Properties

Label 1008.3.cg.b.577.1
Level $1008$
Weight $3$
Character 1008.577
Analytic conductor $27.466$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,3,Mod(145,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.145");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.cg (of order \(6\), 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: \(27.4660106475\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 577.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1008.577
Dual form 1008.3.cg.b.145.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 + 0.866025i) q^{5} +(-6.50000 - 2.59808i) q^{7} +O(q^{10})\) \(q+(-1.50000 + 0.866025i) q^{5} +(-6.50000 - 2.59808i) q^{7} +(1.50000 - 2.59808i) q^{11} +6.92820i q^{13} +(15.0000 + 8.66025i) q^{17} +(-9.00000 + 5.19615i) q^{19} +(-18.0000 - 31.1769i) q^{23} +(-11.0000 + 19.0526i) q^{25} +51.0000 q^{29} +(10.5000 + 6.06218i) q^{31} +(12.0000 - 1.73205i) q^{35} +(-11.0000 - 19.0526i) q^{37} -24.2487i q^{41} -10.0000 q^{43} +(78.0000 - 45.0333i) q^{47} +(35.5000 + 33.7750i) q^{49} +(25.5000 - 44.1673i) q^{53} +5.19615i q^{55} +(64.5000 + 37.2391i) q^{59} +(60.0000 - 34.6410i) q^{61} +(-6.00000 - 10.3923i) q^{65} +(34.0000 - 58.8897i) q^{67} +(-18.0000 - 10.3923i) q^{73} +(-16.5000 + 12.9904i) q^{77} +(62.5000 + 108.253i) q^{79} +154.153i q^{83} -30.0000 q^{85} +(-63.0000 + 36.3731i) q^{89} +(18.0000 - 45.0333i) q^{91} +(9.00000 - 15.5885i) q^{95} -147.224i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} - 13 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{5} - 13 q^{7} + 3 q^{11} + 30 q^{17} - 18 q^{19} - 36 q^{23} - 22 q^{25} + 102 q^{29} + 21 q^{31} + 24 q^{35} - 22 q^{37} - 20 q^{43} + 156 q^{47} + 71 q^{49} + 51 q^{53} + 129 q^{59} + 120 q^{61} - 12 q^{65} + 68 q^{67} - 36 q^{73} - 33 q^{77} + 125 q^{79} - 60 q^{85} - 126 q^{89} + 36 q^{91} + 18 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\)

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 + 0.866025i −0.300000 + 0.173205i −0.642443 0.766334i \(-0.722079\pi\)
0.342443 + 0.939539i \(0.388746\pi\)
\(6\) 0 0
\(7\) −6.50000 2.59808i −0.928571 0.371154i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.50000 2.59808i 0.136364 0.236189i −0.789754 0.613424i \(-0.789792\pi\)
0.926118 + 0.377235i \(0.123125\pi\)
\(12\) 0 0
\(13\) 6.92820i 0.532939i 0.963843 + 0.266469i \(0.0858571\pi\)
−0.963843 + 0.266469i \(0.914143\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 15.0000 + 8.66025i 0.882353 + 0.509427i 0.871433 0.490514i \(-0.163191\pi\)
0.0109194 + 0.999940i \(0.496524\pi\)
\(18\) 0 0
\(19\) −9.00000 + 5.19615i −0.473684 + 0.273482i −0.717781 0.696269i \(-0.754842\pi\)
0.244096 + 0.969751i \(0.421509\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −18.0000 31.1769i −0.782609 1.35552i −0.930417 0.366502i \(-0.880555\pi\)
0.147809 0.989016i \(-0.452778\pi\)
\(24\) 0 0
\(25\) −11.0000 + 19.0526i −0.440000 + 0.762102i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 51.0000 1.75862 0.879310 0.476249i \(-0.158004\pi\)
0.879310 + 0.476249i \(0.158004\pi\)
\(30\) 0 0
\(31\) 10.5000 + 6.06218i 0.338710 + 0.195554i 0.659701 0.751528i \(-0.270683\pi\)
−0.320992 + 0.947082i \(0.604016\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.0000 1.73205i 0.342857 0.0494872i
\(36\) 0 0
\(37\) −11.0000 19.0526i −0.297297 0.514934i 0.678219 0.734859i \(-0.262752\pi\)
−0.975517 + 0.219925i \(0.929419\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 24.2487i 0.591432i −0.955276 0.295716i \(-0.904442\pi\)
0.955276 0.295716i \(-0.0955582\pi\)
\(42\) 0 0
\(43\) −10.0000 −0.232558 −0.116279 0.993217i \(-0.537097\pi\)
−0.116279 + 0.993217i \(0.537097\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 78.0000 45.0333i 1.65957 0.958156i 0.686664 0.726975i \(-0.259075\pi\)
0.972911 0.231180i \(-0.0742588\pi\)
\(48\) 0 0
\(49\) 35.5000 + 33.7750i 0.724490 + 0.689286i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 25.5000 44.1673i 0.481132 0.833345i −0.518634 0.854997i \(-0.673559\pi\)
0.999766 + 0.0216515i \(0.00689241\pi\)
\(54\) 0 0
\(55\) 5.19615i 0.0944755i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 64.5000 + 37.2391i 1.09322 + 0.631171i 0.934432 0.356142i \(-0.115908\pi\)
0.158788 + 0.987313i \(0.449241\pi\)
\(60\) 0 0
\(61\) 60.0000 34.6410i 0.983607 0.567886i 0.0802495 0.996775i \(-0.474428\pi\)
0.903357 + 0.428889i \(0.141095\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.00000 10.3923i −0.0923077 0.159882i
\(66\) 0 0
\(67\) 34.0000 58.8897i 0.507463 0.878951i −0.492500 0.870312i \(-0.663917\pi\)
0.999963 0.00863871i \(-0.00274982\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −18.0000 10.3923i −0.246575 0.142360i 0.371620 0.928385i \(-0.378803\pi\)
−0.618195 + 0.786025i \(0.712136\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −16.5000 + 12.9904i −0.214286 + 0.168706i
