Properties

Label 3564.2.i.j.1189.1
Level $3564$
Weight $2$
Character 3564.1189
Analytic conductor $28.459$
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] = [3564,2,Mod(1189,3564)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3564, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3564.1189");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3564 = 2^{2} \cdot 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3564.i (of order \(3\), degree \(2\), 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: \(28.4586832804\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1189.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3564.1189
Dual form 3564.2.i.j.2377.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 + 2.59808i) q^{5} +(-1.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q+(1.50000 + 2.59808i) q^{5} +(-1.00000 + 1.73205i) q^{7} +(0.500000 - 0.866025i) q^{11} +(2.00000 + 3.46410i) q^{13} +6.00000 q^{17} +8.00000 q^{19} +(1.50000 + 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(-2.50000 - 4.33013i) q^{31} -6.00000 q^{35} -1.00000 q^{37} +(5.00000 - 8.66025i) q^{43} +(1.50000 + 2.59808i) q^{49} -6.00000 q^{53} +3.00000 q^{55} +(-1.50000 - 2.59808i) q^{59} +(2.00000 - 3.46410i) q^{61} +(-6.00000 + 10.3923i) q^{65} +(0.500000 + 0.866025i) q^{67} +15.0000 q^{71} -4.00000 q^{73} +(1.00000 + 1.73205i) q^{77} +(-1.00000 + 1.73205i) q^{79} +(-3.00000 + 5.19615i) q^{83} +(9.00000 + 15.5885i) q^{85} -9.00000 q^{89} -8.00000 q^{91} +(12.0000 + 20.7846i) q^{95} +(3.50000 - 6.06218i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{5} - 2 q^{7} + q^{11} + 4 q^{13} + 12 q^{17} + 16 q^{19} + 3 q^{23} - 4 q^{25} - 5 q^{31} - 12 q^{35} - 2 q^{37} + 10 q^{43} + 3 q^{49} - 12 q^{53} + 6 q^{55} - 3 q^{59} + 4 q^{61} - 12 q^{65} + q^{67} + 30 q^{71} - 8 q^{73} + 2 q^{77} - 2 q^{79} - 6 q^{83} + 18 q^{85} - 18 q^{89} - 16 q^{91} + 24 q^{95} + 7 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3564\mathbb{Z}\right)^\times\).

\(n\) \(1541\) \(1783\) \(2917\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.50000 + 2.59808i 0.670820 + 1.16190i 0.977672 + 0.210138i \(0.0673912\pi\)
−0.306851 + 0.951757i \(0.599275\pi\)
\(6\) 0 0
\(7\) −1.00000 + 1.73205i −0.377964 + 0.654654i −0.990766 0.135583i \(-0.956709\pi\)
0.612801 + 0.790237i \(0.290043\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.500000 0.866025i 0.150756 0.261116i
\(12\) 0 0
\(13\) 2.00000 + 3.46410i 0.554700 + 0.960769i 0.997927 + 0.0643593i \(0.0205004\pi\)
−0.443227 + 0.896410i \(0.646166\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.50000 + 2.59808i 0.312772 + 0.541736i 0.978961 0.204046i \(-0.0654092\pi\)
−0.666190 + 0.745782i \(0.732076\pi\)
\(24\) 0 0
\(25\) −2.00000 + 3.46410i −0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(30\) 0 0
\(31\) −2.50000 4.33013i −0.449013 0.777714i 0.549309 0.835619i \(-0.314891\pi\)
−0.998322 + 0.0579057i \(0.981558\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.00000 −1.01419
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(42\) 0 0
\(43\) 5.00000 8.66025i 0.762493 1.32068i −0.179069 0.983836i \(-0.557309\pi\)
0.941562 0.336840i \(-0.109358\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 1.50000 + 2.59808i 0.214286 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.50000 2.59808i −0.195283 0.338241i 0.751710 0.659494i \(-0.229229\pi\)
−0.946993 + 0.321253i \(0.895896\pi\)
\(60\) 0 0
\(61\) 2.00000 3.46410i 0.256074 0.443533i −0.709113 0.705095i \(-0.750904\pi\)
0.965187 + 0.261562i \(0.0842377\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.00000 + 10.3923i −0.744208 + 1.28901i
\(66\) 0 0
\(67\) 0.500000 + 0.866025i 0.0610847 + 0.105802i 0.894951 0.446165i \(-0.147211\pi\)
−0.833866 + 0.551967i \(0.813877\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 15.0000 1.78017 0.890086 0.455792i \(-0.150644\pi\)
0.890086 + 0.455792i \(0.150644\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.00000 + 1.73205i 0.113961 + 0.197386i
\(78\) 0 0
