Properties

Label 1323.2.c.c.1322.2
Level $1323$
Weight $2$
Character 1323.1322
Analytic conductor $10.564$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,2,Mod(1322,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1323.1322");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1322.2
Root \(1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1323.1322
Dual form 1323.2.c.c.1322.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} +2.44949 q^{5} -2.82843i q^{8} +O(q^{10})\) \(q-1.41421i q^{2} +2.44949 q^{5} -2.82843i q^{8} -3.46410i q^{10} -5.65685i q^{11} -3.46410i q^{13} -4.00000 q^{16} +2.44949 q^{17} +1.73205i q^{19} -8.00000 q^{22} +7.07107i q^{23} +1.00000 q^{25} -4.89898 q^{26} -1.41421i q^{29} +5.19615i q^{31} -3.46410i q^{34} -4.00000 q^{37} +2.44949 q^{38} -6.92820i q^{40} -2.44949 q^{41} -7.00000 q^{43} +10.0000 q^{46} +7.34847 q^{47} -1.41421i q^{50} -1.41421i q^{53} -13.8564i q^{55} -2.00000 q^{58} +14.6969 q^{59} -5.19615i q^{61} +7.34847 q^{62} -8.00000 q^{64} -8.48528i q^{65} +2.00000 q^{67} +2.82843i q^{71} -12.1244i q^{73} +5.65685i q^{74} -4.00000 q^{79} -9.79796 q^{80} +3.46410i q^{82} +9.79796 q^{83} +6.00000 q^{85} +9.89949i q^{86} -16.0000 q^{88} -2.44949 q^{89} -10.3923i q^{94} +4.24264i q^{95} +8.66025i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{16} - 32 q^{22} + 4 q^{25} - 16 q^{37} - 28 q^{43} + 40 q^{46} - 8 q^{58} - 32 q^{64} + 8 q^{67} - 16 q^{79} + 24 q^{85} - 64 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(3\) 0 0
\(4\) 0 0
\(5\) 2.44949 1.09545 0.547723 0.836660i \(-0.315495\pi\)
0.547723 + 0.836660i \(0.315495\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 2.82843i − 1.00000i
\(9\) 0 0
\(10\) − 3.46410i − 1.09545i
\(11\) − 5.65685i − 1.70561i −0.522233 0.852803i \(-0.674901\pi\)
0.522233 0.852803i \(-0.325099\pi\)
\(12\) 0 0
\(13\) − 3.46410i − 0.960769i −0.877058 0.480384i \(-0.840497\pi\)
0.877058 0.480384i \(-0.159503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 2.44949 0.594089 0.297044 0.954864i \(-0.403999\pi\)
0.297044 + 0.954864i \(0.403999\pi\)
\(18\) 0 0
\(19\) 1.73205i 0.397360i 0.980064 + 0.198680i \(0.0636654\pi\)
−0.980064 + 0.198680i \(0.936335\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −8.00000 −1.70561
\(23\) 7.07107i 1.47442i 0.675664 + 0.737210i \(0.263857\pi\)
−0.675664 + 0.737210i \(0.736143\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.89898 −0.960769
\(27\) 0 0
\(28\) 0 0
\(29\) − 1.41421i − 0.262613i −0.991342 0.131306i \(-0.958083\pi\)
0.991342 0.131306i \(-0.0419172\pi\)
\(30\) 0 0
\(31\) 5.19615i 0.933257i 0.884454 + 0.466628i \(0.154531\pi\)
−0.884454 + 0.466628i \(0.845469\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) − 3.46410i − 0.594089i
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 2.44949 0.397360
\(39\) 0 0
\(40\) − 6.92820i − 1.09545i
\(41\) −2.44949 −0.382546 −0.191273 0.981537i \(-0.561262\pi\)
−0.191273 + 0.981537i \(0.561262\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 10.0000 1.47442
\(47\) 7.34847 1.07188 0.535942 0.844255i \(-0.319956\pi\)
0.535942 + 0.844255i \(0.319956\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 1.41421i − 0.200000i
