Properties

Label 3042.3.d.a.3041.3
Level $3042$
Weight $3$
Character 3042.3041
Analytic conductor $82.888$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3042,3,Mod(3041,3042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3042.3041");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3042.d (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: \(82.8884964184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3041.3
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 3042.3041
Dual form 3042.3.d.a.3041.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +2.00000 q^{4} -4.24264 q^{5} -4.00000i q^{7} +2.82843 q^{8} +O(q^{10})\) \(q+1.41421 q^{2} +2.00000 q^{4} -4.24264 q^{5} -4.00000i q^{7} +2.82843 q^{8} -6.00000 q^{10} -16.9706 q^{11} -5.65685i q^{14} +4.00000 q^{16} +12.7279i q^{17} +16.0000i q^{19} -8.48528 q^{20} -24.0000 q^{22} +16.9706i q^{23} -7.00000 q^{25} -8.00000i q^{28} +4.24264i q^{29} -44.0000i q^{31} +5.65685 q^{32} +18.0000i q^{34} +16.9706i q^{35} -34.0000i q^{37} +22.6274i q^{38} -12.0000 q^{40} +46.6690 q^{41} +40.0000 q^{43} -33.9411 q^{44} +24.0000i q^{46} +84.8528 q^{47} +33.0000 q^{49} -9.89949 q^{50} +38.1838i q^{53} +72.0000 q^{55} -11.3137i q^{56} +6.00000i q^{58} -33.9411 q^{59} +50.0000 q^{61} -62.2254i q^{62} +8.00000 q^{64} -8.00000i q^{67} +25.4558i q^{68} +24.0000i q^{70} -50.9117 q^{71} -16.0000i q^{73} -48.0833i q^{74} +32.0000i q^{76} +67.8823i q^{77} -76.0000 q^{79} -16.9706 q^{80} +66.0000 q^{82} +118.794 q^{83} -54.0000i q^{85} +56.5685 q^{86} -48.0000 q^{88} -12.7279 q^{89} +33.9411i q^{92} +120.000 q^{94} -67.8823i q^{95} -176.000i q^{97} +46.6690 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} - 24 q^{10} + 16 q^{16} - 96 q^{22} - 28 q^{25} - 48 q^{40} + 160 q^{43} + 132 q^{49} + 288 q^{55} + 200 q^{61} + 32 q^{64} - 304 q^{79} + 264 q^{82} - 192 q^{88} + 480 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 0.707107
\(3\) 0 0
\(4\) 2.00000 0.500000
\(5\) −4.24264 −0.848528 −0.424264 0.905539i \(-0.639467\pi\)
−0.424264 + 0.905539i \(0.639467\pi\)
\(6\) 0 0
\(7\) − 4.00000i − 0.571429i −0.958315 0.285714i \(-0.907769\pi\)
0.958315 0.285714i \(-0.0922308\pi\)
\(8\) 2.82843 0.353553
\(9\) 0 0
\(10\) −6.00000 −0.600000
\(11\) −16.9706 −1.54278 −0.771389 0.636364i \(-0.780438\pi\)
−0.771389 + 0.636364i \(0.780438\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) − 5.65685i − 0.404061i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 12.7279i 0.748701i 0.927287 + 0.374351i \(0.122134\pi\)
−0.927287 + 0.374351i \(0.877866\pi\)
\(18\) 0 0
\(19\) 16.0000i 0.842105i 0.907036 + 0.421053i \(0.138339\pi\)
−0.907036 + 0.421053i \(0.861661\pi\)
\(20\) −8.48528 −0.424264
\(21\) 0 0
\(22\) −24.0000 −1.09091
\(23\) 16.9706i 0.737851i 0.929459 + 0.368925i \(0.120274\pi\)
−0.929459 + 0.368925i \(0.879726\pi\)
\(24\) 0 0
\(25\) −7.00000 −0.280000
\(26\) 0 0
\(27\) 0 0
\(28\) − 8.00000i − 0.285714i
\(29\) 4.24264i 0.146298i 0.997321 + 0.0731490i \(0.0233049\pi\)
−0.997321 + 0.0731490i \(0.976695\pi\)
\(30\) 0 0
\(31\) − 44.0000i − 1.41935i −0.704527 0.709677i \(-0.748841\pi\)
0.704527 0.709677i \(-0.251159\pi\)
\(32\) 5.65685 0.176777
\(33\) 0 0
\(34\) 18.0000i 0.529412i
\(35\) 16.9706i 0.484873i
\(36\) 0 0
\(37\) − 34.0000i − 0.918919i −0.888199 0.459459i \(-0.848043\pi\)
0.888199 0.459459i \(-0.151957\pi\)
\(38\) 22.6274i 0.595458i
\(39\) 0 0
\(40\) −12.0000 −0.300000
\(41\) 46.6690 1.13827 0.569135 0.822244i \(-0.307278\pi\)
0.569135 + 0.822244i \(0.307278\pi\)
\(42\) 0 0
\(43\) 40.0000 0.930233 0.465116 0.885250i \(-0.346013\pi\)
0.465116 + 0.885250i \(0.346013\pi\)
\(44\) −33.9411 −0.771389
\(45\) 0 0
\(46\) 24.0000i 0.521739i
\(47\) 84.8528 1.80538 0.902690 0.430293i \(-0.141590\pi\)
0.902690 + 0.430293i \(0.141590\pi\)
\(48\) 0 0
\(49\) 33.0000 0.673469
\(50\) −9.89949 −0.197990
\(51\) 0 0
\(52\) 0 0
\(53\) 38.1838i 0.720448i 0.932866 + 0.360224i \(0.117300\pi\)
−0.932866 + 0.360224i \(0.882700\pi\)
\(54\) 0 0
\(55\) 72.0000 1.30909
\(56\) − 11.3137i − 0.202031i
\(57\) 0 0
\(58\) 6.00000i 0.103448i
\(59\) −33.9411 −0.575273 −0.287637 0.957740i \(-0.592870\pi\)
−0.287637 + 0.957740i \(0.592870\pi\)
\(60\) 0 0
\(61\) 50.0000 0.819672 0.409836 0.912159i \(-0.365586\pi\)
0.409836 + 0.912159i \(0.365586\pi\)
\(62\) − 62.2254i − 1.00364i
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 8.00000i − 0.119403i −0.998216 0.0597015i \(-0.980985\pi\)
0.998216 0.0597015i \(-0.0190149\pi\)
\(68\) 25.4558i 0.374351i
\(69\) 0 0
\(70\) 24.0000i 0.342857i
\(71\) −50.9117 −0.717066 −0.358533 0.933517i \(-0.616723\pi\)
−0.358533 + 0.933517i \(0.616723\pi\)
\(72\) 0 0
\(73\) − 16.0000i − 0.219178i −0.993977 0.109589i \(-0.965047\pi\)
0.993977 0.109589i \(-0.0349535\pi\)
\(74\) − 48.0833i − 0.649774i
\(75\) 0 0
\(76\) 32.0000i 0.421053i
\(77\) 67.8823i 0.881588i
\(78\) 0 0
\(79\) −76.0000 −0.962025 −0.481013 0.876714i \(-0.659731\pi\)
−0.481013 + 0.876714i \(0.659731\pi\)
