Properties

Label 3328.2.b.x.1665.2
Level $3328$
Weight $2$
Character 3328.1665
Analytic conductor $26.574$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3328,2,Mod(1665,3328)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3328, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3328.1665");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3328.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.5742137927\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 416)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1665.2
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 3328.1665
Dual form 3328.2.b.x.1665.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.23607i q^{3} +3.00000i q^{5} -2.23607 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-2.23607i q^{3} +3.00000i q^{5} -2.23607 q^{7} -2.00000 q^{9} -4.47214i q^{11} +1.00000i q^{13} +6.70820 q^{15} -3.00000 q^{17} +4.47214i q^{19} +5.00000i q^{21} +8.94427 q^{23} -4.00000 q^{25} -2.23607i q^{27} -10.0000i q^{29} -10.0000 q^{33} -6.70820i q^{35} +3.00000i q^{37} +2.23607 q^{39} +6.70820i q^{43} -6.00000i q^{45} +2.23607 q^{47} -2.00000 q^{49} +6.70820i q^{51} +4.00000i q^{53} +13.4164 q^{55} +10.0000 q^{57} -4.47214i q^{59} +4.47214 q^{63} -3.00000 q^{65} -13.4164i q^{67} -20.0000i q^{69} -6.70820 q^{71} -14.0000 q^{73} +8.94427i q^{75} +10.0000i q^{77} -8.94427 q^{79} -11.0000 q^{81} -17.8885i q^{83} -9.00000i q^{85} -22.3607 q^{87} +10.0000 q^{89} -2.23607i q^{91} -13.4164 q^{95} -2.00000 q^{97} +8.94427i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{9} - 12 q^{17} - 16 q^{25} - 40 q^{33} - 8 q^{49} + 40 q^{57} - 12 q^{65} - 56 q^{73} - 44 q^{81} + 40 q^{89} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 2.23607i − 1.29099i −0.763763 0.645497i \(-0.776650\pi\)
0.763763 0.645497i \(-0.223350\pi\)
\(4\) 0 0
\(5\) 3.00000i 1.34164i 0.741620 + 0.670820i \(0.234058\pi\)
−0.741620 + 0.670820i \(0.765942\pi\)
\(6\) 0 0
\(7\) −2.23607 −0.845154 −0.422577 0.906327i \(-0.638874\pi\)
−0.422577 + 0.906327i \(0.638874\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) − 4.47214i − 1.34840i −0.738549 0.674200i \(-0.764489\pi\)
0.738549 0.674200i \(-0.235511\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 6.70820 1.73205
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 4.47214i 1.02598i 0.858395 + 0.512989i \(0.171462\pi\)
−0.858395 + 0.512989i \(0.828538\pi\)
\(20\) 0 0
\(21\) 5.00000i 1.09109i
\(22\) 0 0
\(23\) 8.94427 1.86501 0.932505 0.361158i \(-0.117618\pi\)
0.932505 + 0.361158i \(0.117618\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) − 2.23607i − 0.430331i
\(28\) 0 0
\(29\) − 10.0000i − 1.85695i −0.371391 0.928477i \(-0.621119\pi\)
0.371391 0.928477i \(-0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −10.0000 −1.74078
\(34\) 0 0
\(35\) − 6.70820i − 1.13389i
\(36\) 0 0
\(37\) 3.00000i 0.493197i 0.969118 + 0.246598i \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) 0 0
\(39\) 2.23607 0.358057
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 6.70820i 1.02299i 0.859286 + 0.511496i \(0.170908\pi\)
−0.859286 + 0.511496i \(0.829092\pi\)
\(44\) 0 0
\(45\) − 6.00000i − 0.894427i
\(46\) 0 0
\(47\) 2.23607 0.326164 0.163082 0.986613i \(-0.447856\pi\)
0.163082 + 0.986613i \(0.447856\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 6.70820i 0.939336i
\(52\) 0 0
\(53\) 4.00000i 0.549442i 0.961524 + 0.274721i \(0.0885855\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) 13.4164 1.80907
\(56\) 0 0
\(57\) 10.0000 1.32453
\(58\) 0 0
\(59\) − 4.47214i − 0.582223i −0.956689 0.291111i \(-0.905975\pi\)
0.956689 0.291111i \(-0.0940250\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 4.47214 0.563436
\(64\) 0 0
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) − 13.4164i − 1.63908i −0.573025 0.819538i \(-0.694230\pi\)
0.573025 0.819538i \(-0.305770\pi\)
\(68\) 0 0
\(69\) − 20.0000i − 2.40772i
\(70\) 0 0