\(51\) 0 0
\(52\) 0 0
\(53\) − 1.41421i − 0.194257i −0.995272 0.0971286i \(-0.969034\pi\)
0.995272 0.0971286i \(-0.0309658\pi\)
\(54\) 0 0
\(55\) − 13.8564i − 1.86840i
\(56\) 0 0
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 14.6969 1.91338 0.956689 0.291111i \(-0.0940250\pi\)
0.956689 + 0.291111i \(0.0940250\pi\)
\(60\) 0 0
\(61\) − 5.19615i − 0.665299i −0.943051 0.332650i \(-0.892057\pi\)
0.943051 0.332650i \(-0.107943\pi\)
\(62\) 7.34847 0.933257
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) − 8.48528i − 1.05247i
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.82843i 0.335673i 0.985815 + 0.167836i \(0.0536780\pi\)
−0.985815 + 0.167836i \(0.946322\pi\)
\(72\) 0 0
\(73\) − 12.1244i − 1.41905i −0.704681 0.709524i \(-0.748910\pi\)
0.704681 0.709524i \(-0.251090\pi\)
\(74\) 5.65685i 0.657596i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −9.79796 −1.09545
\(81\) 0 0
\(82\) 3.46410i 0.382546i
\(83\) 9.79796 1.07547 0.537733 0.843115i \(-0.319281\pi\)
0.537733 + 0.843115i \(0.319281\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 9.89949i 1.06749i
\(87\) 0 0
\(88\) −16.0000 −1.70561
\(89\) −2.44949 −0.259645 −0.129823 0.991537i \(-0.541441\pi\)
−0.129823 + 0.991537i \(0.541441\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) − 10.3923i − 1.07188i
\(95\) 4.24264i 0.435286i
\(96\) 0 0
\(97\) 8.66025i 0.879316i 0.898165 + 0.439658i \(0.144900\pi\)
−0.898165 + 0.439658i \(0.855100\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.79796 −0.974933 −0.487467 0.873142i \(-0.662079\pi\)
−0.487467 + 0.873142i \(0.662079\pi\)
\(102\) 0 0
\(103\) 13.8564i 1.36531i 0.730740 + 0.682656i \(0.239175\pi\)
−0.730740 + 0.682656i \(0.760825\pi\)
\(104\) −9.79796 −0.960769
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) − 14.1421i − 1.36717i −0.729870 0.683586i \(-0.760419\pi\)
0.729870 0.683586i \(-0.239581\pi\)
\(108\) 0 0
\(109\) 17.0000 1.62830 0.814152 0.580651i \(-0.197202\pi\)
0.814152 + 0.580651i \(0.197202\pi\)
\(110\) −19.5959 −1.86840
\(111\) 0 0
\(112\) 0 0
\(113\) 2.82843i 0.266076i 0.991111 + 0.133038i \(0.0424732\pi\)
−0.991111 + 0.133038i \(0.957527\pi\)
\(114\) 0 0
\(115\) 17.3205i 1.61515i
\(116\) 0 0
\(117\) 0 0
\(118\) − 20.7846i − 1.91338i
\(119\) 0 0
\(120\) 0 0
\(121\) −21.0000 −1.90909
\(122\) −7.34847 −0.665299
\(123\) 0 0
\(124\) 0 0
\(125\) −9.79796 −0.876356
\(126\) 0 0
\(127\) −1.00000 −0.0887357 −0.0443678 0.999015i \(-0.514127\pi\)
−0.0443678 + 0.999015i \(0.514127\pi\)
\(128\) 11.3137i 1.00000i
\(129\) 0 0
\(130\) −12.0000 −1.05247
\(131\) 17.1464 1.49809 0.749045 0.662519i \(-0.230513\pi\)
0.749045 + 0.662519i \(0.230513\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 2.82843i − 0.244339i
\(135\) 0 0
\(136\) − 6.92820i − 0.594089i
\(137\) 7.07107i 0.604122i 0.953289 + 0.302061i \(0.0976746\pi\)
−0.953289 + 0.302061i \(0.902325\pi\)
\(138\) 0 0
\(139\) 6.92820i 0.587643i 0.955860 + 0.293821i \(0.0949270\pi\)
−0.955860 + 0.293821i \(0.905073\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.00000 0.335673
\(143\) −19.5959 −1.63869
\(144\) 0 0
\(145\) − 3.46410i − 0.287678i
\(146\) −17.1464 −1.41905
\(147\) 0 0
\(148\) 0 0