\(80\) −16.9706 −0.212132
\(81\) 0 0
\(82\) 66.0000 0.804878
\(83\) 118.794 1.43125 0.715626 0.698484i \(-0.246141\pi\)
0.715626 + 0.698484i \(0.246141\pi\)
\(84\) 0 0
\(85\) − 54.0000i − 0.635294i
\(86\) 56.5685 0.657774
\(87\) 0 0
\(88\) −48.0000 −0.545455
\(89\) −12.7279 −0.143010 −0.0715052 0.997440i \(-0.522780\pi\)
−0.0715052 + 0.997440i \(0.522780\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 33.9411i 0.368925i
\(93\) 0 0
\(94\) 120.000 1.27660
\(95\) − 67.8823i − 0.714550i
\(96\) 0 0
\(97\) − 176.000i − 1.81443i −0.420664 0.907216i \(-0.638203\pi\)
0.420664 0.907216i \(-0.361797\pi\)
\(98\) 46.6690 0.476215
\(99\) 0 0
\(100\) −14.0000 −0.140000
\(101\) − 29.6985i − 0.294044i −0.989133 0.147022i \(-0.953031\pi\)
0.989133 0.147022i \(-0.0469689\pi\)
\(102\) 0 0
\(103\) 28.0000 0.271845 0.135922 0.990719i \(-0.456600\pi\)
0.135922 + 0.990719i \(0.456600\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 54.0000i 0.509434i
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) − 56.0000i − 0.513761i −0.966443 0.256881i \(-0.917305\pi\)
0.966443 0.256881i \(-0.0826948\pi\)
\(110\) 101.823 0.925667
\(111\) 0 0
\(112\) − 16.0000i − 0.142857i
\(113\) − 156.978i − 1.38918i −0.719404 0.694592i \(-0.755585\pi\)
0.719404 0.694592i \(-0.244415\pi\)
\(114\) 0 0
\(115\) − 72.0000i − 0.626087i
\(116\) 8.48528i 0.0731490i
\(117\) 0 0
\(118\) −48.0000 −0.406780
\(119\) 50.9117 0.427829
\(120\) 0 0
\(121\) 167.000 1.38017
\(122\) 70.7107 0.579596
\(123\) 0 0
\(124\) − 88.0000i − 0.709677i
\(125\) 135.765 1.08612
\(126\) 0 0
\(127\) −92.0000 −0.724409 −0.362205 0.932099i \(-0.617976\pi\)
−0.362205 + 0.932099i \(0.617976\pi\)
\(128\) 11.3137 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) − 169.706i − 1.29546i −0.761869 0.647731i \(-0.775718\pi\)
0.761869 0.647731i \(-0.224282\pi\)
\(132\) 0 0
\(133\) 64.0000 0.481203
\(134\) − 11.3137i − 0.0844307i
\(135\) 0 0
\(136\) 36.0000i 0.264706i
\(137\) −156.978 −1.14582 −0.572911 0.819617i \(-0.694186\pi\)
−0.572911 + 0.819617i \(0.694186\pi\)
\(138\) 0 0
\(139\) 152.000 1.09353 0.546763 0.837288i \(-0.315860\pi\)
0.546763 + 0.837288i \(0.315860\pi\)
\(140\) 33.9411i 0.242437i
\(141\) 0 0
\(142\) −72.0000 −0.507042
\(143\) 0 0
\(144\) 0 0
\(145\) − 18.0000i − 0.124138i
\(146\) − 22.6274i − 0.154982i
\(147\) 0 0
\(148\) − 68.0000i − 0.459459i
\(149\) 275.772 1.85082 0.925408 0.378972i \(-0.123722\pi\)
0.925408 + 0.378972i \(0.123722\pi\)
\(150\) 0 0
\(151\) − 148.000i − 0.980132i −0.871685 0.490066i \(-0.836973\pi\)
0.871685 0.490066i \(-0.163027\pi\)
\(152\) 45.2548i 0.297729i
\(153\) 0 0
\(154\) 96.0000i 0.623377i
\(155\) 186.676i 1.20436i
\(156\) 0 0
\(157\) −82.0000 −0.522293 −0.261146 0.965299i \(-0.584101\pi\)
−0.261146 + 0.965299i \(0.584101\pi\)
\(158\) −107.480 −0.680255
\(159\) 0 0
\(160\) −24.0000 −0.150000
\(161\) 67.8823 0.421629
\(162\) 0 0
\(163\) 56.0000i 0.343558i 0.985135 + 0.171779i \(0.0549515\pi\)
−0.985135 + 0.171779i \(0.945048\pi\)
\(164\) 93.3381 0.569135
\(165\) 0 0
\(166\) 168.000 1.01205
\(167\) −33.9411 −0.203240 −0.101620 0.994823i \(-0.532403\pi\)
−0.101620 + 0.994823i \(0.532403\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) − 76.3675i − 0.449221i
\(171\) 0 0
\(172\) 80.0000 0.465116
\(173\) 173.948i 1.00548i 0.864437 + 0.502741i \(0.167675\pi\)
−0.864437 + 0.502741i \(0.832325\pi\)
\(174\) 0 0
\(175\) 28.0000i 0.160000i
\(176\) −67.8823 −0.385695
\(177\) 0 0
\(178\) −18.0000 −0.101124
\(179\) 203.647i 1.13769i 0.822444 + 0.568846i \(0.192610\pi\)
−0.822444 + 0.568846i \(0.807390\pi\)
\(180\) 0 0
\(181\) 232.000 1.28177 0.640884 0.767638i \(-0.278568\pi\)
0.640884 + 0.767638i \(0.278568\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 48.0000i 0.260870i
\(185\) 144.250i 0.779729i
\(186\) 0 0
\(187\) − 216.000i − 1.15508i
\(188\) 169.706 0.902690
\(189\) 0 0
\(190\) − 96.0000i − 0.505263i
\(191\) − 33.9411i − 0.177702i −0.996045 0.0888511i \(-0.971680\pi\)
0.996045 0.0888511i \(-0.0283195\pi\)
\(192\) 0 0
\(193\) 206.000i 1.06736i 0.845687 + 0.533679i \(0.179191\pi\)
−0.845687 + 0.533679i \(0.820809\pi\)
\(194\) − 248.902i − 1.28300i
\(195\) 0 0
\(196\) 66.0000 0.336735
\(197\) 165.463 0.839914 0.419957 0.907544i \(-0.362045\pi\)
0.419957 + 0.907544i \(0.362045\pi\)
\(198\) 0 0
\(199\) −20.0000 −0.100503 −0.0502513 0.998737i \(-0.516002\pi\)
−0.0502513 + 0.998737i \(0.516002\pi\)
\(200\) −19.7990 −0.0989949
\(201\) 0 0
\(202\) − 42.0000i − 0.207921i
\(203\) 16.9706 0.0835988
\(204\) 0 0
\(205\) −198.000 −0.965854
\(206\) 39.5980 0.192223
\(207\) 0 0
\(208\) 0 0
\(209\) − 271.529i − 1.29918i
\(210\) 0 0
\(211\) 296.000 1.40284 0.701422 0.712746i \(-0.252549\pi\)
0.701422 + 0.712746i \(0.252549\pi\)
\(212\) 76.3675i 0.360224i
\(213\) 0 0
\(214\) 0 0
\(215\) −169.706 −0.789328
\(216\) 0 0
\(217\) −176.000 −0.811060
\(218\) − 79.1960i − 0.363284i
\(219\) 0 0
\(220\) 144.000 0.654545
\(221\) 0 0
\(222\) 0 0