\(79\) −1.00000 + 1.73205i −0.112509 + 0.194871i −0.916781 0.399390i \(-0.869222\pi\)
0.804272 + 0.594261i \(0.202555\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.00000 + 5.19615i −0.329293 + 0.570352i −0.982372 0.186938i \(-0.940144\pi\)
0.653079 + 0.757290i \(0.273477\pi\)
\(84\) 0 0
\(85\) 9.00000 + 15.5885i 0.976187 + 1.69081i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 12.0000 + 20.7846i 1.23117 + 2.13246i
\(96\) 0 0
\(97\) 3.50000 6.06218i 0.355371 0.615521i −0.631810 0.775123i \(-0.717688\pi\)
0.987181 + 0.159602i \(0.0510211\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.00000 + 15.5885i −0.895533 + 1.55111i −0.0623905 + 0.998052i \(0.519872\pi\)
−0.833143 + 0.553058i \(0.813461\pi\)
\(102\) 0 0
\(103\) −4.00000 6.92820i −0.394132 0.682656i 0.598858 0.800855i \(-0.295621\pi\)
−0.992990 + 0.118199i \(0.962288\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.50000 + 12.9904i 0.705541 + 1.22203i 0.966496 + 0.256681i \(0.0826291\pi\)
−0.260955 + 0.965351i \(0.584038\pi\)
\(114\) 0 0
\(115\) −4.50000 + 7.79423i −0.419627 + 0.726816i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.00000 + 10.3923i −0.550019 + 0.952661i
\(120\) 0 0
\(121\) −0.500000 0.866025i −0.0454545 0.0787296i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.00000 + 5.19615i 0.262111 + 0.453990i 0.966803 0.255524i \(-0.0822479\pi\)
−0.704692 + 0.709514i \(0.748915\pi\)
\(132\) 0 0
\(133\) −8.00000 + 13.8564i −0.693688 + 1.20150i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.50000 + 7.79423i −0.384461 + 0.665906i −0.991694 0.128618i \(-0.958946\pi\)
0.607233 + 0.794524i \(0.292279\pi\)
\(138\) 0 0
\(139\) −7.00000 12.1244i −0.593732 1.02837i −0.993724 0.111856i \(-0.964321\pi\)
0.399992 0.916519i \(-0.369013\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.00000 5.19615i −0.245770 0.425685i 0.716578 0.697507i \(-0.245707\pi\)
−0.962348 + 0.271821i \(0.912374\pi\)
\(150\) 0 0
\(151\) 5.00000 8.66025i 0.406894 0.704761i −0.587646 0.809118i \(-0.699945\pi\)
0.994540 + 0.104357i \(0.0332784\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 7.50000 12.9904i 0.602414 1.04341i
\(156\) 0 0
\(157\) −2.50000 4.33013i −0.199522 0.345582i 0.748852 0.662738i \(-0.230606\pi\)
−0.948373 + 0.317156i \(0.897272\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.00000 + 10.3923i 0.464294 + 0.804181i 0.999169 0.0407502i \(-0.0129748\pi\)
−0.534875 + 0.844931i \(0.679641\pi\)
\(168\) 0 0
\(169\) −1.50000 + 2.59808i −0.115385 + 0.199852i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.00000 + 15.5885i −0.684257 + 1.18517i 0.289412 + 0.957205i \(0.406540\pi\)
−0.973670 + 0.227964i \(0.926793\pi\)
\(174\) 0 0
\(175\) −4.00000 6.92820i −0.302372 0.523723i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) −13.0000 −0.966282 −0.483141 0.875542i \(-0.660504\pi\)
−0.483141 + 0.875542i \(0.660504\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.50000 2.59808i −0.110282 0.191014i
\(186\) 0 0
\(187\) 3.00000 5.19615i 0.219382 0.379980i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.5000 18.1865i 0.759753 1.31593i −0.183223 0.983071i \(-0.558653\pi\)
0.942976 0.332860i \(-0.108014\pi\)
\(192\) 0 0
\(193\) −10.0000 17.3205i −0.719816 1.24676i −0.961073 0.276296i \(-0.910893\pi\)
0.241257 0.970461i \(-0.422440\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.00000 6.92820i 0.276686 0.479234i
\(210\) 0 0
\(211\) −10.0000 17.3205i −0.688428 1.19239i −0.972346 0.233544i \(-0.924968\pi\)
0.283918 0.958849i \(-0.408366\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 30.0000 2.04598
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 + 20.7846i 0.807207 + 1.39812i
\(222\) 0 0
\(223\) −8.50000 + 14.7224i −0.569202 + 0.985887i 0.427443 + 0.904042i \(0.359414\pi\)
−0.996645 + 0.0818447i \(0.973919\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.00000 + 5.19615i −0.199117 + 0.344881i −0.948242 0.317547i \(-0.897141\pi\)
0.749125 + 0.662428i \(0.230474\pi\)
\(228\) 0 0
\(229\) 6.50000 + 11.2583i 0.429532 + 0.743971i 0.996832 0.0795401i \(-0.0253452\pi\)
