Properties

Label 1183.2.c.a.337.1
Level $1183$
Weight $2$
Character 1183.337
Analytic conductor $9.446$
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] = [1183,2,Mod(337,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1183.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (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: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.a.337.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} +2.00000 q^{4} -3.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +2.00000 q^{4} -3.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} -4.00000 q^{12} +6.00000i q^{15} +4.00000 q^{16} +6.00000 q^{17} -7.00000i q^{19} -6.00000i q^{20} +2.00000i q^{21} -3.00000 q^{23} -4.00000 q^{25} +4.00000 q^{27} -2.00000i q^{28} -9.00000 q^{29} +5.00000i q^{31} -3.00000 q^{35} +2.00000 q^{36} -2.00000i q^{37} -6.00000i q^{41} +1.00000 q^{43} -3.00000i q^{45} -3.00000i q^{47} -8.00000 q^{48} -1.00000 q^{49} -12.0000 q^{51} -9.00000 q^{53} +14.0000i q^{57} +12.0000i q^{60} -10.0000 q^{61} -1.00000i q^{63} +8.00000 q^{64} +14.0000i q^{67} +12.0000 q^{68} +6.00000 q^{69} -6.00000i q^{71} -11.0000i q^{73} +8.00000 q^{75} -14.0000i q^{76} -1.00000 q^{79} -12.0000i q^{80} -11.0000 q^{81} +3.00000i q^{83} +4.00000i q^{84} -18.0000i q^{85} +18.0000 q^{87} -15.0000i q^{89} -6.00000 q^{92} -10.0000i q^{93} -21.0000 q^{95} -1.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 4 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} + 4 q^{4} + 2 q^{9} - 8 q^{12} + 8 q^{16} + 12 q^{17} - 6 q^{23} - 8 q^{25} + 8 q^{27} - 18 q^{29} - 6 q^{35} + 4 q^{36} + 2 q^{43} - 16 q^{48} - 2 q^{49} - 24 q^{51} - 18 q^{53} - 20 q^{61} + 16 q^{64} + 24 q^{68} + 12 q^{69} + 16 q^{75} - 2 q^{79} - 22 q^{81} + 36 q^{87} - 12 q^{92} - 42 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 2.00000 1.00000
\(5\) − 3.00000i − 1.34164i −0.741620 0.670820i \(-0.765942\pi\)
0.741620 0.670820i \(-0.234058\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −4.00000 −1.15470
\(13\) 0 0
\(14\) 0 0
\(15\) 6.00000i 1.54919i
\(16\) 4.00000 1.00000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) − 7.00000i − 1.60591i −0.596040 0.802955i \(-0.703260\pi\)
0.596040 0.802955i \(-0.296740\pi\)
\(20\) − 6.00000i − 1.34164i
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) − 2.00000i − 0.377964i
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 5.00000i 0.898027i 0.893525 + 0.449013i \(0.148224\pi\)
−0.893525 + 0.449013i \(0.851776\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 2.00000 0.333333
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 6.00000i − 0.937043i −0.883452 0.468521i \(-0.844787\pi\)
0.883452 0.468521i \(-0.155213\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) − 3.00000i − 0.447214i
\(46\) 0 0
\(47\) − 3.00000i − 0.437595i −0.975770 0.218797i \(-0.929787\pi\)
0.975770 0.218797i \(-0.0702134\pi\)
\(48\) −8.00000 −1.15470
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) 0 0
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 14.0000i 1.85435i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 12.0000i 1.54919i
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) − 1.00000i − 0.125988i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 14.0000i 1.71037i 0.518321 + 0.855186i \(0.326557\pi\)
−0.518321 + 0.855186i \(0.673443\pi\)
\(68\) 12.0000 1.45521
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) − 6.00000i − 0.712069i −0.934473 0.356034i \(-0.884129\pi\)
0.934473 0.356034i \(-0.115871\pi\)
\(72\) 0 0
\(73\) − 11.0000i − 1.28745i −0.765256 0.643726i \(-0.777388\pi\)
0.765256 0.643726i \(-0.222612\pi\)
