Properties

Label 9702.2.a.dx.1.1
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
Dimension $3$
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] = [9702,2,Mod(1,9702)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9702, 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("9702.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.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: \(77.4708600410\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1304.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 11x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1386)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.182370\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.89219 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.89219 q^{5} -1.00000 q^{8} +2.89219 q^{10} -1.00000 q^{11} +0.364739 q^{13} +1.00000 q^{16} -2.89219 q^{17} -7.25693 q^{19} -2.89219 q^{20} +1.00000 q^{22} -8.14911 q^{23} +3.36474 q^{25} -0.364739 q^{26} +2.36474 q^{29} -10.6766 q^{31} -1.00000 q^{32} +2.89219 q^{34} -3.78437 q^{37} +7.25693 q^{38} +2.89219 q^{40} +2.89219 q^{41} +11.4196 q^{43} -1.00000 q^{44} +8.14911 q^{46} -6.52745 q^{47} -3.36474 q^{50} +0.364739 q^{52} -11.7844 q^{53} +2.89219 q^{55} -2.36474 q^{58} -10.5139 q^{59} +11.9335 q^{61} +10.6766 q^{62} +1.00000 q^{64} -1.05489 q^{65} -6.14911 q^{67} -2.89219 q^{68} -13.9335 q^{71} +1.10781 q^{73} +3.78437 q^{74} -7.25693 q^{76} +10.1491 q^{79} -2.89219 q^{80} -2.89219 q^{82} -3.10781 q^{83} +8.36474 q^{85} -11.4196 q^{86} +1.00000 q^{88} +2.14911 q^{89} -8.14911 q^{92} +6.52745 q^{94} +20.9884 q^{95} -6.72948 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{8} - 2 q^{10} - 3 q^{11} + 3 q^{16} + 2 q^{17} - 10 q^{19} + 2 q^{20} + 3 q^{22} - 2 q^{23} + 9 q^{25} + 6 q^{29} - 3 q^{32} - 2 q^{34} + 10 q^{37} + 10 q^{38} - 2 q^{40} - 2 q^{41} + 14 q^{43} - 3 q^{44} + 2 q^{46} - 10 q^{47} - 9 q^{50} - 14 q^{53} - 2 q^{55} - 6 q^{58} - 8 q^{59} - 8 q^{61} + 3 q^{64} + 16 q^{65} + 4 q^{67} + 2 q^{68} + 2 q^{71} + 14 q^{73} - 10 q^{74} - 10 q^{76} + 8 q^{79} + 2 q^{80} + 2 q^{82} - 20 q^{83} + 24 q^{85} - 14 q^{86} + 3 q^{88} - 16 q^{89} - 2 q^{92} + 10 q^{94} - 18 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.89219 −1.29342 −0.646712 0.762734i \(-0.723857\pi\)
−0.646712 + 0.762734i \(0.723857\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 2.89219 0.914589
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.364739 0.101160 0.0505802 0.998720i \(-0.483893\pi\)
0.0505802 + 0.998720i \(0.483893\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.89219 −0.701458 −0.350729 0.936477i \(-0.614066\pi\)
−0.350729 + 0.936477i \(0.614066\pi\)
\(18\) 0 0
\(19\) −7.25693 −1.66485 −0.832426 0.554136i \(-0.813049\pi\)
−0.832426 + 0.554136i \(0.813049\pi\)
\(20\) −2.89219 −0.646712
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −8.14911 −1.69921 −0.849604 0.527422i \(-0.823159\pi\)
−0.849604 + 0.527422i \(0.823159\pi\)
\(24\) 0 0
\(25\) 3.36474 0.672948
\(26\) −0.364739 −0.0715312
\(27\) 0 0
\(28\) 0 0
\(29\) 2.36474 0.439121 0.219561 0.975599i \(-0.429538\pi\)
0.219561 + 0.975599i \(0.429538\pi\)
\(30\) 0 0
\(31\) −10.6766 −1.91757 −0.958783 0.284139i \(-0.908292\pi\)
−0.958783 + 0.284139i \(0.908292\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.89219 0.496006
\(35\) 0 0
\(36\) 0 0
\(37\) −3.78437 −0.622147 −0.311073 0.950386i \(-0.600689\pi\)
−0.311073 + 0.950386i \(0.600689\pi\)
\(38\) 7.25693 1.17723
\(39\) 0 0
\(40\) 2.89219 0.457295
\(41\) 2.89219 0.451684 0.225842 0.974164i \(-0.427487\pi\)
0.225842 + 0.974164i \(0.427487\pi\)
\(42\) 0 0
\(43\) 11.4196 1.74148 0.870739 0.491746i \(-0.163641\pi\)
0.870739 + 0.491746i \(0.163641\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 8.14911 1.20152
\(47\) −6.52745 −0.952126 −0.476063 0.879411i \(-0.657937\pi\)
−0.476063 + 0.879411i \(0.657937\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −3.36474 −0.475846
\(51\) 0 0
\(52\) 0.364739 0.0505802
\(53\) −11.7844 −1.61871 −0.809354 0.587321i \(-0.800183\pi\)
−0.809354 + 0.587321i \(0.800183\pi\)
\(54\) 0 0
\(55\) 2.89219 0.389982
\(56\) 0 0
\(57\) 0 0
\(58\) −2.36474 −0.310505
\(59\) −10.5139 −1.36879 −0.684393 0.729113i \(-0.739933\pi\)
−0.684393 + 0.729113i \(0.739933\pi\)
\(60\) 0 0
\(61\) 11.9335 1.52793 0.763963 0.645260i \(-0.223251\pi\)
0.763963 + 0.645260i \(0.223251\pi\)
