0 0
\(443\) 23.1684 1.10076 0.550382 0.834913i \(-0.314482\pi\)
0.550382 + 0.834913i \(0.314482\pi\)
\(444\) 0 0
\(445\) 7.23607 0.343023
\(446\) 0 0
\(447\) −25.3480 −1.19892
\(448\) 0 0
\(449\) −37.4164 −1.76579 −0.882895 0.469571i \(-0.844409\pi\)
−0.882895 + 0.469571i \(0.844409\pi\)
\(450\) 0 0
\(451\) −32.7849 −1.54378
\(452\) 0 0
\(453\) −74.0689 −3.48006
\(454\) 0 0
\(455\) −17.5680 −0.823603
\(456\) 0 0
\(457\) −24.8328 −1.16163 −0.580815 0.814036i \(-0.697266\pi\)
−0.580815 + 0.814036i \(0.697266\pi\)
\(458\) 0 0
\(459\) 55.9533 2.61168
\(460\) 0 0
\(461\) −35.8885 −1.67150 −0.835748 0.549113i \(-0.814966\pi\)
−0.835748 + 0.549113i \(0.814966\pi\)
\(462\) 0 0
\(463\) 20.8172 0.967459 0.483730 0.875217i \(-0.339282\pi\)
0.483730 + 0.875217i \(0.339282\pi\)
\(464\) 0 0
\(465\) 9.47214 0.439260
\(466\) 0 0
\(467\) 9.23305 0.427255 0.213627 0.976915i \(-0.431472\pi\)
0.213627 + 0.976915i \(0.431472\pi\)
\(468\) 0 0
\(469\) 28.9443 1.33652
\(470\) 0 0
\(471\) −29.8788 −1.37674
\(472\) 0 0
\(473\) −27.1115 −1.24659
\(474\) 0 0
\(475\) 79.6767 3.65582
\(476\) 0 0
\(477\) 24.3607 1.11540
\(478\) 0 0
\(479\) 40.5649 1.85346 0.926729 0.375730i \(-0.122608\pi\)
0.926729 + 0.375730i \(0.122608\pi\)
\(480\) 0 0
\(481\) −4.36068 −0.198830
\(482\) 0 0
\(483\) 27.5276 1.25255
\(484\) 0 0
\(485\) 41.1246 1.86737
\(486\) 0 0
\(487\) −15.7719 −0.714695 −0.357347 0.933972i \(-0.616319\pi\)
−0.357347 + 0.933972i \(0.616319\pi\)
\(488\) 0 0
\(489\) −3.94427 −0.178366
\(490\) 0 0
\(491\) −6.32688 −0.285528 −0.142764 0.989757i \(-0.545599\pi\)
−0.142764 + 0.989757i \(0.545599\pi\)
\(492\) 0 0
\(493\) 5.23607 0.235821
\(494\) 0 0
\(495\) 124.217 5.58316
\(496\) 0 0
\(497\) −7.63932 −0.342670
\(498\) 0 0
\(499\) 22.6134 1.01231 0.506156 0.862442i \(-0.331066\pi\)
0.506156 + 0.862442i \(0.331066\pi\)
\(500\) 0 0
\(501\) −33.4164 −1.49294
\(502\) 0 0
\(503\) −5.08580 −0.226765 −0.113382 0.993551i \(-0.536168\pi\)
−0.113382 + 0.993551i \(0.536168\pi\)
\(504\) 0 0
\(505\) 65.3050 2.90603
\(506\) 0 0
\(507\) 30.4338 1.35161
\(508\) 0 0
\(509\) −1.18034 −0.0523176 −0.0261588 0.999658i \(-0.508328\pi\)
−0.0261588 + 0.999658i \(0.508328\pi\)
\(510\) 0 0
\(511\) 11.6247 0.514246
\(512\) 0 0
\(513\) −65.7771 −2.90413
\(514\) 0 0
\(515\) −13.7638 −0.606506
\(516\) 0 0
\(517\) 24.5967 1.08176
\(518\) 0 0
\(519\) −63.0068 −2.76569
\(520\) 0 0
\(521\) −44.8885 −1.96660 −0.983301 0.181984i \(-0.941748\pi\)