\(74\) 0 0
\(75\) 8.00000 0.923760
\(76\) − 14.0000i − 1.60591i
\(77\) 0 0
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) − 12.0000i − 1.34164i
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 3.00000i 0.329293i 0.986353 + 0.164646i \(0.0526483\pi\)
−0.986353 + 0.164646i \(0.947352\pi\)
\(84\) 4.00000i 0.436436i
\(85\) − 18.0000i − 1.95237i
\(86\) 0 0
\(87\) 18.0000 1.92980
\(88\) 0 0
\(89\) − 15.0000i − 1.59000i −0.606612 0.794998i \(-0.707472\pi\)
0.606612 0.794998i \(-0.292528\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) − 10.0000i − 1.03695i
\(94\) 0 0
\(95\) −21.0000 −2.15455
\(96\) 0 0
\(97\) − 1.00000i − 0.101535i −0.998711 0.0507673i \(-0.983833\pi\)
0.998711 0.0507673i \(-0.0161667\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −8.00000 −0.800000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 6.00000 0.585540
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 8.00000 0.769800
\(109\) − 16.0000i − 1.53252i −0.642529 0.766261i \(-0.722115\pi\)
0.642529 0.766261i \(-0.277885\pi\)
\(110\) 0 0
\(111\) 4.00000i 0.379663i
\(112\) − 4.00000i − 0.377964i
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) 9.00000i 0.839254i
\(116\) −18.0000 −1.67126
\(117\) 0 0
\(118\) 0 0
\(119\) − 6.00000i − 0.550019i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 12.0000i 1.08200i
\(124\) 10.0000i 0.898027i
\(125\) − 3.00000i − 0.268328i
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 0 0
\(135\) − 12.0000i − 1.03280i
\(136\) 0 0
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −6.00000 −0.507093
\(141\) 6.00000i 0.505291i
\(142\) 0 0
\(143\) 0 0
\(144\) 4.00000 0.333333
\(145\) 27.0000i 2.24223i
\(146\) 0 0
\(147\) 2.00000 0.164957
\(148\) − 4.00000i − 0.328798i
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 10.0000i 0.813788i 0.913475 + 0.406894i \(0.133388\pi\)
−0.913475 + 0.406894i \(0.866612\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 15.0000 1.20483
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 18.0000 1.42749
\(160\) 0 0
\(161\) 3.00000i 0.236433i
\(162\) 0 0
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) − 12.0000i − 0.937043i
\(165\) 0 0
\(166\) 0 0
\(167\) 15.0000i 1.16073i 0.814355 + 0.580367i \(0.197091\pi\)
−0.814355 + 0.580367i \(0.802909\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) − 7.00000i − 0.535303i
\(172\) 2.00000 0.152499
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 4.00000i 0.302372i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) − 6.00000i − 0.447214i
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 0 0
\(183\) 20.0000 1.47844
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) − 6.00000i − 0.437595i
\(189\) − 4.00000i − 0.290957i
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) −16.0000 −1.15470
\(193\) 22.0000i 1.58359i 0.610784 + 0.791797i \(0.290854\pi\)
−0.610784 + 0.791797i \(0.709146\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) − 28.0000i − 1.97497i
\(202\) 0 0
\(203\) 9.00000i 0.631676i
\(204\) −24.0000 −1.68034
\(205\) −18.0000 −1.25717
\(206\) 0 0
\(207\) −3.00000 −0.208514
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 23.0000 1.58339 0.791693 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) −18.0000 −1.23625
\(213\) 12.0000i 0.822226i
\(214\) 0 0
\(215\) − 3.00000i − 0.204598i
\(216\) 0 0
\(217\) 5.00000 0.339422
\(218\) 0 0
\(219\) 22.0000i 1.48662i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 1.00000i − 0.0669650i −0.999439 0.0334825i \(-0.989340\pi\)
0.999439 0.0334825i \(-0.0106598\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) 24.0000i 1.59294i 0.604681 + 0.796468i \(0.293301\pi\)