\(71\) −6.70820 −0.796117 −0.398059 0.917360i \(-0.630316\pi\)
−0.398059 + 0.917360i \(0.630316\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 8.94427i 1.03280i
\(76\) 0 0
\(77\) 10.0000i 1.13961i
\(78\) 0 0
\(79\) −8.94427 −1.00631 −0.503155 0.864196i \(-0.667827\pi\)
−0.503155 + 0.864196i \(0.667827\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) − 17.8885i − 1.96352i −0.190117 0.981761i \(-0.560887\pi\)
0.190117 0.981761i \(-0.439113\pi\)
\(84\) 0 0
\(85\) − 9.00000i − 0.976187i
\(86\) 0 0
\(87\) −22.3607 −2.39732
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) − 2.23607i − 0.234404i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −13.4164 −1.37649
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 8.94427i 0.898933i
\(100\) 0 0
\(101\) − 12.0000i − 1.19404i −0.802225 0.597022i \(-0.796350\pi\)
0.802225 0.597022i \(-0.203650\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) −15.0000 −1.46385
\(106\) 0 0
\(107\) − 17.8885i − 1.72935i −0.502331 0.864675i \(-0.667524\pi\)
0.502331 0.864675i \(-0.332476\pi\)
\(108\) 0 0
\(109\) − 11.0000i − 1.05361i −0.849987 0.526804i \(-0.823390\pi\)
0.849987 0.526804i \(-0.176610\pi\)
\(110\) 0 0
\(111\) 6.70820 0.636715
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 26.8328i 2.50217i
\(116\) 0 0
\(117\) − 2.00000i − 0.184900i
\(118\) 0 0
\(119\) 6.70820 0.614940
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000i 0.268328i
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 15.0000 1.32068
\(130\) 0 0
\(131\) − 15.6525i − 1.36756i −0.729687 0.683782i \(-0.760334\pi\)
0.729687 0.683782i \(-0.239666\pi\)
\(132\) 0 0
\(133\) − 10.0000i − 0.867110i
\(134\) 0 0
\(135\) 6.70820 0.577350
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) − 11.1803i − 0.948304i −0.880443 0.474152i \(-0.842755\pi\)
0.880443 0.474152i \(-0.157245\pi\)
\(140\) 0 0
\(141\) − 5.00000i − 0.421076i
\(142\) 0 0
\(143\) 4.47214 0.373979
\(144\) 0 0
\(145\) 30.0000 2.49136
\(146\) 0 0
\(147\) 4.47214i 0.368856i
\(148\) 0 0
\(149\) 14.0000i 1.14692i 0.819232 + 0.573462i \(0.194400\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) −6.70820 −0.545906 −0.272953 0.962027i \(-0.588000\pi\)
−0.272953 + 0.962027i \(0.588000\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 22.0000i − 1.75579i −0.478852 0.877896i \(-0.658947\pi\)
0.478852 0.877896i \(-0.341053\pi\)
\(158\) 0 0
\(159\) 8.94427 0.709327
\(160\) 0 0
\(161\) −20.0000 −1.57622
\(162\) 0 0
\(163\) − 8.94427i − 0.700569i −0.936643 0.350285i \(-0.886085\pi\)
0.936643 0.350285i \(-0.113915\pi\)
\(164\) 0 0
\(165\) − 30.0000i − 2.33550i
\(166\) 0 0
\(167\) 8.94427 0.692129 0.346064 0.938211i \(-0.387518\pi\)
0.346064 + 0.938211i \(0.387518\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) − 8.94427i − 0.683986i
\(172\) 0 0
\(173\) − 4.00000i − 0.304114i −0.988372 0.152057i \(-0.951410\pi\)
0.988372 0.152057i \(-0.0485898\pi\)
\(174\) 0 0
\(175\) 8.94427 0.676123
\(176\) 0 0
\(177\) −10.0000 −0.751646
\(178\) 0 0
\(179\) 11.1803i 0.835658i 0.908526 + 0.417829i \(0.137209\pi\)
−0.908526 + 0.417829i \(0.862791\pi\)
\(180\) 0 0
\(181\) − 8.00000i − 0.594635i −0.954779 0.297318i \(-0.903908\pi\)
0.954779 0.297318i \(-0.0960920\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −9.00000 −0.661693
\(186\) 0 0
\(187\) 13.4164i 0.981105i
\(188\) 0 0
\(189\) 5.00000i 0.363696i
\(190\) 0 0
\(191\) 4.47214 0.323592 0.161796 0.986824i \(-0.448271\pi\)
0.161796 + 0.986824i \(0.448271\pi\)
\(192\) 0 0
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 0 0
\(195\) 6.70820i 0.480384i
\(196\) 0 0
\(197\) − 7.00000i − 0.498729i −0.968410 0.249365i \(-0.919778\pi\)
0.968410 0.249365i \(-0.0802218\pi\)
\(198\) 0 0
\(199\) −13.4164 −0.951064 −0.475532 0.879698i \(-0.657744\pi\)
−0.475532 + 0.879698i \(0.657744\pi\)
\(200\) 0 0
\(201\) −30.0000 −2.11604
\(202\) 0 0
\(203\) 22.3607i 1.56941i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −17.8885 −1.24334
