Properties

Label 1104.2.a.j.1.1
Level $1104$
Weight $2$
Character 1104.1
Self dual yes
Analytic conductor $8.815$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1104,2,Mod(1,1104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1104, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1104.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1104.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: \(8.81548438315\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1104.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.23607 q^{5} -4.47214 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.23607 q^{5} -4.47214 q^{7} +1.00000 q^{9} +0.763932 q^{11} -4.47214 q^{13} +3.23607 q^{15} -4.00000 q^{17} +7.70820 q^{19} +4.47214 q^{21} -1.00000 q^{23} +5.47214 q^{25} -1.00000 q^{27} +4.47214 q^{29} -6.47214 q^{31} -0.763932 q^{33} +14.4721 q^{35} +6.76393 q^{37} +4.47214 q^{39} -2.00000 q^{41} +9.23607 q^{43} -3.23607 q^{45} -4.00000 q^{47} +13.0000 q^{49} +4.00000 q^{51} +0.763932 q^{53} -2.47214 q^{55} -7.70820 q^{57} -8.94427 q^{59} +5.23607 q^{61} -4.47214 q^{63} +14.4721 q^{65} +3.70820 q^{67} +1.00000 q^{69} +8.94427 q^{71} +4.47214 q^{73} -5.47214 q^{75} -3.41641 q^{77} +4.47214 q^{79} +1.00000 q^{81} -8.76393 q^{83} +12.9443 q^{85} -4.47214 q^{87} -1.52786 q^{89} +20.0000 q^{91} +6.47214 q^{93} -24.9443 q^{95} -8.47214 q^{97} +0.763932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} + 6 q^{11} + 2 q^{15} - 8 q^{17} + 2 q^{19} - 2 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{31} - 6 q^{33} + 20 q^{35} + 18 q^{37} - 4 q^{41} + 14 q^{43} - 2 q^{45} - 8 q^{47} + 26 q^{49} + 8 q^{51} + 6 q^{53} + 4 q^{55} - 2 q^{57} + 6 q^{61} + 20 q^{65} - 6 q^{67} + 2 q^{69} - 2 q^{75} + 20 q^{77} + 2 q^{81} - 22 q^{83} + 8 q^{85} - 12 q^{89} + 40 q^{91} + 4 q^{93} - 32 q^{95} - 8 q^{97} + 6 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −3.23607 −1.44721 −0.723607 0.690212i \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) 0 0
\(7\) −4.47214 −1.69031 −0.845154 0.534522i \(-0.820491\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.763932 0.230334 0.115167 0.993346i \(-0.463260\pi\)
0.115167 + 0.993346i \(0.463260\pi\)
\(12\) 0 0
\(13\) −4.47214 −1.24035 −0.620174 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) 3.23607 0.835549
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 7.70820 1.76838 0.884192 0.467124i \(-0.154710\pi\)
0.884192 + 0.467124i \(0.154710\pi\)
\(20\) 0 0
\(21\) 4.47214 0.975900
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) −6.47214 −1.16243 −0.581215 0.813750i \(-0.697422\pi\)
−0.581215 + 0.813750i \(0.697422\pi\)
\(32\) 0 0
\(33\) −0.763932 −0.132983
\(34\) 0 0
\(35\) 14.4721 2.44624
\(36\) 0 0
\(37\) 6.76393 1.11198 0.555992 0.831188i \(-0.312339\pi\)
0.555992 + 0.831188i \(0.312339\pi\)
\(38\) 0 0
\(39\) 4.47214 0.716115
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 9.23607 1.40849 0.704244 0.709958i \(-0.251286\pi\)
0.704244 + 0.709958i \(0.251286\pi\)
\(44\) 0 0
\(45\) −3.23607 −0.482405
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 13.0000 1.85714
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 0.763932 0.104934 0.0524671 0.998623i \(-0.483292\pi\)
0.0524671 + 0.998623i \(0.483292\pi\)
\(54\) 0 0
\(55\) −2.47214 −0.333343
\(56\) 0 0
\(57\) −7.70820 −1.02098
\(58\) 0 0
\(59\) −8.94427 −1.16445 −0.582223 0.813029i \(-0.697817\pi\)
−0.582223 + 0.813029i \(0.697817\pi\)
\(60\) 0 0
\(61\) 5.23607 0.670410 0.335205 0.942145i \(-0.391194\pi\)
0.335205 + 0.942145i \(0.391194\pi\)
\(62\) 0 0
\(63\) −4.47214 −0.563436
\(64\) 0 0
\(65\) 14.4721 1.79505
\(66\) 0 0
\(67\) 3.70820 0.453029 0.226515 0.974008i \(-0.427267\pi\)
0.226515 + 0.974008i \(0.427267\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 8.94427 1.06149 0.530745 0.847532i \(-0.321912\pi\)
0.530745 + 0.847532i \(0.321912\pi\)
\(72\) 0 0
\(73\) 4.47214 0.523424 0.261712 0.965146i \(-0.415713\pi\)
0.261712 + 0.965146i \(0.415713\pi\)
\(74\) 0 0
\(75\) −5.47214 −0.631868
\(76\) 0 0
