Properties

Label 156.2.b.a.25.2
Level $156$
Weight $2$
Character 156.25
Analytic conductor $1.246$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [156,2,Mod(25,156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(156, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("156.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 156.b (of order \(2\), degree \(1\), 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: \(1.24566627153\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
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: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 25.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 156.25
Dual form 156.2.b.a.25.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.46410i q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.46410i q^{5} +1.00000 q^{9} +3.46410i q^{11} +(-1.00000 + 3.46410i) q^{13} -3.46410i q^{15} +6.00000 q^{17} -6.92820i q^{19} -7.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} -6.92820i q^{31} -3.46410i q^{33} +(1.00000 - 3.46410i) q^{39} -3.46410i q^{41} +8.00000 q^{43} +3.46410i q^{45} +3.46410i q^{47} +7.00000 q^{49} -6.00000 q^{51} +6.00000 q^{53} -12.0000 q^{55} +6.92820i q^{57} -3.46410i q^{59} +10.0000 q^{61} +(-12.0000 - 3.46410i) q^{65} +13.8564i q^{67} +10.3923i q^{71} -6.92820i q^{73} +7.00000 q^{75} -8.00000 q^{79} +1.00000 q^{81} -3.46410i q^{83} +20.7846i q^{85} +6.00000 q^{87} -17.3205i q^{89} +6.92820i q^{93} +24.0000 q^{95} +6.92820i q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{9} - 2 q^{13} + 12 q^{17} - 14 q^{25} - 2 q^{27} - 12 q^{29} + 2 q^{39} + 16 q^{43} + 14 q^{49} - 12 q^{51} + 12 q^{53} - 24 q^{55} + 20 q^{61} - 24 q^{65} + 14 q^{75} - 16 q^{79} + 2 q^{81} + 12 q^{87} + 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 3.46410i 1.54919i 0.632456 + 0.774597i \(0.282047\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) −1.00000 + 3.46410i −0.277350 + 0.960769i
\(14\) 0 0
\(15\) 3.46410i 0.894427i
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 6.92820i 1.58944i −0.606977 0.794719i \(-0.707618\pi\)
0.606977 0.794719i \(-0.292382\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 6.92820i 1.24434i −0.782881 0.622171i \(-0.786251\pi\)
0.782881 0.622171i \(-0.213749\pi\)
\(32\) 0 0
\(33\) 3.46410i 0.603023i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 1.00000 3.46410i 0.160128 0.554700i
\(40\) 0 0
\(41\) 3.46410i 0.541002i −0.962720 0.270501i \(-0.912811\pi\)
0.962720 0.270501i \(-0.0871893\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 3.46410i 0.516398i
\(46\) 0 0
\(47\) 3.46410i 0.505291i 0.967559 + 0.252646i \(0.0813007\pi\)
−0.967559 + 0.252646i \(0.918699\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 6.92820i 0.917663i
\(58\) 0 0
\(59\) 3.46410i 0.450988i −0.974245 0.225494i \(-0.927600\pi\)
0.974245 0.225494i \(-0.0723995\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12.0000 3.46410i −1.48842 0.429669i
\(66\) 0 0
\(67\) 13.8564i 1.69283i 0.532524 + 0.846415i \(0.321244\pi\)
−0.532524 + 0.846415i \(0.678756\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.3923i 1.23334i 0.787222 + 0.616670i \(0.211519\pi\)
−0.787222 + 0.616670i \(0.788481\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i −0.914121 0.405442i \(-0.867117\pi\)
0.914121 0.405442i \(-0.132883\pi\)
\(74\) 0 0
\(75\) 7.00000 0.808290
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.46410i 0.380235i −0.981761 0.190117i \(-0.939113\pi\)
0.981761 0.190117i \(-0.0608868\pi\)