−0.567300 + 0.823511i \(0.692012\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3.00000 5.19615i −0.194054 0.336111i 0.752536 0.658551i \(-0.228830\pi\)
−0.946590 + 0.322440i \(0.895497\pi\)
\(240\) 0 0
\(241\) −4.00000 + 6.92820i −0.257663 + 0.446285i −0.965615 0.259975i \(-0.916286\pi\)
0.707953 + 0.706260i \(0.249619\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.50000 + 7.79423i −0.287494 + 0.497955i
\(246\) 0 0
\(247\) 16.0000 + 27.7128i 1.01806 + 1.76332i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −9.00000 −0.568075 −0.284037 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 0 0
\(253\) 3.00000 0.188608
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.00000 + 15.5885i 0.561405 + 0.972381i 0.997374 + 0.0724199i \(0.0230722\pi\)
−0.435970 + 0.899961i \(0.643595\pi\)
\(258\) 0 0
\(259\) 1.00000 1.73205i 0.0621370 0.107624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −9.00000 + 15.5885i −0.554964 + 0.961225i 0.442943 + 0.896550i \(0.353935\pi\)
−0.997906 + 0.0646755i \(0.979399\pi\)
\(264\) 0 0
\(265\) −9.00000 15.5885i −0.552866 0.957591i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.00000 + 3.46410i 0.120605 + 0.208893i
\(276\) 0 0
\(277\) 5.00000 8.66025i 0.300421 0.520344i −0.675810 0.737075i \(-0.736206\pi\)
0.976231 + 0.216731i \(0.0695395\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.00000 15.5885i 0.536895 0.929929i −0.462174 0.886789i \(-0.652930\pi\)
0.999069 0.0431402i \(-0.0137362\pi\)
\(282\) 0 0
\(283\) 2.00000 + 3.46410i 0.118888 + 0.205919i 0.919327 0.393494i \(-0.128734\pi\)
−0.800439 + 0.599414i \(0.795400\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(294\) 0 0
\(295\) 4.50000 7.79423i 0.262000 0.453798i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.00000 + 10.3923i −0.346989 + 0.601003i
\(300\) 0 0
\(301\) 10.0000 + 17.3205i 0.576390 + 0.998337i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −6.00000 10.3923i −0.340229 0.589294i 0.644246 0.764818i \(-0.277171\pi\)
−0.984475 + 0.175525i \(0.943838\pi\)
\(312\) 0 0
\(313\) 0.500000 0.866025i 0.0282617 0.0489506i −0.851549 0.524276i \(-0.824336\pi\)
0.879810 + 0.475325i \(0.157669\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −16.5000 + 28.5788i −0.926732 + 1.60515i −0.137981 + 0.990435i \(0.544061\pi\)
−0.788751 + 0.614713i \(0.789272\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 48.0000 2.67079
\(324\) 0 0
\(325\) −16.0000 −0.887520
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3.50000 6.06218i 0.192377 0.333207i −0.753660 0.657264i \(-0.771714\pi\)
0.946038 + 0.324057i \(0.105047\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.50000 + 2.59808i −0.0819538 + 0.141948i
\(336\) 0 0
\(337\) −1.00000 1.73205i −0.0544735 0.0943508i 0.837503 0.546433i \(-0.184015\pi\)
−0.891976 + 0.452082i \(0.850681\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −5.00000 −0.270765
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.00000 10.3923i −0.322097 0.557888i 0.658824 0.752297i \(-0.271054\pi\)
−0.980921 + 0.194409i \(0.937721\pi\)
\(348\) 0 0
\(349\) −7.00000 + 12.1244i −0.374701 + 0.649002i −0.990282 0.139072i \(-0.955588\pi\)
0.615581 + 0.788074i \(0.288921\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.5000 18.1865i 0.558859 0.967972i −0.438733 0.898617i \(-0.644573\pi\)
0.997592 0.0693543i \(-0.0220939\pi\)
\(354\) 0 0
\(355\) 22.5000 + 38.9711i 1.19418 + 2.06837i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.00000 10.3923i −0.314054 0.543958i
\(366\) 0 0
\(367\) 9.50000 16.4545i 0.495896 0.858917i −0.504093 0.863649i \(-0.668173\pi\)
0.999989 + 0.00473247i \(0.00150640\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.00000 10.3923i 0.311504 0.539542i
\(372\) 0 0
\(373\) 5.00000 + 8.66025i 0.258890 + 0.448411i 0.965945 0.258748i \(-0.0833099\pi\)
−0.707055 + 0.707159i \(0.749977\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 29.0000 1.48963 0.744815 0.667271i \(-0.232538\pi\)
0.744815 + 0.667271i \(0.232538\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 13.5000 + 23.3827i 0.689818 + 1.19480i 0.971897 + 0.235408i \(0.0756427\pi\)