\(62\) 10.6766 1.35592
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.05489 −0.130843
\(66\) 0 0
\(67\) −6.14911 −0.751233 −0.375617 0.926775i \(-0.622569\pi\)
−0.375617 + 0.926775i \(0.622569\pi\)
\(68\) −2.89219 −0.350729
\(69\) 0 0
\(70\) 0 0
\(71\) −13.9335 −1.65360 −0.826800 0.562496i \(-0.809841\pi\)
−0.826800 + 0.562496i \(0.809841\pi\)
\(72\) 0 0
\(73\) 1.10781 0.129660 0.0648299 0.997896i \(-0.479350\pi\)
0.0648299 + 0.997896i \(0.479350\pi\)
\(74\) 3.78437 0.439924
\(75\) 0 0
\(76\) −7.25693 −0.832426
\(77\) 0 0
\(78\) 0 0
\(79\) 10.1491 1.14186 0.570932 0.820997i \(-0.306582\pi\)
0.570932 + 0.820997i \(0.306582\pi\)
\(80\) −2.89219 −0.323356
\(81\) 0 0
\(82\) −2.89219 −0.319389
\(83\) −3.10781 −0.341127 −0.170563 0.985347i \(-0.554559\pi\)
−0.170563 + 0.985347i \(0.554559\pi\)
\(84\) 0 0
\(85\) 8.36474 0.907283
\(86\) −11.4196 −1.23141
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 2.14911 0.227805 0.113903 0.993492i \(-0.463665\pi\)
0.113903 + 0.993492i \(0.463665\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −8.14911 −0.849604
\(93\) 0 0
\(94\) 6.52745 0.673255
\(95\) 20.9884 2.15336
\(96\) 0 0
\(97\) −6.72948 −0.683275 −0.341638 0.939832i \(-0.610982\pi\)
−0.341638 + 0.939832i \(0.610982\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 3.36474 0.336474
\(101\) −1.27052 −0.126422 −0.0632108 0.998000i \(-0.520134\pi\)
−0.0632108 + 0.998000i \(0.520134\pi\)
\(102\) 0 0
\(103\) 0.892186 0.0879097 0.0439548 0.999034i \(-0.486004\pi\)
0.0439548 + 0.999034i \(0.486004\pi\)
\(104\) −0.364739 −0.0357656
\(105\) 0 0
\(106\) 11.7844 1.14460
\(107\) −5.78437 −0.559196 −0.279598 0.960117i \(-0.590201\pi\)
−0.279598 + 0.960117i \(0.590201\pi\)
\(108\) 0 0
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) −2.89219 −0.275759
\(111\) 0 0
\(112\) 0 0
\(113\) 19.9335 1.87518 0.937592 0.347737i \(-0.113050\pi\)
0.937592 + 0.347737i \(0.113050\pi\)
\(114\) 0 0
\(115\) 23.5687 2.19780
\(116\) 2.36474 0.219561
\(117\) 0 0
\(118\) 10.5139 0.967878
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −11.9335 −1.08041
\(123\) 0 0
\(124\) −10.6766 −0.958783
\(125\) 4.72948 0.423017
\(126\) 0 0
\(127\) 10.1491 0.900588 0.450294 0.892880i \(-0.351319\pi\)
0.450294 + 0.892880i \(0.351319\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 1.05489 0.0925203
\(131\) −20.8922 −1.82536 −0.912679 0.408676i \(-0.865990\pi\)
−0.912679 + 0.408676i \(0.865990\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 6.14911 0.531202
\(135\) 0 0
\(136\) 2.89219 0.248003
\(137\) 7.93348 0.677803 0.338902 0.940822i \(-0.389945\pi\)
0.338902 + 0.940822i \(0.389945\pi\)
\(138\) 0 0
\(139\) −18.8257 −1.59677 −0.798386 0.602146i \(-0.794313\pi\)
−0.798386 + 0.602146i \(0.794313\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 13.9335 1.16927
\(143\) −0.364739 −0.0305010
\(144\) 0 0
\(145\) −6.83927 −0.567970
\(146\) −1.10781 −0.0916833
\(147\) 0 0
\(148\) −3.78437 −0.311073
\(149\) 1.27052 0.104085 0.0520426 0.998645i \(-0.483427\pi\)
0.0520426 + 0.998645i \(0.483427\pi\)
\(150\) 0 0
\(151\) −8.29822 −0.675300 −0.337650 0.941272i \(-0.609632\pi\)
−0.337650 + 0.941272i \(0.609632\pi\)
\(152\) 7.25693 0.588614
\(153\) 0 0
\(154\) 0 0
\(155\) 30.8786 2.48023
\(156\) 0 0
\(157\) 18.8922 1.50776 0.753880 0.657012i \(-0.228180\pi\)
0.753880 + 0.657012i \(0.228180\pi\)
\(158\) −10.1491 −0.807420
\(159\) 0 0
\(160\) 2.89219 0.228647
\(161\) 0 0
\(162\) 0 0
\(163\) −20.2982 −1.58988 −0.794940 0.606688i \(-0.792498\pi\)
−0.794940 + 0.606688i \(0.792498\pi\)
\(164\) 2.89219 0.225842
\(165\) 0 0
\(166\) 3.10781 0.241213
\(167\) 1.05489 0.0816301 0.0408151 0.999167i \(-0.487005\pi\)
0.0408151 + 0.999167i \(0.487005\pi\)
\(168\) 0 0
\(169\) −12.8670 −0.989767
\(170\) −8.36474 −0.641546
\(171\) 0 0
\(172\) 11.4196 0.870739
\(173\) −16.5139 −1.25552 −0.627762 0.778405i \(-0.716029\pi\)
−0.627762 + 0.778405i \(0.716029\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −2.14911 −0.161083
\(179\) −20.9884 −1.56874 −0.784372 0.620290i \(-0.787015\pi\)
−0.784372 + 0.620290i \(0.787015\pi\)
\(180\) 0 0
\(181\) 1.10781 0.0823432 0.0411716 0.999152i \(-0.486891\pi\)
0.0411716 + 0.999152i \(0.486891\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 8.14911 0.600760
\(185\) 10.9451 0.804700
\(186\) 0 0
\(187\) 2.89219 0.211498
\(188\) −6.52745 −0.476063
\(189\) 0 0