\(223\) 436.000i 1.95516i 0.210571 + 0.977578i \(0.432468\pi\)
−0.210571 + 0.977578i \(0.567532\pi\)
\(224\) − 22.6274i − 0.101015i
\(225\) 0 0
\(226\) − 222.000i − 0.982301i
\(227\) 16.9706 0.0747602 0.0373801 0.999301i \(-0.488099\pi\)
0.0373801 + 0.999301i \(0.488099\pi\)
\(228\) 0 0
\(229\) 8.00000i 0.0349345i 0.999847 + 0.0174672i \(0.00556028\pi\)
−0.999847 + 0.0174672i \(0.994440\pi\)
\(230\) − 101.823i − 0.442710i
\(231\) 0 0
\(232\) 12.0000i 0.0517241i
\(233\) 12.7279i 0.0546263i 0.999627 + 0.0273131i \(0.00869512\pi\)
−0.999627 + 0.0273131i \(0.991305\pi\)
\(234\) 0 0
\(235\) −360.000 −1.53191
\(236\) −67.8823 −0.287637
\(237\) 0 0
\(238\) 72.0000 0.302521
\(239\) 135.765 0.568052 0.284026 0.958817i \(-0.408330\pi\)
0.284026 + 0.958817i \(0.408330\pi\)
\(240\) 0 0
\(241\) 32.0000i 0.132780i 0.997794 + 0.0663900i \(0.0211482\pi\)
−0.997794 + 0.0663900i \(0.978852\pi\)
\(242\) 236.174 0.975924
\(243\) 0 0
\(244\) 100.000 0.409836
\(245\) −140.007 −0.571458
\(246\) 0 0
\(247\) 0 0
\(248\) − 124.451i − 0.501818i
\(249\) 0 0
\(250\) 192.000 0.768000
\(251\) − 50.9117i − 0.202835i −0.994844 0.101418i \(-0.967662\pi\)
0.994844 0.101418i \(-0.0323379\pi\)
\(252\) 0 0
\(253\) − 288.000i − 1.13834i
\(254\) −130.108 −0.512235
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 182.434i 0.709858i 0.934893 + 0.354929i \(0.115495\pi\)
−0.934893 + 0.354929i \(0.884505\pi\)
\(258\) 0 0
\(259\) −136.000 −0.525097
\(260\) 0 0
\(261\) 0 0
\(262\) − 240.000i − 0.916031i
\(263\) 373.352i 1.41959i 0.704408 + 0.709795i \(0.251213\pi\)
−0.704408 + 0.709795i \(0.748787\pi\)
\(264\) 0 0
\(265\) − 162.000i − 0.611321i
\(266\) 90.5097 0.340262
\(267\) 0 0
\(268\) − 16.0000i − 0.0597015i
\(269\) − 343.654i − 1.27752i −0.769404 0.638762i \(-0.779447\pi\)
0.769404 0.638762i \(-0.220553\pi\)
\(270\) 0 0
\(271\) 380.000i 1.40221i 0.713056 + 0.701107i \(0.247310\pi\)
−0.713056 + 0.701107i \(0.752690\pi\)
\(272\) 50.9117i 0.187175i
\(273\) 0 0
\(274\) −222.000 −0.810219
\(275\) 118.794 0.431978
\(276\) 0 0
\(277\) 328.000 1.18412 0.592058 0.805896i \(-0.298316\pi\)
0.592058 + 0.805896i \(0.298316\pi\)
\(278\) 214.960 0.773239
\(279\) 0 0
\(280\) 48.0000i 0.171429i
\(281\) −284.257 −1.01159 −0.505795 0.862654i \(-0.668801\pi\)
−0.505795 + 0.862654i \(0.668801\pi\)
\(282\) 0 0
\(283\) 208.000 0.734982 0.367491 0.930027i \(-0.380217\pi\)
0.367491 + 0.930027i \(0.380217\pi\)
\(284\) −101.823 −0.358533
\(285\) 0 0
\(286\) 0 0
\(287\) − 186.676i − 0.650440i
\(288\) 0 0
\(289\) 127.000 0.439446
\(290\) − 25.4558i − 0.0877788i
\(291\) 0 0
\(292\) − 32.0000i − 0.109589i
\(293\) 436.992 1.49144 0.745720 0.666259i \(-0.232106\pi\)
0.745720 + 0.666259i \(0.232106\pi\)
\(294\) 0 0
\(295\) 144.000 0.488136
\(296\) − 96.1665i − 0.324887i
\(297\) 0 0
\(298\) 390.000 1.30872
\(299\) 0 0
\(300\) 0 0
\(301\) − 160.000i − 0.531561i
\(302\) − 209.304i − 0.693058i
\(303\) 0 0
\(304\) 64.0000i 0.210526i
\(305\) −212.132 −0.695515
\(306\) 0 0
\(307\) − 520.000i − 1.69381i −0.531743 0.846906i \(-0.678463\pi\)
0.531743 0.846906i \(-0.321537\pi\)
\(308\) 135.765i 0.440794i
\(309\) 0 0
\(310\) 264.000i 0.851613i
\(311\) 373.352i 1.20049i 0.799816 + 0.600245i \(0.204930\pi\)
−0.799816 + 0.600245i \(0.795070\pi\)
\(312\) 0 0
\(313\) −94.0000 −0.300319 −0.150160 0.988662i \(-0.547979\pi\)
−0.150160 + 0.988662i \(0.547979\pi\)
\(314\) −115.966 −0.369317
\(315\) 0 0
\(316\) −152.000 −0.481013
\(317\) 335.169 1.05731 0.528657 0.848835i \(-0.322696\pi\)
0.528657 + 0.848835i \(0.322696\pi\)
\(318\) 0 0
\(319\) − 72.0000i − 0.225705i
\(320\) −33.9411 −0.106066
\(321\) 0 0
\(322\) 96.0000 0.298137
\(323\) −203.647 −0.630485
\(324\) 0 0
\(325\) 0 0
\(326\) 79.1960i 0.242932i
\(327\) 0 0
\(328\) 132.000 0.402439
\(329\) − 339.411i − 1.03165i
\(330\) 0 0
\(331\) − 536.000i − 1.61934i −0.586889 0.809668i \(-0.699647\pi\)
0.586889 0.809668i \(-0.300353\pi\)
\(332\) 237.588 0.715626
\(333\) 0 0
\(334\) −48.0000 −0.143713
\(335\) 33.9411i 0.101317i
\(336\) 0 0
\(337\) 208.000 0.617211 0.308605 0.951190i \(-0.400138\pi\)
0.308605 + 0.951190i \(0.400138\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) − 108.000i − 0.317647i
\(341\) 746.705i 2.18975i
\(342\) 0 0
\(343\) − 328.000i − 0.956268i
\(344\) 113.137 0.328887
\(345\) 0 0
\(346\) 246.000i 0.710983i
\(347\) 288.500i 0.831411i 0.909499 + 0.415705i \(0.136465\pi\)
−0.909499 + 0.415705i \(0.863535\pi\)
\(348\) 0 0
\(349\) − 238.000i − 0.681948i −0.940073 0.340974i \(-0.889243\pi\)
0.940073 0.340974i \(-0.110757\pi\)
\(350\) 39.5980i 0.113137i
\(351\) 0 0
\(352\) −96.0000 −0.272727
\(353\) −224.860 −0.636997 −0.318499 0.947923i \(-0.603179\pi\)
−0.318499 + 0.947923i \(0.603179\pi\)
\(354\) 0 0
\(355\) 216.000 0.608451
\(356\) −25.4558 −0.0715052
\(357\) 0 0
\(358\) 288.000i 0.804469i
\(359\) 560.029 1.55997 0.779984 0.625799i \(-0.215227\pi\)
0.779984 + 0.625799i \(0.215227\pi\)
\(360\) 0 0
\(361\) 105.000 0.290859
\(362\) 328.098 0.906347