−0.282079 + 0.959391i \(0.591024\pi\)
\(384\) 0 0
\(385\) −3.00000 + 5.19615i −0.152894 + 0.264820i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 13.5000 23.3827i 0.684477 1.18555i −0.289124 0.957292i \(-0.593364\pi\)
0.973601 0.228257i \(-0.0733028\pi\)
\(390\) 0 0
\(391\) 9.00000 + 15.5885i 0.455150 + 0.788342i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −6.00000 −0.301893
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.00000 15.5885i −0.449439 0.778450i 0.548911 0.835881i \(-0.315043\pi\)
−0.998350 + 0.0574304i \(0.981709\pi\)
\(402\) 0 0
\(403\) 10.0000 17.3205i 0.498135 0.862796i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.500000 + 0.866025i −0.0247841 + 0.0429273i
\(408\) 0 0
\(409\) −1.00000 1.73205i −0.0494468 0.0856444i 0.840243 0.542211i \(-0.182412\pi\)
−0.889689 + 0.456566i \(0.849079\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) −18.0000 −0.883585
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6.00000 10.3923i −0.293119 0.507697i 0.681426 0.731887i \(-0.261360\pi\)
−0.974546 + 0.224189i \(0.928027\pi\)
\(420\) 0 0
\(421\) 5.00000 8.66025i 0.243685 0.422075i −0.718076 0.695965i \(-0.754977\pi\)
0.961761 + 0.273890i \(0.0883103\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −12.0000 + 20.7846i −0.582086 + 1.00820i
\(426\) 0 0
\(427\) 4.00000 + 6.92820i 0.193574 + 0.335279i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 29.0000 1.39365 0.696826 0.717241i \(-0.254595\pi\)
0.696826 + 0.717241i \(0.254595\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.0000 + 20.7846i 0.574038 + 0.994263i
\(438\) 0 0
\(439\) −4.00000 + 6.92820i −0.190910 + 0.330665i −0.945552 0.325471i \(-0.894477\pi\)
0.754642 + 0.656136i \(0.227810\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.5000 18.1865i 0.498870 0.864068i −0.501129 0.865373i \(-0.667082\pi\)
0.999999 + 0.00130426i \(0.000415158\pi\)
\(444\) 0 0
\(445\) −13.5000 23.3827i −0.639961 1.10845i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3.00000 0.141579 0.0707894 0.997491i \(-0.477448\pi\)
0.0707894 + 0.997491i \(0.477448\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −12.0000 20.7846i −0.562569 0.974398i
\(456\) 0 0
\(457\) 14.0000 24.2487i 0.654892 1.13431i −0.327028 0.945015i \(-0.606047\pi\)
0.981921 0.189292i \(-0.0606194\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.00000 + 10.3923i −0.279448 + 0.484018i −0.971248 0.238071i \(-0.923485\pi\)
0.691800 + 0.722089i \(0.256818\pi\)
\(462\) 0 0
\(463\) −11.5000 19.9186i −0.534450 0.925695i −0.999190 0.0402476i \(-0.987185\pi\)
0.464739 0.885448i \(-0.346148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.00000 0.138823 0.0694117 0.997588i \(-0.477888\pi\)
0.0694117 + 0.997588i \(0.477888\pi\)
\(468\) 0 0
\(469\) −2.00000 −0.0923514
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.00000 8.66025i −0.229900 0.398199i
\(474\) 0 0
\(475\) −16.0000 + 27.7128i −0.734130 + 1.27155i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6.00000 10.3923i 0.274147 0.474837i −0.695773 0.718262i \(-0.744938\pi\)
0.969920 + 0.243426i \(0.0782712\pi\)
\(480\) 0 0
\(481\) −2.00000 3.46410i −0.0911922 0.157949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 21.0000 0.953561
\(486\) 0 0
\(487\) 29.0000 1.31412 0.657058 0.753840i \(-0.271801\pi\)
0.657058 + 0.753840i \(0.271801\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −12.0000 20.7846i −0.541552 0.937996i −0.998815 0.0486647i \(-0.984503\pi\)
0.457263 0.889332i \(-0.348830\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −15.0000 + 25.9808i −0.672842 + 1.16540i
\(498\) 0 0
\(499\) 2.00000 + 3.46410i 0.0895323 + 0.155074i 0.907314 0.420455i \(-0.138129\pi\)
−0.817781 + 0.575529i \(0.804796\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −30.0000 −1.33763 −0.668817 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) 0 0
\(505\) −54.0000 −2.40297
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 10.5000 + 18.1865i 0.465404 + 0.806104i 0.999220 0.0394971i \(-0.0125756\pi\)