−0.604681 + 0.796468i \(0.706699\pi\)
\(228\) 28.0000i 1.85435i
\(229\) − 14.0000i − 0.925146i −0.886581 0.462573i \(-0.846926\pi\)
0.886581 0.462573i \(-0.153074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.00000 0.589610 0.294805 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) 0 0
\(237\) 2.00000 0.129914
\(238\) 0 0
\(239\) − 12.0000i − 0.776215i −0.921614 0.388108i \(-0.873129\pi\)
0.921614 0.388108i \(-0.126871\pi\)
\(240\) 24.0000i 1.54919i
\(241\) 1.00000i 0.0644157i 0.999481 + 0.0322078i \(0.0102538\pi\)
−0.999481 + 0.0322078i \(0.989746\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) −20.0000 −1.28037
\(245\) 3.00000i 0.191663i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) − 6.00000i − 0.380235i
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) 0 0
\(254\) 0 0
\(255\) 36.0000i 2.25441i
\(256\) 16.0000 1.00000
\(257\) 24.0000 1.49708 0.748539 0.663090i \(-0.230755\pi\)
0.748539 + 0.663090i \(0.230755\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −9.00000 −0.557086
\(262\) 0 0
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 0 0
\(265\) 27.0000i 1.65860i
\(266\) 0 0
\(267\) 30.0000i 1.83597i
\(268\) 28.0000i 1.71037i
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) 16.0000i 0.971931i 0.873978 + 0.485965i \(0.161532\pi\)
−0.873978 + 0.485965i \(0.838468\pi\)
\(272\) 24.0000 1.45521
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) 19.0000 1.14160 0.570800 0.821089i \(-0.306633\pi\)
0.570800 + 0.821089i \(0.306633\pi\)
\(278\) 0 0
\(279\) 5.00000i 0.299342i
\(280\) 0 0
\(281\) − 12.0000i − 0.715860i −0.933748 0.357930i \(-0.883483\pi\)
0.933748 0.357930i \(-0.116517\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) − 12.0000i − 0.712069i
\(285\) 42.0000 2.48787
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 2.00000i 0.117242i
\(292\) − 22.0000i − 1.28745i
\(293\) − 9.00000i − 0.525786i −0.964825 0.262893i \(-0.915323\pi\)
0.964825 0.262893i \(-0.0846766\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 16.0000 0.923760
\(301\) − 1.00000i − 0.0576390i
\(302\) 0 0
\(303\) 0 0
\(304\) − 28.0000i − 1.60591i
\(305\) 30.0000i 1.71780i
\(306\) 0 0
\(307\) − 11.0000i − 0.627803i −0.949456 0.313902i \(-0.898364\pi\)
0.949456 0.313902i \(-0.101636\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) −2.00000 −0.112509
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 24.0000i − 1.34164i
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) − 42.0000i − 2.33694i
\(324\) −22.0000 −1.22222
\(325\) 0 0
\(326\) 0 0
\(327\) 32.0000i 1.76960i
\(328\) 0 0
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) 26.0000i 1.42909i 0.699590 + 0.714545i \(0.253366\pi\)
−0.699590 + 0.714545i \(0.746634\pi\)
\(332\) 6.00000i 0.329293i
\(333\) − 2.00000i − 0.109599i
\(334\) 0 0
\(335\) 42.0000 2.29471
\(336\) 8.00000i 0.436436i
\(337\) −5.00000 −0.272367 −0.136184 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) − 36.0000i − 1.95237i
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) − 18.0000i − 0.969087i
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 36.0000 1.92980
\(349\) 1.00000i 0.0535288i 0.999642 + 0.0267644i \(0.00852039\pi\)
−0.999642 + 0.0267644i \(0.991480\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 18.0000i − 0.958043i −0.877803 0.479022i \(-0.840992\pi\)
0.877803 0.479022i \(-0.159008\pi\)
\(354\) 0 0
\(355\) −18.0000 −0.955341
\(356\) − 30.0000i − 1.59000i
\(357\) 12.0000i 0.635107i
\(358\) 0 0
\(359\) − 18.0000i − 0.950004i −0.879985 0.475002i \(-0.842447\pi\)
0.879985 0.475002i \(-0.157553\pi\)
\(360\) 0 0