\(84\) 0 0
\(85\) 20.7846i 2.25441i
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 17.3205i 1.83597i −0.396615 0.917985i \(-0.629815\pi\)
0.396615 0.917985i \(-0.370185\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.92820i 0.718421i
\(94\) 0 0
\(95\) 24.0000 2.46235
\(96\) 0 0
\(97\) 6.92820i 0.703452i 0.936103 + 0.351726i \(0.114405\pi\)
−0.936103 + 0.351726i \(0.885595\pi\)
\(98\) 0 0
\(99\) 3.46410i 0.348155i
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 6.92820i 0.663602i −0.943349 0.331801i \(-0.892344\pi\)
0.943349 0.331801i \(-0.107656\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.00000 + 3.46410i −0.0924500 + 0.320256i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 3.46410i 0.312348i
\(124\) 0 0
\(125\) 6.92820i 0.619677i
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.46410i 0.298142i
\(136\) 0 0
\(137\) 10.3923i 0.887875i −0.896058 0.443937i \(-0.853581\pi\)
0.896058 0.443937i \(-0.146419\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 3.46410i 0.291730i
\(142\) 0 0
\(143\) −12.0000 3.46410i −1.00349 0.289683i
\(144\) 0 0
\(145\) 20.7846i 1.72607i
\(146\) 0 0
\(147\) −7.00000 −0.577350
\(148\) 0 0
\(149\) 17.3205i 1.41895i −0.704730 0.709476i \(-0.748932\pi\)
0.704730 0.709476i \(-0.251068\pi\)
\(150\) 0 0
\(151\) 6.92820i 0.563809i 0.959442 + 0.281905i \(0.0909662\pi\)
−0.959442 + 0.281905i \(0.909034\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 24.0000 1.92773
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20.7846i 1.62798i −0.580881 0.813988i \(-0.697292\pi\)
0.580881 0.813988i \(-0.302708\pi\)
\(164\) 0 0
\(165\) 12.0000 0.934199
\(166\) 0 0
\(167\) 17.3205i 1.34030i 0.742225 + 0.670151i \(0.233770\pi\)
−0.742225 + 0.670151i \(0.766230\pi\)
\(168\) 0 0
\(169\) −11.0000 6.92820i −0.846154 0.532939i
\(170\) 0 0
\(171\) 6.92820i 0.529813i
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.46410i 0.260378i
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 20.7846i 1.51992i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 13.8564i 0.997406i 0.866773 + 0.498703i \(0.166190\pi\)
−0.866773 + 0.498703i \(0.833810\pi\)
\(194\) 0 0
\(195\) 12.0000 + 3.46410i 0.859338 + 0.248069i
\(196\) 0 0
\(197\) 17.3205i 1.23404i 0.786949 + 0.617018i \(0.211659\pi\)
−0.786949 + 0.617018i \(0.788341\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 13.8564i 0.977356i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 10.3923i 0.712069i
\(214\) 0 0
\(215\) 27.7128i 1.89000i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 6.92820i 0.468165i
\(220\) 0 0
\(221\) −6.00000 + 20.7846i −0.403604 + 1.39812i
\(222\) 0 0
\(223\) 6.92820i 0.463947i −0.972722 0.231973i \(-0.925482\pi\)
0.972722 0.231973i \(-0.0745182\pi\)
\(224\) 0 0
\(225\) −7.00000 −0.466667
\(226\) 0 0
\(227\) 10.3923i 0.689761i −0.938647 0.344881i \(-0.887919\pi\)
0.938647 0.344881i \(-0.112081\pi\)
\(228\) 0 0
\(229\) 6.92820i 0.457829i 0.973447 + 0.228914i \(0.0735176\pi\)
−0.973447 + 0.228914i \(0.926482\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 10.3923i 0.672222i 0.941822 + 0.336111i \(0.109112\pi\)
−0.941822 + 0.336111i \(0.890888\pi\)
\(240\) 0 0
\(241\) 20.7846i 1.33885i −0.742878 0.669427i \(-0.766540\pi\)
0.742878 0.669427i \(-0.233460\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 24.2487i 1.54919i
\(246\) 0 0
\(247\) 24.0000 + 6.92820i 1.52708 + 0.440831i
\(248\) 0 0
\(249\) 3.46410i 0.219529i
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 20.7846i 1.30158i