\(78\) 0 0
\(79\) 62.5000 + 108.253i 0.791139 + 1.37029i 0.925262 + 0.379329i \(0.123845\pi\)
−0.134123 + 0.990965i \(0.542822\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 154.153i 1.85726i 0.371008 + 0.928630i \(0.379012\pi\)
−0.371008 + 0.928630i \(0.620988\pi\)
\(84\) 0 0
\(85\) −30.0000 −0.352941
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −63.0000 + 36.3731i −0.707865 + 0.408686i −0.810270 0.586057i \(-0.800680\pi\)
0.102405 + 0.994743i \(0.467346\pi\)
\(90\) 0 0
\(91\) 18.0000 45.0333i 0.197802 0.494872i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 9.00000 15.5885i 0.0947368 0.164089i
\(96\) 0 0
\(97\) 147.224i 1.51778i −0.651221 0.758888i \(-0.725743\pi\)
0.651221 0.758888i \(-0.274257\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −18.0000 10.3923i −0.178218 0.102894i 0.408237 0.912876i \(-0.366144\pi\)
−0.586455 + 0.809982i \(0.699477\pi\)
\(102\) 0 0
\(103\) 132.000 76.2102i 1.28155 0.739905i 0.304421 0.952538i \(-0.401537\pi\)
0.977132 + 0.212632i \(0.0682037\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.5000 + 28.5788i 0.154206 + 0.267092i 0.932769 0.360473i \(-0.117385\pi\)
−0.778564 + 0.627565i \(0.784052\pi\)
\(108\) 0 0
\(109\) 16.0000 27.7128i 0.146789 0.254246i −0.783250 0.621707i \(-0.786440\pi\)
0.930039 + 0.367461i \(0.119773\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 186.000 1.64602 0.823009 0.568029i \(-0.192294\pi\)
0.823009 + 0.568029i \(0.192294\pi\)
\(114\) 0 0
\(115\) 54.0000 + 31.1769i 0.469565 + 0.271104i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −75.0000 95.2628i −0.630252 0.800528i
\(120\) 0 0
\(121\) 56.0000 + 96.9948i 0.462810 + 0.801610i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 81.4064i 0.651251i
\(126\) 0 0
\(127\) −131.000 −1.03150 −0.515748 0.856740i \(-0.672486\pi\)
−0.515748 + 0.856740i \(0.672486\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 85.5000 49.3634i 0.652672 0.376820i −0.136807 0.990598i \(-0.543684\pi\)
0.789479 + 0.613777i \(0.210351\pi\)
\(132\) 0 0
\(133\) 72.0000 10.3923i 0.541353 0.0781376i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −72.0000 + 124.708i −0.525547 + 0.910275i 0.474010 + 0.880520i \(0.342806\pi\)
−0.999557 + 0.0297553i \(0.990527\pi\)
\(138\) 0 0
\(139\) 148.956i 1.07163i −0.844336 0.535814i \(-0.820005\pi\)
0.844336 0.535814i \(-0.179995\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 18.0000 + 10.3923i 0.125874 + 0.0726735i
\(144\) 0 0
\(145\) −76.5000 + 44.1673i −0.527586 + 0.304602i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −75.0000 129.904i −0.503356 0.871838i −0.999992 0.00387918i \(-0.998765\pi\)
0.496637 0.867958i \(-0.334568\pi\)
\(150\) 0 0
\(151\) −116.500 + 201.784i −0.771523 + 1.33632i 0.165205 + 0.986259i \(0.447171\pi\)
−0.936728 + 0.350058i \(0.886162\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −21.0000 −0.135484
\(156\) 0 0
\(157\) 126.000 + 72.7461i 0.802548 + 0.463351i 0.844361 0.535774i \(-0.179980\pi\)
−0.0418135 + 0.999125i \(0.513314\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 36.0000 + 249.415i 0.223602 + 1.54916i
\(162\) 0 0
\(163\) −74.0000 128.172i −0.453988 0.786330i 0.544642 0.838669i \(-0.316666\pi\)
−0.998629 + 0.0523391i \(0.983332\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 27.7128i 0.165945i 0.996552 + 0.0829725i \(0.0264414\pi\)
−0.996552 + 0.0829725i \(0.973559\pi\)
\(168\) 0 0
\(169\) 121.000 0.715976
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 162.000 93.5307i 0.936416 0.540640i 0.0475811 0.998867i \(-0.484849\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(174\) 0 0
\(175\) 121.000 95.2628i 0.691429 0.544359i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 81.0000 140.296i 0.452514 0.783777i −0.546028 0.837767i \(-0.683861\pi\)
0.998541 + 0.0539901i \(0.0171939\pi\)
\(180\) 0 0
\(181\) 259.808i 1.43540i −0.696352 0.717701i \(-0.745195\pi\)
0.696352 0.717701i \(-0.254805\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 33.0000 + 19.0526i 0.178378 + 0.102987i
\(186\) 0 0
\(187\) 45.0000 25.9808i 0.240642 0.138935i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −48.0000 83.1384i −0.251309 0.435280i 0.712578 0.701593i \(-0.247528\pi\)
−0.963886 + 0.266314i \(0.914194\pi\)
\(192\) 0 0
\(193\) −147.500 + 255.477i −0.764249 + 1.32372i 0.176394 + 0.984320i \(0.443557\pi\)
−0.940643 + 0.339398i \(0.889777\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 330.000 1.67513 0.837563 0.546340i \(-0.183979\pi\)
0.837563 + 0.546340i \(0.183979\pi\)
\(198\) 0 0
\(199\) 102.000 + 58.8897i 0.512563 + 0.295928i 0.733886 0.679272i \(-0.237704\pi\)
−0.221324 + 0.975200i \(0.571038\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −331.500 132.502i −1.63300 0.652719i
\(204\) 0 0