\(361\) −30.0000 −1.57895
\(362\) 0 0
\(363\) −22.0000 −1.15470
\(364\) 0 0
\(365\) −33.0000 −1.72730
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) −12.0000 −0.625543
\(369\) − 6.00000i − 0.312348i
\(370\) 0 0
\(371\) 9.00000i 0.467257i
\(372\) − 20.0000i − 1.03695i
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 6.00000i 0.309839i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 16.0000i − 0.821865i −0.911666 0.410932i \(-0.865203\pi\)
0.911666 0.410932i \(-0.134797\pi\)
\(380\) −42.0000 −2.15455
\(381\) −32.0000 −1.63941
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.00000 0.0508329
\(388\) − 2.00000i − 0.101535i
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 24.0000 1.21064
\(394\) 0 0
\(395\) 3.00000i 0.150946i
\(396\) 0 0
\(397\) − 11.0000i − 0.552074i −0.961147 0.276037i \(-0.910979\pi\)
0.961147 0.276037i \(-0.0890213\pi\)
\(398\) 0 0
\(399\) 14.0000 0.700877
\(400\) −16.0000 −0.800000
\(401\) − 12.0000i − 0.599251i −0.954057 0.299626i \(-0.903138\pi\)
0.954057 0.299626i \(-0.0968618\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 33.0000i 1.63978i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 23.0000i 1.13728i 0.822588 + 0.568638i \(0.192530\pi\)
−0.822588 + 0.568638i \(0.807470\pi\)
\(410\) 0 0
\(411\) 12.0000i 0.591916i
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) 9.00000 0.441793
\(416\) 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 \(0.405307\pi\)
\(420\) 12.0000 0.585540
\(421\) − 10.0000i − 0.487370i −0.969854 0.243685i \(-0.921644\pi\)
0.969854 0.243685i \(-0.0783563\pi\)
\(422\) 0 0
\(423\) − 3.00000i − 0.145865i
\(424\) 0 0
\(425\) −24.0000 −1.16417
\(426\) 0 0
\(427\) 10.0000i 0.483934i
\(428\) −24.0000 −1.16008
\(429\) 0 0
\(430\) 0 0
\(431\) − 30.0000i − 1.44505i −0.691345 0.722525i \(-0.742982\pi\)
0.691345 0.722525i \(-0.257018\pi\)
\(432\) 16.0000 0.769800
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) − 54.0000i − 2.58910i
\(436\) − 32.0000i − 1.53252i
\(437\) 21.0000i 1.00457i
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) 39.0000 1.85295 0.926473 0.376361i \(-0.122825\pi\)
0.926473 + 0.376361i \(0.122825\pi\)
\(444\) 8.00000i 0.379663i
\(445\) −45.0000 −2.13320
\(446\) 0 0
\(447\) 0 0
\(448\) − 8.00000i − 0.377964i
\(449\) 24.0000i 1.13263i 0.824189 + 0.566315i \(0.191631\pi\)
−0.824189 + 0.566315i \(0.808369\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 18.0000 0.846649
\(453\) − 20.0000i − 0.939682i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) 0 0
\(459\) 24.0000 1.12022
\(460\) 18.0000i 0.839254i
\(461\) − 30.0000i − 1.39724i −0.715493 0.698620i \(-0.753798\pi\)
0.715493 0.698620i \(-0.246202\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) −36.0000 −1.67126
\(465\) −30.0000 −1.39122
\(466\) 0 0
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) 14.0000 0.646460
\(470\) 0 0
\(471\) −28.0000 −1.29017
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 28.0000i 1.28473i
\(476\) − 12.0000i − 0.550019i
\(477\) −9.00000 −0.412082
\(478\) 0 0
\(479\) − 9.00000i − 0.411220i −0.978634 0.205610i \(-0.934082\pi\)
0.978634 0.205610i \(-0.0659179\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) − 6.00000i − 0.273009i
\(484\) 22.0000 1.00000
\(485\) −3.00000 −0.136223
\(486\) 0 0
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 0 0
\(489\) − 32.0000i − 1.44709i
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 24.0000i 1.08200i
\(493\) −54.0000 −2.43204
\(494\) 0 0
\(495\) 0 0
\(496\) 20.0000i 0.898027i
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 14.0000i 0.626726i 0.949633 + 0.313363i \(0.101456\pi\)