−0.983301 + 0.181984i \(0.941748\pi\)
\(522\) 0 0
\(523\) −25.7315 −1.12516 −0.562581 0.826742i \(-0.690191\pi\)
−0.562581 + 0.826742i \(0.690191\pi\)
\(524\) 0 0
\(525\) 93.6656 4.08790
\(526\) 0 0
\(527\) 3.80423 0.165715
\(528\) 0 0
\(529\) −8.52786 −0.370777
\(530\) 0 0
\(531\) 43.4306 1.88473
\(532\) 0 0
\(533\) −12.7639 −0.552867
\(534\) 0 0
\(535\) 9.95959 0.430591
\(536\) 0 0
\(537\) 65.7771 2.83849
\(538\) 0 0
\(539\) −6.66991 −0.287293
\(540\) 0 0
\(541\) −40.3607 −1.73524 −0.867621 0.497227i \(-0.834352\pi\)
−0.867621 + 0.497227i \(0.834352\pi\)
\(542\) 0 0
\(543\) −59.0310 −2.53326
\(544\) 0 0
\(545\) 7.47214 0.320071
\(546\) 0 0
\(547\) 22.2703 0.952210 0.476105 0.879388i \(-0.342048\pi\)
0.476105 + 0.879388i \(0.342048\pi\)
\(548\) 0 0
\(549\) 17.8885 0.763464
\(550\) 0 0
\(551\) −6.15537 −0.262227
\(552\) 0 0
\(553\) −5.12461 −0.217921
\(554\) 0 0
\(555\) 32.2299 1.36808
\(556\) 0 0
\(557\) −30.3607 −1.28642 −0.643212 0.765688i \(-0.722398\pi\)
−0.643212 + 0.765688i \(0.722398\pi\)
\(558\) 0 0
\(559\) −10.5551 −0.446434
\(560\) 0 0
\(561\) 73.0132 3.08262
\(562\) 0 0
\(563\) 3.07768 0.129709 0.0648544 0.997895i \(-0.479342\pi\)
0.0648544 + 0.997895i \(0.479342\pi\)
\(564\) 0 0
\(565\) −12.4721 −0.524707
\(566\) 0 0
\(567\) −31.6749 −1.33022
\(568\) 0 0
\(569\) −14.0000 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) 0 0
\(571\) −28.6377 −1.19845 −0.599225 0.800581i \(-0.704525\pi\)
−0.599225 + 0.800581i \(0.704525\pi\)
\(572\) 0 0
\(573\) 74.7214 3.12153
\(574\) 0 0
\(575\) 49.2429 2.05357
\(576\) 0 0
\(577\) 7.12461 0.296601 0.148301 0.988942i \(-0.452620\pi\)
0.148301 + 0.988942i \(0.452620\pi\)
\(578\) 0 0
\(579\) −26.9726 −1.12094
\(580\) 0 0
\(581\) −37.8885 −1.57188
\(582\) 0 0
\(583\) 17.0535 0.706284
\(584\) 0 0
\(585\) 48.3607 1.99947
\(586\) 0 0
\(587\) 26.6296 1.09912 0.549560 0.835454i \(-0.314795\pi\)
0.549560 + 0.835454i \(0.314795\pi\)
\(588\) 0 0
\(589\) −4.47214 −0.184271
\(590\) 0 0
\(591\) −33.6830 −1.38553
\(592\) 0 0
\(593\) 15.8328 0.650176 0.325088 0.945684i \(-0.394606\pi\)
0.325088 + 0.945684i \(0.394606\pi\)
\(594\) 0 0
\(595\) 52.1491 2.13790
\(596\) 0 0
\(597\) −59.5967 −2.43913
\(598\) 0 0
\(599\) 43.8141 1.79020 0.895098 0.445869i \(-0.147105\pi\)
0.895098 + 0.445869i \(0.147105\pi\)
\(600\) 0 0
\(601\) −32.7639 −1.33647 −0.668234 0.743951i \(-0.732950\pi\)
−0.668234 + 0.743951i \(0.732950\pi\)
\(602\) 0 0
\(603\) −79.6767 −3.24469
\(604\) 0 0