\(208\) 0 0
\(209\) 20.0000 1.38343
\(210\) 0 0
\(211\) − 15.6525i − 1.07756i −0.842446 0.538780i \(-0.818885\pi\)
0.842446 0.538780i \(-0.181115\pi\)
\(212\) 0 0
\(213\) 15.0000i 1.02778i
\(214\) 0 0
\(215\) −20.1246 −1.37249
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 31.3050i 2.11539i
\(220\) 0 0
\(221\) − 3.00000i − 0.201802i
\(222\) 0 0
\(223\) −2.23607 −0.149738 −0.0748691 0.997193i \(-0.523854\pi\)
−0.0748691 + 0.997193i \(0.523854\pi\)
\(224\) 0 0
\(225\) 8.00000 0.533333
\(226\) 0 0
\(227\) 17.8885i 1.18730i 0.804722 + 0.593652i \(0.202314\pi\)
−0.804722 + 0.593652i \(0.797686\pi\)
\(228\) 0 0
\(229\) 5.00000i 0.330409i 0.986259 + 0.165205i \(0.0528285\pi\)
−0.986259 + 0.165205i \(0.947172\pi\)
\(230\) 0 0
\(231\) 22.3607 1.47122
\(232\) 0 0
\(233\) −1.00000 −0.0655122 −0.0327561 0.999463i \(-0.510428\pi\)
−0.0327561 + 0.999463i \(0.510428\pi\)
\(234\) 0 0
\(235\) 6.70820i 0.437595i
\(236\) 0 0
\(237\) 20.0000i 1.29914i
\(238\) 0 0
\(239\) −15.6525 −1.01247 −0.506237 0.862394i \(-0.668964\pi\)
−0.506237 + 0.862394i \(0.668964\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 17.8885i 1.14755i
\(244\) 0 0
\(245\) − 6.00000i − 0.383326i
\(246\) 0 0
\(247\) −4.47214 −0.284555
\(248\) 0 0
\(249\) −40.0000 −2.53490
\(250\) 0 0
\(251\) 8.94427i 0.564557i 0.959332 + 0.282279i \(0.0910903\pi\)
−0.959332 + 0.282279i \(0.908910\pi\)
\(252\) 0 0
\(253\) − 40.0000i − 2.51478i
\(254\) 0 0
\(255\) −20.1246 −1.26025
\(256\) 0 0
\(257\) −23.0000 −1.43470 −0.717350 0.696713i \(-0.754645\pi\)
−0.717350 + 0.696713i \(0.754645\pi\)
\(258\) 0 0
\(259\) − 6.70820i − 0.416828i
\(260\) 0 0
\(261\) 20.0000i 1.23797i
\(262\) 0 0
\(263\) −8.94427 −0.551527 −0.275764 0.961225i \(-0.588931\pi\)
−0.275764 + 0.961225i \(0.588931\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) − 22.3607i − 1.36845i
\(268\) 0 0
\(269\) 16.0000i 0.975537i 0.872973 + 0.487769i \(0.162189\pi\)
−0.872973 + 0.487769i \(0.837811\pi\)
\(270\) 0 0
\(271\) 20.1246 1.22248 0.611242 0.791444i \(-0.290670\pi\)
0.611242 + 0.791444i \(0.290670\pi\)
\(272\) 0 0
\(273\) −5.00000 −0.302614
\(274\) 0 0
\(275\) 17.8885i 1.07872i
\(276\) 0 0
\(277\) − 28.0000i − 1.68236i −0.540758 0.841178i \(-0.681862\pi\)
0.540758 0.841178i \(-0.318138\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 17.8885i 1.06336i 0.846944 + 0.531682i \(0.178440\pi\)
−0.846944 + 0.531682i \(0.821560\pi\)
\(284\) 0 0
\(285\) 30.0000i 1.77705i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 4.47214i 0.262161i
\(292\) 0 0
\(293\) 31.0000i 1.81104i 0.424304 + 0.905520i \(0.360519\pi\)
−0.424304 + 0.905520i \(0.639481\pi\)
\(294\) 0 0
\(295\) 13.4164 0.781133
\(296\) 0 0
\(297\) −10.0000 −0.580259
\(298\) 0 0
\(299\) 8.94427i 0.517261i
\(300\) 0 0
\(301\) − 15.0000i − 0.864586i
\(302\) 0 0
\(303\) −26.8328 −1.54150
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.47214i 0.255238i 0.991823 + 0.127619i \(0.0407335\pi\)
−0.991823 + 0.127619i \(0.959266\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.47214 −0.253592 −0.126796 0.991929i \(-0.540469\pi\)
−0.126796 + 0.991929i \(0.540469\pi\)
\(312\) 0 0
\(313\) 29.0000 1.63918 0.819588 0.572953i \(-0.194202\pi\)
0.819588 + 0.572953i \(0.194202\pi\)
\(314\) 0 0
\(315\) 13.4164i 0.755929i
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) −44.7214 −2.50392
\(320\) 0 0
\(321\) −40.0000 −2.23258
\(322\) 0 0
\(323\) − 13.4164i − 0.746509i
\(324\) 0 0
\(325\) − 4.00000i − 0.221880i
\(326\) 0 0
\(327\) −24.5967 −1.36020
\(328\) 0 0
\(329\) −5.00000 −0.275659
\(330\) 0 0
\(331\) − 26.8328i − 1.47486i −0.675421 0.737432i \(-0.736038\pi\)
0.675421 0.737432i \(-0.263962\pi\)
\(332\) 0 0
\(333\) − 6.00000i − 0.328798i
\(334\) 0 0
\(335\) 40.2492 2.19905
\(336\) 0 0
\(337\) 3.00000 0.163420 0.0817102 0.996656i \(-0.473962\pi\)
0.0817102 + 0.996656i \(0.473962\pi\)
\(338\) 0 0
\(339\) 31.3050i 1.70025i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.1246 1.08663