−0.949633 + 0.313363i \(0.898544\pi\)
\(500\) − 6.00000i − 0.268328i
\(501\) − 30.0000i − 1.34030i
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 32.0000 1.41977
\(509\) 21.0000i 0.930809i 0.885098 + 0.465404i \(0.154091\pi\)
−0.885098 + 0.465404i \(0.845909\pi\)
\(510\) 0 0
\(511\) −11.0000 −0.486611
\(512\) 0 0
\(513\) − 28.0000i − 1.23623i
\(514\) 0 0
\(515\) − 12.0000i − 0.528783i
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 2.00000 0.0874539 0.0437269 0.999044i \(-0.486077\pi\)
0.0437269 + 0.999044i \(0.486077\pi\)
\(524\) −24.0000 −1.04844
\(525\) − 8.00000i − 0.349149i
\(526\) 0 0
\(527\) 30.0000i 1.30682i
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) −14.0000 −0.606977
\(533\) 0 0
\(534\) 0 0
\(535\) 36.0000i 1.55642i
\(536\) 0 0
\(537\) 30.0000 1.29460
\(538\) 0 0
\(539\) 0 0
\(540\) − 24.0000i − 1.03280i
\(541\) 34.0000i 1.46177i 0.682498 + 0.730887i \(0.260893\pi\)
−0.682498 + 0.730887i \(0.739107\pi\)
\(542\) 0 0
\(543\) −32.0000 −1.37325
\(544\) 0 0
\(545\) −48.0000 −2.05609
\(546\) 0 0
\(547\) 17.0000 0.726868 0.363434 0.931620i \(-0.381604\pi\)
0.363434 + 0.931620i \(0.381604\pi\)
\(548\) − 12.0000i − 0.512615i
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 63.0000i 2.68389i
\(552\) 0 0
\(553\) 1.00000i 0.0425243i
\(554\) 0 0
\(555\) 12.0000 0.509372
\(556\) −8.00000 −0.339276
\(557\) 12.0000i 0.508456i 0.967144 + 0.254228i \(0.0818214\pi\)
−0.967144 + 0.254228i \(0.918179\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −12.0000 −0.507093
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 12.0000i 0.505291i
\(565\) − 27.0000i − 1.13590i
\(566\) 0 0
\(567\) 11.0000i 0.461957i
\(568\) 0 0
\(569\) −27.0000 −1.13190 −0.565949 0.824440i \(-0.691490\pi\)
−0.565949 + 0.824440i \(0.691490\pi\)
\(570\) 0 0
\(571\) −23.0000 −0.962520 −0.481260 0.876578i \(-0.659821\pi\)
−0.481260 + 0.876578i \(0.659821\pi\)
\(572\) 0 0
\(573\) −48.0000 −2.00523
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 8.00000 0.333333
\(577\) − 34.0000i − 1.41544i −0.706494 0.707719i \(-0.749724\pi\)
0.706494 0.707719i \(-0.250276\pi\)
\(578\) 0 0
\(579\) − 44.0000i − 1.82858i
\(580\) 54.0000i 2.24223i
\(581\) 3.00000 0.124461
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 9.00000i − 0.371470i −0.982600 0.185735i \(-0.940533\pi\)
0.982600 0.185735i \(-0.0594666\pi\)
\(588\) 4.00000 0.164957
\(589\) 35.0000 1.44215
\(590\) 0 0
\(591\) − 12.0000i − 0.493614i
\(592\) − 8.00000i − 0.328798i
\(593\) − 9.00000i − 0.369586i −0.982777 0.184793i \(-0.940839\pi\)
0.982777 0.184793i \(-0.0591614\pi\)
\(594\) 0 0
\(595\) −18.0000 −0.737928
\(596\) 0 0
\(597\) 4.00000 0.163709
\(598\) 0 0
\(599\) 15.0000 0.612883 0.306442 0.951889i \(-0.400862\pi\)
0.306442 + 0.951889i \(0.400862\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 14.0000i 0.570124i
\(604\) 20.0000i 0.813788i
\(605\) − 33.0000i − 1.34164i
\(606\) 0 0
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 0 0
\(609\) − 18.0000i − 0.729397i
\(610\) 0 0
\(611\) 0 0
\(612\) 12.0000 0.485071
\(613\) − 16.0000i − 0.646234i −0.946359 0.323117i \(-0.895269\pi\)
0.946359 0.323117i \(-0.104731\pi\)
\(614\) 0 0
\(615\) 36.0000 1.45166
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 0 0
\(619\) 28.0000i 1.12542i 0.826656 + 0.562708i \(0.190240\pi\)
−0.826656 + 0.562708i \(0.809760\pi\)
\(620\) 30.0000 1.20483
\(621\) −12.0000 −0.481543
\(622\) 0 0
\(623\) −15.0000 −0.600962
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 28.0000 1.11732
\(629\) − 12.0000i − 0.478471i
\(630\) 0 0
\(631\) − 2.00000i − 0.0796187i −0.999207 0.0398094i \(-0.987325\pi\)
0.999207 0.0398094i \(-0.0126751\pi\)