\(149\) − 9.89949i − 0.810998i −0.914095 0.405499i \(-0.867098\pi\)
0.914095 0.405499i \(-0.132902\pi\)
\(150\) 0 0
\(151\) 11.0000 0.895167 0.447584 0.894242i \(-0.352285\pi\)
0.447584 + 0.894242i \(0.352285\pi\)
\(152\) 4.89898 0.397360
\(153\) 0 0
\(154\) 0 0
\(155\) 12.7279i 1.02233i
\(156\) 0 0
\(157\) 10.3923i 0.829396i 0.909959 + 0.414698i \(0.136113\pi\)
−0.909959 + 0.414698i \(0.863887\pi\)
\(158\) 5.65685i 0.450035i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 23.0000 1.80150 0.900750 0.434339i \(-0.143018\pi\)
0.900750 + 0.434339i \(0.143018\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) − 13.8564i − 1.07547i
\(167\) 4.89898 0.379094 0.189547 0.981872i \(-0.439298\pi\)
0.189547 + 0.981872i \(0.439298\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) − 8.48528i − 0.650791i
\(171\) 0 0
\(172\) 0 0
\(173\) −7.34847 −0.558694 −0.279347 0.960190i \(-0.590118\pi\)
−0.279347 + 0.960190i \(0.590118\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 22.6274i 1.70561i
\(177\) 0 0
\(178\) 3.46410i 0.259645i
\(179\) − 9.89949i − 0.739923i −0.929047 0.369961i \(-0.879371\pi\)
0.929047 0.369961i \(-0.120629\pi\)
\(180\) 0 0
\(181\) 15.5885i 1.15868i 0.815086 + 0.579340i \(0.196690\pi\)
−0.815086 + 0.579340i \(0.803310\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 20.0000 1.47442
\(185\) −9.79796 −0.720360
\(186\) 0 0
\(187\) − 13.8564i − 1.01328i
\(188\) 0 0
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) − 1.41421i − 0.102329i −0.998690 0.0511645i \(-0.983707\pi\)
0.998690 0.0511645i \(-0.0162933\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 12.2474 0.879316
\(195\) 0 0
\(196\) 0 0
\(197\) 2.82843i 0.201517i 0.994911 + 0.100759i \(0.0321270\pi\)
−0.994911 + 0.100759i \(0.967873\pi\)
\(198\) 0 0
\(199\) − 1.73205i − 0.122782i −0.998114 0.0613909i \(-0.980446\pi\)
0.998114 0.0613909i \(-0.0195536\pi\)
\(200\) − 2.82843i − 0.200000i
\(201\) 0 0
\(202\) 13.8564i 0.974933i
\(203\) 0 0
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 19.5959 1.36531
\(207\) 0 0
\(208\) 13.8564i 0.960769i
\(209\) 9.79796 0.677739
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −20.0000 −1.36717
\(215\) −17.1464 −1.16938
\(216\) 0 0
\(217\) 0 0
\(218\) − 24.0416i − 1.62830i
\(219\) 0 0
\(220\) 0 0
\(221\) − 8.48528i − 0.570782i
\(222\) 0 0
\(223\) 13.8564i 0.927894i 0.885863 + 0.463947i \(0.153567\pi\)
−0.885863 + 0.463947i \(0.846433\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 4.00000 0.266076
\(227\) −2.44949 −0.162578 −0.0812892 0.996691i \(-0.525904\pi\)
−0.0812892 + 0.996691i \(0.525904\pi\)
\(228\) 0 0
\(229\) − 1.73205i − 0.114457i −0.998361 0.0572286i \(-0.981774\pi\)
0.998361 0.0572286i \(-0.0182264\pi\)
\(230\) 24.4949 1.61515
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) 15.5563i 1.01913i 0.860432 + 0.509565i \(0.170194\pi\)
−0.860432 + 0.509565i \(0.829806\pi\)
\(234\) 0 0
\(235\) 18.0000 1.17419
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 9.89949i − 0.640345i −0.947359 0.320173i \(-0.896259\pi\)
0.947359 0.320173i \(-0.103741\pi\)
\(240\) 0 0
\(241\) 12.1244i 0.780998i 0.920603 + 0.390499i \(0.127698\pi\)
−0.920603 + 0.390499i \(0.872302\pi\)