\(190\) −20.9884 −1.52266
\(191\) 10.6902 0.773512 0.386756 0.922182i \(-0.373595\pi\)
0.386756 + 0.922182i \(0.373595\pi\)
\(192\) 0 0
\(193\) −21.5687 −1.55255 −0.776276 0.630393i \(-0.782894\pi\)
−0.776276 + 0.630393i \(0.782894\pi\)
\(194\) 6.72948 0.483148
\(195\) 0 0
\(196\) 0 0
\(197\) 4.90578 0.349523 0.174761 0.984611i \(-0.444085\pi\)
0.174761 + 0.984611i \(0.444085\pi\)
\(198\) 0 0
\(199\) 7.10781 0.503860 0.251930 0.967746i \(-0.418935\pi\)
0.251930 + 0.967746i \(0.418935\pi\)
\(200\) −3.36474 −0.237923
\(201\) 0 0
\(202\) 1.27052 0.0893936
\(203\) 0 0
\(204\) 0 0
\(205\) −8.36474 −0.584219
\(206\) −0.892186 −0.0621615
\(207\) 0 0
\(208\) 0.364739 0.0252901
\(209\) 7.25693 0.501972
\(210\) 0 0
\(211\) 8.87859 0.611227 0.305614 0.952156i \(-0.401138\pi\)
0.305614 + 0.952156i \(0.401138\pi\)
\(212\) −11.7844 −0.809354
\(213\) 0 0
\(214\) 5.78437 0.395412
\(215\) −33.0277 −2.25247
\(216\) 0 0
\(217\) 0 0
\(218\) −8.00000 −0.541828
\(219\) 0 0
\(220\) 2.89219 0.194991
\(221\) −1.05489 −0.0709598
\(222\) 0 0
\(223\) −22.2453 −1.48966 −0.744828 0.667257i \(-0.767468\pi\)
−0.744828 + 0.667257i \(0.767468\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −19.9335 −1.32596
\(227\) 8.46093 0.561572 0.280786 0.959770i \(-0.409405\pi\)
0.280786 + 0.959770i \(0.409405\pi\)
\(228\) 0 0
\(229\) 2.16271 0.142916 0.0714579 0.997444i \(-0.477235\pi\)
0.0714579 + 0.997444i \(0.477235\pi\)
\(230\) −23.5687 −1.55408
\(231\) 0 0
\(232\) −2.36474 −0.155253
\(233\) 3.45896 0.226604 0.113302 0.993561i \(-0.463857\pi\)
0.113302 + 0.993561i \(0.463857\pi\)
\(234\) 0 0
\(235\) 18.8786 1.23150
\(236\) −10.5139 −0.684393
\(237\) 0 0
\(238\) 0 0
\(239\) −16.2982 −1.05424 −0.527122 0.849790i \(-0.676729\pi\)
−0.527122 + 0.849790i \(0.676729\pi\)
\(240\) 0 0
\(241\) −4.67656 −0.301244 −0.150622 0.988591i \(-0.548128\pi\)
−0.150622 + 0.988591i \(0.548128\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 11.9335 0.763963
\(245\) 0 0
\(246\) 0 0
\(247\) −2.64688 −0.168417
\(248\) 10.6766 0.677962
\(249\) 0 0
\(250\) −4.72948 −0.299118
\(251\) −10.0826 −0.636408 −0.318204 0.948022i \(-0.603080\pi\)
−0.318204 + 0.948022i \(0.603080\pi\)
\(252\) 0 0
\(253\) 8.14911 0.512330
\(254\) −10.1491 −0.636812
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 9.41963 0.587581 0.293790 0.955870i \(-0.405083\pi\)
0.293790 + 0.955870i \(0.405083\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.05489 −0.0654217
\(261\) 0 0
\(262\) 20.8922 1.29072
\(263\) −6.87859 −0.424152 −0.212076 0.977253i \(-0.568022\pi\)
−0.212076 + 0.977253i \(0.568022\pi\)
\(264\) 0 0
\(265\) 34.0826 2.09368
\(266\) 0 0
\(267\) 0 0
\(268\) −6.14911 −0.375617
\(269\) 26.8922 1.63965 0.819823 0.572617i \(-0.194072\pi\)
0.819823 + 0.572617i \(0.194072\pi\)
\(270\) 0 0
\(271\) 21.7844 1.32331 0.661653 0.749810i \(-0.269855\pi\)
0.661653 + 0.749810i \(0.269855\pi\)
\(272\) −2.89219 −0.175365
\(273\) 0 0
\(274\) −7.93348 −0.479279
\(275\) −3.36474 −0.202901
\(276\) 0 0
\(277\) 3.27052 0.196507 0.0982533 0.995161i \(-0.468674\pi\)
0.0982533 + 0.995161i \(0.468674\pi\)
\(278\) 18.8257 1.12909
\(279\) 0 0
\(280\) 0 0
\(281\) −10.2982 −0.614340 −0.307170 0.951655i \(-0.599382\pi\)
−0.307170 + 0.951655i \(0.599382\pi\)
\(282\) 0 0
\(283\) 28.3118 1.68296 0.841481 0.540286i \(-0.181684\pi\)
0.841481 + 0.540286i \(0.181684\pi\)
\(284\) −13.9335 −0.826800
\(285\) 0 0
\(286\) 0.364739 0.0215675
\(287\) 0 0
\(288\) 0 0
\(289\) −8.63526 −0.507957
\(290\) 6.83927 0.401615
\(291\) 0 0
\(292\) 1.10781 0.0648299
\(293\) 11.7844 0.688450 0.344225 0.938887i \(-0.388142\pi\)
0.344225 + 0.938887i \(0.388142\pi\)
\(294\) 0 0
\(295\) 30.4080 1.77042
\(296\) 3.78437 0.219962
\(297\) 0 0
\(298\) −1.27052 −0.0735993
\(299\) −2.97230 −0.171893
\(300\) 0 0
\(301\) 0 0
\(302\) 8.29822 0.477509
\(303\) 0 0
\(304\) −7.25693 −0.416213
\(305\) −34.5139 −1.97626
\(306\) 0 0
\(307\) −4.01360 −0.229068 −0.114534 0.993419i \(-0.536537\pi\)
−0.114534 + 0.993419i \(0.536537\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −30.8786 −1.75379
\(311\) −6.52745 −0.370138 −0.185069 0.982726i \(-0.559251\pi\)
−0.185069 + 0.982726i \(0.559251\pi\)
\(312\) 0 0
\(313\) −0.513850 −0.0290445 −0.0145223 0.999895i \(-0.504623\pi\)
−0.0145223 + 0.999895i \(0.504623\pi\)
\(314\) −18.8922 −1.06615
\(315\) 0 0
\(316\) 10.1491 0.570932
\(317\) −7.48615 −0.420464 −0.210232 0.977652i \(-0.567422\pi\)