\(363\) 0 0
\(364\) 0 0
\(365\) 67.8823i 0.185979i
\(366\) 0 0
\(367\) 284.000 0.773842 0.386921 0.922113i \(-0.373539\pi\)
0.386921 + 0.922113i \(0.373539\pi\)
\(368\) 67.8823i 0.184463i
\(369\) 0 0
\(370\) 204.000i 0.551351i
\(371\) 152.735 0.411685
\(372\) 0 0
\(373\) −190.000 −0.509383 −0.254692 0.967022i \(-0.581974\pi\)
−0.254692 + 0.967022i \(0.581974\pi\)
\(374\) − 305.470i − 0.816765i
\(375\) 0 0
\(376\) 240.000 0.638298
\(377\) 0 0
\(378\) 0 0
\(379\) 160.000i 0.422164i 0.977468 + 0.211082i \(0.0676986\pi\)
−0.977468 + 0.211082i \(0.932301\pi\)
\(380\) − 135.765i − 0.357275i
\(381\) 0 0
\(382\) − 48.0000i − 0.125654i
\(383\) −271.529 −0.708953 −0.354477 0.935065i \(-0.615341\pi\)
−0.354477 + 0.935065i \(0.615341\pi\)
\(384\) 0 0
\(385\) − 288.000i − 0.748052i
\(386\) 291.328i 0.754736i
\(387\) 0 0
\(388\) − 352.000i − 0.907216i
\(389\) 403.051i 1.03612i 0.855344 + 0.518060i \(0.173346\pi\)
−0.855344 + 0.518060i \(0.826654\pi\)
\(390\) 0 0
\(391\) −216.000 −0.552430
\(392\) 93.3381 0.238107
\(393\) 0 0
\(394\) 234.000 0.593909
\(395\) 322.441 0.816306
\(396\) 0 0
\(397\) 146.000i 0.367758i 0.982949 + 0.183879i \(0.0588655\pi\)
−0.982949 + 0.183879i \(0.941135\pi\)
\(398\) −28.2843 −0.0710660
\(399\) 0 0
\(400\) −28.0000 −0.0700000
\(401\) −326.683 −0.814672 −0.407336 0.913278i \(-0.633542\pi\)
−0.407336 + 0.913278i \(0.633542\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) − 59.3970i − 0.147022i
\(405\) 0 0
\(406\) 24.0000 0.0591133
\(407\) 576.999i 1.41769i
\(408\) 0 0
\(409\) − 368.000i − 0.899756i −0.893090 0.449878i \(-0.851468\pi\)
0.893090 0.449878i \(-0.148532\pi\)
\(410\) −280.014 −0.682962
\(411\) 0 0
\(412\) 56.0000 0.135922
\(413\) 135.765i 0.328728i
\(414\) 0 0
\(415\) −504.000 −1.21446
\(416\) 0 0
\(417\) 0 0
\(418\) − 384.000i − 0.918660i
\(419\) 390.323i 0.931558i 0.884901 + 0.465779i \(0.154226\pi\)
−0.884901 + 0.465779i \(0.845774\pi\)
\(420\) 0 0
\(421\) 40.0000i 0.0950119i 0.998871 + 0.0475059i \(0.0151273\pi\)
−0.998871 + 0.0475059i \(0.984873\pi\)
\(422\) 418.607 0.991960
\(423\) 0 0
\(424\) 108.000i 0.254717i
\(425\) − 89.0955i − 0.209636i
\(426\) 0 0
\(427\) − 200.000i − 0.468384i
\(428\) 0 0
\(429\) 0 0
\(430\) −240.000 −0.558140
\(431\) −152.735 −0.354374 −0.177187 0.984177i \(-0.556700\pi\)
−0.177187 + 0.984177i \(0.556700\pi\)
\(432\) 0 0
\(433\) −542.000 −1.25173 −0.625866 0.779931i \(-0.715254\pi\)
−0.625866 + 0.779931i \(0.715254\pi\)
\(434\) −248.902 −0.573506
\(435\) 0 0
\(436\) − 112.000i − 0.256881i
\(437\) −271.529 −0.621348
\(438\) 0 0
\(439\) 4.00000 0.00911162 0.00455581 0.999990i \(-0.498550\pi\)
0.00455581 + 0.999990i \(0.498550\pi\)
\(440\) 203.647 0.462834
\(441\) 0 0
\(442\) 0 0
\(443\) 322.441i 0.727857i 0.931427 + 0.363929i \(0.118565\pi\)
−0.931427 + 0.363929i \(0.881435\pi\)
\(444\) 0 0
\(445\) 54.0000 0.121348
\(446\) 616.597i 1.38250i
\(447\) 0 0
\(448\) − 32.0000i − 0.0714286i
\(449\) −216.375 −0.481904 −0.240952 0.970537i \(-0.577460\pi\)
−0.240952 + 0.970537i \(0.577460\pi\)
\(450\) 0 0
\(451\) −792.000 −1.75610
\(452\) − 313.955i − 0.694592i
\(453\) 0 0
\(454\) 24.0000 0.0528634
\(455\) 0 0
\(456\) 0 0
\(457\) 400.000i 0.875274i 0.899152 + 0.437637i \(0.144184\pi\)
−0.899152 + 0.437637i \(0.855816\pi\)
\(458\) 11.3137i 0.0247024i
\(459\) 0 0
\(460\) − 144.000i − 0.313043i
\(461\) −301.227 −0.653422 −0.326711 0.945124i \(-0.605940\pi\)
−0.326711 + 0.945124i \(0.605940\pi\)
\(462\) 0 0
\(463\) − 604.000i − 1.30454i −0.757989 0.652268i \(-0.773818\pi\)
0.757989 0.652268i \(-0.226182\pi\)
\(464\) 16.9706i 0.0365745i
\(465\) 0 0
\(466\) 18.0000i 0.0386266i
\(467\) − 356.382i − 0.763130i −0.924342 0.381565i \(-0.875385\pi\)
0.924342 0.381565i \(-0.124615\pi\)
\(468\) 0 0
\(469\) −32.0000 −0.0682303
\(470\) −509.117 −1.08323
\(471\) 0 0
\(472\) −96.0000 −0.203390
\(473\) −678.823 −1.43514
\(474\) 0 0
\(475\) − 112.000i − 0.235789i
\(476\) 101.823 0.213915
\(477\) 0 0
\(478\) 192.000 0.401674
\(479\) −526.087 −1.09830 −0.549152 0.835723i \(-0.685049\pi\)
−0.549152 + 0.835723i \(0.685049\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 45.2548i 0.0938897i
\(483\) 0 0
\(484\) 334.000 0.690083
\(485\) 746.705i 1.53960i
\(486\) 0 0
\(487\) − 596.000i − 1.22382i −0.790928 0.611910i \(-0.790402\pi\)
0.790928 0.611910i \(-0.209598\pi\)
\(488\) 141.421 0.289798
\(489\) 0 0
\(490\) −198.000 −0.404082
\(491\) 271.529i 0.553012i 0.961012 + 0.276506i \(0.0891766\pi\)
−0.961012 + 0.276506i \(0.910823\pi\)
\(492\) 0 0
\(493\) −54.0000 −0.109533
\(494\) 0 0
\(495\) 0 0
\(496\) − 176.000i − 0.354839i
\(497\) 203.647i 0.409752i
\(498\) 0 0
\(499\) − 224.000i − 0.448898i −0.974486 0.224449i \(-0.927942\pi\)
0.974486 0.224449i \(-0.0720582\pi\)
\(500\) 271.529 0.543058
\(501\) 0 0
\(502\) − 72.0000i − 0.143426i
\(503\) 865.499i 1.72067i 0.509726 + 0.860337i \(0.329747\pi\)
−0.509726 + 0.860337i \(0.670253\pi\)
\(504\) 0 0
\(505\) 126.000i 0.249505i