\(242\) 29.6985i 1.90909i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 14.6969 0.933257
\(249\) 0 0
\(250\) 13.8564i 0.876356i
\(251\) 7.34847 0.463831 0.231916 0.972736i \(-0.425501\pi\)
0.231916 + 0.972736i \(0.425501\pi\)
\(252\) 0 0
\(253\) 40.0000 2.51478
\(254\) 1.41421i 0.0887357i
\(255\) 0 0
\(256\) 0 0
\(257\) −19.5959 −1.22236 −0.611180 0.791492i \(-0.709305\pi\)
−0.611180 + 0.791492i \(0.709305\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) − 24.2487i − 1.49809i
\(263\) 19.7990i 1.22086i 0.792071 + 0.610429i \(0.209003\pi\)
−0.792071 + 0.610429i \(0.790997\pi\)
\(264\) 0 0
\(265\) − 3.46410i − 0.212798i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 24.4949 1.49348 0.746740 0.665116i \(-0.231618\pi\)
0.746740 + 0.665116i \(0.231618\pi\)
\(270\) 0 0
\(271\) 22.5167i 1.36779i 0.729581 + 0.683895i \(0.239715\pi\)
−0.729581 + 0.683895i \(0.760285\pi\)
\(272\) −9.79796 −0.594089
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) − 5.65685i − 0.341121i
\(276\) 0 0
\(277\) −25.0000 −1.50210 −0.751052 0.660243i \(-0.770453\pi\)
−0.751052 + 0.660243i \(0.770453\pi\)
\(278\) 9.79796 0.587643
\(279\) 0 0
\(280\) 0 0
\(281\) − 5.65685i − 0.337460i −0.985662 0.168730i \(-0.946033\pi\)
0.985662 0.168730i \(-0.0539665\pi\)
\(282\) 0 0
\(283\) 25.9808i 1.54440i 0.635382 + 0.772198i \(0.280843\pi\)
−0.635382 + 0.772198i \(0.719157\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 27.7128i 1.63869i
\(287\) 0 0
\(288\) 0 0
\(289\) −11.0000 −0.647059
\(290\) −4.89898 −0.287678
\(291\) 0 0
\(292\) 0 0
\(293\) −24.4949 −1.43101 −0.715504 0.698609i \(-0.753803\pi\)
−0.715504 + 0.698609i \(0.753803\pi\)
\(294\) 0 0
\(295\) 36.0000 2.09600
\(296\) 11.3137i 0.657596i
\(297\) 0 0
\(298\) −14.0000 −0.810998
\(299\) 24.4949 1.41658
\(300\) 0 0
\(301\) 0 0
\(302\) − 15.5563i − 0.895167i
\(303\) 0 0
\(304\) − 6.92820i − 0.397360i
\(305\) − 12.7279i − 0.728799i
\(306\) 0 0
\(307\) − 5.19615i − 0.296560i −0.988945 0.148280i \(-0.952626\pi\)
0.988945 0.148280i \(-0.0473737\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 18.0000 1.02233
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 5.19615i 0.293704i 0.989158 + 0.146852i \(0.0469141\pi\)
−0.989158 + 0.146852i \(0.953086\pi\)
\(314\) 14.6969 0.829396
\(315\) 0 0
\(316\) 0 0
\(317\) 2.82843i 0.158860i 0.996840 + 0.0794301i \(0.0253101\pi\)
−0.996840 + 0.0794301i \(0.974690\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) −19.5959 −1.09545
\(321\) 0 0
\(322\) 0 0
\(323\) 4.24264i 0.236067i
\(324\) 0 0
\(325\) − 3.46410i − 0.192154i
\(326\) − 32.5269i − 1.80150i
\(327\) 0 0
\(328\) 6.92820i 0.382546i
\(329\) 0 0
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) − 6.92820i − 0.379094i
\(335\) 4.89898 0.267660
\(336\) 0 0
\(337\) −1.00000 −0.0544735 −0.0272367 0.999629i \(-0.508671\pi\)
−0.0272367 + 0.999629i \(0.508671\pi\)
\(338\) − 1.41421i − 0.0769231i
\(339\) 0 0
\(340\) 0 0
\(341\) 29.3939 1.59177
\(342\) 0 0
\(343\) 0 0
\(344\) 19.7990i 1.06749i
\(345\) 0 0
\(346\) 10.3923i 0.558694i
\(347\) − 31.1127i − 1.67022i −0.550085 0.835109i \(-0.685405\pi\)
0.550085 0.835109i \(-0.314595\pi\)
\(348\) 0 0
\(349\) − 1.73205i − 0.0927146i −0.998925 0.0463573i \(-0.985239\pi\)