\(632\) 0 0
\(633\) −46.0000 −1.82834
\(634\) 0 0
\(635\) − 48.0000i − 1.90482i
\(636\) 36.0000 1.42749
\(637\) 0 0
\(638\) 0 0
\(639\) − 6.00000i − 0.237356i
\(640\) 0 0
\(641\) 39.0000 1.54041 0.770204 0.637798i \(-0.220155\pi\)
0.770204 + 0.637798i \(0.220155\pi\)
\(642\) 0 0
\(643\) − 4.00000i − 0.157745i −0.996885 0.0788723i \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) 6.00000i 0.236433i
\(645\) 6.00000i 0.236250i
\(646\) 0 0
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −10.0000 −0.391931
\(652\) 32.0000i 1.25322i
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 36.0000i 1.40664i
\(656\) − 24.0000i − 0.937043i
\(657\) − 11.0000i − 0.429151i
\(658\) 0 0
\(659\) −3.00000 −0.116863 −0.0584317 0.998291i \(-0.518610\pi\)
−0.0584317 + 0.998291i \(0.518610\pi\)
\(660\) 0 0
\(661\) 13.0000i 0.505641i 0.967513 + 0.252821i \(0.0813583\pi\)
−0.967513 + 0.252821i \(0.918642\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 21.0000i 0.814345i
\(666\) 0 0
\(667\) 27.0000 1.04544
\(668\) 30.0000i 1.16073i
\(669\) 2.00000i 0.0773245i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 0 0
\(675\) −16.0000 −0.615840
\(676\) 0 0
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 0 0
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) − 48.0000i − 1.83936i
\(682\) 0 0
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) − 14.0000i − 0.535303i
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) 28.0000i 1.06827i
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) − 37.0000i − 1.40755i −0.710425 0.703773i \(-0.751497\pi\)
0.710425 0.703773i \(-0.248503\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) 0 0
\(695\) 12.0000i 0.455186i
\(696\) 0 0
\(697\) − 36.0000i − 1.36360i
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 8.00000i 0.302372i
\(701\) −9.00000 −0.339925 −0.169963 0.985451i \(-0.554365\pi\)
−0.169963 + 0.985451i \(0.554365\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) 0 0
\(705\) 18.0000 0.677919
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) − 8.00000i − 0.300446i −0.988652 0.150223i \(-0.952001\pi\)
0.988652 0.150223i \(-0.0479992\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) 0 0
\(713\) − 15.0000i − 0.561754i
\(714\) 0 0
\(715\) 0 0
\(716\) −30.0000 −1.12115
\(717\) 24.0000i 0.896296i
\(718\) 0 0
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) − 12.0000i − 0.447214i
\(721\) − 4.00000i − 0.148968i
\(722\) 0 0
\(723\) − 2.00000i − 0.0743808i
\(724\) 32.0000 1.18927
\(725\) 36.0000 1.33701
\(726\) 0 0
\(727\) −26.0000 −0.964287 −0.482143 0.876092i \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 6.00000 0.221918
\(732\) 40.0000 1.47844
\(733\) − 49.0000i − 1.80986i −0.425564 0.904928i \(-0.639924\pi\)
0.425564 0.904928i \(-0.360076\pi\)
\(734\) 0 0
\(735\) − 6.00000i − 0.221313i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 20.0000i − 0.735712i −0.929883 0.367856i \(-0.880092\pi\)
0.929883 0.367856i \(-0.119908\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) 0 0
\(743\) 48.0000i 1.76095i 0.474093 + 0.880475i \(0.342776\pi\)
−0.474093 + 0.880475i \(0.657224\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3.00000i 0.109764i
\(748\) 0 0
\(749\) 12.0000i 0.438470i
\(750\) 0 0
\(751\) 13.0000 0.474377 0.237188 0.971464i \(-0.423774\pi\)
0.237188 + 0.971464i \(0.423774\pi\)
\(752\) − 12.0000i − 0.437595i
\(753\) 24.0000 0.874609
\(754\) 0 0
\(755\) 30.0000 1.09181
\(756\) − 8.00000i − 0.290957i
\(757\) −43.0000 −1.56286 −0.781431 0.623992i \(-0.785510\pi\)
−0.781431 + 0.623992i \(0.785510\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 27.0000i 0.978749i 0.872074 + 0.489375i \(0.162775\pi\)