−0.533815 + 0.845601i \(0.679242\pi\)
\(510\) 0 0
\(511\) 4.00000 6.92820i 0.176950 0.306486i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 12.0000 20.7846i 0.528783 0.915879i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −27.0000 −1.18289 −0.591446 0.806345i \(-0.701443\pi\)
−0.591446 + 0.806345i \(0.701443\pi\)
\(522\) 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
\(527\) −15.0000 25.9808i −0.653410 1.13174i
\(528\) 0 0
\(529\) 7.00000 12.1244i 0.304348 0.527146i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 9.00000 + 15.5885i 0.389104 + 0.673948i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) −16.0000 −0.687894 −0.343947 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.00000 + 5.19615i 0.128506 + 0.222579i
\(546\) 0 0
\(547\) −4.00000 + 6.92820i −0.171028 + 0.296229i −0.938779 0.344519i \(-0.888042\pi\)
0.767752 + 0.640747i \(0.221375\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −2.00000 3.46410i −0.0850487 0.147309i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 40.0000 1.69182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18.0000 + 31.1769i 0.758610 + 1.31395i 0.943560 + 0.331202i \(0.107454\pi\)
−0.184950 + 0.982748i \(0.559212\pi\)
\(564\) 0 0
\(565\) −22.5000 + 38.9711i −0.946582 + 1.63953i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) −22.0000 38.1051i −0.920671 1.59465i −0.798379 0.602155i \(-0.794309\pi\)
−0.122292 0.992494i \(-0.539025\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) 17.0000 0.707719 0.353860 0.935299i \(-0.384869\pi\)
0.353860 + 0.935299i \(0.384869\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.00000 10.3923i −0.248922 0.431145i
\(582\) 0 0
\(583\) −3.00000 + 5.19615i −0.124247 + 0.215203i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.00000 10.3923i 0.247647 0.428936i −0.715226 0.698893i \(-0.753676\pi\)
0.962872 + 0.269957i \(0.0870095\pi\)
\(588\) 0 0
\(589\) −20.0000 34.6410i −0.824086 1.42736i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −36.0000 −1.47834 −0.739171 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) 0 0
\(595\) −36.0000 −1.47586
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) −13.0000 + 22.5167i −0.530281 + 0.918474i 0.469095 + 0.883148i \(0.344580\pi\)
−0.999376 + 0.0353259i \(0.988753\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.50000 2.59808i 0.0609837 0.105627i
\(606\) 0 0
\(607\) −7.00000 12.1244i −0.284121 0.492112i 0.688274 0.725450i \(-0.258368\pi\)
−0.972396 + 0.233338i \(0.925035\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 32.0000 1.29247 0.646234 0.763139i \(-0.276343\pi\)
0.646234 + 0.763139i \(0.276343\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −9.00000 15.5885i −0.362326 0.627568i 0.626017 0.779809i \(-0.284684\pi\)
−0.988343 + 0.152242i \(0.951351\pi\)
\(618\) 0 0
\(619\) −8.50000 + 14.7224i −0.341644 + 0.591744i −0.984738 0.174042i \(-0.944317\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9.00000 15.5885i 0.360577 0.624538i
\(624\) 0 0
\(625\) 14.5000 + 25.1147i 0.580000 + 1.00459i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −43.0000 −1.71180 −0.855901 0.517139i \(-0.826997\pi\)
−0.855901 + 0.517139i \(0.826997\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −24.0000 41.5692i −0.952411 1.64962i
\(636\) 0 0
\(637\) −6.00000 + 10.3923i −0.237729 + 0.411758i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −19.5000 + 33.7750i −0.770204 + 1.33403i 0.167247 + 0.985915i \(0.446512\pi\)
−0.937451 + 0.348117i \(0.886821\pi\)
\(642\) 0 0
\(643\) 6.50000 + 11.2583i 0.256335 + 0.443985i 0.965257 0.261301i \(-0.0841516\pi\)
−0.708922 + 0.705287i \(0.750818\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.00000 0.117942 0.0589711 0.998260i \(-0.481218\pi\)
0.0589711 + 0.998260i \(0.481218\pi\)
\(648\) 0 0
\(649\) −3.00000 −0.117760
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.50000 2.59808i −0.0586995 0.101671i 0.835182 0.549973i \(-0.185362\pi\)
−0.893882 + 0.448303i \(0.852029\pi\)