\(205\) 21.0000 + 36.3731i 0.102439 + 0.177430i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 31.1769i 0.149172i
\(210\) 0 0
\(211\) −220.000 −1.04265 −0.521327 0.853357i \(-0.674563\pi\)
−0.521327 + 0.853357i \(0.674563\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 15.0000 8.66025i 0.0697674 0.0402803i
\(216\) 0 0
\(217\) −52.5000 66.6840i −0.241935 0.307299i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −60.0000 + 103.923i −0.271493 + 0.470240i
\(222\) 0 0
\(223\) 150.688i 0.675733i 0.941194 + 0.337866i \(0.109705\pi\)
−0.941194 + 0.337866i \(0.890295\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 94.5000 + 54.5596i 0.416300 + 0.240351i 0.693493 0.720464i \(-0.256071\pi\)
−0.277193 + 0.960814i \(0.589404\pi\)
\(228\) 0 0
\(229\) −327.000 + 188.794i −1.42795 + 0.824426i −0.996959 0.0779313i \(-0.975169\pi\)
−0.430989 + 0.902357i \(0.641835\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −105.000 181.865i −0.450644 0.780538i 0.547782 0.836621i \(-0.315472\pi\)
−0.998426 + 0.0560830i \(0.982139\pi\)
\(234\) 0 0
\(235\) −78.0000 + 135.100i −0.331915 + 0.574893i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 120.000 0.502092 0.251046 0.967975i \(-0.419225\pi\)
0.251046 + 0.967975i \(0.419225\pi\)
\(240\) 0 0
\(241\) −10.5000 6.06218i −0.0435685 0.0251543i 0.478058 0.878329i \(-0.341341\pi\)
−0.521626 + 0.853174i \(0.674674\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −82.5000 19.9186i −0.336735 0.0813003i
\(246\) 0 0
\(247\) −36.0000 62.3538i −0.145749 0.252445i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 136.832i 0.545147i 0.962135 + 0.272574i \(0.0878749\pi\)
−0.962135 + 0.272574i \(0.912125\pi\)
\(252\) 0 0
\(253\) −108.000 −0.426877
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −117.000 + 67.5500i −0.455253 + 0.262840i −0.710046 0.704155i \(-0.751326\pi\)
0.254793 + 0.966996i \(0.417993\pi\)
\(258\) 0 0
\(259\) 22.0000 + 152.420i 0.0849421 + 0.588496i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −81.0000 + 140.296i −0.307985 + 0.533445i −0.977921 0.208973i \(-0.932988\pi\)
0.669937 + 0.742418i \(0.266321\pi\)
\(264\) 0 0
\(265\) 88.3346i 0.333338i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −334.500 193.124i −1.24349 0.717932i −0.273691 0.961818i \(-0.588244\pi\)
−0.969804 + 0.243886i \(0.921578\pi\)
\(270\) 0 0
\(271\) −187.500 + 108.253i −0.691882 + 0.399458i −0.804317 0.594201i \(-0.797468\pi\)
0.112435 + 0.993659i \(0.464135\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 33.0000 + 57.1577i 0.120000 + 0.207846i
\(276\) 0 0
\(277\) 50.0000 86.6025i 0.180505 0.312645i −0.761547 0.648109i \(-0.775560\pi\)
0.942053 + 0.335465i \(0.108893\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −228.000 −0.811388 −0.405694 0.914009i \(-0.632970\pi\)
−0.405694 + 0.914009i \(0.632970\pi\)
\(282\) 0 0
\(283\) −15.0000 8.66025i −0.0530035 0.0306016i 0.473264 0.880921i \(-0.343076\pi\)
−0.526268 + 0.850319i \(0.676409\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −63.0000 + 157.617i −0.219512 + 0.549187i
\(288\) 0 0
\(289\) 5.50000 + 9.52628i 0.0190311 + 0.0329629i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 278.860i 0.951741i −0.879515 0.475871i \(-0.842133\pi\)
0.879515 0.475871i \(-0.157867\pi\)
\(294\) 0 0
\(295\) −129.000 −0.437288
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 216.000 124.708i 0.722408 0.417082i
\(300\) 0 0
\(301\) 65.0000 + 25.9808i 0.215947 + 0.0863148i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −60.0000 + 103.923i −0.196721 + 0.340731i
\(306\) 0 0
\(307\) 27.7128i 0.0902697i −0.998981 0.0451349i \(-0.985628\pi\)
0.998981 0.0451349i \(-0.0143718\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 57.0000 + 32.9090i 0.183280 + 0.105817i 0.588833 0.808255i \(-0.299588\pi\)
−0.405553 + 0.914072i \(0.632921\pi\)
\(312\) 0 0
\(313\) 103.500 59.7558i 0.330671 0.190913i −0.325468 0.945553i \(-0.605522\pi\)
0.656139 + 0.754640i \(0.272189\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −100.500 174.071i −0.317035 0.549120i 0.662833 0.748767i \(-0.269354\pi\)
−0.979868 + 0.199647i \(0.936021\pi\)
\(318\) 0 0
\(319\) 76.5000 132.502i 0.239812 0.415366i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −180.000 −0.557276
\(324\) 0 0
\(325\) −132.000 76.2102i −0.406154 0.234493i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −624.000 + 90.0666i −1.89666 + 0.273759i
\(330\) 0 0
\(331\) 224.000 + 387.979i 0.676737 + 1.17214i 0.975958 + 0.217959i \(0.0699400\pi\)
−0.299221 + 0.954184i \(0.596727\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 117.779i 0.351580i
\(336\) 0 0
\(337\) −293.000 −0.869436 −0.434718 0.900567i \(-0.643152\pi\)
−0.434718 + 0.900567i \(0.643152\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 31.5000 18.1865i 0.0923754 0.0533329i