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 20.7846i 1.27679i
\(266\) 0 0
\(267\) 17.3205i 1.06000i
\(268\) 0 0
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) 27.7128i 1.68343i −0.539919 0.841717i \(-0.681545\pi\)
0.539919 0.841717i \(-0.318455\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 24.2487i 1.46225i
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 6.92820i 0.414781i
\(280\) 0 0
\(281\) 17.3205i 1.03325i 0.856210 + 0.516627i \(0.172813\pi\)
−0.856210 + 0.516627i \(0.827187\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) −24.0000 −1.42164
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 6.92820i 0.406138i
\(292\) 0 0
\(293\) 3.46410i 0.202375i 0.994867 + 0.101187i \(0.0322642\pi\)
−0.994867 + 0.101187i \(0.967736\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) 3.46410i 0.201008i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) 34.6410i 1.98354i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.2487i 1.36194i −0.732310 0.680972i \(-0.761558\pi\)
0.732310 0.680972i \(-0.238442\pi\)
\(318\) 0 0
\(319\) 20.7846i 1.16371i
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 41.5692i 2.31297i
\(324\) 0 0
\(325\) 7.00000 24.2487i 0.388290 1.34508i
\(326\) 0 0
\(327\) 6.92820i 0.383131i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 13.8564i 0.761617i 0.924654 + 0.380808i \(0.124354\pi\)
−0.924654 + 0.380808i \(0.875646\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −48.0000 −2.62252
\(336\) 0 0
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 0 0
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 1.00000 3.46410i 0.0533761 0.184900i
\(352\) 0 0
\(353\) 10.3923i 0.553127i 0.960996 + 0.276563i \(0.0891955\pi\)
−0.960996 + 0.276563i \(0.910804\pi\)
\(354\) 0 0
\(355\) −36.0000 −1.91068
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 24.2487i 1.27980i −0.768459 0.639899i \(-0.778976\pi\)
0.768459 0.639899i \(-0.221024\pi\)
\(360\) 0 0
\(361\) −29.0000 −1.52632
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 24.0000 1.25622
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) 3.46410i 0.180334i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) 6.92820i 0.357771i
\(376\) 0 0
\(377\) 6.00000 20.7846i 0.309016 1.07046i
\(378\) 0 0
\(379\) 20.7846i 1.06763i 0.845600 + 0.533817i \(0.179243\pi\)
−0.845600 + 0.533817i \(0.820757\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 24.2487i 1.23905i −0.784976 0.619526i \(-0.787325\pi\)
0.784976 0.619526i \(-0.212675\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.00000 0.406663
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 27.7128i 1.39438i
\(396\) 0 0
\(397\) 13.8564i 0.695433i 0.937600 + 0.347717i \(0.113043\pi\)
−0.937600 + 0.347717i \(0.886957\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.3923i 0.518967i 0.965748 + 0.259483i \(0.0835523\pi\)
−0.965748 + 0.259483i \(0.916448\pi\)
\(402\) 0 0
\(403\) 24.0000 + 6.92820i 1.19553 + 0.345118i
\(404\) 0 0
\(405\) 3.46410i 0.172133i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 20.7846i 1.02773i 0.857870 + 0.513866i \(0.171787\pi\)
−0.857870 + 0.513866i \(0.828213\pi\)
\(410\) 0 0
\(411\) 10.3923i 0.512615i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 20.7846i 1.01298i −0.862246 0.506490i \(-0.830943\pi\)
0.862246 0.506490i \(-0.169057\pi\)
\(422\) 0 0
\(423\) 3.46410i 0.168430i
\(424\) 0 0
\(425\) −42.0000 −2.03730
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 12.0000 + 3.46410i 0.579365 + 0.167248i