\(605\) 40.3607 1.64089
\(606\) 0 0
\(607\) 28.9402 1.17465 0.587324 0.809352i \(-0.300181\pi\)
0.587324 + 0.809352i \(0.300181\pi\)
\(608\) 0 0
\(609\) −7.23607 −0.293220
\(610\) 0 0
\(611\) 9.57608 0.387407
\(612\) 0 0
\(613\) −1.65248 −0.0667429 −0.0333714 0.999443i \(-0.510624\pi\)
−0.0333714 + 0.999443i \(0.510624\pi\)
\(614\) 0 0
\(615\) 94.3386 3.80410
\(616\) 0 0
\(617\) 24.3607 0.980724 0.490362 0.871519i \(-0.336865\pi\)
0.490362 + 0.871519i \(0.336865\pi\)
\(618\) 0 0
\(619\) 37.1037 1.49132 0.745662 0.666324i \(-0.232133\pi\)
0.745662 + 0.666324i \(0.232133\pi\)
\(620\) 0 0
\(621\) −40.6525 −1.63133
\(622\) 0 0
\(623\) −4.01623 −0.160907
\(624\) 0 0
\(625\) 77.8328 3.11331
\(626\) 0 0
\(627\) −85.8321 −3.42780
\(628\) 0 0
\(629\) 12.9443 0.516122
\(630\) 0 0
\(631\) −36.5892 −1.45659 −0.728296 0.685263i \(-0.759687\pi\)
−0.728296 + 0.685263i \(0.759687\pi\)
\(632\) 0 0
\(633\) 13.9443 0.554235
\(634\) 0 0
\(635\) −26.0746 −1.03474
\(636\) 0 0
\(637\) −2.59675 −0.102887
\(638\) 0 0
\(639\) 21.0292 0.831904
\(640\) 0 0
\(641\) −17.2361 −0.680784 −0.340392 0.940284i \(-0.610560\pi\)
−0.340392 + 0.940284i \(0.610560\pi\)
\(642\) 0 0
\(643\) −4.14725 −0.163552 −0.0817758 0.996651i \(-0.526059\pi\)
−0.0817758 + 0.996651i \(0.526059\pi\)
\(644\) 0 0
\(645\) 78.0132 3.07177
\(646\) 0 0
\(647\) 37.8303 1.48726 0.743630 0.668591i \(-0.233102\pi\)
0.743630 + 0.668591i \(0.233102\pi\)
\(648\) 0 0
\(649\) 30.4033 1.19343
\(650\) 0 0
\(651\) −5.25731 −0.206050
\(652\) 0 0
\(653\) −11.2361 −0.439701 −0.219851 0.975534i \(-0.570557\pi\)
−0.219851 + 0.975534i \(0.570557\pi\)
\(654\) 0 0
\(655\) −18.4661 −0.721530
\(656\) 0 0
\(657\) −32.0000 −1.24844
\(658\) 0 0
\(659\) −25.9030 −1.00904 −0.504520 0.863400i \(-0.668330\pi\)
−0.504520 + 0.863400i \(0.668330\pi\)
\(660\) 0 0
\(661\) −31.3050 −1.21762 −0.608811 0.793315i \(-0.708353\pi\)
−0.608811 + 0.793315i \(0.708353\pi\)
\(662\) 0 0
\(663\) 28.4257 1.10396
\(664\) 0 0
\(665\) −61.3050 −2.37730
\(666\) 0 0
\(667\) −3.80423 −0.147300
\(668\) 0 0
\(669\) 85.7771 3.31633
\(670\) 0 0
\(671\) 12.5227 0.483435
\(672\) 0 0
\(673\) −17.8328 −0.687405 −0.343702 0.939079i \(-0.611681\pi\)
−0.343702 + 0.939079i \(0.611681\pi\)
\(674\) 0 0
\(675\) −138.324 −5.32410
\(676\) 0 0
\(677\) −10.3607 −0.398193 −0.199097 0.979980i \(-0.563801\pi\)
−0.199097 + 0.979980i \(0.563801\pi\)
\(678\) 0 0
\(679\) −22.8254 −0.875957
\(680\) 0 0
\(681\) 33.4164 1.28052
\(682\) 0 0