\(342\) 0 0
\(343\) −143.000 311.769i −0.416910 0.908948i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −195.000 + 337.750i −0.561960 + 0.973343i 0.435366 + 0.900254i \(0.356619\pi\)
−0.997325 + 0.0730890i \(0.976714\pi\)
\(348\) 0 0
\(349\) 138.564i 0.397032i −0.980098 0.198516i \(-0.936388\pi\)
0.980098 0.198516i \(-0.0636121\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 438.000 + 252.879i 1.24079 + 0.716372i 0.969256 0.246056i \(-0.0791346\pi\)
0.271537 + 0.962428i \(0.412468\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −150.000 259.808i −0.417827 0.723698i 0.577893 0.816112i \(-0.303875\pi\)
−0.995721 + 0.0924142i \(0.970542\pi\)
\(360\) 0 0
\(361\) −126.500 + 219.104i −0.350416 + 0.606937i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 36.0000 0.0986301
\(366\) 0 0
\(367\) 175.500 + 101.325i 0.478202 + 0.276090i 0.719667 0.694320i \(-0.244295\pi\)
−0.241465 + 0.970410i \(0.577628\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −280.500 + 220.836i −0.756065 + 0.595247i
\(372\) 0 0
\(373\) −205.000 355.070i −0.549598 0.951931i −0.998302 0.0582510i \(-0.981448\pi\)
0.448704 0.893680i \(-0.351886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 353.338i 0.937237i
\(378\) 0 0
\(379\) 374.000 0.986807 0.493404 0.869800i \(-0.335753\pi\)
0.493404 + 0.869800i \(0.335753\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −249.000 + 143.760i −0.650131 + 0.375353i −0.788506 0.615027i \(-0.789145\pi\)
0.138376 + 0.990380i \(0.455812\pi\)
\(384\) 0 0
\(385\) 13.5000 33.7750i 0.0350649 0.0877272i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −69.0000 + 119.512i −0.177378 + 0.307228i −0.940982 0.338458i \(-0.890095\pi\)
0.763604 + 0.645685i \(0.223428\pi\)
\(390\) 0 0
\(391\) 623.538i 1.59473i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −187.500 108.253i −0.474684 0.274059i
\(396\) 0 0
\(397\) 534.000 308.305i 1.34509 0.776587i 0.357539 0.933898i \(-0.383616\pi\)
0.987549 + 0.157311i \(0.0502826\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 258.000 + 446.869i 0.643392 + 1.11439i 0.984671 + 0.174425i \(0.0558066\pi\)
−0.341279 + 0.939962i \(0.610860\pi\)
\(402\) 0 0
\(403\) −42.0000 + 72.7461i −0.104218 + 0.180511i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −66.0000 −0.162162
\(408\) 0 0
\(409\) 226.500 + 130.770i 0.553790 + 0.319731i 0.750649 0.660701i \(-0.229741\pi\)
−0.196859 + 0.980432i \(0.563074\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −322.500 409.630i −0.780872 0.991840i
\(414\) 0 0
\(415\) −133.500 231.229i −0.321687 0.557178i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 533.472i 1.27320i −0.771193 0.636601i \(-0.780340\pi\)
0.771193 0.636601i \(-0.219660\pi\)
\(420\) 0 0
\(421\) −76.0000 −0.180523 −0.0902613 0.995918i \(-0.528770\pi\)
−0.0902613 + 0.995918i \(0.528770\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −330.000 + 190.526i −0.776471 + 0.448296i
\(426\) 0 0
\(427\) −480.000 + 69.2820i −1.12412 + 0.162253i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −303.000 + 524.811i −0.703016 + 1.21766i 0.264386 + 0.964417i \(0.414831\pi\)
−0.967403 + 0.253243i \(0.918503\pi\)
\(432\) 0 0
\(433\) 34.6410i 0.0800023i 0.999200 + 0.0400012i \(0.0127362\pi\)
−0.999200 + 0.0400012i \(0.987264\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 324.000 + 187.061i 0.741419 + 0.428058i
\(438\) 0 0
\(439\) −85.5000 + 49.3634i −0.194761 + 0.112445i −0.594209 0.804310i \(-0.702535\pi\)
0.399449 + 0.916756i \(0.369202\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −310.500 537.802i −0.700903 1.21400i −0.968150 0.250371i \(-0.919447\pi\)
0.267247 0.963628i \(-0.413886\pi\)
\(444\) 0 0
\(445\) 63.0000 109.119i 0.141573 0.245212i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −504.000 −1.12249 −0.561247 0.827648i \(-0.689678\pi\)
−0.561247 + 0.827648i \(0.689678\pi\)
\(450\) 0 0
\(451\) −63.0000 36.3731i −0.139690 0.0806498i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 12.0000 + 83.1384i 0.0263736 + 0.182722i
\(456\) 0 0
\(457\) 221.500 + 383.649i 0.484683 + 0.839495i 0.999845 0.0175975i \(-0.00560174\pi\)
−0.515162 + 0.857093i \(0.672268\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 180.133i 0.390745i −0.980729 0.195372i \(-0.937408\pi\)
0.980729 0.195372i \(-0.0625915\pi\)
\(462\) 0 0
\(463\) −158.000 −0.341253 −0.170626 0.985336i \(-0.554579\pi\)
−0.170626 + 0.985336i \(0.554579\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.0000 10.3923i 0.0385439 0.0222533i −0.480604 0.876938i \(-0.659583\pi\)
0.519148 + 0.854684i \(0.326249\pi\)
\(468\) 0 0
\(469\) −374.000 + 294.449i −0.797441 + 0.627822i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −15.0000 + 25.9808i −0.0317125 + 0.0549276i
\(474\) 0 0