\(430\) 0 0
\(431\) 31.1769i 1.50174i 0.660451 + 0.750870i \(0.270365\pi\)
−0.660451 + 0.750870i \(0.729635\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 20.7846i 0.996546i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 60.0000 2.84427
\(446\) 0 0
\(447\) 17.3205i 0.819232i
\(448\) 0 0
\(449\) 3.46410i 0.163481i −0.996654 0.0817405i \(-0.973952\pi\)
0.996654 0.0817405i \(-0.0260479\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 6.92820i 0.325515i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.92820i 0.324088i 0.986784 + 0.162044i \(0.0518086\pi\)
−0.986784 + 0.162044i \(0.948191\pi\)
\(458\) 0 0
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 10.3923i 0.484018i −0.970274 0.242009i \(-0.922194\pi\)
0.970274 0.242009i \(-0.0778063\pi\)
\(462\) 0 0
\(463\) 13.8564i 0.643962i 0.946746 + 0.321981i \(0.104349\pi\)
−0.946746 + 0.321981i \(0.895651\pi\)
\(464\) 0 0
\(465\) −24.0000 −1.11297
\(466\) 0 0
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 27.7128i 1.27424i
\(474\) 0 0
\(475\) 48.4974i 2.22521i
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 3.46410i 0.158279i −0.996864 0.0791394i \(-0.974783\pi\)
0.996864 0.0791394i \(-0.0252172\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −24.0000 −1.08978
\(486\) 0 0
\(487\) 6.92820i 0.313947i 0.987603 + 0.156973i \(0.0501737\pi\)
−0.987603 + 0.156973i \(0.949826\pi\)
\(488\) 0 0
\(489\) 20.7846i 0.939913i
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 17.3205i 0.773823i
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 20.7846i 0.924903i
\(506\) 0 0
\(507\) 11.0000 + 6.92820i 0.488527 + 0.307692i
\(508\) 0 0
\(509\) 24.2487i 1.07481i −0.843326 0.537403i \(-0.819406\pi\)
0.843326 0.537403i \(-0.180594\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 6.92820i 0.305888i
\(514\) 0 0
\(515\) 13.8564i 0.610586i
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 8.00000 0.349816 0.174908 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 41.5692i 1.81078i
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 3.46410i 0.150329i
\(532\) 0 0
\(533\) 12.0000 + 3.46410i 0.519778 + 0.150047i
\(534\) 0 0
\(535\) 41.5692i 1.79719i
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) 24.2487i 1.04447i
\(540\) 0 0
\(541\) 20.7846i 0.893600i −0.894634 0.446800i \(-0.852564\pi\)
0.894634 0.446800i \(-0.147436\pi\)
\(542\) 0 0
\(543\) 22.0000 0.944110
\(544\) 0 0
\(545\) 24.0000 1.02805
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 41.5692i 1.77091i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 24.2487i 1.02745i −0.857955 0.513725i \(-0.828265\pi\)
0.857955 0.513725i \(-0.171735\pi\)
\(558\) 0 0
\(559\) −8.00000 + 27.7128i −0.338364 + 1.17213i
\(560\) 0 0
\(561\) 20.7846i 0.877527i
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 62.3538i 2.62325i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 13.8564i 0.576850i −0.957503 0.288425i \(-0.906868\pi\)
0.957503 0.288425i \(-0.0931316\pi\)
\(578\) 0 0
\(579\) 13.8564i 0.575853i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 20.7846i 0.860811i
\(584\) 0 0
\(585\) −12.0000 3.46410i −0.496139 0.143223i
\(586\) 0 0
\(587\) 31.1769i 1.28681i −0.765526 0.643404i \(-0.777521\pi\)
0.765526 0.643404i \(-0.222479\pi\)
\(588\) 0 0
\(589\) −48.0000 −1.97781
\(590\) 0 0
\(591\) 17.3205i 0.712470i
\(592\) 0 0
\(593\) 24.2487i 0.995775i 0.867242 + 0.497888i \(0.165891\pi\)
−0.867242 + 0.497888i \(0.834109\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 13.8564i 0.564276i