\(344\) 0 0
\(345\) 60.0000 3.23029
\(346\) 0 0
\(347\) − 11.1803i − 0.600192i −0.953909 0.300096i \(-0.902981\pi\)
0.953909 0.300096i \(-0.0970187\pi\)
\(348\) 0 0
\(349\) 5.00000i 0.267644i 0.991005 + 0.133822i \(0.0427250\pi\)
−0.991005 + 0.133822i \(0.957275\pi\)
\(350\) 0 0
\(351\) 2.23607 0.119352
\(352\) 0 0
\(353\) 4.00000 0.212899 0.106449 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) 0 0
\(355\) − 20.1246i − 1.06810i
\(356\) 0 0
\(357\) − 15.0000i − 0.793884i
\(358\) 0 0
\(359\) −8.94427 −0.472061 −0.236030 0.971746i \(-0.575846\pi\)
−0.236030 + 0.971746i \(0.575846\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 20.1246i 1.05627i
\(364\) 0 0
\(365\) − 42.0000i − 2.19838i
\(366\) 0 0
\(367\) −4.47214 −0.233444 −0.116722 0.993165i \(-0.537239\pi\)
−0.116722 + 0.993165i \(0.537239\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 8.94427i − 0.464363i
\(372\) 0 0
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 0 0
\(375\) 6.70820 0.346410
\(376\) 0 0
\(377\) 10.0000 0.515026
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 15.6525 0.799804 0.399902 0.916558i \(-0.369044\pi\)
0.399902 + 0.916558i \(0.369044\pi\)
\(384\) 0 0
\(385\) −30.0000 −1.52894
\(386\) 0 0
\(387\) − 13.4164i − 0.681994i
\(388\) 0 0
\(389\) 26.0000i 1.31825i 0.752032 + 0.659126i \(0.229074\pi\)
−0.752032 + 0.659126i \(0.770926\pi\)
\(390\) 0 0
\(391\) −26.8328 −1.35699
\(392\) 0 0
\(393\) −35.0000 −1.76552
\(394\) 0 0
\(395\) − 26.8328i − 1.35011i
\(396\) 0 0
\(397\) 22.0000i 1.10415i 0.833795 + 0.552074i \(0.186163\pi\)
−0.833795 + 0.552074i \(0.813837\pi\)
\(398\) 0 0
\(399\) −22.3607 −1.11943
\(400\) 0 0
\(401\) −8.00000 −0.399501 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 33.0000i − 1.63978i
\(406\) 0 0
\(407\) 13.4164 0.665027
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) − 26.8328i − 1.32357i
\(412\) 0 0
\(413\) 10.0000i 0.492068i
\(414\) 0 0
\(415\) 53.6656 2.63434
\(416\) 0 0
\(417\) −25.0000 −1.22426
\(418\) 0 0
\(419\) − 2.23607i − 0.109239i −0.998507 0.0546195i \(-0.982605\pi\)
0.998507 0.0546195i \(-0.0173946\pi\)
\(420\) 0 0
\(421\) 15.0000i 0.731055i 0.930800 + 0.365528i \(0.119111\pi\)
−0.930800 + 0.365528i \(0.880889\pi\)
\(422\) 0 0
\(423\) −4.47214 −0.217443
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) − 10.0000i − 0.482805i
\(430\) 0 0
\(431\) −15.6525 −0.753953 −0.376977 0.926223i \(-0.623036\pi\)
−0.376977 + 0.926223i \(0.623036\pi\)
\(432\) 0 0
\(433\) −21.0000 −1.00920 −0.504598 0.863355i \(-0.668359\pi\)
−0.504598 + 0.863355i \(0.668359\pi\)
\(434\) 0 0
\(435\) − 67.0820i − 3.21634i
\(436\) 0 0
\(437\) 40.0000i 1.91346i
\(438\) 0 0
\(439\) −4.47214 −0.213443 −0.106722 0.994289i \(-0.534035\pi\)
−0.106722 + 0.994289i \(0.534035\pi\)
\(440\) 0 0
\(441\) 4.00000 0.190476
\(442\) 0 0
\(443\) − 15.6525i − 0.743672i −0.928299 0.371836i \(-0.878728\pi\)
0.928299 0.371836i \(-0.121272\pi\)
\(444\) 0 0
\(445\) 30.0000i 1.42214i
\(446\) 0 0
\(447\) 31.3050 1.48067
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 15.0000i 0.704761i
\(454\) 0 0
\(455\) 6.70820 0.314485
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) 0 0
\(459\) 6.70820i 0.313112i
\(460\) 0 0
\(461\) − 3.00000i − 0.139724i −0.997557 0.0698620i \(-0.977744\pi\)
0.997557 0.0698620i \(-0.0222559\pi\)
\(462\) 0 0
\(463\) 8.94427 0.415676 0.207838 0.978163i \(-0.433357\pi\)
0.207838 + 0.978163i \(0.433357\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 30.0000i 1.38527i
\(470\) 0 0
\(471\) −49.1935 −2.26672
\(472\) 0 0
\(473\) 30.0000 1.37940
\(474\) 0 0
\(475\) − 17.8885i − 0.820783i
\(476\) 0 0
\(477\) − 8.00000i − 0.366295i
\(478\) 0 0
\(479\) 20.1246 0.919517 0.459758 0.888044i \(-0.347936\pi\)
0.459758 + 0.888044i \(0.347936\pi\)
\(480\) 0 0
\(481\) −3.00000 −0.136788
\(482\) 0 0
\(483\) 44.7214i 2.03489i
\(484\) 0 0
\(485\) − 6.00000i − 0.272446i
\(486\) 0 0