−0.872074 + 0.489375i \(0.837225\pi\)
\(762\) 0 0
\(763\) −16.0000 −0.579239
\(764\) 48.0000 1.73658
\(765\) − 18.0000i − 0.650791i
\(766\) 0 0
\(767\) 0 0
\(768\) −32.0000 −1.15470
\(769\) 5.00000i 0.180305i 0.995928 + 0.0901523i \(0.0287354\pi\)
−0.995928 + 0.0901523i \(0.971265\pi\)
\(770\) 0 0
\(771\) −48.0000 −1.72868
\(772\) 44.0000i 1.58359i
\(773\) 30.0000i 1.07903i 0.841978 + 0.539513i \(0.181391\pi\)
−0.841978 + 0.539513i \(0.818609\pi\)
\(774\) 0 0
\(775\) − 20.0000i − 0.718421i
\(776\) 0 0
\(777\) 4.00000 0.143499
\(778\) 0 0
\(779\) −42.0000 −1.50481
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −36.0000 −1.28654
\(784\) −4.00000 −0.142857
\(785\) − 42.0000i − 1.49904i
\(786\) 0 0
\(787\) − 5.00000i − 0.178231i −0.996021 0.0891154i \(-0.971596\pi\)
0.996021 0.0891154i \(-0.0284040\pi\)
\(788\) 12.0000i 0.427482i
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) − 9.00000i − 0.320003i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) − 54.0000i − 1.91518i
\(796\) −4.00000 −0.141776
\(797\) −54.0000 −1.91278 −0.956389 0.292096i \(-0.905647\pi\)
−0.956389 + 0.292096i \(0.905647\pi\)
\(798\) 0 0
\(799\) − 18.0000i − 0.636794i
\(800\) 0 0
\(801\) − 15.0000i − 0.529999i
\(802\) 0 0
\(803\) 0 0
\(804\) − 56.0000i − 1.97497i
\(805\) 9.00000 0.317208
\(806\) 0 0
\(807\) 48.0000 1.68968
\(808\) 0 0
\(809\) −3.00000 −0.105474 −0.0527372 0.998608i \(-0.516795\pi\)
−0.0527372 + 0.998608i \(0.516795\pi\)
\(810\) 0 0
\(811\) 20.0000i 0.702295i 0.936320 + 0.351147i \(0.114208\pi\)
−0.936320 + 0.351147i \(0.885792\pi\)
\(812\) 18.0000i 0.631676i
\(813\) − 32.0000i − 1.12229i
\(814\) 0 0
\(815\) 48.0000 1.68137
\(816\) −48.0000 −1.68034
\(817\) − 7.00000i − 0.244899i
\(818\) 0 0
\(819\) 0 0
\(820\) −36.0000 −1.25717
\(821\) − 54.0000i − 1.88461i −0.334751 0.942306i \(-0.608652\pi\)
0.334751 0.942306i \(-0.391348\pi\)
\(822\) 0 0
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −6.00000 −0.208514
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 0 0
\(831\) −38.0000 −1.31821
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 45.0000 1.55729
\(836\) 0 0
\(837\) 20.0000i 0.691301i
\(838\) 0 0
\(839\) 12.0000i 0.414286i 0.978311 + 0.207143i \(0.0664165\pi\)
−0.978311 + 0.207143i \(0.933583\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 24.0000i 0.826604i
\(844\) 46.0000 1.58339
\(845\) 0 0
\(846\) 0 0
\(847\) − 11.0000i − 0.377964i
\(848\) −36.0000 −1.23625
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) 6.00000i 0.205677i
\(852\) 24.0000i 0.822226i
\(853\) − 17.0000i − 0.582069i −0.956713 0.291034i \(-0.906001\pi\)
0.956713 0.291034i \(-0.0939994\pi\)
\(854\) 0 0
\(855\) −21.0000 −0.718185
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) − 6.00000i − 0.204598i
\(861\) 12.0000 0.408959
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 18.0000i 0.612018i
\(866\) 0 0
\(867\) −38.0000 −1.29055
\(868\) 10.0000 0.339422
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) − 1.00000i − 0.0338449i
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 44.0000i 1.48662i
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) 0 0
\(879\) 18.0000i 0.607125i
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) − 16.0000i − 0.536623i
\(890\) 0 0
\(891\) 0 0
\(892\) − 2.00000i − 0.0669650i
\(893\) −21.0000 −0.702738
\(894\) 0 0
\(895\) 45.0000i 1.50418i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 45.0000i − 1.50083i
\(900\) −8.00000 −0.266667
\(901\) −54.0000 −1.79900
\(902\) 0 0
\(903\) 2.00000i 0.0665558i
\(904\) 0 0
\(905\) − 48.0000i − 1.59557i
\(906\) 0 0