\(604\) 0 0
\(605\) 3.46410i 0.140836i
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −12.0000 3.46410i −0.485468 0.140143i
\(612\) 0 0
\(613\) 13.8564i 0.559655i 0.960050 + 0.279827i \(0.0902773\pi\)
−0.960050 + 0.279827i \(0.909723\pi\)
\(614\) 0 0
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) 31.1769i 1.25514i −0.778562 0.627568i \(-0.784051\pi\)
0.778562 0.627568i \(-0.215949\pi\)
\(618\) 0 0
\(619\) 27.7128i 1.11387i −0.830555 0.556936i \(-0.811977\pi\)
0.830555 0.556936i \(-0.188023\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) −24.0000 −0.958468
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 20.7846i 0.827422i 0.910408 + 0.413711i \(0.135768\pi\)
−0.910408 + 0.413711i \(0.864232\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) 55.4256i 2.19950i
\(636\) 0 0
\(637\) −7.00000 + 24.2487i −0.277350 + 0.960769i
\(638\) 0 0
\(639\) 10.3923i 0.411113i
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 41.5692i 1.63933i −0.572843 0.819665i \(-0.694160\pi\)
0.572843 0.819665i \(-0.305840\pi\)
\(644\) 0 0
\(645\) 27.7128i 1.09119i
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 41.5692i 1.62424i
\(656\) 0 0
\(657\) 6.92820i 0.270295i
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 13.8564i 0.538952i −0.963007 0.269476i \(-0.913150\pi\)
0.963007 0.269476i \(-0.0868504\pi\)
\(662\) 0 0
\(663\) 6.00000 20.7846i 0.233021 0.807207i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 6.92820i 0.267860i
\(670\) 0 0
\(671\) 34.6410i 1.33730i
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 0 0
\(675\) 7.00000 0.269430
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 10.3923i 0.398234i
\(682\) 0 0
\(683\) 3.46410i 0.132550i 0.997801 + 0.0662751i \(0.0211115\pi\)
−0.997801 + 0.0662751i \(0.978889\pi\)
\(684\) 0 0
\(685\) 36.0000 1.37549
\(686\) 0 0
\(687\) 6.92820i 0.264327i
\(688\) 0 0
\(689\) −6.00000 + 20.7846i −0.228582 + 0.791831i
\(690\) 0 0
\(691\) 27.7128i 1.05425i 0.849789 + 0.527123i \(0.176729\pi\)
−0.849789 + 0.527123i \(0.823271\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 27.7128i 1.05121i
\(696\) 0 0
\(697\) 20.7846i 0.787273i
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6.92820i 0.260194i −0.991501 0.130097i \(-0.958471\pi\)
0.991501 0.130097i \(-0.0415289\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 12.0000 41.5692i 0.448775 1.55460i
\(716\) 0 0
\(717\) 10.3923i 0.388108i
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 20.7846i 0.772988i
\(724\) 0 0
\(725\) 42.0000 1.55984
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) 0 0
\(733\) 6.92820i 0.255899i 0.991781 + 0.127950i \(0.0408395\pi\)
−0.991781 + 0.127950i \(0.959160\pi\)
\(734\) 0 0
\(735\) 24.2487i 0.894427i
\(736\) 0 0
\(737\) −48.0000 −1.76810
\(738\) 0 0
\(739\) 27.7128i 1.01943i 0.860343 + 0.509716i \(0.170250\pi\)
−0.860343 + 0.509716i \(0.829750\pi\)
\(740\) 0 0
\(741\) −24.0000 6.92820i −0.881662 0.254514i
\(742\) 0 0
\(743\) 10.3923i 0.381257i 0.981662 + 0.190628i \(0.0610525\pi\)
−0.981662 + 0.190628i \(0.938947\pi\)
\(744\) 0 0
\(745\) 60.0000 2.19823
\(746\) 0 0
\(747\) 3.46410i 0.126745i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) −24.0000 −0.873449
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 31.1769i 1.13016i 0.825035 + 0.565081i \(0.191155\pi\)
−0.825035 + 0.565081i \(0.808845\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 20.7846i 0.751469i
\(766\) 0 0
\(767\) 12.0000 + 3.46410i 0.433295 + 0.125081i