\(77\) −3.41641 −0.389336
\(78\) 0 0
\(79\) 4.47214 0.503155 0.251577 0.967837i \(-0.419051\pi\)
0.251577 + 0.967837i \(0.419051\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.76393 −0.961967 −0.480983 0.876730i \(-0.659720\pi\)
−0.480983 + 0.876730i \(0.659720\pi\)
\(84\) 0 0
\(85\) 12.9443 1.40400
\(86\) 0 0
\(87\) −4.47214 −0.479463
\(88\) 0 0
\(89\) −1.52786 −0.161953 −0.0809766 0.996716i \(-0.525804\pi\)
−0.0809766 + 0.996716i \(0.525804\pi\)
\(90\) 0 0
\(91\) 20.0000 2.09657
\(92\) 0 0
\(93\) 6.47214 0.671129
\(94\) 0 0
\(95\) −24.9443 −2.55923
\(96\) 0 0
\(97\) −8.47214 −0.860215 −0.430108 0.902778i \(-0.641524\pi\)
−0.430108 + 0.902778i \(0.641524\pi\)
\(98\) 0 0
\(99\) 0.763932 0.0767781
\(100\) 0 0
\(101\) −4.47214 −0.444994 −0.222497 0.974933i \(-0.571421\pi\)
−0.222497 + 0.974933i \(0.571421\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) −14.4721 −1.41234
\(106\) 0 0
\(107\) −18.6525 −1.80320 −0.901601 0.432568i \(-0.857608\pi\)
−0.901601 + 0.432568i \(0.857608\pi\)
\(108\) 0 0
\(109\) 9.23607 0.884655 0.442327 0.896854i \(-0.354153\pi\)
0.442327 + 0.896854i \(0.354153\pi\)
\(110\) 0 0
\(111\) −6.76393 −0.642004
\(112\) 0 0
\(113\) −14.4721 −1.36142 −0.680712 0.732551i \(-0.738329\pi\)
−0.680712 + 0.732551i \(0.738329\pi\)
\(114\) 0 0
\(115\) 3.23607 0.301765
\(116\) 0 0
\(117\) −4.47214 −0.413449
\(118\) 0 0
\(119\) 17.8885 1.63984
\(120\) 0 0
\(121\) −10.4164 −0.946946
\(122\) 0 0
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) −1.52786 −0.136656
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) −9.23607 −0.813190
\(130\) 0 0
\(131\) 18.4721 1.61392 0.806959 0.590607i \(-0.201112\pi\)
0.806959 + 0.590607i \(0.201112\pi\)
\(132\) 0 0
\(133\) −34.4721 −2.98911
\(134\) 0 0
\(135\) 3.23607 0.278516
\(136\) 0 0
\(137\) 20.9443 1.78939 0.894695 0.446678i \(-0.147393\pi\)
0.894695 + 0.446678i \(0.147393\pi\)
\(138\) 0 0
\(139\) −0.944272 −0.0800921 −0.0400460 0.999198i \(-0.512750\pi\)
−0.0400460 + 0.999198i \(0.512750\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 0 0
\(143\) −3.41641 −0.285694
\(144\) 0 0
\(145\) −14.4721 −1.20185
\(146\) 0 0
\(147\) −13.0000 −1.07222
\(148\) 0 0
\(149\) 1.70820 0.139942 0.0699708 0.997549i \(-0.477709\pi\)
0.0699708 + 0.997549i \(0.477709\pi\)
\(150\) 0 0
\(151\) 5.52786 0.449851 0.224926 0.974376i \(-0.427786\pi\)
0.224926 + 0.974376i \(0.427786\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) 20.9443 1.68228
\(156\) 0 0
\(157\) 24.6525 1.96748 0.983741 0.179594i \(-0.0574783\pi\)
0.983741 + 0.179594i \(0.0574783\pi\)
\(158\) 0 0
\(159\) −0.763932 −0.0605838
\(160\) 0 0
\(161\) 4.47214 0.352454
\(162\) 0 0
\(163\) −6.47214 −0.506937 −0.253468 0.967344i \(-0.581571\pi\)
−0.253468 + 0.967344i \(0.581571\pi\)
\(164\) 0 0
\(165\) 2.47214 0.192456
\(166\) 0 0
\(167\) 0.944272 0.0730700 0.0365350 0.999332i \(-0.488368\pi\)
0.0365350 + 0.999332i \(0.488368\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 7.70820 0.589461
\(172\) 0 0
\(173\) 9.41641 0.715916 0.357958 0.933738i \(-0.383473\pi\)
0.357958 + 0.933738i \(0.383473\pi\)
\(174\) 0 0
\(175\) −24.4721 −1.84992
\(176\) 0 0
\(177\) 8.94427 0.672293
\(178\) 0 0
\(179\) 7.41641 0.554328 0.277164 0.960823i \(-0.410605\pi\)
0.277164 + 0.960823i \(0.410605\pi\)
\(180\) 0 0
\(181\) −6.76393 −0.502759 −0.251380 0.967889i \(-0.580884\pi\)
−0.251380 + 0.967889i \(0.580884\pi\)
\(182\) 0 0
\(183\) −5.23607 −0.387061
\(184\) 0 0
\(185\) −21.8885 −1.60928
\(186\) 0 0
\(187\) −3.05573 −0.223457
\(188\) 0 0
\(189\) 4.47214 0.325300
\(190\) 0 0
\(191\) 2.47214 0.178877 0.0894387 0.995992i \(-0.471493\pi\)
0.0894387 + 0.995992i \(0.471493\pi\)
\(192\) 0 0
\(193\) −11.8885 −0.855756 −0.427878 0.903836i \(-0.640739\pi\)
−0.427878 + 0.903836i \(0.640739\pi\)
\(194\) 0 0
\(195\) −14.4721 −1.03637
\(196\) 0 0
\(197\) −14.9443 −1.06474 −0.532368 0.846513i \(-0.678698\pi\)
−0.532368 + 0.846513i \(0.678698\pi\)
\(198\) 0 0
\(199\) 9.41641 0.667511 0.333756 0.942660i \(-0.391684\pi\)
0.333756 + 0.942660i \(0.391684\pi\)
\(200\) 0 0
\(201\) −3.70820 −0.261557