\(475\) 228.631i 0.481328i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 357.000 + 206.114i 0.745303 + 0.430301i 0.823994 0.566598i \(-0.191741\pi\)
−0.0786914 + 0.996899i \(0.525074\pi\)
\(480\) 0 0
\(481\) 132.000 76.2102i 0.274428 0.158441i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 127.500 + 220.836i 0.262887 + 0.455333i
\(486\) 0 0
\(487\) 57.5000 99.5929i 0.118070 0.204503i −0.800933 0.598754i \(-0.795663\pi\)
0.919003 + 0.394251i \(0.128996\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 273.000 0.556008 0.278004 0.960580i \(-0.410327\pi\)
0.278004 + 0.960580i \(0.410327\pi\)
\(492\) 0 0
\(493\) 765.000 + 441.673i 1.55172 + 0.895888i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 233.000 + 403.568i 0.466934 + 0.808753i 0.999286 0.0377695i \(-0.0120253\pi\)
−0.532353 + 0.846523i \(0.678692\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 723.997i 1.43936i −0.694307 0.719679i \(-0.744289\pi\)
0.694307 0.719679i \(-0.255711\pi\)
\(504\) 0 0
\(505\) 36.0000 0.0712871
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 307.500 177.535i 0.604126 0.348792i −0.166537 0.986035i \(-0.553259\pi\)
0.770663 + 0.637243i \(0.219925\pi\)
\(510\) 0 0
\(511\) 90.0000 + 114.315i 0.176125 + 0.223709i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −132.000 + 228.631i −0.256311 + 0.443943i
\(516\) 0 0
\(517\) 270.200i 0.522630i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 795.000 + 458.993i 1.52591 + 0.880986i 0.999527 + 0.0307387i \(0.00978599\pi\)
0.526384 + 0.850247i \(0.323547\pi\)
\(522\) 0 0
\(523\) −420.000 + 242.487i −0.803059 + 0.463646i −0.844540 0.535493i \(-0.820126\pi\)
0.0414805 + 0.999139i \(0.486793\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 105.000 + 181.865i 0.199241 + 0.345096i
\(528\) 0 0
\(529\) −383.500 + 664.241i −0.724953 + 1.25565i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 168.000 0.315197
\(534\) 0 0
\(535\) −49.5000 28.5788i −0.0925234 0.0534184i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 141.000 41.5692i 0.261596 0.0771229i
\(540\) 0 0
\(541\) −31.0000 53.6936i −0.0573013 0.0992488i 0.835952 0.548803i \(-0.184916\pi\)
−0.893253 + 0.449554i \(0.851583\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 55.4256i 0.101698i
\(546\) 0 0
\(547\) 802.000 1.46618 0.733090 0.680132i \(-0.238078\pi\)
0.733090 + 0.680132i \(0.238078\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −459.000 + 265.004i −0.833031 + 0.480951i
\(552\) 0 0
\(553\) −125.000 866.025i −0.226040 1.56605i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −55.5000 + 96.1288i −0.0996409 + 0.172583i −0.911536 0.411220i \(-0.865103\pi\)
0.811895 + 0.583803i \(0.198436\pi\)
\(558\) 0 0
\(559\) 69.2820i 0.123939i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 166.500 + 96.1288i 0.295737 + 0.170744i 0.640526 0.767936i \(-0.278716\pi\)
−0.344789 + 0.938680i \(0.612050\pi\)
\(564\) 0 0
\(565\) −279.000 + 161.081i −0.493805 + 0.285099i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −213.000 368.927i −0.374341 0.648378i 0.615887 0.787834i \(-0.288798\pi\)
−0.990228 + 0.139457i \(0.955464\pi\)
\(570\) 0 0
\(571\) −290.000 + 502.295i −0.507881 + 0.879676i 0.492077 + 0.870551i \(0.336238\pi\)
−0.999958 + 0.00912412i \(0.997096\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 792.000 1.37739
\(576\) 0 0
\(577\) −835.500 482.376i −1.44801 0.836007i −0.449644 0.893208i \(-0.648449\pi\)
−0.998363 + 0.0572006i \(0.981783\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 400.500 1001.99i 0.689329 1.72460i
\(582\) 0 0
\(583\) −76.5000 132.502i −0.131218 0.227276i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1002.86i 1.70845i −0.519907 0.854223i \(-0.674034\pi\)
0.519907 0.854223i \(-0.325966\pi\)
\(588\) 0 0
\(589\) −126.000 −0.213922
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 762.000 439.941i 1.28499 0.741890i 0.307235 0.951634i \(-0.400596\pi\)
0.977756 + 0.209743i \(0.0672629\pi\)
\(594\) 0 0
\(595\) 195.000 + 77.9423i 0.327731 + 0.130995i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 78.0000 135.100i 0.130217 0.225543i −0.793543 0.608514i \(-0.791766\pi\)
0.923760 + 0.382972i \(0.125099\pi\)
\(600\) 0 0
\(601\) 1124.10i 1.87038i 0.354141 + 0.935192i \(0.384773\pi\)
−0.354141 + 0.935192i \(0.615227\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −168.000 96.9948i −0.277686 0.160322i
\(606\) 0 0
\(607\) 346.500 200.052i 0.570840 0.329575i −0.186645 0.982427i \(-0.559761\pi\)
0.757485 + 0.652853i \(0.226428\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 312.000 + 540.400i 0.510638 + 0.884451i
\(612\) 0 0
\(613\) 428.000 741.318i 0.698206 1.20933i −0.270883 0.962612i \(-0.587315\pi\)
0.969088 0.246715i \(-0.0793512\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −444.000 −0.719611 −0.359806 0.933027i \(-0.617157\pi\)
−0.359806 + 0.933027i \(0.617157\pi\)