\(768\) 0 0
\(769\) 27.7128i 0.999350i 0.866213 + 0.499675i \(0.166547\pi\)
−0.866213 + 0.499675i \(0.833453\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) 51.9615i 1.86893i 0.356060 + 0.934463i \(0.384120\pi\)
−0.356060 + 0.934463i \(0.615880\pi\)
\(774\) 0 0
\(775\) 48.4974i 1.74208i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 6.92820i 0.247278i
\(786\) 0 0
\(787\) 27.7128i 0.987855i −0.869503 0.493928i \(-0.835561\pi\)
0.869503 0.493928i \(-0.164439\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −10.0000 + 34.6410i −0.355110 + 1.23014i
\(794\) 0 0
\(795\) 20.7846i 0.737154i
\(796\) 0 0
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 20.7846i 0.735307i
\(800\) 0 0
\(801\) 17.3205i 0.611990i
\(802\) 0 0
\(803\) 24.0000 0.846942
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 30.0000 1.05605
\(808\) 0 0
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 6.92820i 0.243282i 0.992574 + 0.121641i \(0.0388157\pi\)
−0.992574 + 0.121641i \(0.961184\pi\)
\(812\) 0 0
\(813\) 27.7128i 0.971931i
\(814\) 0 0
\(815\) 72.0000 2.52205
\(816\) 0 0
\(817\) 55.4256i 1.93910i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.46410i 0.120898i −0.998171 0.0604490i \(-0.980747\pi\)
0.998171 0.0604490i \(-0.0192532\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) 24.2487i 0.844232i
\(826\) 0 0
\(827\) 17.3205i 0.602293i 0.953578 + 0.301147i \(0.0973693\pi\)
−0.953578 + 0.301147i \(0.902631\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 42.0000 1.45521
\(834\) 0 0
\(835\) −60.0000 −2.07639
\(836\) 0 0
\(837\) 6.92820i 0.239474i
\(838\) 0 0
\(839\) 10.3923i 0.358782i 0.983778 + 0.179391i \(0.0574128\pi\)
−0.983778 + 0.179391i \(0.942587\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 17.3205i 0.596550i
\(844\) 0 0
\(845\) 24.0000 38.1051i 0.825625 1.31086i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 13.8564i 0.474434i 0.971457 + 0.237217i \(0.0762353\pi\)
−0.971457 + 0.237217i \(0.923765\pi\)
\(854\) 0 0
\(855\) 24.0000 0.820783
\(856\) 0 0
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 38.1051i 1.29711i −0.761166 0.648557i \(-0.775373\pi\)
0.761166 0.648557i \(-0.224627\pi\)
\(864\) 0 0
\(865\) 20.7846i 0.706698i
\(866\) 0 0
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 27.7128i 0.940093i
\(870\) 0 0
\(871\) −48.0000 13.8564i −1.62642 0.469506i
\(872\) 0 0
\(873\) 6.92820i 0.234484i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 41.5692i 1.40369i 0.712328 + 0.701846i \(0.247641\pi\)
−0.712328 + 0.701846i \(0.752359\pi\)
\(878\) 0 0
\(879\) 3.46410i 0.116841i
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) 0 0
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 3.46410i 0.116052i
\(892\) 0 0
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) 41.5692i 1.38951i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 41.5692i 1.38641i
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 76.2102i 2.53331i
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 34.6410i 1.14520i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −36.0000 10.3923i −1.18495 0.342067i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) 31.1769i 1.02288i −0.859319 0.511441i \(-0.829112\pi\)
0.859319 0.511441i \(-0.170888\pi\)
\(930\) 0 0
\(931\) 48.4974i 1.58944i
\(932\) 0 0
\(933\) −24.0000 −0.785725
\(934\) 0 0
\(935\) −72.0000 −2.35465
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 10.3923i 0.338779i 0.985549 + 0.169390i \(0.0541797\pi\)