0.998925 0.0463573i \(-0.0147613\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 34.2929 1.82522 0.912612 0.408826i \(-0.134062\pi\)
0.912612 + 0.408826i \(0.134062\pi\)
\(354\) 0 0
\(355\) 6.92820i 0.367711i
\(356\) 0 0
\(357\) 0 0
\(358\) −14.0000 −0.739923
\(359\) 11.3137i 0.597115i 0.954392 + 0.298557i \(0.0965054\pi\)
−0.954392 + 0.298557i \(0.903495\pi\)
\(360\) 0 0
\(361\) 16.0000 0.842105
\(362\) 22.0454 1.15868
\(363\) 0 0
\(364\) 0 0
\(365\) − 29.6985i − 1.55449i
\(366\) 0 0
\(367\) − 8.66025i − 0.452062i −0.974120 0.226031i \(-0.927425\pi\)
0.974120 0.226031i \(-0.0725750\pi\)
\(368\) − 28.2843i − 1.47442i
\(369\) 0 0
\(370\) 13.8564i 0.720360i
\(371\) 0 0
\(372\) 0 0
\(373\) −1.00000 −0.0517780 −0.0258890 0.999665i \(-0.508242\pi\)
−0.0258890 + 0.999665i \(0.508242\pi\)
\(374\) −19.5959 −1.01328
\(375\) 0 0
\(376\) − 20.7846i − 1.07188i
\(377\) −4.89898 −0.252310
\(378\) 0 0
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2.00000 −0.102329
\(383\) −34.2929 −1.75228 −0.876142 0.482054i \(-0.839891\pi\)
−0.876142 + 0.482054i \(0.839891\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5.65685i 0.287926i
\(387\) 0 0
\(388\) 0 0
\(389\) − 26.8701i − 1.36237i −0.732113 0.681183i \(-0.761466\pi\)
0.732113 0.681183i \(-0.238534\pi\)
\(390\) 0 0
\(391\) 17.3205i 0.875936i
\(392\) 0 0
\(393\) 0 0
\(394\) 4.00000 0.201517
\(395\) −9.79796 −0.492989
\(396\) 0 0
\(397\) − 19.0526i − 0.956221i −0.878300 0.478110i \(-0.841322\pi\)
0.878300 0.478110i \(-0.158678\pi\)
\(398\) −2.44949 −0.122782
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) − 22.6274i − 1.12996i −0.825105 0.564980i \(-0.808884\pi\)
0.825105 0.564980i \(-0.191116\pi\)
\(402\) 0 0
\(403\) 18.0000 0.896644
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 22.6274i 1.12160i
\(408\) 0 0
\(409\) 10.3923i 0.513866i 0.966429 + 0.256933i \(0.0827120\pi\)
−0.966429 + 0.256933i \(0.917288\pi\)
\(410\) 8.48528i 0.419058i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) 0 0
\(417\) 0 0
\(418\) − 13.8564i − 0.677739i
\(419\) −2.44949 −0.119665 −0.0598327 0.998208i \(-0.519057\pi\)
−0.0598327 + 0.998208i \(0.519057\pi\)
\(420\) 0 0
\(421\) 11.0000 0.536107 0.268054 0.963404i \(-0.413620\pi\)
0.268054 + 0.963404i \(0.413620\pi\)
\(422\) 1.41421i 0.0688428i
\(423\) 0 0
\(424\) −4.00000 −0.194257
\(425\) 2.44949 0.118818
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 24.2487i 1.16938i
\(431\) 28.2843i 1.36241i 0.732095 + 0.681203i \(0.238543\pi\)
−0.732095 + 0.681203i \(0.761457\pi\)
\(432\) 0 0
\(433\) − 5.19615i − 0.249711i −0.992175 0.124856i \(-0.960153\pi\)
0.992175 0.124856i \(-0.0398468\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12.2474 −0.585875
\(438\) 0 0
\(439\) 20.7846i 0.991995i 0.868324 + 0.495998i \(0.165198\pi\)
−0.868324 + 0.495998i \(0.834802\pi\)
\(440\) −39.1918 −1.86840
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) 15.5563i 0.739104i 0.929210 + 0.369552i \(0.120489\pi\)
−0.929210 + 0.369552i \(0.879511\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 19.5959 0.927894
\(447\) 0 0
\(448\) 0 0
\(449\) − 35.3553i − 1.66852i −0.551370 0.834261i \(-0.685895\pi\)