\(487\) 26.8328 1.21591 0.607955 0.793971i \(-0.291990\pi\)
0.607955 + 0.793971i \(0.291990\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) − 11.1803i − 0.504562i −0.967654 0.252281i \(-0.918819\pi\)
0.967654 0.252281i \(-0.0811807\pi\)
\(492\) 0 0
\(493\) 30.0000i 1.35113i
\(494\) 0 0
\(495\) −26.8328 −1.20605
\(496\) 0 0
\(497\) 15.0000 0.672842
\(498\) 0 0
\(499\) 35.7771i 1.60160i 0.598930 + 0.800801i \(0.295593\pi\)
−0.598930 + 0.800801i \(0.704407\pi\)
\(500\) 0 0
\(501\) − 20.0000i − 0.893534i
\(502\) 0 0
\(503\) 31.3050 1.39582 0.697909 0.716186i \(-0.254114\pi\)
0.697909 + 0.716186i \(0.254114\pi\)
\(504\) 0 0
\(505\) 36.0000 1.60198
\(506\) 0 0
\(507\) 2.23607i 0.0993073i
\(508\) 0 0
\(509\) 30.0000i 1.32973i 0.746965 + 0.664863i \(0.231510\pi\)
−0.746965 + 0.664863i \(0.768490\pi\)
\(510\) 0 0
\(511\) 31.3050 1.38485
\(512\) 0 0
\(513\) 10.0000 0.441511
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 10.0000i − 0.439799i
\(518\) 0 0
\(519\) −8.94427 −0.392610
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) − 17.8885i − 0.782211i −0.920346 0.391106i \(-0.872093\pi\)
0.920346 0.391106i \(-0.127907\pi\)
\(524\) 0 0
\(525\) − 20.0000i − 0.872872i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 57.0000 2.47826
\(530\) 0 0
\(531\) 8.94427i 0.388148i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 53.6656 2.32017
\(536\) 0 0
\(537\) 25.0000 1.07883
\(538\) 0 0
\(539\) 8.94427i 0.385257i
\(540\) 0 0
\(541\) − 33.0000i − 1.41878i −0.704816 0.709390i \(-0.748970\pi\)
0.704816 0.709390i \(-0.251030\pi\)
\(542\) 0 0
\(543\) −17.8885 −0.767671
\(544\) 0 0
\(545\) 33.0000 1.41356
\(546\) 0 0
\(547\) − 38.0132i − 1.62533i −0.582735 0.812663i \(-0.698017\pi\)
0.582735 0.812663i \(-0.301983\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 44.7214 1.90519
\(552\) 0 0
\(553\) 20.0000 0.850487
\(554\) 0 0
\(555\) 20.1246i 0.854242i
\(556\) 0 0
\(557\) 7.00000i 0.296600i 0.988942 + 0.148300i \(0.0473800\pi\)
−0.988942 + 0.148300i \(0.952620\pi\)
\(558\) 0 0
\(559\) −6.70820 −0.283727
\(560\) 0 0
\(561\) 30.0000 1.26660
\(562\) 0 0
\(563\) − 15.6525i − 0.659673i −0.944038 0.329837i \(-0.893006\pi\)
0.944038 0.329837i \(-0.106994\pi\)
\(564\) 0 0
\(565\) − 42.0000i − 1.76695i
\(566\) 0 0
\(567\) 24.5967 1.03297
\(568\) 0 0
\(569\) 9.00000 0.377300 0.188650 0.982044i \(-0.439589\pi\)
0.188650 + 0.982044i \(0.439589\pi\)
\(570\) 0 0
\(571\) − 24.5967i − 1.02934i −0.857388 0.514671i \(-0.827914\pi\)
0.857388 0.514671i \(-0.172086\pi\)
\(572\) 0 0
\(573\) − 10.0000i − 0.417756i
\(574\) 0 0
\(575\) −35.7771 −1.49201
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) − 35.7771i − 1.48685i
\(580\) 0 0
\(581\) 40.0000i 1.65948i
\(582\) 0 0
\(583\) 17.8885 0.740868
\(584\) 0 0
\(585\) 6.00000 0.248069
\(586\) 0 0
\(587\) 44.7214i 1.84585i 0.384982 + 0.922924i \(0.374208\pi\)
−0.384982 + 0.922924i \(0.625792\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −15.6525 −0.643857
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 20.1246i 0.825029i
\(596\) 0 0
\(597\) 30.0000i 1.22782i
\(598\) 0 0
\(599\) −31.3050 −1.27909 −0.639543 0.768755i \(-0.720876\pi\)
−0.639543 + 0.768755i \(0.720876\pi\)
\(600\) 0 0
\(601\) −5.00000 −0.203954 −0.101977 0.994787i \(-0.532517\pi\)
−0.101977 + 0.994787i \(0.532517\pi\)
\(602\) 0 0
\(603\) 26.8328i 1.09272i
\(604\) 0 0
\(605\) − 27.0000i − 1.09771i
\(606\) 0 0
\(607\) 13.4164 0.544555 0.272278 0.962219i \(-0.412223\pi\)
0.272278 + 0.962219i \(0.412223\pi\)
\(608\) 0 0
\(609\) 50.0000 2.02610
\(610\) 0 0
\(611\) 2.23607i 0.0904616i
\(612\) 0 0
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.00000 0.322068 0.161034 0.986949i \(-0.448517\pi\)
0.161034 + 0.986949i \(0.448517\pi\)
\(618\) 0 0
\(619\) 26.8328i 1.07850i 0.842145 + 0.539251i \(0.181293\pi\)
−0.842145 + 0.539251i \(0.818707\pi\)
\(620\) 0 0
\(621\) − 20.0000i − 0.802572i
\(622\) 0 0
\(623\) −22.3607 −0.895862