\(506\) − 407.294i − 0.804928i
\(507\) 0 0
\(508\) −184.000 −0.362205
\(509\) 479.418 0.941883 0.470941 0.882164i \(-0.343914\pi\)
0.470941 + 0.882164i \(0.343914\pi\)
\(510\) 0 0
\(511\) −64.0000 −0.125245
\(512\) 22.6274 0.0441942
\(513\) 0 0
\(514\) 258.000i 0.501946i
\(515\) −118.794 −0.230668
\(516\) 0 0
\(517\) −1440.00 −2.78530
\(518\) −192.333 −0.371299
\(519\) 0 0
\(520\) 0 0
\(521\) − 521.845i − 1.00162i −0.865557 0.500811i \(-0.833035\pi\)
0.865557 0.500811i \(-0.166965\pi\)
\(522\) 0 0
\(523\) −736.000 −1.40727 −0.703633 0.710564i \(-0.748440\pi\)
−0.703633 + 0.710564i \(0.748440\pi\)
\(524\) − 339.411i − 0.647731i
\(525\) 0 0
\(526\) 528.000i 1.00380i
\(527\) 560.029 1.06267
\(528\) 0 0
\(529\) 241.000 0.455577
\(530\) − 229.103i − 0.432269i
\(531\) 0 0
\(532\) 128.000 0.240602
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) − 22.6274i − 0.0422153i
\(537\) 0 0
\(538\) − 486.000i − 0.903346i
\(539\) −560.029 −1.03901
\(540\) 0 0
\(541\) − 808.000i − 1.49353i −0.665088 0.746765i \(-0.731606\pi\)
0.665088 0.746765i \(-0.268394\pi\)
\(542\) 537.401i 0.991515i
\(543\) 0 0
\(544\) 72.0000i 0.132353i
\(545\) 237.588i 0.435941i
\(546\) 0 0
\(547\) 536.000 0.979890 0.489945 0.871753i \(-0.337017\pi\)
0.489945 + 0.871753i \(0.337017\pi\)
\(548\) −313.955 −0.572911
\(549\) 0 0
\(550\) 168.000 0.305455
\(551\) −67.8823 −0.123198
\(552\) 0 0
\(553\) 304.000i 0.549729i
\(554\) 463.862 0.837296
\(555\) 0 0
\(556\) 304.000 0.546763
\(557\) 165.463 0.297061 0.148531 0.988908i \(-0.452546\pi\)
0.148531 + 0.988908i \(0.452546\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 67.8823i 0.121218i
\(561\) 0 0
\(562\) −402.000 −0.715302
\(563\) 322.441i 0.572719i 0.958122 + 0.286359i \(0.0924451\pi\)
−0.958122 + 0.286359i \(0.907555\pi\)
\(564\) 0 0
\(565\) 666.000i 1.17876i
\(566\) 294.156 0.519711
\(567\) 0 0
\(568\) −144.000 −0.253521
\(569\) − 156.978i − 0.275883i −0.990440 0.137942i \(-0.955951\pi\)
0.990440 0.137942i \(-0.0440487\pi\)
\(570\) 0 0
\(571\) −368.000 −0.644483 −0.322242 0.946657i \(-0.604436\pi\)
−0.322242 + 0.946657i \(0.604436\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 264.000i − 0.459930i
\(575\) − 118.794i − 0.206598i
\(576\) 0 0
\(577\) 142.000i 0.246101i 0.992400 + 0.123050i \(0.0392676\pi\)
−0.992400 + 0.123050i \(0.960732\pi\)
\(578\) 179.605 0.310736
\(579\) 0 0
\(580\) − 36.0000i − 0.0620690i
\(581\) − 475.176i − 0.817858i
\(582\) 0 0
\(583\) − 648.000i − 1.11149i
\(584\) − 45.2548i − 0.0774912i
\(585\) 0 0
\(586\) 618.000 1.05461
\(587\) 373.352 0.636035 0.318017 0.948085i \(-0.396983\pi\)
0.318017 + 0.948085i \(0.396983\pi\)
\(588\) 0 0
\(589\) 704.000 1.19525
\(590\) 203.647 0.345164
\(591\) 0 0
\(592\) − 136.000i − 0.229730i
\(593\) −1107.33 −1.86733 −0.933667 0.358142i \(-0.883410\pi\)
−0.933667 + 0.358142i \(0.883410\pi\)
\(594\) 0 0
\(595\) −216.000 −0.363025
\(596\) 551.543 0.925408
\(597\) 0 0
\(598\) 0 0
\(599\) 797.616i 1.33158i 0.746139 + 0.665790i \(0.231905\pi\)
−0.746139 + 0.665790i \(0.768095\pi\)
\(600\) 0 0
\(601\) 158.000 0.262895 0.131448 0.991323i \(-0.458037\pi\)
0.131448 + 0.991323i \(0.458037\pi\)
\(602\) − 226.274i − 0.375871i
\(603\) 0 0
\(604\) − 296.000i − 0.490066i
\(605\) −708.521 −1.17111
\(606\) 0 0
\(607\) 332.000 0.546952 0.273476 0.961879i \(-0.411827\pi\)
0.273476 + 0.961879i \(0.411827\pi\)
\(608\) 90.5097i 0.148865i
\(609\) 0 0
\(610\) −300.000 −0.491803
\(611\) 0 0
\(612\) 0 0
\(613\) − 578.000i − 0.942904i −0.881892 0.471452i \(-0.843730\pi\)
0.881892 0.471452i \(-0.156270\pi\)
\(614\) − 735.391i − 1.19771i
\(615\) 0 0
\(616\) 192.000i 0.311688i
\(617\) −55.1543 −0.0893911 −0.0446956 0.999001i \(-0.514232\pi\)
−0.0446956 + 0.999001i \(0.514232\pi\)
\(618\) 0 0
\(619\) 896.000i 1.44750i 0.690064 + 0.723748i \(0.257582\pi\)
−0.690064 + 0.723748i \(0.742418\pi\)
\(620\) 373.352i 0.602181i
\(621\) 0 0
\(622\) 528.000i 0.848875i
\(623\) 50.9117i 0.0817202i
\(624\) 0 0
\(625\) −401.000 −0.641600
\(626\) −132.936 −0.212358
\(627\) 0 0
\(628\) −164.000 −0.261146
\(629\) 432.749 0.687996
\(630\) 0 0
\(631\) 20.0000i 0.0316957i 0.999874 + 0.0158479i \(0.00504474\pi\)
−0.999874 + 0.0158479i \(0.994955\pi\)
\(632\) −214.960 −0.340127
\(633\) 0 0
\(634\) 474.000 0.747634
\(635\) 390.323 0.614682
\(636\) 0 0
\(637\) 0 0
\(638\) − 101.823i − 0.159598i
\(639\) 0 0
\(640\) −48.0000 −0.0750000
\(641\) − 258.801i − 0.403746i −0.979412 0.201873i \(-0.935297\pi\)
0.979412 0.201873i \(-0.0647028\pi\)
\(642\) 0 0
\(643\) − 728.000i − 1.13219i −0.824339 0.566096i \(-0.808453\pi\)
0.824339 0.566096i \(-0.191547\pi\)
\(644\) 135.765 0.210814
\(645\) 0 0
\(646\) −288.000 −0.445820
\(647\) 458.205i 0.708200i 0.935208 + 0.354100i \(0.115213\pi\)
−0.935208 + 0.354100i \(0.884787\pi\)
\(648\) 0 0
\(649\) 576.000 0.887519
\(650\) 0 0
\(651\) 0 0
\(652\) 112.000i 0.171779i
\(653\) 301.227i 0.461298i 0.973037 + 0.230649i \(0.0740849\pi\)
−0.973037 + 0.230649i \(0.925915\pi\)