\(618\) 0 0
\(619\) 222.000 + 128.172i 0.358643 + 0.207063i 0.668485 0.743725i \(-0.266943\pi\)
−0.309842 + 0.950788i \(0.600276\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 504.000 72.7461i 0.808989 0.116767i
\(624\) 0 0
\(625\) −204.500 354.204i −0.327200 0.566727i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 381.051i 0.605805i
\(630\) 0 0
\(631\) 85.0000 0.134707 0.0673534 0.997729i \(-0.478545\pi\)
0.0673534 + 0.997729i \(0.478545\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 196.500 113.449i 0.309449 0.178660i
\(636\) 0 0
\(637\) −234.000 + 245.951i −0.367347 + 0.386109i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −66.0000 + 114.315i −0.102964 + 0.178339i −0.912905 0.408173i \(-0.866166\pi\)
0.809940 + 0.586512i \(0.199499\pi\)
\(642\) 0 0
\(643\) 713.605i 1.10981i 0.831915 + 0.554903i \(0.187245\pi\)
−0.831915 + 0.554903i \(0.812755\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 864.000 + 498.831i 1.33539 + 0.770990i 0.986121 0.166031i \(-0.0530951\pi\)
0.349274 + 0.937021i \(0.386428\pi\)
\(648\) 0 0
\(649\) 193.500 111.717i 0.298151 0.172138i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −187.500 324.760i −0.287136 0.497335i 0.685989 0.727612i \(-0.259370\pi\)
−0.973125 + 0.230278i \(0.926037\pi\)
\(654\) 0 0
\(655\) −85.5000 + 148.090i −0.130534 + 0.226092i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1038.00 1.57511 0.787557 0.616242i \(-0.211346\pi\)
0.787557 + 0.616242i \(0.211346\pi\)
\(660\) 0 0
\(661\) 387.000 + 223.435i 0.585477 + 0.338025i 0.763307 0.646036i \(-0.223574\pi\)
−0.177830 + 0.984061i \(0.556908\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −99.0000 + 77.9423i −0.148872 + 0.117206i
\(666\) 0 0
\(667\) −918.000 1590.02i −1.37631 2.38384i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 207.846i 0.309756i
\(672\) 0 0
\(673\) 1093.00 1.62407 0.812036 0.583608i \(-0.198359\pi\)
0.812036 + 0.583608i \(0.198359\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1048.50 605.352i 1.54874 0.894168i 0.550506 0.834831i \(-0.314435\pi\)
0.998238 0.0593371i \(-0.0188987\pi\)
\(678\) 0 0
\(679\) −382.500 + 956.958i −0.563328 + 1.40936i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.50000 2.59808i 0.00219619 0.00380392i −0.864925 0.501901i \(-0.832634\pi\)
0.867121 + 0.498097i \(0.165968\pi\)
\(684\) 0 0
\(685\) 249.415i 0.364110i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 306.000 + 176.669i 0.444122 + 0.256414i
\(690\) 0 0
\(691\) 987.000 569.845i 1.42836 0.824667i 0.431373 0.902174i \(-0.358029\pi\)
0.996992 + 0.0775070i \(0.0246960\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 129.000 + 223.435i 0.185612 + 0.321489i
\(696\) 0 0
\(697\) 210.000 363.731i 0.301291 0.521852i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −447.000 −0.637660 −0.318830 0.947812i \(-0.603290\pi\)
−0.318830 + 0.947812i \(0.603290\pi\)
\(702\) 0 0
\(703\) 198.000 + 114.315i 0.281650 + 0.162611i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 90.0000 + 114.315i 0.127298 + 0.161691i
\(708\) 0 0
\(709\) 211.000 + 365.463i 0.297602 + 0.515462i 0.975587 0.219614i \(-0.0704797\pi\)
−0.677985 + 0.735076i \(0.737146\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 436.477i 0.612169i
\(714\) 0 0
\(715\) −36.0000 −0.0503497
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 147.000 84.8705i 0.204451 0.118040i −0.394279 0.918991i \(-0.629006\pi\)
0.598730 + 0.800951i \(0.295672\pi\)
\(720\) 0 0
\(721\) −1056.00 + 152.420i −1.46463 + 0.211401i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −561.000 + 971.681i −0.773793 + 1.34025i
\(726\) 0 0
\(727\) 50.2295i 0.0690914i −0.999403 0.0345457i \(-0.989002\pi\)
0.999403 0.0345457i \(-0.0109984\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −150.000 86.6025i −0.205198 0.118471i
\(732\) 0 0
\(733\) 255.000 147.224i 0.347885 0.200852i −0.315868 0.948803i \(-0.602296\pi\)
0.663754 + 0.747951i \(0.268962\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −102.000 176.669i −0.138399 0.239714i
\(738\) 0 0
\(739\) 49.0000 84.8705i 0.0663058 0.114845i −0.830967 0.556322i \(-0.812212\pi\)
0.897272 + 0.441477i \(0.145545\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −180.000 −0.242261 −0.121131 0.992637i \(-0.538652\pi\)
−0.121131 + 0.992637i \(0.538652\pi\)
\(744\) 0 0
\(745\) 225.000 + 129.904i 0.302013 + 0.174368i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −33.0000 228.631i −0.0440587 0.305248i
\(750\) 0 0
\(751\) 360.500 + 624.404i 0.480027 + 0.831431i 0.999737 0.0229117i \(-0.00729366\pi\)
−0.519711 + 0.854342i \(0.673960\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 403.568i 0.534527i
\(756\) 0 0
\(757\) −644.000 −0.850727 −0.425363 0.905023i \(-0.639854\pi\)