\(202\) 0 0
\(203\) −20.0000 −1.40372
\(204\) 0 0
\(205\) 6.47214 0.452034
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 5.88854 0.407319
\(210\) 0 0
\(211\) −3.41641 −0.235195 −0.117598 0.993061i \(-0.537519\pi\)
−0.117598 + 0.993061i \(0.537519\pi\)
\(212\) 0 0
\(213\) −8.94427 −0.612851
\(214\) 0 0
\(215\) −29.8885 −2.03838
\(216\) 0 0
\(217\) 28.9443 1.96487
\(218\) 0 0
\(219\) −4.47214 −0.302199
\(220\) 0 0
\(221\) 17.8885 1.20331
\(222\) 0 0
\(223\) 7.41641 0.496639 0.248320 0.968678i \(-0.420122\pi\)
0.248320 + 0.968678i \(0.420122\pi\)
\(224\) 0 0
\(225\) 5.47214 0.364809
\(226\) 0 0
\(227\) 4.76393 0.316193 0.158097 0.987424i \(-0.449464\pi\)
0.158097 + 0.987424i \(0.449464\pi\)
\(228\) 0 0
\(229\) 4.29180 0.283610 0.141805 0.989895i \(-0.454709\pi\)
0.141805 + 0.989895i \(0.454709\pi\)
\(230\) 0 0
\(231\) 3.41641 0.224783
\(232\) 0 0
\(233\) −15.8885 −1.04089 −0.520447 0.853894i \(-0.674235\pi\)
−0.520447 + 0.853894i \(0.674235\pi\)
\(234\) 0 0
\(235\) 12.9443 0.844391
\(236\) 0 0
\(237\) −4.47214 −0.290496
\(238\) 0 0
\(239\) −12.9443 −0.837295 −0.418648 0.908149i \(-0.637496\pi\)
−0.418648 + 0.908149i \(0.637496\pi\)
\(240\) 0 0
\(241\) −3.52786 −0.227250 −0.113625 0.993524i \(-0.536246\pi\)
−0.113625 + 0.993524i \(0.536246\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −42.0689 −2.68768
\(246\) 0 0
\(247\) −34.4721 −2.19341
\(248\) 0 0
\(249\) 8.76393 0.555392
\(250\) 0 0
\(251\) 6.29180 0.397135 0.198567 0.980087i \(-0.436371\pi\)
0.198567 + 0.980087i \(0.436371\pi\)
\(252\) 0 0
\(253\) −0.763932 −0.0480280
\(254\) 0 0
\(255\) −12.9443 −0.810602
\(256\) 0 0
\(257\) 23.8885 1.49013 0.745063 0.666994i \(-0.232419\pi\)
0.745063 + 0.666994i \(0.232419\pi\)
\(258\) 0 0
\(259\) −30.2492 −1.87960
\(260\) 0 0
\(261\) 4.47214 0.276818
\(262\) 0 0
\(263\) −7.05573 −0.435075 −0.217537 0.976052i \(-0.569802\pi\)
−0.217537 + 0.976052i \(0.569802\pi\)
\(264\) 0 0
\(265\) −2.47214 −0.151862
\(266\) 0 0
\(267\) 1.52786 0.0935038
\(268\) 0 0
\(269\) −30.9443 −1.88671 −0.943353 0.331791i \(-0.892347\pi\)
−0.943353 + 0.331791i \(0.892347\pi\)
\(270\) 0 0
\(271\) −0.944272 −0.0573604 −0.0286802 0.999589i \(-0.509130\pi\)
−0.0286802 + 0.999589i \(0.509130\pi\)
\(272\) 0 0
\(273\) −20.0000 −1.21046
\(274\) 0 0
\(275\) 4.18034 0.252084
\(276\) 0 0
\(277\) −11.5279 −0.692642 −0.346321 0.938116i \(-0.612569\pi\)
−0.346321 + 0.938116i \(0.612569\pi\)
\(278\) 0 0
\(279\) −6.47214 −0.387477
\(280\) 0 0
\(281\) −22.4721 −1.34058 −0.670288 0.742101i \(-0.733829\pi\)
−0.670288 + 0.742101i \(0.733829\pi\)
\(282\) 0 0
\(283\) 26.1803 1.55626 0.778130 0.628103i \(-0.216169\pi\)
0.778130 + 0.628103i \(0.216169\pi\)
\(284\) 0 0
\(285\) 24.9443 1.47757
\(286\) 0 0
\(287\) 8.94427 0.527964
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 8.47214 0.496645
\(292\) 0 0
\(293\) 13.7082 0.800842 0.400421 0.916331i \(-0.368864\pi\)
0.400421 + 0.916331i \(0.368864\pi\)
\(294\) 0 0
\(295\) 28.9443 1.68520
\(296\) 0 0
\(297\) −0.763932 −0.0443278
\(298\) 0 0
\(299\) 4.47214 0.258630
\(300\) 0 0
\(301\) −41.3050 −2.38078
\(302\) 0 0
\(303\) 4.47214 0.256917
\(304\) 0 0
\(305\) −16.9443 −0.970226
\(306\) 0 0
\(307\) 11.4164 0.651569 0.325784 0.945444i \(-0.394372\pi\)
0.325784 + 0.945444i \(0.394372\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) 3.05573 0.173274 0.0866372 0.996240i \(-0.472388\pi\)
0.0866372 + 0.996240i \(0.472388\pi\)
\(312\) 0 0
\(313\) 24.4721 1.38325 0.691623 0.722258i \(-0.256896\pi\)
0.691623 + 0.722258i \(0.256896\pi\)
\(314\) 0 0
\(315\) 14.4721 0.815412
\(316\) 0 0
\(317\) 28.4721 1.59915 0.799577 0.600563i \(-0.205057\pi\)
0.799577 + 0.600563i \(0.205057\pi\)
\(318\) 0 0
\(319\) 3.41641 0.191282
\(320\) 0 0
\(321\) 18.6525 1.04108
\(322\) 0 0
\(323\) −30.8328 −1.71558
\(324\) 0 0
\(325\) −24.4721 −1.35747
\(326\) 0 0
\(327\) −9.23607 −0.510756
\(328\) 0 0
\(329\) 17.8885 0.986227
\(330\) 0 0
\(331\) 1.52786 0.0839790 0.0419895 0.999118i \(-0.486630\pi\)
0.0419895 + 0.999118i \(0.486630\pi\)
\(332\) 0 0