\(654\) 0 0
\(655\) −9.00000 + 15.5885i −0.351659 + 0.609091i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 21.0000 36.3731i 0.818044 1.41689i −0.0890776 0.996025i \(-0.528392\pi\)
0.907122 0.420869i \(-0.138275\pi\)
\(660\) 0 0
\(661\) −8.50000 14.7224i −0.330612 0.572636i 0.652020 0.758202i \(-0.273922\pi\)
−0.982632 + 0.185565i \(0.940588\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −48.0000 −1.86136
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.00000 3.46410i −0.0772091 0.133730i
\(672\) 0 0
\(673\) 17.0000 29.4449i 0.655302 1.13502i −0.326516 0.945192i \(-0.605875\pi\)
0.981818 0.189824i \(-0.0607919\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 21.0000 36.3731i 0.807096 1.39793i −0.107772 0.994176i \(-0.534372\pi\)
0.914867 0.403755i \(-0.132295\pi\)
\(678\) 0 0
\(679\) 7.00000 + 12.1244i 0.268635 + 0.465290i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 48.0000 1.83667 0.918334 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) 0 0
\(685\) −27.0000 −1.03162
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.0000 20.7846i −0.457164 0.791831i
\(690\) 0 0
\(691\) 0.500000 0.866025i 0.0190209 0.0329452i −0.856358 0.516382i \(-0.827278\pi\)
0.875379 + 0.483437i \(0.160612\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 21.0000 36.3731i 0.796575 1.37971i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −18.0000 31.1769i −0.676960 1.17253i
\(708\) 0 0
\(709\) 18.5000 32.0429i 0.694782 1.20340i −0.275472 0.961309i \(-0.588834\pi\)
0.970254 0.242089i \(-0.0778325\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.50000 12.9904i 0.280877 0.486494i
\(714\) 0 0
\(715\) 6.00000 + 10.3923i 0.224387 + 0.388650i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 45.0000 1.67822 0.839108 0.543964i \(-0.183077\pi\)
0.839108 + 0.543964i \(0.183077\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −8.50000 + 14.7224i −0.315248 + 0.546025i −0.979490 0.201492i \(-0.935421\pi\)
0.664243 + 0.747517i \(0.268754\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 30.0000 51.9615i 1.10959 1.92187i
\(732\) 0 0
\(733\) 2.00000 + 3.46410i 0.0738717 + 0.127950i 0.900595 0.434659i \(-0.143131\pi\)
−0.826723 + 0.562609i \(0.809798\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.00000 0.0368355
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −6.00000 10.3923i −0.220119 0.381257i 0.734725 0.678365i \(-0.237311\pi\)
−0.954844 + 0.297108i \(0.903978\pi\)
\(744\) 0 0
\(745\) 9.00000 15.5885i 0.329734 0.571117i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −6.00000 + 10.3923i −0.219235 + 0.379727i
\(750\) 0 0
\(751\) −17.5000 30.3109i −0.638584 1.10606i −0.985744 0.168254i \(-0.946187\pi\)
0.347160 0.937806i \(-0.387146\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 30.0000 1.09181
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 31.1769i −0.652499 1.13016i −0.982514 0.186187i \(-0.940387\pi\)
0.330015 0.943976i \(-0.392946\pi\)
\(762\) 0 0
\(763\) −2.00000 + 3.46410i −0.0724049 + 0.125409i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.00000 10.3923i 0.216647 0.375244i
\(768\) 0 0
\(769\) −22.0000 38.1051i −0.793340 1.37411i −0.923888 0.382664i \(-0.875007\pi\)
0.130547 0.991442i \(-0.458327\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 0 0
\(775\) 20.0000 0.718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 7.50000 12.9904i 0.268371 0.464832i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7.50000 12.9904i 0.267686 0.463647i
\(786\) 0 0
\(787\) −16.0000 27.7128i −0.570338 0.987855i −0.996531 0.0832226i \(-0.973479\pi\)
0.426193 0.904632i \(-0.359855\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −30.0000 −1.06668
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.50000 7.79423i −0.159398 0.276086i 0.775254 0.631650i \(-0.217622\pi\)
−0.934652 + 0.355564i \(0.884289\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.00000 + 3.46410i −0.0705785 + 0.122245i
\(804\) 0 0
\(805\) −9.00000 15.5885i −0.317208 0.549421i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −24.0000 −0.843795 −0.421898 0.906644i \(-0.638636\pi\)