−0.425363 + 0.905023i \(0.639854\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 492.000 284.056i 0.646518 0.373267i −0.140603 0.990066i \(-0.544904\pi\)
0.787121 + 0.616799i \(0.211571\pi\)
\(762\) 0 0
\(763\) −176.000 + 138.564i −0.230668 + 0.181604i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −258.000 + 446.869i −0.336375 + 0.582619i
\(768\) 0 0
\(769\) 275.396i 0.358122i −0.983838 0.179061i \(-0.942694\pi\)
0.983838 0.179061i \(-0.0573060\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 870.000 + 502.295i 1.12549 + 0.649799i 0.942796 0.333372i \(-0.108186\pi\)
0.182690 + 0.983171i \(0.441520\pi\)
\(774\) 0 0
\(775\) −231.000 + 133.368i −0.298065 + 0.172088i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 126.000 + 218.238i 0.161746 + 0.280152i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −252.000 −0.321019
\(786\) 0 0
\(787\) −690.000 398.372i −0.876747 0.506190i −0.00716264 0.999974i \(-0.502280\pi\)
−0.869585 + 0.493784i \(0.835613\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1209.00 483.242i −1.52845 0.610926i
\(792\) 0 0
\(793\) 240.000 + 415.692i 0.302648 + 0.524202i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 545.596i 0.684562i 0.939598 + 0.342281i \(0.111199\pi\)
−0.939598 + 0.342281i \(0.888801\pi\)
\(798\) 0 0
\(799\) 1560.00 1.95244
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −54.0000 + 31.1769i −0.0672478 + 0.0388255i
\(804\) 0 0
\(805\) −270.000 342.946i −0.335404 0.426020i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 24.0000 41.5692i 0.0296663 0.0513835i −0.850811 0.525472i \(-0.823889\pi\)
0.880477 + 0.474088i \(0.157222\pi\)
\(810\) 0 0
\(811\) 436.477i 0.538196i −0.963113 0.269098i \(-0.913274\pi\)
0.963113 0.269098i \(-0.0867255\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 222.000 + 128.172i 0.272393 + 0.157266i
\(816\) 0 0
\(817\) 90.0000 51.9615i 0.110159 0.0636004i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 595.500 + 1031.44i 0.725335 + 1.25632i 0.958836 + 0.283960i \(0.0916484\pi\)
−0.233501 + 0.972357i \(0.575018\pi\)
\(822\) 0 0
\(823\) −125.000 + 216.506i −0.151883 + 0.263070i −0.931920 0.362664i \(-0.881867\pi\)
0.780036 + 0.625734i \(0.215200\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1137.00 −1.37485 −0.687424 0.726256i \(-0.741259\pi\)
−0.687424 + 0.726256i \(0.741259\pi\)
\(828\) 0 0
\(829\) 63.0000 + 36.3731i 0.0759952 + 0.0438758i 0.537516 0.843253i \(-0.319363\pi\)
−0.461521 + 0.887129i \(0.652696\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 240.000 + 814.064i 0.288115 + 0.977268i
\(834\) 0 0
\(835\) −24.0000 41.5692i −0.0287425 0.0497835i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1416.82i 1.68870i −0.535794 0.844349i \(-0.679988\pi\)
0.535794 0.844349i \(-0.320012\pi\)
\(840\) 0 0
\(841\) 1760.00 2.09275
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −181.500 + 104.789i −0.214793 + 0.124011i
\(846\) 0 0
\(847\) −112.000 775.959i −0.132231 0.916126i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −396.000 + 685.892i −0.465335 + 0.805984i
\(852\) 0 0
\(853\) 872.954i 1.02339i 0.859166 + 0.511696i \(0.170983\pi\)
−0.859166 + 0.511696i \(0.829017\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 600.000 + 346.410i 0.700117 + 0.404213i 0.807391 0.590017i \(-0.200879\pi\)
−0.107274 + 0.994229i \(0.534212\pi\)
\(858\) 0 0
\(859\) 771.000 445.137i 0.897555 0.518204i 0.0211491 0.999776i \(-0.493268\pi\)
0.876406 + 0.481573i \(0.159934\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −315.000 545.596i −0.365006 0.632209i 0.623771 0.781607i \(-0.285600\pi\)
−0.988777 + 0.149398i \(0.952266\pi\)
\(864\) 0 0
\(865\) −162.000 + 280.592i −0.187283 + 0.324384i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 375.000 0.431530
\(870\) 0 0
\(871\) 408.000 + 235.559i 0.468427 + 0.270447i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −211.500 + 529.142i −0.241714 + 0.604733i
\(876\) 0 0
\(877\) −820.000 1420.28i −0.935006 1.61948i −0.774624 0.632422i \(-0.782061\pi\)
−0.160382 0.987055i \(-0.551273\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1267.86i 1.43912i −0.694432 0.719558i \(-0.744344\pi\)
0.694432 0.719558i \(-0.255656\pi\)
\(882\) 0 0
\(883\) −226.000 −0.255946 −0.127973 0.991778i \(-0.540847\pi\)
−0.127973 + 0.991778i \(0.540847\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.00000 3.46410i 0.00676437 0.00390541i −0.496614 0.867972i \(-0.665424\pi\)
0.503378 + 0.864066i \(0.332090\pi\)
\(888\) 0 0
\(889\) 851.500 + 340.348i 0.957818 + 0.382844i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −468.000 + 810.600i −0.524076 + 0.907727i
\(894\) 0 0
\(895\) 280.592i 0.313511i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 535.500 + 309.171i 0.595662 + 0.343906i
\(900\) 0 0
\(901\) 765.000 441.673i 0.849057 0.490203i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 225.000 + 389.711i 0.248619 + 0.430620i