\(333\) 6.76393 0.370661
\(334\) 0 0
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) 15.8885 0.865504 0.432752 0.901513i \(-0.357543\pi\)
0.432752 + 0.901513i \(0.357543\pi\)
\(338\) 0 0
\(339\) 14.4721 0.786019
\(340\) 0 0
\(341\) −4.94427 −0.267747
\(342\) 0 0
\(343\) −26.8328 −1.44884
\(344\) 0 0
\(345\) −3.23607 −0.174224
\(346\) 0 0
\(347\) −21.5279 −1.15568 −0.577838 0.816151i \(-0.696104\pi\)
−0.577838 + 0.816151i \(0.696104\pi\)
\(348\) 0 0
\(349\) −31.8885 −1.70695 −0.853477 0.521130i \(-0.825511\pi\)
−0.853477 + 0.521130i \(0.825511\pi\)
\(350\) 0 0
\(351\) 4.47214 0.238705
\(352\) 0 0
\(353\) 31.8885 1.69726 0.848628 0.528990i \(-0.177429\pi\)
0.848628 + 0.528990i \(0.177429\pi\)
\(354\) 0 0
\(355\) −28.9443 −1.53620
\(356\) 0 0
\(357\) −17.8885 −0.946762
\(358\) 0 0
\(359\) 33.3050 1.75777 0.878884 0.477035i \(-0.158289\pi\)
0.878884 + 0.477035i \(0.158289\pi\)
\(360\) 0 0
\(361\) 40.4164 2.12718
\(362\) 0 0
\(363\) 10.4164 0.546720
\(364\) 0 0
\(365\) −14.4721 −0.757506
\(366\) 0 0
\(367\) 17.4164 0.909129 0.454565 0.890714i \(-0.349795\pi\)
0.454565 + 0.890714i \(0.349795\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −3.41641 −0.177371
\(372\) 0 0
\(373\) 13.5967 0.704013 0.352006 0.935998i \(-0.385500\pi\)
0.352006 + 0.935998i \(0.385500\pi\)
\(374\) 0 0
\(375\) 1.52786 0.0788986
\(376\) 0 0
\(377\) −20.0000 −1.03005
\(378\) 0 0
\(379\) −0.291796 −0.0149886 −0.00749428 0.999972i \(-0.502386\pi\)
−0.00749428 + 0.999972i \(0.502386\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) −24.9443 −1.27459 −0.637296 0.770619i \(-0.719947\pi\)
−0.637296 + 0.770619i \(0.719947\pi\)
\(384\) 0 0
\(385\) 11.0557 0.563452
\(386\) 0 0
\(387\) 9.23607 0.469496
\(388\) 0 0
\(389\) 10.2918 0.521815 0.260907 0.965364i \(-0.415978\pi\)
0.260907 + 0.965364i \(0.415978\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) −18.4721 −0.931796
\(394\) 0 0
\(395\) −14.4721 −0.728172
\(396\) 0 0
\(397\) 9.05573 0.454494 0.227247 0.973837i \(-0.427028\pi\)
0.227247 + 0.973837i \(0.427028\pi\)
\(398\) 0 0
\(399\) 34.4721 1.72577
\(400\) 0 0
\(401\) −8.94427 −0.446656 −0.223328 0.974743i \(-0.571692\pi\)
−0.223328 + 0.974743i \(0.571692\pi\)
\(402\) 0 0
\(403\) 28.9443 1.44182
\(404\) 0 0
\(405\) −3.23607 −0.160802
\(406\) 0 0
\(407\) 5.16718 0.256128
\(408\) 0 0
\(409\) −0.111456 −0.00551115 −0.00275558 0.999996i \(-0.500877\pi\)
−0.00275558 + 0.999996i \(0.500877\pi\)
\(410\) 0 0
\(411\) −20.9443 −1.03310
\(412\) 0 0
\(413\) 40.0000 1.96827
\(414\) 0 0
\(415\) 28.3607 1.39217
\(416\) 0 0
\(417\) 0.944272 0.0462412
\(418\) 0 0
\(419\) 30.0689 1.46896 0.734481 0.678630i \(-0.237426\pi\)
0.734481 + 0.678630i \(0.237426\pi\)
\(420\) 0 0
\(421\) −5.23607 −0.255190 −0.127595 0.991826i \(-0.540726\pi\)
−0.127595 + 0.991826i \(0.540726\pi\)
\(422\) 0 0
\(423\) −4.00000 −0.194487
\(424\) 0 0
\(425\) −21.8885 −1.06175
\(426\) 0 0
\(427\) −23.4164 −1.13320
\(428\) 0 0
\(429\) 3.41641 0.164946
\(430\) 0 0
\(431\) −3.41641 −0.164563 −0.0822813 0.996609i \(-0.526221\pi\)
−0.0822813 + 0.996609i \(0.526221\pi\)
\(432\) 0 0
\(433\) 5.41641 0.260296 0.130148 0.991495i \(-0.458455\pi\)
0.130148 + 0.991495i \(0.458455\pi\)
\(434\) 0 0
\(435\) 14.4721 0.693886
\(436\) 0 0
\(437\) −7.70820 −0.368733
\(438\) 0 0
\(439\) −36.9443 −1.76325 −0.881627 0.471947i \(-0.843551\pi\)
−0.881627 + 0.471947i \(0.843551\pi\)
\(440\) 0 0
\(441\) 13.0000 0.619048
\(442\) 0 0
\(443\) 32.9443 1.56523 0.782615 0.622506i \(-0.213885\pi\)
0.782615 + 0.622506i \(0.213885\pi\)
\(444\) 0 0
\(445\) 4.94427 0.234381
\(446\) 0 0
\(447\) −1.70820 −0.0807953
\(448\) 0 0
\(449\) −19.8885 −0.938598 −0.469299 0.883039i \(-0.655493\pi\)
−0.469299 + 0.883039i \(0.655493\pi\)
\(450\) 0 0
\(451\) −1.52786 −0.0719443
\(452\) 0 0
\(453\) −5.52786 −0.259722
\(454\) 0 0
\(455\) −64.7214 −3.03418
\(456\) 0 0
\(457\) 36.4721 1.70609 0.853047 0.521834i \(-0.174752\pi\)
0.853047 + 0.521834i \(0.174752\pi\)
\(458\) 0 0
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 0 0
\(465\) −20.9443 −0.971267
\(466\) 0 0