−0.985549 + 0.169390i \(0.945820\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 45.0333i 1.46339i −0.681634 0.731693i \(-0.738730\pi\)
0.681634 0.731693i \(-0.261270\pi\)
\(948\) 0 0
\(949\) 24.0000 + 6.92820i 0.779073 + 0.224899i
\(950\) 0 0
\(951\) 24.2487i 0.786318i
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 20.7846i 0.671871i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −17.0000 −0.548387
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) −48.0000 −1.54517
\(966\) 0 0
\(967\) 34.6410i 1.11398i −0.830519 0.556990i \(-0.811956\pi\)
0.830519 0.556990i \(-0.188044\pi\)
\(968\) 0 0
\(969\) 41.5692i 1.33540i
\(970\) 0 0
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −7.00000 + 24.2487i −0.224179 + 0.776580i
\(976\) 0 0
\(977\) 24.2487i 0.775785i −0.921705 0.387893i \(-0.873203\pi\)
0.921705 0.387893i \(-0.126797\pi\)
\(978\) 0 0
\(979\) 60.0000 1.91761
\(980\) 0 0
\(981\) 6.92820i 0.221201i
\(982\) 0 0
\(983\) 31.1769i 0.994389i −0.867639 0.497195i \(-0.834364\pi\)
0.867639 0.497195i \(-0.165636\pi\)
\(984\) 0 0
\(985\) −60.0000 −1.91176
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 0 0
\(993\) 13.8564i 0.439720i
\(994\) 0 0
\(995\) 27.7128i 0.878555i
\(996\) 0 0
\(997\) 38.0000 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) 0 0
\(999\) 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 156.2.b.a.25.2 yes 2
3.2 odd 2 468.2.b.a.181.1 2
4.3 odd 2 624.2.c.f.337.2 2
5.2 odd 4 3900.2.j.h.649.4 4
5.3 odd 4 3900.2.j.h.649.2 4
5.4 even 2 3900.2.c.c.3301.2 2
7.6 odd 2 7644.2.e.g.4705.1 2
8.3 odd 2 2496.2.c.e.961.1 2
8.5 even 2 2496.2.c.l.961.1 2
12.11 even 2 1872.2.c.c.1585.1 2
13.2 odd 12 2028.2.i.i.529.2 4
13.3 even 3 2028.2.q.b.1837.1 2
13.4 even 6 2028.2.q.b.361.1 2
13.5 odd 4 2028.2.a.g.1.2 2
13.6 odd 12 2028.2.i.i.2005.2 4
13.7 odd 12 2028.2.i.i.2005.1 4
13.8 odd 4 2028.2.a.g.1.1 2
13.9 even 3 2028.2.q.c.361.1 2
13.10 even 6 2028.2.q.c.1837.1 2
13.11 odd 12 2028.2.i.i.529.1 4
13.12 even 2 inner 156.2.b.a.25.1 2
39.5 even 4 6084.2.a.v.1.1 2
39.8 even 4 6084.2.a.v.1.2 2
39.38 odd 2 468.2.b.a.181.2 2
52.31 even 4 8112.2.a.bs.1.2 2
52.47 even 4 8112.2.a.bs.1.1 2
52.51 odd 2 624.2.c.f.337.1 2
65.12 odd 4 3900.2.j.h.649.3 4
65.38 odd 4 3900.2.j.h.649.1 4
65.64 even 2 3900.2.c.c.3301.1 2
91.90 odd 2 7644.2.e.g.4705.2 2
104.51 odd 2 2496.2.c.e.961.2 2
104.77 even 2 2496.2.c.l.961.2 2
156.155 even 2 1872.2.c.c.1585.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
156.2.b.a.25.1 2 13.12 even 2 inner
156.2.b.a.25.2 yes 2 1.1 even 1 trivial
468.2.b.a.181.1 2 3.2 odd 2
468.2.b.a.181.2 2 39.38 odd 2
624.2.c.f.337.1 2 52.51 odd 2
624.2.c.f.337.2 2 4.3 odd 2
1872.2.c.c.1585.1 2 12.11 even 2
1872.2.c.c.1585.2 2 156.155 even 2
2028.2.a.g.1.1 2 13.8 odd 4
2028.2.a.g.1.2 2 13.5 odd 4
2028.2.i.i.529.1 4 13.11 odd 12
2028.2.i.i.529.2 4 13.2 odd 12
2028.2.i.i.2005.1 4 13.7 odd 12
2028.2.i.i.2005.2 4 13.6 odd 12
2028.2.q.b.361.1 2 13.4 even 6
2028.2.q.b.1837.1 2 13.3 even 3
2028.2.q.c.361.1 2 13.9 even 3
2028.2.q.c.1837.1 2 13.10 even 6
2496.2.c.e.961.1 2 8.3 odd 2
2496.2.c.e.961.2 2 104.51 odd 2
2496.2.c.l.961.1 2 8.5 even 2
2496.2.c.l.961.2 2 104.77 even 2
3900.2.c.c.3301.1 2 65.64 even 2
3900.2.c.c.3301.2 2 5.4 even 2
3900.2.j.h.649.1 4 65.38 odd 4
3900.2.j.h.649.2 4 5.3 odd 4
3900.2.j.h.649.3 4 65.12 odd 4
3900.2.j.h.649.4 4 5.2 odd 4
6084.2.a.v.1.1 2 39.5 even 4
6084.2.a.v.1.2 2 39.8 even 4
7644.2.e.g.4705.1 2 7.6 odd 2
7644.2.e.g.4705.2 2 91.90 odd 2
8112.2.a.bs.1.1 2 52.47 even 4
8112.2.a.bs.1.2 2 52.31 even 4