\(683\) −3.59222 −0.137453 −0.0687263 0.997636i \(-0.521894\pi\)
−0.0687263 + 0.997636i \(0.521894\pi\)
\(684\) 0 0
\(685\) 48.3607 1.84777
\(686\) 0 0
\(687\) 66.8110 2.54900
\(688\) 0 0
\(689\) 6.63932 0.252938
\(690\) 0 0
\(691\) −18.8091 −0.715533 −0.357766 0.933811i \(-0.616462\pi\)
−0.357766 + 0.933811i \(0.616462\pi\)
\(692\) 0 0
\(693\) −68.9443 −2.61898
\(694\) 0 0
\(695\) 3.80423 0.144303
\(696\) 0 0
\(697\) 37.8885 1.43513
\(698\) 0 0
\(699\) 55.2268 2.08887
\(700\) 0 0
\(701\) 48.0132 1.81343 0.906716 0.421742i \(-0.138581\pi\)
0.906716 + 0.421742i \(0.138581\pi\)
\(702\) 0 0
\(703\) −15.2169 −0.573916
\(704\) 0 0
\(705\) −70.7771 −2.66562
\(706\) 0 0
\(707\) −36.2461 −1.36318
\(708\) 0 0
\(709\) 35.6525 1.33896 0.669478 0.742832i \(-0.266518\pi\)
0.669478 + 0.742832i \(0.266518\pi\)
\(710\) 0 0
\(711\) 14.1068 0.529048
\(712\) 0 0
\(713\) −2.76393 −0.103510
\(714\) 0 0
\(715\) 33.8545 1.26609
\(716\) 0 0
\(717\) 2.76393 0.103221
\(718\) 0 0
\(719\) −51.8061 −1.93204 −0.966020 0.258466i \(-0.916783\pi\)
−0.966020 + 0.258466i \(0.916783\pi\)
\(720\) 0 0
\(721\) 7.63932 0.284503
\(722\) 0 0
\(723\) 12.1392 0.451462
\(724\) 0 0
\(725\) −12.9443 −0.480738
\(726\) 0 0
\(727\) −24.2784 −0.900438 −0.450219 0.892918i \(-0.648654\pi\)
−0.450219 + 0.892918i \(0.648654\pi\)
\(728\) 0 0
\(729\) −11.4721 −0.424894
\(730\) 0 0
\(731\) 31.3319 1.15885
\(732\) 0 0
\(733\) 4.94427 0.182621 0.0913104 0.995822i \(-0.470894\pi\)
0.0913104 + 0.995822i \(0.470894\pi\)
\(734\) 0 0
\(735\) 19.1926 0.707931
\(736\) 0 0
\(737\) −55.7771 −2.05458
\(738\) 0 0
\(739\) 39.6669 1.45917 0.729585 0.683891i \(-0.239713\pi\)
0.729585 + 0.683891i \(0.239713\pi\)
\(740\) 0 0
\(741\) −33.4164 −1.22758
\(742\) 0 0
\(743\) −47.1038 −1.72807 −0.864035 0.503431i \(-0.832071\pi\)
−0.864035 + 0.503431i \(0.832071\pi\)
\(744\) 0 0
\(745\) 34.8885 1.27822
\(746\) 0 0
\(747\) 104.298 3.81607
\(748\) 0 0
\(749\) −5.52786 −0.201984
\(750\) 0 0
\(751\) 26.0746 0.951474 0.475737 0.879588i \(-0.342181\pi\)
0.475737 + 0.879588i \(0.342181\pi\)
\(752\) 0 0
\(753\) 27.3607 0.997079
\(754\) 0 0
\(755\) 101.947 3.71023
\(756\) 0 0
\(757\) 21.8197 0.793049 0.396525 0.918024i \(-0.370216\pi\)
0.396525 + 0.918024i \(0.370216\pi\)
\(758\) 0 0
\(759\) −53.0472 −1.92549
\(760\) 0 0
\(761\) −15.8885 −0.575959 −0.287980 0.957637i \(-0.592984\pi\)
−0.287980 + 0.957637i \(0.592984\pi\)
\(762\) 0 0
\(763\) −4.14725 −0.150141
\(764\) 0 0