\(467\) 40.1803 1.85932 0.929662 0.368413i \(-0.120099\pi\)
0.929662 + 0.368413i \(0.120099\pi\)
\(468\) 0 0
\(469\) −16.5836 −0.765759
\(470\) 0 0
\(471\) −24.6525 −1.13593
\(472\) 0 0
\(473\) 7.05573 0.324423
\(474\) 0 0
\(475\) 42.1803 1.93537
\(476\) 0 0
\(477\) 0.763932 0.0349780
\(478\) 0 0
\(479\) 11.0557 0.505149 0.252575 0.967577i \(-0.418723\pi\)
0.252575 + 0.967577i \(0.418723\pi\)
\(480\) 0 0
\(481\) −30.2492 −1.37925
\(482\) 0 0
\(483\) −4.47214 −0.203489
\(484\) 0 0
\(485\) 27.4164 1.24491
\(486\) 0 0
\(487\) 25.8885 1.17312 0.586561 0.809905i \(-0.300481\pi\)
0.586561 + 0.809905i \(0.300481\pi\)
\(488\) 0 0
\(489\) 6.47214 0.292680
\(490\) 0 0
\(491\) 33.3050 1.50303 0.751516 0.659715i \(-0.229323\pi\)
0.751516 + 0.659715i \(0.229323\pi\)
\(492\) 0 0
\(493\) −17.8885 −0.805659
\(494\) 0 0
\(495\) −2.47214 −0.111114
\(496\) 0 0
\(497\) −40.0000 −1.79425
\(498\) 0 0
\(499\) −27.4164 −1.22733 −0.613663 0.789568i \(-0.710305\pi\)
−0.613663 + 0.789568i \(0.710305\pi\)
\(500\) 0 0
\(501\) −0.944272 −0.0421870
\(502\) 0 0
\(503\) −1.52786 −0.0681241 −0.0340620 0.999420i \(-0.510844\pi\)
−0.0340620 + 0.999420i \(0.510844\pi\)
\(504\) 0 0
\(505\) 14.4721 0.644002
\(506\) 0 0
\(507\) −7.00000 −0.310881
\(508\) 0 0
\(509\) −6.58359 −0.291813 −0.145906 0.989298i \(-0.546610\pi\)
−0.145906 + 0.989298i \(0.546610\pi\)
\(510\) 0 0
\(511\) −20.0000 −0.884748
\(512\) 0 0
\(513\) −7.70820 −0.340326
\(514\) 0 0
\(515\) −19.4164 −0.855589
\(516\) 0 0
\(517\) −3.05573 −0.134391
\(518\) 0 0
\(519\) −9.41641 −0.413334
\(520\) 0 0
\(521\) 16.0000 0.700973 0.350486 0.936568i \(-0.386016\pi\)
0.350486 + 0.936568i \(0.386016\pi\)
\(522\) 0 0
\(523\) −12.2918 −0.537483 −0.268741 0.963212i \(-0.586608\pi\)
−0.268741 + 0.963212i \(0.586608\pi\)
\(524\) 0 0
\(525\) 24.4721 1.06805
\(526\) 0 0
\(527\) 25.8885 1.12772
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −8.94427 −0.388148
\(532\) 0 0
\(533\) 8.94427 0.387419
\(534\) 0 0
\(535\) 60.3607 2.60962
\(536\) 0 0
\(537\) −7.41641 −0.320042
\(538\) 0 0
\(539\) 9.93112 0.427763
\(540\) 0 0
\(541\) 43.8885 1.88692 0.943458 0.331492i \(-0.107552\pi\)
0.943458 + 0.331492i \(0.107552\pi\)
\(542\) 0 0
\(543\) 6.76393 0.290268
\(544\) 0 0
\(545\) −29.8885 −1.28028
\(546\) 0 0
\(547\) 21.3050 0.910934 0.455467 0.890253i \(-0.349472\pi\)
0.455467 + 0.890253i \(0.349472\pi\)
\(548\) 0 0
\(549\) 5.23607 0.223470
\(550\) 0 0
\(551\) 34.4721 1.46856
\(552\) 0 0
\(553\) −20.0000 −0.850487
\(554\) 0 0
\(555\) 21.8885 0.929117
\(556\) 0 0
\(557\) 25.1246 1.06456 0.532282 0.846567i \(-0.321335\pi\)
0.532282 + 0.846567i \(0.321335\pi\)
\(558\) 0 0
\(559\) −41.3050 −1.74701
\(560\) 0 0
\(561\) 3.05573 0.129013
\(562\) 0 0
\(563\) 6.65248 0.280368 0.140184 0.990125i \(-0.455231\pi\)
0.140184 + 0.990125i \(0.455231\pi\)
\(564\) 0 0
\(565\) 46.8328 1.97027
\(566\) 0 0
\(567\) −4.47214 −0.187812
\(568\) 0 0
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) 27.7082 1.15955 0.579776 0.814776i \(-0.303140\pi\)
0.579776 + 0.814776i \(0.303140\pi\)
\(572\) 0 0
\(573\) −2.47214 −0.103275
\(574\) 0 0
\(575\) −5.47214 −0.228204
\(576\) 0 0
\(577\) 10.3607 0.431321 0.215660 0.976468i \(-0.430810\pi\)
0.215660 + 0.976468i \(0.430810\pi\)
\(578\) 0 0
\(579\) 11.8885 0.494071
\(580\) 0 0
\(581\) 39.1935 1.62602
\(582\) 0 0
\(583\) 0.583592 0.0241699
\(584\) 0 0
\(585\) 14.4721 0.598349
\(586\) 0 0
\(587\) 2.47214 0.102036 0.0510180 0.998698i \(-0.483753\pi\)
0.0510180 + 0.998698i \(0.483753\pi\)
\(588\) 0 0
\(589\) −49.8885 −2.05562
\(590\) 0 0
\(591\) 14.9443 0.614725
\(592\) 0 0
\(593\) −37.7771 −1.55132 −0.775660 0.631152i \(-0.782583\pi\)
−0.775660 + 0.631152i \(0.782583\pi\)
\(594\) 0 0
\(595\) −57.8885 −2.37320
\(596\) 0 0
\(597\) −9.41641 −0.385388
\(598\) 0 0
\(599\) 1.88854 0.0771638 0.0385819 0.999255i \(-0.487716\pi\)
0.0385819 + 0.999255i \(0.487716\pi\)
\(600\) 0 0
\(601\) 34.3607 1.40160 0.700801 0.713357i \(-0.252826\pi\)
0.700801 + 0.713357i \(0.252826\pi\)
\(602\) 0 0
\(603\) 3.70820 0.151010
\(604\) 0 0
\(605\) 33.7082 1.37043