−0.210232 + 0.977652i \(0.567422\pi\)
\(318\) 0 0
\(319\) −2.36474 −0.132400
\(320\) −2.89219 −0.161678
\(321\) 0 0
\(322\) 0 0
\(323\) 20.9884 1.16782
\(324\) 0 0
\(325\) 1.22725 0.0680757
\(326\) 20.2982 1.12421
\(327\) 0 0
\(328\) −2.89219 −0.159694
\(329\) 0 0
\(330\) 0 0
\(331\) −13.4196 −0.737610 −0.368805 0.929507i \(-0.620233\pi\)
−0.368805 + 0.929507i \(0.620233\pi\)
\(332\) −3.10781 −0.170563
\(333\) 0 0
\(334\) −1.05489 −0.0577212
\(335\) 17.7844 0.971664
\(336\) 0 0
\(337\) −8.94511 −0.487271 −0.243636 0.969867i \(-0.578340\pi\)
−0.243636 + 0.969867i \(0.578340\pi\)
\(338\) 12.8670 0.699871
\(339\) 0 0
\(340\) 8.36474 0.453642
\(341\) 10.6766 0.578168
\(342\) 0 0
\(343\) 0 0
\(344\) −11.4196 −0.615705
\(345\) 0 0
\(346\) 16.5139 0.887790
\(347\) −5.78437 −0.310521 −0.155261 0.987874i \(-0.549622\pi\)
−0.155261 + 0.987874i \(0.549622\pi\)
\(348\) 0 0
\(349\) −22.1491 −1.18561 −0.592807 0.805344i \(-0.701980\pi\)
−0.592807 + 0.805344i \(0.701980\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 9.41963 0.501356 0.250678 0.968070i \(-0.419346\pi\)
0.250678 + 0.968070i \(0.419346\pi\)
\(354\) 0 0
\(355\) 40.2982 2.13881
\(356\) 2.14911 0.113903
\(357\) 0 0
\(358\) 20.9884 1.10927
\(359\) 18.4473 0.973613 0.486806 0.873510i \(-0.338162\pi\)
0.486806 + 0.873510i \(0.338162\pi\)
\(360\) 0 0
\(361\) 33.6630 1.77173
\(362\) −1.10781 −0.0582254
\(363\) 0 0
\(364\) 0 0
\(365\) −3.20400 −0.167705
\(366\) 0 0
\(367\) 18.6766 0.974908 0.487454 0.873149i \(-0.337926\pi\)
0.487454 + 0.873149i \(0.337926\pi\)
\(368\) −8.14911 −0.424802
\(369\) 0 0
\(370\) −10.9451 −0.569009
\(371\) 0 0
\(372\) 0 0
\(373\) −19.2040 −0.994346 −0.497173 0.867652i \(-0.665629\pi\)
−0.497173 + 0.867652i \(0.665629\pi\)
\(374\) −2.89219 −0.149551
\(375\) 0 0
\(376\) 6.52745 0.336627
\(377\) 0.862513 0.0444217
\(378\) 0 0
\(379\) −31.8670 −1.63690 −0.818448 0.574581i \(-0.805165\pi\)
−0.818448 + 0.574581i \(0.805165\pi\)
\(380\) 20.9884 1.07668
\(381\) 0 0
\(382\) −10.6902 −0.546956
\(383\) 29.0413 1.48394 0.741970 0.670433i \(-0.233891\pi\)
0.741970 + 0.670433i \(0.233891\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 21.5687 1.09782
\(387\) 0 0
\(388\) −6.72948 −0.341638
\(389\) 26.5964 1.34849 0.674247 0.738506i \(-0.264468\pi\)
0.674247 + 0.738506i \(0.264468\pi\)
\(390\) 0 0
\(391\) 23.5687 1.19192
\(392\) 0 0
\(393\) 0 0
\(394\) −4.90578 −0.247150
\(395\) −29.3531 −1.47692
\(396\) 0 0
\(397\) 8.37834 0.420497 0.210248 0.977648i \(-0.432573\pi\)
0.210248 + 0.977648i \(0.432573\pi\)
\(398\) −7.10781 −0.356283
\(399\) 0 0
\(400\) 3.36474 0.168237
\(401\) 7.93348 0.396179 0.198090 0.980184i \(-0.436526\pi\)
0.198090 + 0.980184i \(0.436526\pi\)
\(402\) 0 0
\(403\) −3.89416 −0.193982
\(404\) −1.27052 −0.0632108
\(405\) 0 0
\(406\) 0 0
\(407\) 3.78437 0.187584
\(408\) 0 0
\(409\) 24.6766 1.22018 0.610089 0.792333i \(-0.291134\pi\)
0.610089 + 0.792333i \(0.291134\pi\)
\(410\) 8.36474 0.413105
\(411\) 0 0
\(412\) 0.892186 0.0439548
\(413\) 0 0
\(414\) 0 0
\(415\) 8.98838 0.441222
\(416\) −0.364739 −0.0178828
\(417\) 0 0
\(418\) −7.25693 −0.354948
\(419\) −2.54104 −0.124138 −0.0620690 0.998072i \(-0.519770\pi\)
−0.0620690 + 0.998072i \(0.519770\pi\)
\(420\) 0 0
\(421\) 2.43126 0.118492 0.0592461 0.998243i \(-0.481130\pi\)
0.0592461 + 0.998243i \(0.481130\pi\)
\(422\) −8.87859 −0.432203
\(423\) 0 0
\(424\) 11.7844 0.572300
\(425\) −9.73145 −0.472045
\(426\) 0 0
\(427\) 0 0
\(428\) −5.78437 −0.279598
\(429\) 0 0
\(430\) 33.0277 1.59274
\(431\) −2.58037 −0.124292 −0.0621460 0.998067i \(-0.519794\pi\)
−0.0621460 + 0.998067i \(0.519794\pi\)
\(432\) 0 0
\(433\) −37.1375 −1.78471 −0.892357 0.451331i \(-0.850950\pi\)
−0.892357 + 0.451331i \(0.850950\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) 59.1375 2.82893
\(438\) 0 0
\(439\) −18.5139 −0.883618 −0.441809 0.897109i \(-0.645663\pi\)
−0.441809 + 0.897109i \(0.645663\pi\)
\(440\) −2.89219 −0.137880
\(441\) 0 0
\(442\) 1.05489 0.0501762
\(443\) 21.4196 1.01768 0.508839 0.860862i \(-0.330075\pi\)
0.508839 + 0.860862i \(0.330075\pi\)
\(444\) 0 0
\(445\) −6.21563 −0.294649
\(446\) 22.2453 1.05335
\(447\) 0 0
\(448\) 0 0
\(449\) −4.06652 −0.191911 −0.0959554 0.995386i \(-0.530591\pi\)
−0.0959554 + 0.995386i \(0.530591\pi\)
\(450\) 0 0
\(451\) −2.89219 −0.136188
\(452\) 19.9335 0.937592
\(453\) 0 0
\(454\) −8.46093 −0.397091