\(765\) −143.554 −5.19021
\(766\) 0 0
\(767\) 11.8367 0.427398
\(768\) 0 0
\(769\) 17.2361 0.621549 0.310774 0.950484i \(-0.399412\pi\)
0.310774 + 0.950484i \(0.399412\pi\)
\(770\) 0 0
\(771\) −4.53077 −0.163172
\(772\) 0 0
\(773\) 31.2361 1.12348 0.561742 0.827313i \(-0.310132\pi\)
0.561742 + 0.827313i \(0.310132\pi\)
\(774\) 0 0
\(775\) −9.40456 −0.337822
\(776\) 0 0
\(777\) −17.8885 −0.641748
\(778\) 0 0
\(779\) −44.5407 −1.59583
\(780\) 0 0
\(781\) 14.7214 0.526772
\(782\) 0 0
\(783\) 10.6861 0.381891
\(784\) 0 0
\(785\) 41.1246 1.46780
\(786\) 0 0
\(787\) −39.4953 −1.40786 −0.703929 0.710271i \(-0.748572\pi\)
−0.703929 + 0.710271i \(0.748572\pi\)
\(788\) 0 0
\(789\) 2.23607 0.0796061
\(790\) 0 0
\(791\) 6.92240 0.246132
\(792\) 0 0
\(793\) 4.87539 0.173130
\(794\) 0 0
\(795\) −49.0714 −1.74038
\(796\) 0 0
\(797\) −29.2361 −1.03559 −0.517797 0.855503i \(-0.673248\pi\)
−0.517797 + 0.855503i \(0.673248\pi\)
\(798\) 0 0
\(799\) −28.4257 −1.00563
\(800\) 0 0
\(801\) 11.0557 0.390635
\(802\) 0 0
\(803\) −22.4014 −0.790527
\(804\) 0 0
\(805\) −37.8885 −1.33540
\(806\) 0 0
\(807\) −23.7234 −0.835104
\(808\) 0 0
\(809\) 11.0557 0.388699 0.194349 0.980932i \(-0.437740\pi\)
0.194349 + 0.980932i \(0.437740\pi\)
\(810\) 0 0
\(811\) −31.6749 −1.11226 −0.556128 0.831097i \(-0.687714\pi\)
−0.556128 + 0.831097i \(0.687714\pi\)
\(812\) 0 0
\(813\) 41.1803 1.44426
\(814\) 0 0
\(815\) 5.42882 0.190163
\(816\) 0 0
\(817\) −36.8328 −1.28862
\(818\) 0 0
\(819\) −26.8416 −0.937921
\(820\) 0 0
\(821\) −42.2361 −1.47405 −0.737024 0.675866i \(-0.763770\pi\)
−0.737024 + 0.675866i \(0.763770\pi\)
\(822\) 0 0
\(823\) 9.06154 0.315865 0.157933 0.987450i \(-0.449517\pi\)
0.157933 + 0.987450i \(0.449517\pi\)
\(824\) 0 0
\(825\) −180.498 −6.28415
\(826\) 0 0
\(827\) 50.5245 1.75691 0.878455 0.477826i \(-0.158575\pi\)
0.878455 + 0.477826i \(0.158575\pi\)
\(828\) 0 0
\(829\) −15.7082 −0.545568 −0.272784 0.962075i \(-0.587945\pi\)
−0.272784 + 0.962075i \(0.587945\pi\)
\(830\) 0 0
\(831\) −48.8999 −1.69632
\(832\) 0 0
\(833\) 7.70820 0.267073
\(834\) 0 0
\(835\) 45.9937 1.59168
\(836\) 0 0
\(837\) 7.76393 0.268361
\(838\) 0 0
\(839\) 23.5519 0.813102 0.406551 0.913628i \(-0.366731\pi\)
0.406551 + 0.913628i \(0.366731\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 32.4014 1.11596
\(844\) 0 0
\(845\) −41.8885 −1.44101
\(846\) 0 0
\(847\) −22.4014 −0.769720
\(848\) 0 0
\(849\) −2.76393 −0.0948579
\(850\) 0 0
\(851\) −9.40456 −0.322384