\(606\) 0 0
\(607\) 26.4721 1.07447 0.537235 0.843432i \(-0.319469\pi\)
0.537235 + 0.843432i \(0.319469\pi\)
\(608\) 0 0
\(609\) 20.0000 0.810441
\(610\) 0 0
\(611\) 17.8885 0.723693
\(612\) 0 0
\(613\) 15.1246 0.610877 0.305439 0.952212i \(-0.401197\pi\)
0.305439 + 0.952212i \(0.401197\pi\)
\(614\) 0 0
\(615\) −6.47214 −0.260982
\(616\) 0 0
\(617\) −24.3607 −0.980724 −0.490362 0.871519i \(-0.663135\pi\)
−0.490362 + 0.871519i \(0.663135\pi\)
\(618\) 0 0
\(619\) −31.7082 −1.27446 −0.637230 0.770674i \(-0.719920\pi\)
−0.637230 + 0.770674i \(0.719920\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 6.83282 0.273751
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) 0 0
\(627\) −5.88854 −0.235166
\(628\) 0 0
\(629\) −27.0557 −1.07878
\(630\) 0 0
\(631\) 6.00000 0.238856 0.119428 0.992843i \(-0.461894\pi\)
0.119428 + 0.992843i \(0.461894\pi\)
\(632\) 0 0
\(633\) 3.41641 0.135790
\(634\) 0 0
\(635\) −12.9443 −0.513678
\(636\) 0 0
\(637\) −58.1378 −2.30350
\(638\) 0 0
\(639\) 8.94427 0.353830
\(640\) 0 0
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) 0 0
\(643\) −34.5410 −1.36216 −0.681082 0.732207i \(-0.738490\pi\)
−0.681082 + 0.732207i \(0.738490\pi\)
\(644\) 0 0
\(645\) 29.8885 1.17686
\(646\) 0 0
\(647\) 4.94427 0.194379 0.0971897 0.995266i \(-0.469015\pi\)
0.0971897 + 0.995266i \(0.469015\pi\)
\(648\) 0 0
\(649\) −6.83282 −0.268211
\(650\) 0 0
\(651\) −28.9443 −1.13442
\(652\) 0 0
\(653\) −17.4164 −0.681557 −0.340778 0.940144i \(-0.610691\pi\)
−0.340778 + 0.940144i \(0.610691\pi\)
\(654\) 0 0
\(655\) −59.7771 −2.33568
\(656\) 0 0
\(657\) 4.47214 0.174475
\(658\) 0 0
\(659\) 1.34752 0.0524921 0.0262460 0.999656i \(-0.491645\pi\)
0.0262460 + 0.999656i \(0.491645\pi\)
\(660\) 0 0
\(661\) 26.1803 1.01830 0.509149 0.860679i \(-0.329960\pi\)
0.509149 + 0.860679i \(0.329960\pi\)
\(662\) 0 0
\(663\) −17.8885 −0.694733
\(664\) 0 0
\(665\) 111.554 4.32589
\(666\) 0 0
\(667\) −4.47214 −0.173162
\(668\) 0 0
\(669\) −7.41641 −0.286735
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) −4.47214 −0.172388 −0.0861941 0.996278i \(-0.527471\pi\)
−0.0861941 + 0.996278i \(0.527471\pi\)
\(674\) 0 0
\(675\) −5.47214 −0.210623
\(676\) 0 0
\(677\) −21.1246 −0.811885 −0.405942 0.913899i \(-0.633057\pi\)
−0.405942 + 0.913899i \(0.633057\pi\)
\(678\) 0 0
\(679\) 37.8885 1.45403
\(680\) 0 0
\(681\) −4.76393 −0.182554
\(682\) 0 0
\(683\) −5.52786 −0.211518 −0.105759 0.994392i \(-0.533727\pi\)
−0.105759 + 0.994392i \(0.533727\pi\)
\(684\) 0 0
\(685\) −67.7771 −2.58963
\(686\) 0 0
\(687\) −4.29180 −0.163742
\(688\) 0 0
\(689\) −3.41641 −0.130155
\(690\) 0 0
\(691\) −43.4164 −1.65164 −0.825819 0.563935i \(-0.809287\pi\)
−0.825819 + 0.563935i \(0.809287\pi\)
\(692\) 0 0
\(693\) −3.41641 −0.129779
\(694\) 0 0
\(695\) 3.05573 0.115910
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) 0 0
\(699\) 15.8885 0.600960
\(700\) 0 0
\(701\) 37.1246 1.40218 0.701089 0.713074i \(-0.252698\pi\)
0.701089 + 0.713074i \(0.252698\pi\)
\(702\) 0 0
\(703\) 52.1378 1.96641
\(704\) 0 0
\(705\) −12.9443 −0.487509
\(706\) 0 0
\(707\) 20.0000 0.752177
\(708\) 0 0
\(709\) −41.0132 −1.54028 −0.770141 0.637874i \(-0.779814\pi\)
−0.770141 + 0.637874i \(0.779814\pi\)
\(710\) 0 0
\(711\) 4.47214 0.167718
\(712\) 0 0
\(713\) 6.47214 0.242383
\(714\) 0 0
\(715\) 11.0557 0.413461
\(716\) 0 0
\(717\) 12.9443 0.483413
\(718\) 0 0
\(719\) 20.9443 0.781090 0.390545 0.920584i \(-0.372287\pi\)
0.390545 + 0.920584i \(0.372287\pi\)
\(720\) 0 0
\(721\) −26.8328 −0.999306
\(722\) 0 0
\(723\) 3.52786 0.131203
\(724\) 0 0
\(725\) 24.4721 0.908872
\(726\) 0 0
\(727\) −43.3050 −1.60609 −0.803046 0.595917i \(-0.796789\pi\)
−0.803046 + 0.595917i \(0.796789\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −36.9443 −1.36643
\(732\) 0 0
\(733\) −8.29180 −0.306264 −0.153132 0.988206i \(-0.548936\pi\)
−0.153132 + 0.988206i \(0.548936\pi\)
\(734\) 0 0
\(735\) 42.0689 1.55173
\(736\) 0 0
\(737\) 2.83282 0.104348
\(738\) 0 0
\(739\) 15.0557 0.553834 0.276917 0.960894i \(-0.410687\pi\)