\(455\) 0 0
\(456\) 0 0
\(457\) −16.8393 −0.787708 −0.393854 0.919173i \(-0.628858\pi\)
−0.393854 + 0.919173i \(0.628858\pi\)
\(458\) −2.16271 −0.101057
\(459\) 0 0
\(460\) 23.5687 1.09890
\(461\) −15.4590 −0.719995 −0.359998 0.932953i \(-0.617223\pi\)
−0.359998 + 0.932953i \(0.617223\pi\)
\(462\) 0 0
\(463\) 19.5687 0.909437 0.454718 0.890635i \(-0.349740\pi\)
0.454718 + 0.890635i \(0.349740\pi\)
\(464\) 2.36474 0.109780
\(465\) 0 0
\(466\) −3.45896 −0.160233
\(467\) 16.2982 0.754192 0.377096 0.926174i \(-0.376923\pi\)
0.377096 + 0.926174i \(0.376923\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −18.8786 −0.870804
\(471\) 0 0
\(472\) 10.5139 0.483939
\(473\) −11.4196 −0.525075
\(474\) 0 0
\(475\) −24.4177 −1.12036
\(476\) 0 0
\(477\) 0 0
\(478\) 16.2982 0.745463
\(479\) −0.431256 −0.0197046 −0.00985231 0.999951i \(-0.503136\pi\)
−0.00985231 + 0.999951i \(0.503136\pi\)
\(480\) 0 0
\(481\) −1.38031 −0.0629367
\(482\) 4.67656 0.213011
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 19.4629 0.883765
\(486\) 0 0
\(487\) −14.0826 −0.638143 −0.319072 0.947731i \(-0.603371\pi\)
−0.319072 + 0.947731i \(0.603371\pi\)
\(488\) −11.9335 −0.540203
\(489\) 0 0
\(490\) 0 0
\(491\) 23.5687 1.06364 0.531821 0.846857i \(-0.321508\pi\)
0.531821 + 0.846857i \(0.321508\pi\)
\(492\) 0 0
\(493\) −6.83927 −0.308025
\(494\) 2.64688 0.119089
\(495\) 0 0
\(496\) −10.6766 −0.479392
\(497\) 0 0
\(498\) 0 0
\(499\) 28.9884 1.29770 0.648849 0.760917i \(-0.275251\pi\)
0.648849 + 0.760917i \(0.275251\pi\)
\(500\) 4.72948 0.211509
\(501\) 0 0
\(502\) 10.0826 0.450008
\(503\) 25.0549 1.11714 0.558571 0.829457i \(-0.311350\pi\)
0.558571 + 0.829457i \(0.311350\pi\)
\(504\) 0 0
\(505\) 3.67458 0.163517
\(506\) −8.14911 −0.362272
\(507\) 0 0
\(508\) 10.1491 0.450294
\(509\) 29.0020 1.28549 0.642745 0.766080i \(-0.277796\pi\)
0.642745 + 0.766080i \(0.277796\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −9.41963 −0.415482
\(515\) −2.58037 −0.113705
\(516\) 0 0
\(517\) 6.52745 0.287077
\(518\) 0 0
\(519\) 0 0
\(520\) 1.05489 0.0462601
\(521\) 20.9884 0.919517 0.459759 0.888044i \(-0.347936\pi\)
0.459759 + 0.888044i \(0.347936\pi\)
\(522\) 0 0
\(523\) 15.8806 0.694408 0.347204 0.937790i \(-0.387131\pi\)
0.347204 + 0.937790i \(0.387131\pi\)
\(524\) −20.8922 −0.912679
\(525\) 0 0
\(526\) 6.87859 0.299921
\(527\) 30.8786 1.34509
\(528\) 0 0
\(529\) 43.4080 1.88730
\(530\) −34.0826 −1.48045
\(531\) 0 0
\(532\) 0 0
\(533\) 1.05489 0.0456925
\(534\) 0 0
\(535\) 16.7295 0.723278
\(536\) 6.14911 0.265601
\(537\) 0 0
\(538\) −26.8922 −1.15940
\(539\) 0 0
\(540\) 0 0
\(541\) −14.4745 −0.622308 −0.311154 0.950359i \(-0.600716\pi\)
−0.311154 + 0.950359i \(0.600716\pi\)
\(542\) −21.7844 −0.935719
\(543\) 0 0
\(544\) 2.89219 0.124001
\(545\) −23.1375 −0.991101
\(546\) 0 0
\(547\) −29.0710 −1.24298 −0.621492 0.783420i \(-0.713473\pi\)
−0.621492 + 0.783420i \(0.713473\pi\)
\(548\) 7.93348 0.338902
\(549\) 0 0
\(550\) 3.36474 0.143473
\(551\) −17.1607 −0.731072
\(552\) 0 0
\(553\) 0 0
\(554\) −3.27052 −0.138951
\(555\) 0 0
\(556\) −18.8257 −0.798386
\(557\) −37.5022 −1.58902 −0.794510 0.607251i \(-0.792272\pi\)
−0.794510 + 0.607251i \(0.792272\pi\)
\(558\) 0 0
\(559\) 4.16519 0.176169
\(560\) 0 0
\(561\) 0 0
\(562\) 10.2982 0.434404
\(563\) −27.1078 −1.14246 −0.571229 0.820791i \(-0.693533\pi\)
−0.571229 + 0.820791i \(0.693533\pi\)
\(564\) 0 0
\(565\) −57.6513 −2.42541
\(566\) −28.3118 −1.19003
\(567\) 0 0
\(568\) 13.9335 0.584636
\(569\) −46.2982 −1.94092 −0.970461 0.241257i \(-0.922440\pi\)
−0.970461 + 0.241257i \(0.922440\pi\)
\(570\) 0 0
\(571\) 5.63526 0.235828 0.117914 0.993024i \(-0.462379\pi\)
0.117914 + 0.993024i \(0.462379\pi\)
\(572\) −0.364739 −0.0152505
\(573\) 0 0
\(574\) 0 0
\(575\) −27.4196 −1.14348
\(576\) 0 0
\(577\) −0.513850 −0.0213919 −0.0106959 0.999943i \(-0.503405\pi\)
−0.0106959 + 0.999943i \(0.503405\pi\)
\(578\) 8.63526 0.359179
\(579\) 0 0
\(580\) −6.83927 −0.283985
\(581\) 0 0
\(582\) 0 0
\(583\) 11.7844 0.488059
\(584\) −1.10781 −0.0458417
\(585\) 0 0
\(586\) −11.7844 −0.486808
\(587\) 17.7844 0.734040 0.367020 0.930213i \(-0.380378\pi\)
0.367020 + 0.930213i \(0.380378\pi\)
\(588\) 0 0
\(589\) 77.4790 3.19247
\(590\) −30.4080 −1.25188
\(591\) 0 0
\(592\) −3.78437 −0.155537
\(593\) 26.0297 1.06891 0.534455 0.845197i \(-0.320517\pi\)