\(852\) 0 0
\(853\) 10.5836 0.362375 0.181188 0.983449i \(-0.442006\pi\)
0.181188 + 0.983449i \(0.442006\pi\)
\(854\) 0 0
\(855\) 168.758 5.77140
\(856\) 0 0
\(857\) −28.0557 −0.958365 −0.479183 0.877715i \(-0.659067\pi\)
−0.479183 + 0.877715i \(0.659067\pi\)
\(858\) 0 0
\(859\) −27.3561 −0.933379 −0.466689 0.884421i \(-0.654553\pi\)
−0.466689 + 0.884421i \(0.654553\pi\)
\(860\) 0 0
\(861\) −52.3607 −1.78445
\(862\) 0 0
\(863\) −32.5729 −1.10880 −0.554398 0.832252i \(-0.687051\pi\)
−0.554398 + 0.832252i \(0.687051\pi\)
\(864\) 0 0
\(865\) 86.7214 2.94861
\(866\) 0 0
\(867\) −32.0584 −1.08876
\(868\) 0 0
\(869\) 9.87539 0.335000
\(870\) 0 0
\(871\) −21.7153 −0.735795
\(872\) 0 0
\(873\) 62.8328 2.12657
\(874\) 0 0
\(875\) −79.1217 −2.67480
\(876\) 0 0
\(877\) 6.81966 0.230284 0.115142 0.993349i \(-0.463268\pi\)
0.115142 + 0.993349i \(0.463268\pi\)
\(878\) 0 0
\(879\) 28.9807 0.977496
\(880\) 0 0
\(881\) −22.3607 −0.753350 −0.376675 0.926345i \(-0.622933\pi\)
−0.376675 + 0.926345i \(0.622933\pi\)
\(882\) 0 0
\(883\) 13.9758 0.470324 0.235162 0.971956i \(-0.424438\pi\)
0.235162 + 0.971956i \(0.424438\pi\)
\(884\) 0 0
\(885\) −87.4853 −2.94079
\(886\) 0 0
\(887\) −21.3318 −0.716251 −0.358126 0.933673i \(-0.616584\pi\)
−0.358126 + 0.933673i \(0.616584\pi\)
\(888\) 0 0
\(889\) 14.4721 0.485380
\(890\) 0 0
\(891\) 61.0391 2.04489
\(892\) 0 0
\(893\) 33.4164 1.11824
\(894\) 0 0
\(895\) −90.5344 −3.02623
\(896\) 0 0
\(897\) −20.6525 −0.689566
\(898\) 0 0
\(899\) 0.726543 0.0242315
\(900\) 0 0
\(901\) −19.7082 −0.656575
\(902\) 0 0
\(903\) −43.2996 −1.44092
\(904\) 0 0
\(905\) 81.2492 2.70082
\(906\) 0 0
\(907\) −35.4791 −1.17806 −0.589032 0.808109i \(-0.700491\pi\)
−0.589032 + 0.808109i \(0.700491\pi\)
\(908\) 0 0
\(909\) 99.7771 3.30940
\(910\) 0 0
\(911\) −43.8141 −1.45163 −0.725813 0.687892i \(-0.758536\pi\)
−0.725813 + 0.687892i \(0.758536\pi\)
\(912\) 0 0
\(913\) 73.0132 2.41638
\(914\) 0 0
\(915\) −36.0341 −1.19125
\(916\) 0 0
\(917\) 10.2492 0.338459
\(918\) 0 0
\(919\) 7.95148 0.262295 0.131148 0.991363i \(-0.458134\pi\)
0.131148 + 0.991363i \(0.458134\pi\)
\(920\) 0 0
\(921\) 85.2492 2.80906
\(922\) 0 0
\(923\) 5.73136 0.188650
\(924\) 0 0
\(925\) −32.0000 −1.05215
\(926\) 0 0
\(927\) −21.0292 −0.690691
\(928\) 0 0
\(929\) 54.0000 1.77168 0.885841 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 0 0
\(931\) −9.06154 −0.296980
\(932\) 0 0
\(933\) −47.8885 −1.56780
\(934\) 0 0