0.276917 + 0.960894i \(0.410687\pi\)
\(740\) 0 0
\(741\) 34.4721 1.26637
\(742\) 0 0
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) −5.52786 −0.202525
\(746\) 0 0
\(747\) −8.76393 −0.320656
\(748\) 0 0
\(749\) 83.4164 3.04797
\(750\) 0 0
\(751\) 18.0000 0.656829 0.328415 0.944534i \(-0.393486\pi\)
0.328415 + 0.944534i \(0.393486\pi\)
\(752\) 0 0
\(753\) −6.29180 −0.229286
\(754\) 0 0
\(755\) −17.8885 −0.651031
\(756\) 0 0
\(757\) −32.6525 −1.18677 −0.593387 0.804917i \(-0.702210\pi\)
−0.593387 + 0.804917i \(0.702210\pi\)
\(758\) 0 0
\(759\) 0.763932 0.0277290
\(760\) 0 0
\(761\) −22.9443 −0.831729 −0.415865 0.909427i \(-0.636521\pi\)
−0.415865 + 0.909427i \(0.636521\pi\)
\(762\) 0 0
\(763\) −41.3050 −1.49534
\(764\) 0 0
\(765\) 12.9443 0.468001
\(766\) 0 0
\(767\) 40.0000 1.44432
\(768\) 0 0
\(769\) 3.52786 0.127218 0.0636090 0.997975i \(-0.479739\pi\)
0.0636090 + 0.997975i \(0.479739\pi\)
\(770\) 0 0
\(771\) −23.8885 −0.860325
\(772\) 0 0
\(773\) −15.8197 −0.568994 −0.284497 0.958677i \(-0.591827\pi\)
−0.284497 + 0.958677i \(0.591827\pi\)
\(774\) 0 0
\(775\) −35.4164 −1.27219
\(776\) 0 0
\(777\) 30.2492 1.08518
\(778\) 0 0
\(779\) −15.4164 −0.552350
\(780\) 0 0
\(781\) 6.83282 0.244497
\(782\) 0 0
\(783\) −4.47214 −0.159821
\(784\) 0 0
\(785\) −79.7771 −2.84737
\(786\) 0 0
\(787\) −30.7639 −1.09662 −0.548308 0.836277i \(-0.684728\pi\)
−0.548308 + 0.836277i \(0.684728\pi\)
\(788\) 0 0
\(789\) 7.05573 0.251191
\(790\) 0 0
\(791\) 64.7214 2.30123
\(792\) 0 0
\(793\) −23.4164 −0.831541
\(794\) 0 0
\(795\) 2.47214 0.0876776
\(796\) 0 0
\(797\) −7.59675 −0.269091 −0.134545 0.990907i \(-0.542957\pi\)
−0.134545 + 0.990907i \(0.542957\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −1.52786 −0.0539844
\(802\) 0 0
\(803\) 3.41641 0.120562
\(804\) 0 0
\(805\) −14.4721 −0.510076
\(806\) 0 0
\(807\) 30.9443 1.08929
\(808\) 0 0
\(809\) 25.0557 0.880912 0.440456 0.897774i \(-0.354817\pi\)
0.440456 + 0.897774i \(0.354817\pi\)
\(810\) 0 0
\(811\) 21.3050 0.748118 0.374059 0.927405i \(-0.377966\pi\)
0.374059 + 0.927405i \(0.377966\pi\)
\(812\) 0 0
\(813\) 0.944272 0.0331171
\(814\) 0 0
\(815\) 20.9443 0.733646
\(816\) 0 0
\(817\) 71.1935 2.49075
\(818\) 0 0
\(819\) 20.0000 0.698857
\(820\) 0 0
\(821\) 48.4721 1.69169 0.845845 0.533429i \(-0.179097\pi\)
0.845845 + 0.533429i \(0.179097\pi\)
\(822\) 0 0
\(823\) 7.63932 0.266290 0.133145 0.991097i \(-0.457492\pi\)
0.133145 + 0.991097i \(0.457492\pi\)
\(824\) 0 0
\(825\) −4.18034 −0.145541
\(826\) 0 0
\(827\) −27.5967 −0.959633 −0.479816 0.877369i \(-0.659297\pi\)
−0.479816 + 0.877369i \(0.659297\pi\)
\(828\) 0 0
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) 0 0
\(831\) 11.5279 0.399897
\(832\) 0 0
\(833\) −52.0000 −1.80169
\(834\) 0 0
\(835\) −3.05573 −0.105748
\(836\) 0 0
\(837\) 6.47214 0.223710
\(838\) 0 0
\(839\) 10.1115 0.349086 0.174543 0.984650i \(-0.444155\pi\)
0.174543 + 0.984650i \(0.444155\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) 22.4721 0.773981
\(844\) 0 0
\(845\) −22.6525 −0.779269
\(846\) 0 0
\(847\) 46.5836 1.60063
\(848\) 0 0
\(849\) −26.1803 −0.898507
\(850\) 0 0
\(851\) −6.76393 −0.231865
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) −24.9443 −0.853076
\(856\) 0 0
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) −8.94427 −0.304820
\(862\) 0 0
\(863\) 38.8328 1.32188 0.660942 0.750437i \(-0.270157\pi\)
0.660942 + 0.750437i \(0.270157\pi\)
\(864\) 0 0
\(865\) −30.4721 −1.03608
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 3.41641 0.115894
\(870\) 0 0
\(871\) −16.5836 −0.561914
\(872\) 0 0
\(873\) −8.47214 −0.286738
\(874\) 0 0
\(875\) 6.83282 0.230991
\(876\) 0 0
\(877\) 32.2492 1.08898 0.544489 0.838768i \(-0.316723\pi\)
0.544489 + 0.838768i \(0.316723\pi\)
\(878\) 0 0
\(879\) −13.7082 −0.462366
\(880\) 0 0
\(881\) 12.5836 0.423952 0.211976 0.977275i \(-0.432010\pi\)
0.211976 + 0.977275i \(0.432010\pi\)
\(882\) 0 0
\(883\) −22.4721 −0.756248 −0.378124 0.925755i \(-0.623431\pi\)
−0.378124 + 0.925755i \(0.623431\pi\)
\(884\) 0 0
\(885\) −28.9443 −0.972951