0.534455 + 0.845197i \(0.320517\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.27052 0.0520426
\(597\) 0 0
\(598\) 2.97230 0.121546
\(599\) −25.9335 −1.05961 −0.529807 0.848118i \(-0.677736\pi\)
−0.529807 + 0.848118i \(0.677736\pi\)
\(600\) 0 0
\(601\) 33.2730 1.35723 0.678617 0.734492i \(-0.262580\pi\)
0.678617 + 0.734492i \(0.262580\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −8.29822 −0.337650
\(605\) −2.89219 −0.117584
\(606\) 0 0
\(607\) 34.4080 1.39658 0.698289 0.715816i \(-0.253945\pi\)
0.698289 + 0.715816i \(0.253945\pi\)
\(608\) 7.25693 0.294307
\(609\) 0 0
\(610\) 34.5139 1.39742
\(611\) −2.38082 −0.0963175
\(612\) 0 0
\(613\) 7.33704 0.296340 0.148170 0.988962i \(-0.452662\pi\)
0.148170 + 0.988962i \(0.452662\pi\)
\(614\) 4.01360 0.161976
\(615\) 0 0
\(616\) 0 0
\(617\) 7.70178 0.310062 0.155031 0.987910i \(-0.450452\pi\)
0.155031 + 0.987910i \(0.450452\pi\)
\(618\) 0 0
\(619\) 20.2982 0.815854 0.407927 0.913014i \(-0.366252\pi\)
0.407927 + 0.913014i \(0.366252\pi\)
\(620\) 30.8786 1.24011
\(621\) 0 0
\(622\) 6.52745 0.261727
\(623\) 0 0
\(624\) 0 0
\(625\) −30.5022 −1.22009
\(626\) 0.513850 0.0205376
\(627\) 0 0
\(628\) 18.8922 0.753880
\(629\) 10.9451 0.436410
\(630\) 0 0
\(631\) 35.2433 1.40301 0.701507 0.712662i \(-0.252511\pi\)
0.701507 + 0.712662i \(0.252511\pi\)
\(632\) −10.1491 −0.403710
\(633\) 0 0
\(634\) 7.48615 0.297313
\(635\) −29.3531 −1.16484
\(636\) 0 0
\(637\) 0 0
\(638\) 2.36474 0.0936209
\(639\) 0 0
\(640\) 2.89219 0.114324
\(641\) −25.7572 −1.01735 −0.508674 0.860959i \(-0.669864\pi\)
−0.508674 + 0.860959i \(0.669864\pi\)
\(642\) 0 0
\(643\) 5.48615 0.216353 0.108176 0.994132i \(-0.465499\pi\)
0.108176 + 0.994132i \(0.465499\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −20.9884 −0.825777
\(647\) 17.4726 0.686917 0.343458 0.939168i \(-0.388402\pi\)
0.343458 + 0.939168i \(0.388402\pi\)
\(648\) 0 0
\(649\) 10.5139 0.412705
\(650\) −1.22725 −0.0481368
\(651\) 0 0
\(652\) −20.2982 −0.794940
\(653\) 8.54104 0.334237 0.167118 0.985937i \(-0.446554\pi\)
0.167118 + 0.985937i \(0.446554\pi\)
\(654\) 0 0
\(655\) 60.4241 2.36096
\(656\) 2.89219 0.112921
\(657\) 0 0
\(658\) 0 0
\(659\) 43.5416 1.69614 0.848069 0.529886i \(-0.177765\pi\)
0.848069 + 0.529886i \(0.177765\pi\)
\(660\) 0 0
\(661\) −18.8650 −0.733763 −0.366882 0.930268i \(-0.619575\pi\)
−0.366882 + 0.930268i \(0.619575\pi\)
\(662\) 13.4196 0.521569
\(663\) 0 0
\(664\) 3.10781 0.120607
\(665\) 0 0
\(666\) 0 0
\(667\) −19.2705 −0.746158
\(668\) 1.05489 0.0408151
\(669\) 0 0
\(670\) −17.7844 −0.687070
\(671\) −11.9335 −0.460687
\(672\) 0 0
\(673\) −8.08259 −0.311561 −0.155781 0.987792i \(-0.549789\pi\)
−0.155781 + 0.987792i \(0.549789\pi\)
\(674\) 8.94511 0.344553
\(675\) 0 0
\(676\) −12.8670 −0.494883
\(677\) −6.62364 −0.254567 −0.127284 0.991866i \(-0.540626\pi\)
−0.127284 + 0.991866i \(0.540626\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −8.36474 −0.320773
\(681\) 0 0
\(682\) −10.6766 −0.408827
\(683\) −13.7179 −0.524899 −0.262450 0.964946i \(-0.584530\pi\)
−0.262450 + 0.964946i \(0.584530\pi\)
\(684\) 0 0
\(685\) −22.9451 −0.876687
\(686\) 0 0
\(687\) 0 0
\(688\) 11.4196 0.435369
\(689\) −4.29822 −0.163749
\(690\) 0 0
\(691\) 18.3808 0.699239 0.349620 0.936892i \(-0.386311\pi\)
0.349620 + 0.936892i \(0.386311\pi\)
\(692\) −16.5139 −0.627762
\(693\) 0 0
\(694\) 5.78437 0.219572
\(695\) 54.4473 2.06531
\(696\) 0 0
\(697\) −8.36474 −0.316837
\(698\) 22.1491 0.838356
\(699\) 0 0
\(700\) 0 0
\(701\) 31.3259 1.18316 0.591582 0.806245i \(-0.298504\pi\)
0.591582 + 0.806245i \(0.298504\pi\)
\(702\) 0 0
\(703\) 27.4629 1.03578
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −9.41963 −0.354513
\(707\) 0 0
\(708\) 0 0
\(709\) −41.1103 −1.54393 −0.771965 0.635665i \(-0.780726\pi\)
−0.771965 + 0.635665i \(0.780726\pi\)
\(710\) −40.2982 −1.51237
\(711\) 0 0
\(712\) −2.14911 −0.0805413
\(713\) 87.0045 3.25834
\(714\) 0 0
\(715\) 1.05489 0.0394508
\(716\) −20.9884 −0.784372
\(717\) 0 0
\(718\) −18.4473 −0.688448
\(719\) 1.17433 0.0437952 0.0218976 0.999760i \(-0.493029\pi\)
0.0218976 + 0.999760i \(0.493029\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −33.6630 −1.25281
\(723\) 0 0
\(724\) 1.10781 0.0411716
\(725\) 7.95673 0.295506
\(726\) 0 0
\(727\) −43.2730 −1.60491 −0.802453 0.596715i \(-0.796472\pi\)
−0.802453 + 0.596715i \(0.796472\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 3.20400 0.118586