\(935\) −100.494 −3.28650
\(936\) 0 0
\(937\) 34.3607 1.12251 0.561257 0.827641i \(-0.310318\pi\)
0.561257 + 0.827641i \(0.310318\pi\)
\(938\) 0 0
\(939\) −44.3691 −1.44793
\(940\) 0 0
\(941\) −50.0132 −1.63038 −0.815191 0.579192i \(-0.803368\pi\)
−0.815191 + 0.579192i \(0.803368\pi\)
\(942\) 0 0
\(943\) −27.5276 −0.896423
\(944\) 0 0
\(945\) 106.430 3.46215
\(946\) 0 0
\(947\) 25.5600 0.830589 0.415294 0.909687i \(-0.363679\pi\)
0.415294 + 0.909687i \(0.363679\pi\)
\(948\) 0 0
\(949\) −8.72136 −0.283107
\(950\) 0 0
\(951\) 0.898056 0.0291215
\(952\) 0 0
\(953\) −12.4164 −0.402207 −0.201103 0.979570i \(-0.564453\pi\)
−0.201103 + 0.979570i \(0.564453\pi\)
\(954\) 0 0
\(955\) −102.845 −3.32799
\(956\) 0 0
\(957\) 13.9443 0.450754
\(958\) 0 0
\(959\) −26.8416 −0.866760
\(960\) 0 0
\(961\) −30.4721 −0.982972
\(962\) 0 0
\(963\) 15.2169 0.490358
\(964\) 0 0
\(965\) 37.1246 1.19508
\(966\) 0 0
\(967\) −40.2219 −1.29345 −0.646724 0.762724i \(-0.723862\pi\)
−0.646724 + 0.762724i \(0.723862\pi\)
\(968\) 0 0
\(969\) 99.1935 3.18655
\(970\) 0 0
\(971\) 1.45309 0.0466317 0.0233159 0.999728i \(-0.492578\pi\)
0.0233159 + 0.999728i \(0.492578\pi\)
\(972\) 0 0
\(973\) −2.11146 −0.0676902
\(974\) 0 0
\(975\) −70.2722 −2.25051
\(976\) 0 0
\(977\) −43.8328 −1.40234 −0.701168 0.712996i \(-0.747338\pi\)
−0.701168 + 0.712996i \(0.747338\pi\)
\(978\) 0 0
\(979\) 7.73948 0.247355
\(980\) 0 0
\(981\) 11.4164 0.364498
\(982\) 0 0
\(983\) 2.52265 0.0804602 0.0402301 0.999190i \(-0.487191\pi\)
0.0402301 + 0.999190i \(0.487191\pi\)
\(984\) 0 0
\(985\) 46.3607 1.47717
\(986\) 0 0
\(987\) 39.2833 1.25040
\(988\) 0 0
\(989\) −22.7639 −0.723851
\(990\) 0 0
\(991\) −10.8576 −0.344905 −0.172452 0.985018i \(-0.555169\pi\)
−0.172452 + 0.985018i \(0.555169\pi\)
\(992\) 0 0
\(993\) −0.527864 −0.0167513
\(994\) 0 0
\(995\) 82.0279 2.60046
\(996\) 0 0
\(997\) −0.944272 −0.0299054 −0.0149527 0.999888i \(-0.504760\pi\)
−0.0149527 + 0.999888i \(0.504760\pi\)
\(998\) 0 0
\(999\) 26.4176 0.835815
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 1856.2.a.z.1.1 4
4.3 odd 2 inner 1856.2.a.z.1.4 4
8.3 odd 2 928.2.a.f.1.1 4
8.5 even 2 928.2.a.f.1.4 yes 4
24.5 odd 2 8352.2.a.ba.1.3 4
24.11 even 2 8352.2.a.ba.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
928.2.a.f.1.1 4 8.3 odd 2
928.2.a.f.1.4 yes 4 8.5 even 2
1856.2.a.z.1.1 4 1.1 even 1 trivial
1856.2.a.z.1.4 4 4.3 odd 2 inner
8352.2.a.ba.1.3 4 24.5 odd 2
8352.2.a.ba.1.4 4 24.11 even 2