\(654\) 0 0
\(655\) 720.000i 1.09924i
\(656\) 186.676 0.284567
\(657\) 0 0
\(658\) − 480.000i − 0.729483i
\(659\) − 1052.17i − 1.59662i −0.602244 0.798312i \(-0.705727\pi\)
0.602244 0.798312i \(-0.294273\pi\)
\(660\) 0 0
\(661\) 62.0000i 0.0937973i 0.998900 + 0.0468986i \(0.0149338\pi\)
−0.998900 + 0.0468986i \(0.985066\pi\)
\(662\) − 758.018i − 1.14504i
\(663\) 0 0
\(664\) 336.000 0.506024
\(665\) −271.529 −0.408314
\(666\) 0 0
\(667\) −72.0000 −0.107946
\(668\) −67.8823 −0.101620
\(669\) 0 0
\(670\) 48.0000i 0.0716418i
\(671\) −848.528 −1.26457
\(672\) 0 0
\(673\) 670.000 0.995542 0.497771 0.867308i \(-0.334152\pi\)
0.497771 + 0.867308i \(0.334152\pi\)
\(674\) 294.156 0.436434
\(675\) 0 0
\(676\) 0 0
\(677\) − 1294.01i − 1.91138i −0.294372 0.955691i \(-0.595111\pi\)
0.294372 0.955691i \(-0.404889\pi\)
\(678\) 0 0
\(679\) −704.000 −1.03682
\(680\) − 152.735i − 0.224610i
\(681\) 0 0
\(682\) 1056.00i 1.54839i
\(683\) 560.029 0.819954 0.409977 0.912096i \(-0.365537\pi\)
0.409977 + 0.912096i \(0.365537\pi\)
\(684\) 0 0
\(685\) 666.000 0.972263
\(686\) − 463.862i − 0.676184i
\(687\) 0 0
\(688\) 160.000 0.232558
\(689\) 0 0
\(690\) 0 0
\(691\) 40.0000i 0.0578871i 0.999581 + 0.0289436i \(0.00921431\pi\)
−0.999581 + 0.0289436i \(0.990786\pi\)
\(692\) 347.897i 0.502741i
\(693\) 0 0
\(694\) 408.000i 0.587896i
\(695\) −644.881 −0.927887
\(696\) 0 0
\(697\) 594.000i 0.852224i
\(698\) − 336.583i − 0.482210i
\(699\) 0 0
\(700\) 56.0000i 0.0800000i
\(701\) − 954.594i − 1.36176i −0.732395 0.680880i \(-0.761597\pi\)
0.732395 0.680880i \(-0.238403\pi\)
\(702\) 0 0
\(703\) 544.000 0.773826
\(704\) −135.765 −0.192847
\(705\) 0 0
\(706\) −318.000 −0.450425
\(707\) −118.794 −0.168025
\(708\) 0 0
\(709\) 968.000i 1.36530i 0.730744 + 0.682652i \(0.239173\pi\)
−0.730744 + 0.682652i \(0.760827\pi\)
\(710\) 305.470 0.430240
\(711\) 0 0
\(712\) −36.0000 −0.0505618
\(713\) 746.705 1.04727
\(714\) 0 0
\(715\) 0 0
\(716\) 407.294i 0.568846i
\(717\) 0 0
\(718\) 792.000 1.10306
\(719\) − 1170.97i − 1.62861i −0.580439 0.814304i \(-0.697119\pi\)
0.580439 0.814304i \(-0.302881\pi\)
\(720\) 0 0
\(721\) − 112.000i − 0.155340i
\(722\) 148.492 0.205668
\(723\) 0 0
\(724\) 464.000 0.640884
\(725\) − 29.6985i − 0.0409634i
\(726\) 0 0
\(727\) 508.000 0.698762 0.349381 0.936981i \(-0.386392\pi\)
0.349381 + 0.936981i \(0.386392\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 96.0000i 0.131507i
\(731\) 509.117i 0.696466i
\(732\) 0 0
\(733\) 1144.00i 1.56071i 0.625337 + 0.780355i \(0.284961\pi\)
−0.625337 + 0.780355i \(0.715039\pi\)
\(734\) 401.637 0.547189
\(735\) 0 0
\(736\) 96.0000i 0.130435i
\(737\) 135.765i 0.184212i
\(738\) 0 0
\(739\) − 304.000i − 0.411367i −0.978619 0.205683i \(-0.934058\pi\)
0.978619 0.205683i \(-0.0659417\pi\)
\(740\) 288.500i 0.389864i
\(741\) 0 0
\(742\) 216.000 0.291105
\(743\) 848.528 1.14203 0.571015 0.820940i \(-0.306550\pi\)
0.571015 + 0.820940i \(0.306550\pi\)
\(744\) 0 0
\(745\) −1170.00 −1.57047
\(746\) −268.701 −0.360188
\(747\) 0 0
\(748\) − 432.000i − 0.577540i
\(749\) 0 0
\(750\) 0 0
\(751\) −188.000 −0.250333 −0.125166 0.992136i \(-0.539946\pi\)
−0.125166 + 0.992136i \(0.539946\pi\)
\(752\) 339.411 0.451345
\(753\) 0 0
\(754\) 0 0
\(755\) 627.911i 0.831670i
\(756\) 0 0
\(757\) −1240.00 −1.63804 −0.819022 0.573761i \(-0.805484\pi\)
−0.819022 + 0.573761i \(0.805484\pi\)
\(758\) 226.274i 0.298515i
\(759\) 0 0
\(760\) − 192.000i − 0.252632i
\(761\) 156.978 0.206278 0.103139 0.994667i \(-0.467111\pi\)
0.103139 + 0.994667i \(0.467111\pi\)
\(762\) 0 0
\(763\) −224.000 −0.293578
\(764\) − 67.8823i − 0.0888511i
\(765\) 0 0
\(766\) −384.000 −0.501305
\(767\) 0 0
\(768\) 0 0
\(769\) 910.000i 1.18336i 0.806175 + 0.591678i \(0.201534\pi\)
−0.806175 + 0.591678i \(0.798466\pi\)
\(770\) − 407.294i − 0.528953i
\(771\) 0 0
\(772\) 412.000i 0.533679i
\(773\) −1387.34 −1.79475 −0.897376 0.441266i \(-0.854529\pi\)
−0.897376 + 0.441266i \(0.854529\pi\)
\(774\) 0 0
\(775\) 308.000i 0.397419i
\(776\) − 497.803i − 0.641499i
\(777\) 0 0
\(778\) 570.000i 0.732648i
\(779\) 746.705i 0.958543i
\(780\) 0 0
\(781\) 864.000 1.10627
\(782\) −305.470 −0.390627
\(783\) 0 0
\(784\) 132.000 0.168367
\(785\) 347.897 0.443180
\(786\) 0 0
\(787\) − 1360.00i − 1.72808i −0.503422 0.864041i \(-0.667926\pi\)
0.503422 0.864041i \(-0.332074\pi\)
\(788\) 330.926 0.419957
\(789\) 0 0
\(790\) 456.000 0.577215
\(791\) −627.911 −0.793819
\(792\) 0 0
\(793\) 0 0
\(794\) 206.475i 0.260044i
\(795\) 0 0
\(796\) −40.0000 −0.0502513
\(797\) 106.066i 0.133082i 0.997784 + 0.0665408i \(0.0211963\pi\)
−0.997784 + 0.0665408i \(0.978804\pi\)
\(798\) 0 0
\(799\) 1080.00i 1.35169i
\(800\) −39.5980 −0.0494975
\(801\) 0 0
\(802\) −462.000 −0.576060
\(803\) 271.529i 0.338143i
\(804\) 0 0
\(805\) −288.000 −0.357764
\(806\) 0 0
\(807\) 0 0
\(808\) − 84.0000i − 0.103960i
\(809\) − 1107.33i − 1.36876i −0.729124 0.684381i \(-0.760072\pi\)
0.729124 0.684381i \(-0.239928\pi\)
\(810\) 0 0