\(731\) −33.0277 −1.22157
\(732\) 0 0
\(733\) 14.9058 0.550558 0.275279 0.961364i \(-0.411230\pi\)
0.275279 + 0.961364i \(0.411230\pi\)
\(734\) −18.6766 −0.689364
\(735\) 0 0
\(736\) 8.14911 0.300380
\(737\) 6.14911 0.226505
\(738\) 0 0
\(739\) 24.9058 0.916174 0.458087 0.888907i \(-0.348535\pi\)
0.458087 + 0.888907i \(0.348535\pi\)
\(740\) 10.9451 0.402350
\(741\) 0 0
\(742\) 0 0
\(743\) 11.5687 0.424416 0.212208 0.977225i \(-0.431935\pi\)
0.212208 + 0.977225i \(0.431935\pi\)
\(744\) 0 0
\(745\) −3.67458 −0.134626
\(746\) 19.2040 0.703109
\(747\) 0 0
\(748\) 2.89219 0.105749
\(749\) 0 0
\(750\) 0 0
\(751\) 34.3808 1.25457 0.627287 0.778788i \(-0.284165\pi\)
0.627287 + 0.778788i \(0.284165\pi\)
\(752\) −6.52745 −0.238031
\(753\) 0 0
\(754\) −0.862513 −0.0314109
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) 2.62364 0.0953577 0.0476789 0.998863i \(-0.484818\pi\)
0.0476789 + 0.998863i \(0.484818\pi\)
\(758\) 31.8670 1.15746
\(759\) 0 0
\(760\) −20.9884 −0.761328
\(761\) −45.7315 −1.65776 −0.828882 0.559424i \(-0.811023\pi\)
−0.828882 + 0.559424i \(0.811023\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 10.6902 0.386756
\(765\) 0 0
\(766\) −29.0413 −1.04930
\(767\) −3.83481 −0.138467
\(768\) 0 0
\(769\) −14.5668 −0.525291 −0.262646 0.964892i \(-0.584595\pi\)
−0.262646 + 0.964892i \(0.584595\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −21.5687 −0.776276
\(773\) −38.0297 −1.36783 −0.683916 0.729561i \(-0.739725\pi\)
−0.683916 + 0.729561i \(0.739725\pi\)
\(774\) 0 0
\(775\) −35.9238 −1.29042
\(776\) 6.72948 0.241574
\(777\) 0 0
\(778\) −26.5964 −0.953529
\(779\) −20.9884 −0.751987
\(780\) 0 0
\(781\) 13.9335 0.498579
\(782\) −23.5687 −0.842817
\(783\) 0 0
\(784\) 0 0
\(785\) −54.6397 −1.95017
\(786\) 0 0
\(787\) 26.2020 0.934002 0.467001 0.884257i \(-0.345335\pi\)
0.467001 + 0.884257i \(0.345335\pi\)
\(788\) 4.90578 0.174761
\(789\) 0 0
\(790\) 29.3531 1.04434
\(791\) 0 0
\(792\) 0 0
\(793\) 4.35261 0.154566
\(794\) −8.37834 −0.297336
\(795\) 0 0
\(796\) 7.10781 0.251930
\(797\) 13.8373 0.490142 0.245071 0.969505i \(-0.421189\pi\)
0.245071 + 0.969505i \(0.421189\pi\)
\(798\) 0 0
\(799\) 18.8786 0.667876
\(800\) −3.36474 −0.118961
\(801\) 0 0
\(802\) −7.93348 −0.280141
\(803\) −1.10781 −0.0390939
\(804\) 0 0
\(805\) 0 0
\(806\) 3.89416 0.137166
\(807\) 0 0
\(808\) 1.27052 0.0446968
\(809\) 28.0826 0.987331 0.493666 0.869652i \(-0.335657\pi\)
0.493666 + 0.869652i \(0.335657\pi\)
\(810\) 0 0
\(811\) −56.1516 −1.97175 −0.985875 0.167486i \(-0.946435\pi\)
−0.985875 + 0.167486i \(0.946435\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −3.78437 −0.132642
\(815\) 58.7062 2.05639
\(816\) 0 0
\(817\) −82.8714 −2.89930
\(818\) −24.6766 −0.862796
\(819\) 0 0
\(820\) −8.36474 −0.292109
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 35.2433 1.22851 0.614253 0.789109i \(-0.289458\pi\)
0.614253 + 0.789109i \(0.289458\pi\)
\(824\) −0.892186 −0.0310808
\(825\) 0 0
\(826\) 0 0
\(827\) 0.431256 0.0149963 0.00749813 0.999972i \(-0.497613\pi\)
0.00749813 + 0.999972i \(0.497613\pi\)
\(828\) 0 0
\(829\) 2.86499 0.0995053 0.0497527 0.998762i \(-0.484157\pi\)
0.0497527 + 0.998762i \(0.484157\pi\)
\(830\) −8.98838 −0.311991
\(831\) 0 0
\(832\) 0.364739 0.0126451
\(833\) 0 0
\(834\) 0 0
\(835\) −3.05095 −0.105582
\(836\) 7.25693 0.250986
\(837\) 0 0
\(838\) 2.54104 0.0877789
\(839\) 1.17433 0.0405424 0.0202712 0.999795i \(-0.493547\pi\)
0.0202712 + 0.999795i \(0.493547\pi\)
\(840\) 0 0
\(841\) −23.4080 −0.807173
\(842\) −2.43126 −0.0837866
\(843\) 0 0
\(844\) 8.87859 0.305614
\(845\) 37.2137 1.28019
\(846\) 0 0
\(847\) 0 0
\(848\) −11.7844 −0.404677
\(849\) 0 0
\(850\) 9.73145 0.333786
\(851\) 30.8393 1.05716
\(852\) 0 0
\(853\) 51.8004 1.77361 0.886807 0.462140i \(-0.152918\pi\)
0.886807 + 0.462140i \(0.152918\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 5.78437 0.197706
\(857\) 12.7824 0.436638 0.218319 0.975877i \(-0.429943\pi\)
0.218319 + 0.975877i \(0.429943\pi\)
\(858\) 0 0
\(859\) −16.7567 −0.571730 −0.285865 0.958270i \(-0.592281\pi\)
−0.285865 + 0.958270i \(0.592281\pi\)
\(860\) −33.0277 −1.12624
\(861\) 0 0
\(862\) 2.58037 0.0878877
\(863\) −26.5571 −0.904015 −0.452007 0.892014i \(-0.649292\pi\)
−0.452007 + 0.892014i \(0.649292\pi\)
\(864\) 0 0
\(865\) 47.7611 1.62393
\(866\) 37.1375 1.26198
\(867\) 0 0
\(868\) 0 0