−0.421898 + 0.906644i \(0.638636\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −6.00000 10.3923i −0.210171 0.364027i
\(816\) 0 0
\(817\) 40.0000 69.2820i 1.39942 2.42387i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −15.0000 + 25.9808i −0.523504 + 0.906735i 0.476122 + 0.879379i \(0.342042\pi\)
−0.999626 + 0.0273557i \(0.991291\pi\)
\(822\) 0 0
\(823\) 21.5000 + 37.2391i 0.749443 + 1.29807i 0.948090 + 0.318002i \(0.103012\pi\)
−0.198647 + 0.980071i \(0.563655\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −19.0000 −0.659897 −0.329949 0.943999i \(-0.607031\pi\)
−0.329949 + 0.943999i \(0.607031\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 9.00000 + 15.5885i 0.311832 + 0.540108i
\(834\) 0 0
\(835\) −18.0000 + 31.1769i −0.622916 + 1.07892i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 19.5000 33.7750i 0.673215 1.16604i −0.303773 0.952745i \(-0.598246\pi\)
0.976987 0.213298i \(-0.0684204\pi\)
\(840\) 0 0
\(841\) 14.5000 + 25.1147i 0.500000 + 0.866025i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1.50000 2.59808i −0.0514193 0.0890609i
\(852\) 0 0
\(853\) −19.0000 + 32.9090i −0.650548 + 1.12678i 0.332443 + 0.943123i \(0.392127\pi\)
−0.982990 + 0.183658i \(0.941206\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −12.0000 + 20.7846i −0.409912 + 0.709989i −0.994880 0.101068i \(-0.967774\pi\)
0.584967 + 0.811057i \(0.301107\pi\)
\(858\) 0 0
\(859\) 12.5000 + 21.6506i 0.426494 + 0.738710i 0.996559 0.0828900i \(-0.0264150\pi\)
−0.570064 + 0.821600i \(0.693082\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) −54.0000 −1.83606
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.00000 + 1.73205i 0.0339227 + 0.0587558i
\(870\) 0 0
\(871\) −2.00000 + 3.46410i −0.0677674 + 0.117377i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.00000 + 5.19615i −0.101419 + 0.175662i
\(876\) 0 0
\(877\) 26.0000 + 45.0333i 0.877958 + 1.52067i 0.853578 + 0.520964i \(0.174428\pi\)
0.0243792 + 0.999703i \(0.492239\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −27.0000 −0.909653 −0.454827 0.890580i \(-0.650299\pi\)
−0.454827 + 0.890580i \(0.650299\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −15.0000 25.9808i −0.503651 0.872349i −0.999991 0.00422062i \(-0.998657\pi\)
0.496340 0.868128i \(-0.334677\pi\)
\(888\) 0 0
\(889\) 16.0000 27.7128i 0.536623 0.929458i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −13.5000 23.3827i −0.451255 0.781597i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −19.5000 33.7750i −0.648202 1.12272i
\(906\) 0 0
\(907\) 2.00000 3.46410i 0.0664089 0.115024i −0.830909 0.556408i \(-0.812179\pi\)
0.897318 + 0.441384i \(0.145512\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6.00000 + 10.3923i −0.198789 + 0.344312i −0.948136 0.317865i \(-0.897034\pi\)
0.749347 + 0.662177i \(0.230367\pi\)
\(912\) 0 0
\(913\) 3.00000 + 5.19615i 0.0992855 + 0.171968i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 30.0000 + 51.9615i 0.987462 + 1.71033i
\(924\) 0 0
\(925\) 2.00000 3.46410i 0.0657596 0.113899i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −9.00000 + 15.5885i −0.295280 + 0.511441i −0.975050 0.221985i \(-0.928746\pi\)
0.679770 + 0.733426i \(0.262080\pi\)
\(930\) 0 0
\(931\) 12.0000 + 20.7846i 0.393284 + 0.681188i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 18.0000 0.588663
\(936\) 0 0
\(937\) 32.0000 1.04539 0.522697 0.852518i \(-0.324926\pi\)
0.522697 + 0.852518i \(0.324926\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 15.0000 + 25.9808i 0.488986 + 0.846949i 0.999920 0.0126715i \(-0.00403357\pi\)
−0.510934 + 0.859620i \(0.670700\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −13.5000 + 23.3827i −0.438691 + 0.759835i −0.997589 0.0694014i \(-0.977891\pi\)
0.558898 + 0.829237i \(0.311224\pi\)
\(948\) 0 0
\(949\) −8.00000 13.8564i −0.259691 0.449798i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 0 0
\(955\) 63.0000 2.03863
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −9.00000 15.5885i −0.290625 0.503378i