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) − 44.7214i − 1.78600i
\(628\) 0 0
\(629\) − 9.00000i − 0.358854i
\(630\) 0 0
\(631\) −33.5410 −1.33525 −0.667623 0.744499i \(-0.732688\pi\)
−0.667623 + 0.744499i \(0.732688\pi\)
\(632\) 0 0
\(633\) −35.0000 −1.39113
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 2.00000i − 0.0792429i
\(638\) 0 0
\(639\) 13.4164 0.530745
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 22.3607i 0.881819i 0.897552 + 0.440910i \(0.145344\pi\)
−0.897552 + 0.440910i \(0.854656\pi\)
\(644\) 0 0
\(645\) 45.0000i 1.77187i
\(646\) 0 0
\(647\) −40.2492 −1.58236 −0.791180 0.611583i \(-0.790533\pi\)
−0.791180 + 0.611583i \(0.790533\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 24.0000i − 0.939193i −0.882881 0.469596i \(-0.844399\pi\)
0.882881 0.469596i \(-0.155601\pi\)
\(654\) 0 0
\(655\) 46.9574 1.83478
\(656\) 0 0
\(657\) 28.0000 1.09238
\(658\) 0 0
\(659\) − 17.8885i − 0.696839i −0.937339 0.348419i \(-0.886719\pi\)
0.937339 0.348419i \(-0.113281\pi\)
\(660\) 0 0
\(661\) − 10.0000i − 0.388955i −0.980907 0.194477i \(-0.937699\pi\)
0.980907 0.194477i \(-0.0623011\pi\)
\(662\) 0 0
\(663\) −6.70820 −0.260525
\(664\) 0 0
\(665\) 30.0000 1.16335
\(666\) 0 0
\(667\) − 89.4427i − 3.46324i
\(668\) 0 0
\(669\) 5.00000i 0.193311i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 29.0000 1.11787 0.558934 0.829212i \(-0.311211\pi\)
0.558934 + 0.829212i \(0.311211\pi\)
\(674\) 0 0
\(675\) 8.94427i 0.344265i
\(676\) 0 0
\(677\) − 12.0000i − 0.461197i −0.973049 0.230599i \(-0.925932\pi\)
0.973049 0.230599i \(-0.0740685\pi\)
\(678\) 0 0
\(679\) 4.47214 0.171625
\(680\) 0 0
\(681\) 40.0000 1.53280
\(682\) 0 0
\(683\) − 26.8328i − 1.02673i −0.858171 0.513365i \(-0.828399\pi\)
0.858171 0.513365i \(-0.171601\pi\)
\(684\) 0 0
\(685\) 36.0000i 1.37549i
\(686\) 0 0
\(687\) 11.1803 0.426557
\(688\) 0 0
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) − 17.8885i − 0.680512i −0.940333 0.340256i \(-0.889486\pi\)
0.940333 0.340256i \(-0.110514\pi\)
\(692\) 0 0
\(693\) − 20.0000i − 0.759737i
\(694\) 0 0
\(695\) 33.5410 1.27228
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 2.23607i 0.0845759i
\(700\) 0 0
\(701\) 20.0000i 0.755390i 0.925930 + 0.377695i \(0.123283\pi\)
−0.925930 + 0.377695i \(0.876717\pi\)
\(702\) 0 0
\(703\) −13.4164 −0.506009
\(704\) 0 0
\(705\) 15.0000 0.564933
\(706\) 0 0
\(707\) 26.8328i 1.00915i
\(708\) 0 0
\(709\) − 10.0000i − 0.375558i −0.982211 0.187779i \(-0.939871\pi\)
0.982211 0.187779i \(-0.0601289\pi\)
\(710\) 0 0
\(711\) 17.8885 0.670873
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 13.4164i 0.501745i
\(716\) 0 0
\(717\) 35.0000i 1.30710i
\(718\) 0 0
\(719\) 40.2492 1.50104 0.750521 0.660846i \(-0.229802\pi\)
0.750521 + 0.660846i \(0.229802\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 22.3607i − 0.831603i
\(724\) 0 0
\(725\) 40.0000i 1.48556i
\(726\) 0 0
\(727\) 31.3050 1.16104 0.580518 0.814247i \(-0.302850\pi\)
0.580518 + 0.814247i \(0.302850\pi\)
\(728\) 0 0
\(729\) 7.00000 0.259259
\(730\) 0 0
\(731\) − 20.1246i − 0.744336i
\(732\) 0 0
\(733\) − 9.00000i − 0.332423i −0.986090 0.166211i \(-0.946847\pi\)
0.986090 0.166211i \(-0.0531534\pi\)
\(734\) 0 0
\(735\) −13.4164 −0.494872
\(736\) 0 0
\(737\) −60.0000 −2.21013
\(738\) 0 0
\(739\) − 26.8328i − 0.987061i −0.869728 0.493531i \(-0.835706\pi\)
0.869728 0.493531i \(-0.164294\pi\)
\(740\) 0 0
\(741\) 10.0000i 0.367359i
\(742\) 0 0
\(743\) −29.0689 −1.06643 −0.533217 0.845979i \(-0.679017\pi\)
−0.533217 + 0.845979i \(0.679017\pi\)
\(744\) 0 0
\(745\) −42.0000 −1.53876
\(746\) 0 0
\(747\) 35.7771i 1.30902i
\(748\) 0 0
\(749\) 40.0000i 1.46157i
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 20.0000 0.728841
\(754\) 0 0
\(755\) − 20.1246i − 0.732410i
\(756\) 0 0
\(757\) − 28.0000i − 1.01768i −0.860862 0.508839i \(-0.830075\pi\)
0.860862 0.508839i \(-0.169925\pi\)
\(758\) 0 0
\(759\) −89.4427 −3.24657