\(869\) −10.1491 −0.344285
\(870\) 0 0
\(871\) −2.24282 −0.0759951
\(872\) −8.00000 −0.270914
\(873\) 0 0
\(874\) −59.1375 −2.00036
\(875\) 0 0
\(876\) 0 0
\(877\) −15.5687 −0.525719 −0.262860 0.964834i \(-0.584666\pi\)
−0.262860 + 0.964834i \(0.584666\pi\)
\(878\) 18.5139 0.624812
\(879\) 0 0
\(880\) 2.89219 0.0974956
\(881\) −45.6907 −1.53936 −0.769679 0.638431i \(-0.779584\pi\)
−0.769679 + 0.638431i \(0.779584\pi\)
\(882\) 0 0
\(883\) −55.0045 −1.85105 −0.925524 0.378690i \(-0.876375\pi\)
−0.925524 + 0.378690i \(0.876375\pi\)
\(884\) −1.05489 −0.0354799
\(885\) 0 0
\(886\) −21.4196 −0.719607
\(887\) 30.8393 1.03548 0.517741 0.855538i \(-0.326773\pi\)
0.517741 + 0.855538i \(0.326773\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 6.21563 0.208348
\(891\) 0 0
\(892\) −22.2453 −0.744828
\(893\) 47.3692 1.58515
\(894\) 0 0
\(895\) 60.7023 2.02905
\(896\) 0 0
\(897\) 0 0
\(898\) 4.06652 0.135701
\(899\) −25.2473 −0.842044
\(900\) 0 0
\(901\) 34.0826 1.13546
\(902\) 2.89219 0.0962993
\(903\) 0 0
\(904\) −19.9335 −0.662978
\(905\) −3.20400 −0.106505
\(906\) 0 0
\(907\) −25.4196 −0.844045 −0.422023 0.906585i \(-0.638680\pi\)
−0.422023 + 0.906585i \(0.638680\pi\)
\(908\) 8.46093 0.280786
\(909\) 0 0
\(910\) 0 0
\(911\) −50.5571 −1.67503 −0.837516 0.546413i \(-0.815993\pi\)
−0.837516 + 0.546413i \(0.815993\pi\)
\(912\) 0 0
\(913\) 3.10781 0.102854
\(914\) 16.8393 0.556993
\(915\) 0 0
\(916\) 2.16271 0.0714579
\(917\) 0 0
\(918\) 0 0
\(919\) 12.7295 0.419907 0.209953 0.977711i \(-0.432669\pi\)
0.209953 + 0.977711i \(0.432669\pi\)
\(920\) −23.5687 −0.777038
\(921\) 0 0
\(922\) 15.4590 0.509114
\(923\) −5.08209 −0.167279
\(924\) 0 0
\(925\) −12.7334 −0.418672
\(926\) −19.5687 −0.643069
\(927\) 0 0
\(928\) −2.36474 −0.0776264
\(929\) −6.44733 −0.211530 −0.105765 0.994391i \(-0.533729\pi\)
−0.105765 + 0.994391i \(0.533729\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 3.45896 0.113302
\(933\) 0 0
\(934\) −16.2982 −0.533294
\(935\) −8.36474 −0.273556
\(936\) 0 0
\(937\) −39.1904 −1.28029 −0.640147 0.768252i \(-0.721127\pi\)
−0.640147 + 0.768252i \(0.721127\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 18.8786 0.615752
\(941\) −37.1103 −1.20976 −0.604881 0.796316i \(-0.706779\pi\)
−0.604881 + 0.796316i \(0.706779\pi\)
\(942\) 0 0
\(943\) −23.5687 −0.767504
\(944\) −10.5139 −0.342197
\(945\) 0 0
\(946\) 11.4196 0.371284
\(947\) 49.3259 1.60288 0.801439 0.598077i \(-0.204068\pi\)
0.801439 + 0.598077i \(0.204068\pi\)
\(948\) 0 0
\(949\) 0.404063 0.0131164
\(950\) 24.4177 0.792213
\(951\) 0 0
\(952\) 0 0
\(953\) −24.2156 −0.784421 −0.392211 0.919875i \(-0.628290\pi\)
−0.392211 + 0.919875i \(0.628290\pi\)
\(954\) 0 0
\(955\) −30.9179 −1.00048
\(956\) −16.2982 −0.527122
\(957\) 0 0
\(958\) 0.431256 0.0139333
\(959\) 0 0
\(960\) 0 0
\(961\) 82.9889 2.67706
\(962\) 1.38031 0.0445029
\(963\) 0 0
\(964\) −4.67656 −0.150622
\(965\) 62.3808 2.00811
\(966\) 0 0
\(967\) −43.8670 −1.41067 −0.705333 0.708876i \(-0.749203\pi\)
−0.705333 + 0.708876i \(0.749203\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −19.4629 −0.624916
\(971\) 28.2982 0.908133 0.454067 0.890968i \(-0.349973\pi\)
0.454067 + 0.890968i \(0.349973\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 14.0826 0.451235
\(975\) 0 0
\(976\) 11.9335 0.381981
\(977\) 28.2982 0.905340 0.452670 0.891678i \(-0.350472\pi\)
0.452670 + 0.891678i \(0.350472\pi\)
\(978\) 0 0
\(979\) −2.14911 −0.0686859
\(980\) 0 0
\(981\) 0 0
\(982\) −23.5687 −0.752109
\(983\) −32.6372 −1.04097 −0.520483 0.853872i \(-0.674248\pi\)
−0.520483 + 0.853872i \(0.674248\pi\)
\(984\) 0 0
\(985\) −14.1884 −0.452081
\(986\) 6.83927 0.217807
\(987\) 0 0
\(988\) −2.64688 −0.0842086
\(989\) −93.0599 −2.95913
\(990\) 0 0
\(991\) −31.8670 −1.01229 −0.506144 0.862449i \(-0.668929\pi\)
−0.506144 + 0.862449i \(0.668929\pi\)
\(992\) 10.6766 0.338981
\(993\) 0 0
\(994\) 0 0
\(995\) −20.5571 −0.651705
\(996\) 0 0
\(997\) 37.4196 1.18509 0.592546 0.805537i \(-0.298123\pi\)
0.592546 + 0.805537i \(0.298123\pi\)
\(998\) −28.9884 −0.917611
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9702.2.a.dx.1.1 3
3.2 odd 2 9702.2.a.dy.1.3 3
7.6 odd 2 1386.2.a.q.1.3 3
21.20 even 2 1386.2.a.r.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1386.2.a.q.1.3 3 7.6 odd 2
1386.2.a.r.1.1 yes 3 21.20 even 2
9702.2.a.dx.1.1 3 1.1 even 1 trivial
9702.2.a.dy.1.3 3 3.2 odd 2