\(811\) 160.000i 0.197287i 0.995123 + 0.0986436i \(0.0314504\pi\)
−0.995123 + 0.0986436i \(0.968550\pi\)
\(812\) 33.9411 0.0417994
\(813\) 0 0
\(814\) 816.000i 1.00246i
\(815\) − 237.588i − 0.291519i
\(816\) 0 0
\(817\) 640.000i 0.783354i
\(818\) − 520.431i − 0.636223i
\(819\) 0 0
\(820\) −396.000 −0.482927
\(821\) 436.992 0.532268 0.266134 0.963936i \(-0.414254\pi\)
0.266134 + 0.963936i \(0.414254\pi\)
\(822\) 0 0
\(823\) −332.000 −0.403402 −0.201701 0.979447i \(-0.564647\pi\)
−0.201701 + 0.979447i \(0.564647\pi\)
\(824\) 79.1960 0.0961116
\(825\) 0 0
\(826\) 192.000i 0.232446i
\(827\) −101.823 −0.123124 −0.0615619 0.998103i \(-0.519608\pi\)
−0.0615619 + 0.998103i \(0.519608\pi\)
\(828\) 0 0
\(829\) −632.000 −0.762364 −0.381182 0.924500i \(-0.624483\pi\)
−0.381182 + 0.924500i \(0.624483\pi\)
\(830\) −712.764 −0.858751
\(831\) 0 0
\(832\) 0 0
\(833\) 420.021i 0.504227i
\(834\) 0 0
\(835\) 144.000 0.172455
\(836\) − 543.058i − 0.649591i
\(837\) 0 0
\(838\) 552.000i 0.658711i
\(839\) −729.734 −0.869767 −0.434883 0.900487i \(-0.643210\pi\)
−0.434883 + 0.900487i \(0.643210\pi\)
\(840\) 0 0
\(841\) 823.000 0.978597
\(842\) 56.5685i 0.0671835i
\(843\) 0 0
\(844\) 592.000 0.701422
\(845\) 0 0
\(846\) 0 0
\(847\) − 668.000i − 0.788666i
\(848\) 152.735i 0.180112i
\(849\) 0 0
\(850\) − 126.000i − 0.148235i
\(851\) 576.999 0.678025
\(852\) 0 0
\(853\) 446.000i 0.522860i 0.965222 + 0.261430i \(0.0841941\pi\)
−0.965222 + 0.261430i \(0.915806\pi\)
\(854\) − 282.843i − 0.331198i
\(855\) 0 0
\(856\) 0 0
\(857\) 428.507i 0.500008i 0.968245 + 0.250004i \(0.0804319\pi\)
−0.968245 + 0.250004i \(0.919568\pi\)
\(858\) 0 0
\(859\) 728.000 0.847497 0.423749 0.905780i \(-0.360714\pi\)
0.423749 + 0.905780i \(0.360714\pi\)
\(860\) −339.411 −0.394664
\(861\) 0 0
\(862\) −216.000 −0.250580
\(863\) −916.410 −1.06189 −0.530945 0.847407i \(-0.678163\pi\)
−0.530945 + 0.847407i \(0.678163\pi\)
\(864\) 0 0
\(865\) − 738.000i − 0.853179i
\(866\) −766.504 −0.885108
\(867\) 0 0
\(868\) −352.000 −0.405530
\(869\) 1289.76 1.48419
\(870\) 0 0
\(871\) 0 0
\(872\) − 158.392i − 0.181642i
\(873\) 0 0
\(874\) −384.000 −0.439359
\(875\) − 543.058i − 0.620638i
\(876\) 0 0
\(877\) 910.000i 1.03763i 0.854887 + 0.518814i \(0.173626\pi\)
−0.854887 + 0.518814i \(0.826374\pi\)
\(878\) 5.65685 0.00644289
\(879\) 0 0
\(880\) 288.000 0.327273
\(881\) 929.138i 1.05464i 0.849667 + 0.527320i \(0.176803\pi\)
−0.849667 + 0.527320i \(0.823197\pi\)
\(882\) 0 0
\(883\) −1064.00 −1.20498 −0.602492 0.798125i \(-0.705825\pi\)
−0.602492 + 0.798125i \(0.705825\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 456.000i 0.514673i
\(887\) − 1391.59i − 1.56887i −0.620212 0.784434i \(-0.712953\pi\)
0.620212 0.784434i \(-0.287047\pi\)
\(888\) 0 0
\(889\) 368.000i 0.413948i
\(890\) 76.3675 0.0858062
\(891\) 0 0
\(892\) 872.000i 0.977578i
\(893\) 1357.65i 1.52032i
\(894\) 0 0
\(895\) − 864.000i − 0.965363i
\(896\) − 45.2548i − 0.0505076i
\(897\) 0 0
\(898\) −306.000 −0.340757
\(899\) 186.676 0.207649
\(900\) 0 0
\(901\) −486.000 −0.539401
\(902\) −1120.06 −1.24175
\(903\) 0 0
\(904\) − 444.000i − 0.491150i
\(905\) −984.293 −1.08762
\(906\) 0 0
\(907\) 1768.00 1.94928 0.974642 0.223771i \(-0.0718369\pi\)
0.974642 + 0.223771i \(0.0718369\pi\)
\(908\) 33.9411 0.0373801
\(909\) 0 0
\(910\) 0 0
\(911\) − 237.588i − 0.260799i −0.991462 0.130399i \(-0.958374\pi\)
0.991462 0.130399i \(-0.0416260\pi\)
\(912\) 0 0
\(913\) −2016.00 −2.20811
\(914\) 565.685i 0.618912i
\(915\) 0 0
\(916\) 16.0000i 0.0174672i
\(917\) −678.823 −0.740264
\(918\) 0 0
\(919\) 380.000 0.413493 0.206746 0.978395i \(-0.433712\pi\)
0.206746 + 0.978395i \(0.433712\pi\)
\(920\) − 203.647i − 0.221355i
\(921\) 0 0
\(922\) −426.000 −0.462039
\(923\) 0 0
\(924\) 0 0
\(925\) 238.000i 0.257297i
\(926\) − 854.185i − 0.922446i
\(927\) 0 0
\(928\) 24.0000i 0.0258621i
\(929\) 666.095 0.717002 0.358501 0.933529i \(-0.383288\pi\)
0.358501 + 0.933529i \(0.383288\pi\)
\(930\) 0 0
\(931\) 528.000i 0.567132i
\(932\) 25.4558i 0.0273131i
\(933\) 0 0
\(934\) − 504.000i − 0.539615i
\(935\) 916.410i 0.980118i
\(936\) 0 0
\(937\) −178.000 −0.189968 −0.0949840 0.995479i \(-0.530280\pi\)
−0.0949840 + 0.995479i \(0.530280\pi\)
\(938\) −45.2548 −0.0482461
\(939\) 0 0
\(940\) −720.000 −0.765957
\(941\) −436.992 −0.464391 −0.232196 0.972669i \(-0.574591\pi\)
−0.232196 + 0.972669i \(0.574591\pi\)
\(942\) 0 0
\(943\) 792.000i 0.839873i
\(944\) −135.765 −0.143818
\(945\) 0 0
\(946\) −960.000 −1.01480
\(947\) −1798.88 −1.89956 −0.949778 0.312924i \(-0.898691\pi\)
−0.949778 + 0.312924i \(0.898691\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) − 158.392i − 0.166728i
\(951\) 0 0
\(952\) 144.000 0.151261
\(953\) − 1310.98i − 1.37563i −0.725886 0.687815i \(-0.758570\pi\)
0.725886 0.687815i \(-0.241430\pi\)
\(954\) 0 0
\(955\) 144.000i 0.150785i
\(956\) 271.529 0.284026
\(957\) 0 0
\(958\) −744.000 −0.776618
\(959\) 627.911i 0.654756i
\(960\) 0 0