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) 24.5967i 0.890462i
\(764\) 0 0
\(765\) 18.0000i 0.650791i
\(766\) 0 0
\(767\) 4.47214 0.161479
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 51.4296i 1.85219i
\(772\) 0 0
\(773\) − 9.00000i − 0.323708i −0.986815 0.161854i \(-0.948253\pi\)
0.986815 0.161854i \(-0.0517473\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −15.0000 −0.538122
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 30.0000i 1.07348i
\(782\) 0 0
\(783\) −22.3607 −0.799106
\(784\) 0 0
\(785\) 66.0000 2.35564
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 20.0000i 0.712019i
\(790\) 0 0
\(791\) 31.3050 1.11308
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 26.8328i 0.951662i
\(796\) 0 0
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) 0 0
\(799\) −6.70820 −0.237319
\(800\) 0 0
\(801\) −20.0000 −0.706665
\(802\) 0 0
\(803\) 62.6099i 2.20946i
\(804\) 0 0
\(805\) − 60.0000i − 2.11472i
\(806\) 0 0
\(807\) 35.7771 1.25941
\(808\) 0 0
\(809\) −35.0000 −1.23053 −0.615267 0.788319i \(-0.710952\pi\)
−0.615267 + 0.788319i \(0.710952\pi\)
\(810\) 0 0
\(811\) 8.94427i 0.314076i 0.987593 + 0.157038i \(0.0501945\pi\)
−0.987593 + 0.157038i \(0.949806\pi\)
\(812\) 0 0
\(813\) − 45.0000i − 1.57822i
\(814\) 0 0
\(815\) 26.8328 0.939913
\(816\) 0 0
\(817\) −30.0000 −1.04957
\(818\) 0 0
\(819\) 4.47214i 0.156269i
\(820\) 0 0
\(821\) − 45.0000i − 1.57051i −0.619172 0.785255i \(-0.712532\pi\)
0.619172 0.785255i \(-0.287468\pi\)
\(822\) 0 0
\(823\) −31.3050 −1.09122 −0.545611 0.838039i \(-0.683702\pi\)
−0.545611 + 0.838039i \(0.683702\pi\)
\(824\) 0 0
\(825\) 40.0000 1.39262
\(826\) 0 0
\(827\) − 22.3607i − 0.777557i −0.921331 0.388779i \(-0.872897\pi\)
0.921331 0.388779i \(-0.127103\pi\)
\(828\) 0 0
\(829\) 46.0000i 1.59765i 0.601566 + 0.798823i \(0.294544\pi\)
−0.601566 + 0.798823i \(0.705456\pi\)
\(830\) 0 0
\(831\) −62.6099 −2.17191
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 26.8328i 0.928588i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 26.8328 0.926372 0.463186 0.886261i \(-0.346706\pi\)
0.463186 + 0.886261i \(0.346706\pi\)
\(840\) 0 0
\(841\) −71.0000 −2.44828
\(842\) 0 0
\(843\) − 22.3607i − 0.770143i
\(844\) 0 0
\(845\) − 3.00000i − 0.103203i
\(846\) 0 0
\(847\) 20.1246 0.691490
\(848\) 0 0
\(849\) 40.0000 1.37280
\(850\) 0 0
\(851\) 26.8328i 0.919817i
\(852\) 0 0
\(853\) − 19.0000i − 0.650548i −0.945620 0.325274i \(-0.894544\pi\)
0.945620 0.325274i \(-0.105456\pi\)
\(854\) 0 0
\(855\) 26.8328 0.917663
\(856\) 0 0
\(857\) 2.00000 0.0683187 0.0341593 0.999416i \(-0.489125\pi\)
0.0341593 + 0.999416i \(0.489125\pi\)
\(858\) 0 0
\(859\) − 35.7771i − 1.22070i −0.792132 0.610349i \(-0.791029\pi\)
0.792132 0.610349i \(-0.208971\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 29.0689 0.989516 0.494758 0.869031i \(-0.335257\pi\)
0.494758 + 0.869031i \(0.335257\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) 0 0
\(867\) 17.8885i 0.607527i
\(868\) 0 0
\(869\) 40.0000i 1.35691i
\(870\) 0 0
\(871\) 13.4164 0.454598
\(872\) 0 0
\(873\) 4.00000 0.135379
\(874\) 0 0
\(875\) − 6.70820i − 0.226779i
\(876\) 0 0
\(877\) 3.00000i 0.101303i 0.998716 + 0.0506514i \(0.0161297\pi\)
−0.998716 + 0.0506514i \(0.983870\pi\)
\(878\) 0 0
\(879\) 69.3181 2.33804
\(880\) 0 0
\(881\) 5.00000 0.168454 0.0842271 0.996447i \(-0.473158\pi\)
0.0842271 + 0.996447i \(0.473158\pi\)
\(882\) 0 0
\(883\) − 2.23607i − 0.0752497i −0.999292 0.0376248i \(-0.988021\pi\)
0.999292 0.0376248i \(-0.0119792\pi\)
\(884\) 0 0
\(885\) − 30.0000i − 1.00844i
\(886\) 0 0
\(887\) 17.8885 0.600639 0.300319 0.953839i \(-0.402907\pi\)
0.300319 + 0.953839i \(0.402907\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 49.1935i 1.64804i
\(892\) 0 0
\(893\) 10.0000i 0.334637i
\(894\) 0 0
\(895\) −33.5410 −1.12115
\(896\) 0 0
\(897\) 20.0000 0.667781
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) − 12.0000i − 0.399778i