\(907\) 37.0000 1.22856 0.614282 0.789086i \(-0.289446\pi\)
0.614282 + 0.789086i \(0.289446\pi\)
\(908\) 48.0000i 1.59294i
\(909\) 0 0
\(910\) 0 0
\(911\) −15.0000 −0.496972 −0.248486 0.968635i \(-0.579933\pi\)
−0.248486 + 0.968635i \(0.579933\pi\)
\(912\) 56.0000i 1.85435i
\(913\) 0 0
\(914\) 0 0
\(915\) − 60.0000i − 1.98354i
\(916\) − 28.0000i − 0.925146i
\(917\) 12.0000i 0.396275i
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 22.0000i 0.724925i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000i 0.263038i
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) − 3.00000i − 0.0984268i −0.998788 0.0492134i \(-0.984329\pi\)
0.998788 0.0492134i \(-0.0156714\pi\)
\(930\) 0 0
\(931\) 7.00000i 0.229416i
\(932\) 18.0000 0.589610
\(933\) −24.0000 −0.785725
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) −16.0000 −0.522140
\(940\) −18.0000 −0.587095
\(941\) 15.0000i 0.488986i 0.969651 + 0.244493i \(0.0786215\pi\)
−0.969651 + 0.244493i \(0.921378\pi\)
\(942\) 0 0
\(943\) 18.0000i 0.586161i
\(944\) 0 0
\(945\) −12.0000 −0.390360
\(946\) 0 0
\(947\) − 54.0000i − 1.75476i −0.479792 0.877382i \(-0.659288\pi\)
0.479792 0.877382i \(-0.340712\pi\)
\(948\) 4.00000 0.129914
\(949\) 0 0
\(950\) 0 0
\(951\) − 24.0000i − 0.778253i
\(952\) 0 0
\(953\) −21.0000 −0.680257 −0.340128 0.940379i \(-0.610471\pi\)
−0.340128 + 0.940379i \(0.610471\pi\)
\(954\) 0 0
\(955\) − 72.0000i − 2.32987i
\(956\) − 24.0000i − 0.776215i
\(957\) 0 0
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 48.0000i 1.54919i
\(961\) 6.00000 0.193548
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 2.00000i 0.0644157i
\(965\) 66.0000 2.12462
\(966\) 0 0
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) 0 0
\(969\) 84.0000i 2.69847i
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 20.0000 0.641500
\(973\) 4.00000i 0.128234i
\(974\) 0 0
\(975\) 0 0
\(976\) −40.0000 −1.28037
\(977\) 48.0000i 1.53566i 0.640656 + 0.767828i \(0.278662\pi\)
−0.640656 + 0.767828i \(0.721338\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 6.00000i 0.191663i
\(981\) − 16.0000i − 0.510841i
\(982\) 0 0
\(983\) − 57.0000i − 1.81802i −0.416777 0.909009i \(-0.636840\pi\)
0.416777 0.909009i \(-0.363160\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 0 0
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) −3.00000 −0.0953945
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) − 52.0000i − 1.65017i
\(994\) 0 0
\(995\) 6.00000i 0.190213i
\(996\) − 12.0000i − 0.380235i
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) 0 0
\(999\) − 8.00000i − 0.253109i
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 1183.2.c.a.337.1 2
13.5 odd 4 91.2.a.b.1.1 1
13.8 odd 4 1183.2.a.a.1.1 1
13.12 even 2 inner 1183.2.c.a.337.2 2
39.5 even 4 819.2.a.c.1.1 1
52.31 even 4 1456.2.a.k.1.1 1
65.44 odd 4 2275.2.a.d.1.1 1
91.5 even 12 637.2.e.b.508.1 2
91.18 odd 12 637.2.e.c.79.1 2
91.31 even 12 637.2.e.b.79.1 2
91.34 even 4 8281.2.a.h.1.1 1
91.44 odd 12 637.2.e.c.508.1 2
91.83 even 4 637.2.a.b.1.1 1
104.5 odd 4 5824.2.a.bd.1.1 1
104.83 even 4 5824.2.a.f.1.1 1
273.83 odd 4 5733.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.b.1.1 1 13.5 odd 4
637.2.a.b.1.1 1 91.83 even 4
637.2.e.b.79.1 2 91.31 even 12
637.2.e.b.508.1 2 91.5 even 12
637.2.e.c.79.1 2 91.18 odd 12
637.2.e.c.508.1 2 91.44 odd 12
819.2.a.c.1.1 1 39.5 even 4
1183.2.a.a.1.1 1 13.8 odd 4
1183.2.c.a.337.1 2 1.1 even 1 trivial
1183.2.c.a.337.2 2 13.12 even 2 inner
1456.2.a.k.1.1 1 52.31 even 4
2275.2.a.d.1.1 1 65.44 odd 4
5733.2.a.f.1.1 1 273.83 odd 4
5824.2.a.f.1.1 1 104.83 even 4
5824.2.a.bd.1.1 1 104.5 odd 4
8281.2.a.h.1.1 1 91.34 even 4