\(961\) −975.000 −1.01457
\(962\) 0 0
\(963\) 0 0
\(964\) 64.0000i 0.0663900i
\(965\) − 873.984i − 0.905683i
\(966\) 0 0
\(967\) − 1700.00i − 1.75801i −0.476808 0.879007i \(-0.658206\pi\)
0.476808 0.879007i \(-0.341794\pi\)
\(968\) 472.347 0.487962
\(969\) 0 0
\(970\) 1056.00i 1.08866i
\(971\) 458.205i 0.471890i 0.971766 + 0.235945i \(0.0758185\pi\)
−0.971766 + 0.235945i \(0.924181\pi\)
\(972\) 0 0
\(973\) − 608.000i − 0.624872i
\(974\) − 842.871i − 0.865371i
\(975\) 0 0
\(976\) 200.000 0.204918
\(977\) 759.433 0.777311 0.388655 0.921383i \(-0.372940\pi\)
0.388655 + 0.921383i \(0.372940\pi\)
\(978\) 0 0
\(979\) 216.000 0.220633
\(980\) −280.014 −0.285729
\(981\) 0 0
\(982\) 384.000i 0.391039i
\(983\) 1052.17 1.07037 0.535186 0.844734i \(-0.320242\pi\)
0.535186 + 0.844734i \(0.320242\pi\)
\(984\) 0 0
\(985\) −702.000 −0.712690
\(986\) −76.3675 −0.0774519
\(987\) 0 0
\(988\) 0 0
\(989\) 678.823i 0.686373i
\(990\) 0 0
\(991\) −772.000 −0.779011 −0.389506 0.921024i \(-0.627354\pi\)
−0.389506 + 0.921024i \(0.627354\pi\)
\(992\) − 248.902i − 0.250909i
\(993\) 0 0
\(994\) 288.000i 0.289738i
\(995\) 84.8528 0.0852792
\(996\) 0 0
\(997\) 194.000 0.194584 0.0972919 0.995256i \(-0.468982\pi\)
0.0972919 + 0.995256i \(0.468982\pi\)
\(998\) − 316.784i − 0.317419i
\(999\) 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 3042.3.d.a.3041.3 4
3.2 odd 2 inner 3042.3.d.a.3041.1 4
13.5 odd 4 3042.3.c.e.1691.1 2
13.8 odd 4 18.3.b.a.17.2 yes 2
13.12 even 2 inner 3042.3.d.a.3041.2 4
39.5 even 4 3042.3.c.e.1691.2 2
39.8 even 4 18.3.b.a.17.1 2
39.38 odd 2 inner 3042.3.d.a.3041.4 4
52.47 even 4 144.3.e.b.17.1 2
65.8 even 4 450.3.b.b.449.4 4
65.34 odd 4 450.3.d.f.251.1 2
65.47 even 4 450.3.b.b.449.1 4
91.34 even 4 882.3.b.a.197.2 2
91.47 even 12 882.3.s.d.557.2 4
91.60 odd 12 882.3.s.b.863.1 4
91.73 even 12 882.3.s.d.863.1 4
91.86 odd 12 882.3.s.b.557.2 4
104.21 odd 4 576.3.e.c.449.2 2
104.99 even 4 576.3.e.f.449.2 2
117.34 odd 12 162.3.d.b.53.2 4
117.47 even 12 162.3.d.b.53.1 4
117.86 even 12 162.3.d.b.107.2 4
117.112 odd 12 162.3.d.b.107.1 4
143.21 even 4 2178.3.c.d.485.1 2
156.47 odd 4 144.3.e.b.17.2 2
195.8 odd 4 450.3.b.b.449.2 4
195.47 odd 4 450.3.b.b.449.3 4
195.164 even 4 450.3.d.f.251.2 2
208.21 odd 4 2304.3.h.f.2177.3 4
208.99 even 4 2304.3.h.c.2177.2 4
208.125 odd 4 2304.3.h.f.2177.2 4
208.203 even 4 2304.3.h.c.2177.3 4
260.47 odd 4 3600.3.c.b.449.3 4
260.99 even 4 3600.3.l.d.1601.1 2
260.203 odd 4 3600.3.c.b.449.1 4
273.47 odd 12 882.3.s.d.557.1 4
273.86 even 12 882.3.s.b.557.1 4
273.125 odd 4 882.3.b.a.197.1 2
273.164 odd 12 882.3.s.d.863.2 4
273.242 even 12 882.3.s.b.863.2 4
312.125 even 4 576.3.e.c.449.1 2
312.203 odd 4 576.3.e.f.449.1 2
429.164 odd 4 2178.3.c.d.485.2 2
468.47 odd 12 1296.3.q.f.1025.1 4
468.151 even 12 1296.3.q.f.1025.2 4
468.203 odd 12 1296.3.q.f.593.2 4
468.463 even 12 1296.3.q.f.593.1 4
624.125 even 4 2304.3.h.f.2177.4 4
624.203 odd 4 2304.3.h.c.2177.1 4
624.437 even 4 2304.3.h.f.2177.1 4
624.515 odd 4 2304.3.h.c.2177.4 4
780.47 even 4 3600.3.c.b.449.4 4
780.203 even 4 3600.3.c.b.449.2 4
780.359 odd 4 3600.3.l.d.1601.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.3.b.a.17.1 2 39.8 even 4
18.3.b.a.17.2 yes 2 13.8 odd 4
144.3.e.b.17.1 2 52.47 even 4
144.3.e.b.17.2 2 156.47 odd 4
162.3.d.b.53.1 4 117.47 even 12
162.3.d.b.53.2 4 117.34 odd 12
162.3.d.b.107.1 4 117.112 odd 12
162.3.d.b.107.2 4 117.86 even 12
450.3.b.b.449.1 4 65.47 even 4
450.3.b.b.449.2 4 195.8 odd 4
450.3.b.b.449.3 4 195.47 odd 4
450.3.b.b.449.4 4 65.8 even 4
450.3.d.f.251.1 2 65.34 odd 4
450.3.d.f.251.2 2 195.164 even 4
576.3.e.c.449.1 2 312.125 even 4
576.3.e.c.449.2 2 104.21 odd 4
576.3.e.f.449.1 2 312.203 odd 4
576.3.e.f.449.2 2 104.99 even 4
882.3.b.a.197.1 2 273.125 odd 4
882.3.b.a.197.2 2 91.34 even 4
882.3.s.b.557.1 4 273.86 even 12
882.3.s.b.557.2 4 91.86 odd 12
882.3.s.b.863.1 4 91.60 odd 12
882.3.s.b.863.2 4 273.242 even 12
882.3.s.d.557.1 4 273.47 odd 12
882.3.s.d.557.2 4 91.47 even 12
882.3.s.d.863.1 4 91.73 even 12
882.3.s.d.863.2 4 273.164 odd 12
1296.3.q.f.593.1 4 468.463 even 12
1296.3.q.f.593.2 4 468.203 odd 12
1296.3.q.f.1025.1 4 468.47 odd 12
1296.3.q.f.1025.2 4 468.151 even 12
2178.3.c.d.485.1 2 143.21 even 4
2178.3.c.d.485.2 2 429.164 odd 4
2304.3.h.c.2177.1 4 624.203 odd 4
2304.3.h.c.2177.2 4 208.99 even 4
2304.3.h.c.2177.3 4 208.203 even 4
2304.3.h.c.2177.4 4 624.515 odd 4
2304.3.h.f.2177.1 4 624.437 even 4
2304.3.h.f.2177.2 4 208.125 odd 4
2304.3.h.f.2177.3 4 208.21 odd 4
2304.3.h.f.2177.4 4 624.125 even 4
3042.3.c.e.1691.1 2 13.5 odd 4
3042.3.c.e.1691.2 2 39.5 even 4
3042.3.d.a.3041.1 4 3.2 odd 2 inner
3042.3.d.a.3041.2 4 13.12 even 2 inner
3042.3.d.a.3041.3 4 1.1 even 1 trivial
3042.3.d.a.3041.4 4 39.38 odd 2 inner
3600.3.c.b.449.1 4 260.203 odd 4
3600.3.c.b.449.2 4 780.203 even 4
3600.3.c.b.449.3 4 260.47 odd 4
3600.3.c.b.449.4 4 780.47 even 4
3600.3.l.d.1601.1 2 260.99 even 4
3600.3.l.d.1601.2 2 780.359 odd 4