\(902\) 0 0
\(903\) −33.5410 −1.11618
\(904\) 0 0
\(905\) 24.0000 0.797787
\(906\) 0 0
\(907\) 33.5410i 1.11371i 0.830609 + 0.556856i \(0.187992\pi\)
−0.830609 + 0.556856i \(0.812008\pi\)
\(908\) 0 0
\(909\) 24.0000i 0.796030i
\(910\) 0 0
\(911\) 4.47214 0.148168 0.0740842 0.997252i \(-0.476397\pi\)
0.0740842 + 0.997252i \(0.476397\pi\)
\(912\) 0 0
\(913\) −80.0000 −2.64761
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 35.0000i 1.15580i
\(918\) 0 0
\(919\) −35.7771 −1.18018 −0.590089 0.807338i \(-0.700907\pi\)
−0.590089 + 0.807338i \(0.700907\pi\)
\(920\) 0 0
\(921\) 10.0000 0.329511
\(922\) 0 0
\(923\) − 6.70820i − 0.220803i
\(924\) 0 0
\(925\) − 12.0000i − 0.394558i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.00000 0.131236 0.0656179 0.997845i \(-0.479098\pi\)
0.0656179 + 0.997845i \(0.479098\pi\)
\(930\) 0 0
\(931\) − 8.94427i − 0.293137i
\(932\) 0 0
\(933\) 10.0000i 0.327385i
\(934\) 0 0
\(935\) −40.2492 −1.31629
\(936\) 0 0
\(937\) 58.0000 1.89478 0.947389 0.320085i \(-0.103712\pi\)
0.947389 + 0.320085i \(0.103712\pi\)
\(938\) 0 0
\(939\) − 64.8460i − 2.11617i
\(940\) 0 0
\(941\) 3.00000i 0.0977972i 0.998804 + 0.0488986i \(0.0155711\pi\)
−0.998804 + 0.0488986i \(0.984429\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −15.0000 −0.487950
\(946\) 0 0
\(947\) 4.47214i 0.145325i 0.997357 + 0.0726624i \(0.0231496\pi\)
−0.997357 + 0.0726624i \(0.976850\pi\)
\(948\) 0 0
\(949\) − 14.0000i − 0.454459i
\(950\) 0 0
\(951\) 40.2492 1.30517
\(952\) 0 0
\(953\) 9.00000 0.291539 0.145769 0.989319i \(-0.453434\pi\)
0.145769 + 0.989319i \(0.453434\pi\)
\(954\) 0 0
\(955\) 13.4164i 0.434145i
\(956\) 0 0
\(957\) 100.000i 3.23254i
\(958\) 0 0
\(959\) −26.8328 −0.866477
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 35.7771i 1.15290i
\(964\) 0 0
\(965\) 48.0000i 1.54517i
\(966\) 0 0
\(967\) 38.0132 1.22242 0.611210 0.791468i \(-0.290683\pi\)
0.611210 + 0.791468i \(0.290683\pi\)
\(968\) 0 0
\(969\) −30.0000 −0.963739
\(970\) 0 0
\(971\) 55.9017i 1.79397i 0.442060 + 0.896985i \(0.354248\pi\)
−0.442060 + 0.896985i \(0.645752\pi\)
\(972\) 0 0
\(973\) 25.0000i 0.801463i
\(974\) 0 0
\(975\) −8.94427 −0.286446
\(976\) 0 0
\(977\) 58.0000 1.85558 0.927792 0.373097i \(-0.121704\pi\)
0.927792 + 0.373097i \(0.121704\pi\)
\(978\) 0 0
\(979\) − 44.7214i − 1.42930i
\(980\) 0 0
\(981\) 22.0000i 0.702406i
\(982\) 0 0
\(983\) −2.23607 −0.0713195 −0.0356597 0.999364i \(-0.511353\pi\)
−0.0356597 + 0.999364i \(0.511353\pi\)
\(984\) 0 0
\(985\) 21.0000 0.669116
\(986\) 0 0
\(987\) 11.1803i 0.355874i
\(988\) 0 0
\(989\) 60.0000i 1.90789i
\(990\) 0 0
\(991\) −31.3050 −0.994435 −0.497217 0.867626i \(-0.665645\pi\)
−0.497217 + 0.867626i \(0.665645\pi\)
\(992\) 0 0
\(993\) −60.0000 −1.90404
\(994\) 0 0
\(995\) − 40.2492i − 1.27599i
\(996\) 0 0
\(997\) 22.0000i 0.696747i 0.937356 + 0.348373i \(0.113266\pi\)
−0.937356 + 0.348373i \(0.886734\pi\)
\(998\) 0 0
\(999\) 6.70820 0.212238
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 3328.2.b.x.1665.2 4
4.3 odd 2 inner 3328.2.b.x.1665.4 4
8.3 odd 2 inner 3328.2.b.x.1665.1 4
8.5 even 2 inner 3328.2.b.x.1665.3 4
16.3 odd 4 416.2.a.d.1.1 2
16.5 even 4 832.2.a.m.1.1 2
16.11 odd 4 832.2.a.m.1.2 2
16.13 even 4 416.2.a.d.1.2 yes 2
48.5 odd 4 7488.2.a.cw.1.2 2
48.11 even 4 7488.2.a.cw.1.1 2
48.29 odd 4 3744.2.a.q.1.2 2
48.35 even 4 3744.2.a.q.1.1 2
208.51 odd 4 5408.2.a.q.1.1 2
208.77 even 4 5408.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
416.2.a.d.1.1 2 16.3 odd 4
416.2.a.d.1.2 yes 2 16.13 even 4
832.2.a.m.1.1 2 16.5 even 4
832.2.a.m.1.2 2 16.11 odd 4
3328.2.b.x.1665.1 4 8.3 odd 2 inner
3328.2.b.x.1665.2 4 1.1 even 1 trivial
3328.2.b.x.1665.3 4 8.5 even 2 inner
3328.2.b.x.1665.4 4 4.3 odd 2 inner
3744.2.a.q.1.1 2 48.35 even 4
3744.2.a.q.1.2 2 48.29 odd 4
5408.2.a.q.1.1 2 208.51 odd 4
5408.2.a.q.1.2 2 208.77 even 4
7488.2.a.cw.1.1 2 48.11 even 4
7488.2.a.cw.1.2 2 48.5 odd 4