\(960\) 0 0
\(961\) 3.00000 5.19615i 0.0967742 0.167618i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 30.0000 51.9615i 0.965734 1.67270i
\(966\) 0 0
\(967\) 8.00000 + 13.8564i 0.257263 + 0.445592i 0.965508 0.260375i \(-0.0838461\pi\)
−0.708245 + 0.705967i \(0.750513\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) 28.0000 0.897639
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −22.5000 38.9711i −0.719839 1.24680i −0.961063 0.276328i \(-0.910882\pi\)
0.241225 0.970469i \(-0.422451\pi\)
\(978\) 0 0
\(979\) −4.50000 + 7.79423i −0.143821 + 0.249105i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −22.5000 + 38.9711i −0.717639 + 1.24299i 0.244294 + 0.969701i \(0.421444\pi\)
−0.961933 + 0.273285i \(0.911890\pi\)
\(984\) 0 0
\(985\) 9.00000 + 15.5885i 0.286764 + 0.496690i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 30.0000 0.953945
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 12.0000 + 20.7846i 0.380426 + 0.658916i
\(996\) 0 0
\(997\) 5.00000 8.66025i 0.158352 0.274273i −0.775923 0.630828i \(-0.782715\pi\)
0.934274 + 0.356555i \(0.116049\pi\)
\(998\) 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 3564.2.i.j.1189.1 2
3.2 odd 2 3564.2.i.a.1189.1 2
9.2 odd 6 396.2.a.c.1.1 1
9.4 even 3 inner 3564.2.i.j.2377.1 2
9.5 odd 6 3564.2.i.a.2377.1 2
9.7 even 3 44.2.a.a.1.1 1
36.7 odd 6 176.2.a.a.1.1 1
36.11 even 6 1584.2.a.p.1.1 1
45.2 even 12 9900.2.c.g.5149.2 2
45.7 odd 12 1100.2.b.c.749.1 2
45.29 odd 6 9900.2.a.h.1.1 1
45.34 even 6 1100.2.a.b.1.1 1
45.38 even 12 9900.2.c.g.5149.1 2
45.43 odd 12 1100.2.b.c.749.2 2
63.16 even 3 2156.2.i.b.1145.1 2
63.25 even 3 2156.2.i.b.177.1 2
63.34 odd 6 2156.2.a.a.1.1 1
63.52 odd 6 2156.2.i.c.177.1 2
63.61 odd 6 2156.2.i.c.1145.1 2
72.11 even 6 6336.2.a.i.1.1 1
72.29 odd 6 6336.2.a.j.1.1 1
72.43 odd 6 704.2.a.i.1.1 1
72.61 even 6 704.2.a.f.1.1 1
99.7 odd 30 484.2.e.b.269.1 4
99.16 even 15 484.2.e.a.245.1 4
99.25 even 15 484.2.e.a.9.1 4
99.43 odd 6 484.2.a.a.1.1 1
99.52 odd 30 484.2.e.b.9.1 4
99.61 odd 30 484.2.e.b.245.1 4
99.65 even 6 4356.2.a.j.1.1 1
99.70 even 15 484.2.e.a.269.1 4
99.79 odd 30 484.2.e.b.81.1 4
99.97 even 15 484.2.e.a.81.1 4
117.25 even 6 7436.2.a.d.1.1 1
144.43 odd 12 2816.2.c.k.1409.2 2
144.61 even 12 2816.2.c.e.1409.2 2
144.115 odd 12 2816.2.c.k.1409.1 2
144.133 even 12 2816.2.c.e.1409.1 2
180.7 even 12 4400.2.b.k.4049.2 2
180.43 even 12 4400.2.b.k.4049.1 2
180.79 odd 6 4400.2.a.v.1.1 1
252.223 even 6 8624.2.a.w.1.1 1
396.43 even 6 1936.2.a.c.1.1 1
792.43 even 6 7744.2.a.bc.1.1 1
792.637 odd 6 7744.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
44.2.a.a.1.1 1 9.7 even 3
176.2.a.a.1.1 1 36.7 odd 6
396.2.a.c.1.1 1 9.2 odd 6
484.2.a.a.1.1 1 99.43 odd 6
484.2.e.a.9.1 4 99.25 even 15
484.2.e.a.81.1 4 99.97 even 15
484.2.e.a.245.1 4 99.16 even 15
484.2.e.a.269.1 4 99.70 even 15
484.2.e.b.9.1 4 99.52 odd 30
484.2.e.b.81.1 4 99.79 odd 30
484.2.e.b.245.1 4 99.61 odd 30
484.2.e.b.269.1 4 99.7 odd 30
704.2.a.f.1.1 1 72.61 even 6
704.2.a.i.1.1 1 72.43 odd 6
1100.2.a.b.1.1 1 45.34 even 6
1100.2.b.c.749.1 2 45.7 odd 12
1100.2.b.c.749.2 2 45.43 odd 12
1584.2.a.p.1.1 1 36.11 even 6
1936.2.a.c.1.1 1 396.43 even 6
2156.2.a.a.1.1 1 63.34 odd 6
2156.2.i.b.177.1 2 63.25 even 3
2156.2.i.b.1145.1 2 63.16 even 3
2156.2.i.c.177.1 2 63.52 odd 6
2156.2.i.c.1145.1 2 63.61 odd 6
2816.2.c.e.1409.1 2 144.133 even 12
2816.2.c.e.1409.2 2 144.61 even 12
2816.2.c.k.1409.1 2 144.115 odd 12
2816.2.c.k.1409.2 2 144.43 odd 12
3564.2.i.a.1189.1 2 3.2 odd 2
3564.2.i.a.2377.1 2 9.5 odd 6
3564.2.i.j.1189.1 2 1.1 even 1 trivial
3564.2.i.j.2377.1 2 9.4 even 3 inner
4356.2.a.j.1.1 1 99.65 even 6
4400.2.a.v.1.1 1 180.79 odd 6
4400.2.b.k.4049.1 2 180.43 even 12
4400.2.b.k.4049.2 2 180.7 even 12
6336.2.a.i.1.1 1 72.11 even 6
6336.2.a.j.1.1 1 72.29 odd 6
7436.2.a.d.1.1 1 117.25 even 6
7744.2.a.m.1.1 1 792.637 odd 6
7744.2.a.bc.1.1 1 792.43 even 6
8624.2.a.w.1.1 1 252.223 even 6
9900.2.a.h.1.1 1 45.29 odd 6
9900.2.c.g.5149.1 2 45.38 even 12
9900.2.c.g.5149.2 2 45.2 even 12