\(906\) 0 0
\(907\) 572.000 990.733i 0.630650 1.09232i −0.356768 0.934193i \(-0.616121\pi\)
0.987419 0.158126i \(-0.0505452\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −480.000 −0.526894 −0.263447 0.964674i \(-0.584859\pi\)
−0.263447 + 0.964674i \(0.584859\pi\)
\(912\) 0 0
\(913\) 400.500 + 231.229i 0.438664 + 0.253263i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −684.000 + 98.7269i −0.745911 + 0.107663i
\(918\) 0 0
\(919\) −871.000 1508.62i −0.947769 1.64158i −0.750109 0.661314i \(-0.769999\pi\)
−0.197660 0.980271i \(-0.563334\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 484.000 0.523243
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1005.00 + 580.237i −1.08181 + 0.624582i −0.931384 0.364039i \(-0.881397\pi\)
−0.150425 + 0.988621i \(0.548064\pi\)
\(930\) 0 0
\(931\) −495.000 119.512i −0.531686 0.128369i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −45.0000 + 77.9423i −0.0481283 + 0.0833607i
\(936\) 0 0
\(937\) 1331.95i 1.42150i −0.703444 0.710751i \(-0.748355\pi\)
0.703444 0.710751i \(-0.251645\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −703.500 406.166i −0.747609 0.431632i 0.0772204 0.997014i \(-0.475395\pi\)
−0.824829 + 0.565382i \(0.808729\pi\)
\(942\) 0 0
\(943\) −756.000 + 436.477i −0.801697 + 0.462860i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 645.000 + 1117.17i 0.681098 + 1.17970i 0.974646 + 0.223752i \(0.0718307\pi\)
−0.293548 + 0.955944i \(0.594836\pi\)
\(948\) 0 0
\(949\) 72.0000 124.708i 0.0758693 0.131410i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −174.000 −0.182581 −0.0912907 0.995824i \(-0.529099\pi\)
−0.0912907 + 0.995824i \(0.529099\pi\)
\(954\) 0 0
\(955\) 144.000 + 83.1384i 0.150785 + 0.0870560i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 792.000 623.538i 0.825860 0.650196i
\(960\) 0 0
\(961\) −407.000 704.945i −0.423517 0.733553i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 510.955i 0.529487i
\(966\) 0 0
\(967\) −41.0000 −0.0423992 −0.0211996 0.999775i \(-0.506749\pi\)
−0.0211996 + 0.999775i \(0.506749\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 853.500 492.768i 0.878991 0.507486i 0.00866523 0.999962i \(-0.497242\pi\)
0.870326 + 0.492477i \(0.163908\pi\)
\(972\) 0 0
\(973\) −387.000 + 968.216i −0.397739 + 0.995084i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 639.000 1106.78i 0.654043 1.13284i −0.328090 0.944647i \(-0.606405\pi\)
0.982133 0.188189i \(-0.0602618\pi\)
\(978\) 0 0
\(979\) 218.238i 0.222920i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −201.000 116.047i −0.204476 0.118054i 0.394266 0.918997i \(-0.370999\pi\)
−0.598742 + 0.800942i \(0.704332\pi\)
\(984\) 0 0
\(985\) −495.000 + 285.788i −0.502538 + 0.290140i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 180.000 + 311.769i 0.182002 + 0.315237i
\(990\) 0 0
\(991\) −416.500 + 721.399i −0.420283 + 0.727951i −0.995967 0.0897213i \(-0.971402\pi\)
0.575684 + 0.817672i \(0.304736\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −204.000 −0.205025
\(996\) 0 0
\(997\) 1014.00 + 585.433i 1.01705 + 0.587195i 0.913248 0.407403i \(-0.133566\pi\)
0.103803 + 0.994598i \(0.466899\pi\)
\(998\) 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 1008.3.cg.b.577.1 2
3.2 odd 2 336.3.bh.b.241.1 2
4.3 odd 2 252.3.z.b.73.1 2
7.5 odd 6 inner 1008.3.cg.b.145.1 2
12.11 even 2 84.3.m.a.73.1 yes 2
21.5 even 6 336.3.bh.b.145.1 2
21.11 odd 6 2352.3.f.c.97.2 2
21.17 even 6 2352.3.f.c.97.1 2
28.3 even 6 1764.3.d.c.685.2 2
28.11 odd 6 1764.3.d.c.685.1 2
28.19 even 6 252.3.z.b.145.1 2
28.23 odd 6 1764.3.z.e.901.1 2
28.27 even 2 1764.3.z.e.325.1 2
60.23 odd 4 2100.3.be.c.1249.1 4
60.47 odd 4 2100.3.be.c.1249.2 4
60.59 even 2 2100.3.bd.b.1501.1 2
84.11 even 6 588.3.d.a.97.1 2
84.23 even 6 588.3.m.a.313.1 2
84.47 odd 6 84.3.m.a.61.1 2
84.59 odd 6 588.3.d.a.97.2 2
84.83 odd 2 588.3.m.a.325.1 2
420.47 even 12 2100.3.be.c.649.1 4
420.299 odd 6 2100.3.bd.b.901.1 2
420.383 even 12 2100.3.be.c.649.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.3.m.a.61.1 2 84.47 odd 6
84.3.m.a.73.1 yes 2 12.11 even 2
252.3.z.b.73.1 2 4.3 odd 2
252.3.z.b.145.1 2 28.19 even 6
336.3.bh.b.145.1 2 21.5 even 6
336.3.bh.b.241.1 2 3.2 odd 2
588.3.d.a.97.1 2 84.11 even 6
588.3.d.a.97.2 2 84.59 odd 6
588.3.m.a.313.1 2 84.23 even 6
588.3.m.a.325.1 2 84.83 odd 2
1008.3.cg.b.145.1 2 7.5 odd 6 inner
1008.3.cg.b.577.1 2 1.1 even 1 trivial
1764.3.d.c.685.1 2 28.11 odd 6
1764.3.d.c.685.2 2 28.3 even 6
1764.3.z.e.325.1 2 28.27 even 2
1764.3.z.e.901.1 2 28.23 odd 6
2100.3.bd.b.901.1 2 420.299 odd 6
2100.3.bd.b.1501.1 2 60.59 even 2
2100.3.be.c.649.1 4 420.47 even 12
2100.3.be.c.649.2 4 420.383 even 12
2100.3.be.c.1249.1 4 60.23 odd 4
2100.3.be.c.1249.2 4 60.47 odd 4
2352.3.f.c.97.1 2 21.17 even 6
2352.3.f.c.97.2 2 21.11 odd 6