\(886\) 0 0
\(887\) −0.944272 −0.0317055 −0.0158528 0.999874i \(-0.505046\pi\)
−0.0158528 + 0.999874i \(0.505046\pi\)
\(888\) 0 0
\(889\) −17.8885 −0.599963
\(890\) 0 0
\(891\) 0.763932 0.0255927
\(892\) 0 0
\(893\) −30.8328 −1.03178
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) −4.47214 −0.149320
\(898\) 0 0
\(899\) −28.9443 −0.965346
\(900\) 0 0
\(901\) −3.05573 −0.101801
\(902\) 0 0
\(903\) 41.3050 1.37454
\(904\) 0 0
\(905\) 21.8885 0.727600
\(906\) 0 0
\(907\) 18.1803 0.603668 0.301834 0.953360i \(-0.402401\pi\)
0.301834 + 0.953360i \(0.402401\pi\)
\(908\) 0 0
\(909\) −4.47214 −0.148331
\(910\) 0 0
\(911\) −43.4164 −1.43845 −0.719225 0.694777i \(-0.755503\pi\)
−0.719225 + 0.694777i \(0.755503\pi\)
\(912\) 0 0
\(913\) −6.69505 −0.221574
\(914\) 0 0
\(915\) 16.9443 0.560160
\(916\) 0 0
\(917\) −82.6099 −2.72802
\(918\) 0 0
\(919\) 7.52786 0.248321 0.124161 0.992262i \(-0.460376\pi\)
0.124161 + 0.992262i \(0.460376\pi\)
\(920\) 0 0
\(921\) −11.4164 −0.376183
\(922\) 0 0
\(923\) −40.0000 −1.31662
\(924\) 0 0
\(925\) 37.0132 1.21699
\(926\) 0 0
\(927\) 6.00000 0.197066
\(928\) 0 0
\(929\) 28.8328 0.945974 0.472987 0.881069i \(-0.343176\pi\)
0.472987 + 0.881069i \(0.343176\pi\)
\(930\) 0 0
\(931\) 100.207 3.28414
\(932\) 0 0
\(933\) −3.05573 −0.100040
\(934\) 0 0
\(935\) 9.88854 0.323390
\(936\) 0 0
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 0 0
\(939\) −24.4721 −0.798618
\(940\) 0 0
\(941\) 15.8197 0.515706 0.257853 0.966184i \(-0.416985\pi\)
0.257853 + 0.966184i \(0.416985\pi\)
\(942\) 0 0
\(943\) 2.00000 0.0651290
\(944\) 0 0
\(945\) −14.4721 −0.470779
\(946\) 0 0
\(947\) −51.1935 −1.66357 −0.831783 0.555102i \(-0.812679\pi\)
−0.831783 + 0.555102i \(0.812679\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) −28.4721 −0.923272
\(952\) 0 0
\(953\) 23.7771 0.770215 0.385108 0.922872i \(-0.374164\pi\)
0.385108 + 0.922872i \(0.374164\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) 0 0
\(957\) −3.41641 −0.110437
\(958\) 0 0
\(959\) −93.6656 −3.02462
\(960\) 0 0
\(961\) 10.8885 0.351243
\(962\) 0 0
\(963\) −18.6525 −0.601068
\(964\) 0 0
\(965\) 38.4721 1.23846
\(966\) 0 0
\(967\) 11.4164 0.367127 0.183563 0.983008i \(-0.441237\pi\)
0.183563 + 0.983008i \(0.441237\pi\)
\(968\) 0 0
\(969\) 30.8328 0.990493
\(970\) 0 0
\(971\) −21.1246 −0.677921 −0.338961 0.940801i \(-0.610075\pi\)
−0.338961 + 0.940801i \(0.610075\pi\)
\(972\) 0 0
\(973\) 4.22291 0.135380
\(974\) 0 0
\(975\) 24.4721 0.783736
\(976\) 0 0
\(977\) 54.8328 1.75426 0.877129 0.480256i \(-0.159456\pi\)
0.877129 + 0.480256i \(0.159456\pi\)
\(978\) 0 0
\(979\) −1.16718 −0.0373034
\(980\) 0 0
\(981\) 9.23607 0.294885
\(982\) 0 0
\(983\) −27.4164 −0.874448 −0.437224 0.899353i \(-0.644038\pi\)
−0.437224 + 0.899353i \(0.644038\pi\)
\(984\) 0 0
\(985\) 48.3607 1.54090
\(986\) 0 0
\(987\) −17.8885 −0.569399
\(988\) 0 0
\(989\) −9.23607 −0.293690
\(990\) 0 0
\(991\) 1.52786 0.0485342 0.0242671 0.999706i \(-0.492275\pi\)
0.0242671 + 0.999706i \(0.492275\pi\)
\(992\) 0 0
\(993\) −1.52786 −0.0484853
\(994\) 0 0
\(995\) −30.4721 −0.966032
\(996\) 0 0
\(997\) 22.9443 0.726652 0.363326 0.931662i \(-0.381641\pi\)
0.363326 + 0.931662i \(0.381641\pi\)
\(998\) 0 0
\(999\) −6.76393 −0.214001
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 1104.2.a.j.1.1 2
3.2 odd 2 3312.2.a.bc.1.2 2
4.3 odd 2 138.2.a.d.1.1 2
8.3 odd 2 4416.2.a.bh.1.2 2
8.5 even 2 4416.2.a.bl.1.2 2
12.11 even 2 414.2.a.f.1.2 2
20.3 even 4 3450.2.d.x.2899.1 4
20.7 even 4 3450.2.d.x.2899.4 4
20.19 odd 2 3450.2.a.be.1.1 2
28.27 even 2 6762.2.a.cb.1.2 2
92.91 even 2 3174.2.a.s.1.2 2
276.275 odd 2 9522.2.a.q.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.d.1.1 2 4.3 odd 2
414.2.a.f.1.2 2 12.11 even 2
1104.2.a.j.1.1 2 1.1 even 1 trivial
3174.2.a.s.1.2 2 92.91 even 2
3312.2.a.bc.1.2 2 3.2 odd 2
3450.2.a.be.1.1 2 20.19 odd 2
3450.2.d.x.2899.1 4 20.3 even 4
3450.2.d.x.2899.4 4 20.7 even 4
4416.2.a.bh.1.2 2 8.3 odd 2
4416.2.a.bl.1.2 2 8.5 even 2
6762.2.a.cb.1.2 2 28.27 even 2
9522.2.a.q.1.1 2 276.275 odd 2