0.551370 0.834261i \(-0.314105\pi\)
\(450\) 0 0
\(451\) 13.8564i 0.652473i
\(452\) 0 0
\(453\) 0 0
\(454\) 3.46410i 0.162578i
\(455\) 0 0
\(456\) 0 0
\(457\) 5.00000 0.233890 0.116945 0.993138i \(-0.462690\pi\)
0.116945 + 0.993138i \(0.462690\pi\)
\(458\) −2.44949 −0.114457
\(459\) 0 0
\(460\) 0 0
\(461\) 24.4949 1.14084 0.570421 0.821353i \(-0.306780\pi\)
0.570421 + 0.821353i \(0.306780\pi\)
\(462\) 0 0
\(463\) −1.00000 −0.0464739 −0.0232370 0.999730i \(-0.507397\pi\)
−0.0232370 + 0.999730i \(0.507397\pi\)
\(464\) 5.65685i 0.262613i
\(465\) 0 0
\(466\) 22.0000 1.01913
\(467\) 4.89898 0.226698 0.113349 0.993555i \(-0.463842\pi\)
0.113349 + 0.993555i \(0.463842\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 25.4558i − 1.17419i
\(471\) 0 0
\(472\) − 41.5692i − 1.91338i
\(473\) 39.5980i 1.82072i
\(474\) 0 0
\(475\) 1.73205i 0.0794719i
\(476\) 0 0
\(477\) 0 0
\(478\) −14.0000 −0.640345
\(479\) −2.44949 −0.111920 −0.0559600 0.998433i \(-0.517822\pi\)
−0.0559600 + 0.998433i \(0.517822\pi\)
\(480\) 0 0
\(481\) 13.8564i 0.631798i
\(482\) 17.1464 0.780998
\(483\) 0 0
\(484\) 0 0
\(485\) 21.2132i 0.963242i
\(486\) 0 0
\(487\) −13.0000 −0.589086 −0.294543 0.955638i \(-0.595167\pi\)
−0.294543 + 0.955638i \(0.595167\pi\)
\(488\) −14.6969 −0.665299
\(489\) 0 0
\(490\) 0 0
\(491\) − 5.65685i − 0.255290i −0.991820 0.127645i \(-0.959258\pi\)
0.991820 0.127645i \(-0.0407419\pi\)
\(492\) 0 0
\(493\) − 3.46410i − 0.156015i
\(494\) − 8.48528i − 0.381771i
\(495\) 0 0
\(496\) − 20.7846i − 0.933257i
\(497\) 0 0
\(498\) 0 0
\(499\) −25.0000 −1.11915 −0.559577 0.828778i \(-0.689036\pi\)
−0.559577 + 0.828778i \(0.689036\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 10.3923i − 0.463831i
\(503\) −22.0454 −0.982956 −0.491478 0.870890i \(-0.663543\pi\)
−0.491478 + 0.870890i \(0.663543\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) − 56.5685i − 2.51478i
\(507\) 0 0
\(508\) 0 0
\(509\) −19.5959 −0.868574 −0.434287 0.900775i \(-0.643000\pi\)
−0.434287 + 0.900775i \(0.643000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) 0 0
\(514\) 27.7128i 1.22236i
\(515\) 33.9411i 1.49562i
\(516\) 0 0
\(517\) − 41.5692i − 1.82821i
\(518\) 0 0
\(519\) 0 0
\(520\) −24.0000 −1.05247
\(521\) 24.4949 1.07314 0.536570 0.843856i \(-0.319720\pi\)
0.536570 + 0.843856i \(0.319720\pi\)
\(522\) 0 0
\(523\) − 3.46410i − 0.151475i −0.997128 0.0757373i \(-0.975869\pi\)
0.997128 0.0757373i \(-0.0241310\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 28.0000 1.22086
\(527\) 12.7279i 0.554437i
\(528\) 0 0
\(529\) −27.0000 −1.17391
\(530\) −4.89898 −0.212798
\(531\) 0 0
\(532\) 0 0
\(533\) 8.48528i 0.367538i
\(534\) 0 0
\(535\) − 34.6410i − 1.49766i
\(536\) − 5.65685i − 0.244339i
\(537\) 0 0
\(538\) − 34.6410i − 1.49348i
\(539\) 0 0
\(540\) 0 0
\(541\) −13.0000 −0.558914 −0.279457 0.960158i \(-0.590154\pi\)
−0.279457 + 0.960158i \(0.590154\pi\)
\(542\) 31.8434 1.36779
\(543\) 0 0
\(544\) 0 0
\(545\) 41.6413 1.78372
\(546\) 0 0
\(547\) −25.0000 −1.06892 −0.534461 0.845193i \(-0.679486\pi\)
−0.534461 + 0.845193i \(0.679486\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −8.00000 −0.341121
\(551\) 2.44949 0.104352
\(552\) 0 0
\(553\) 0 0
\(554\) 35.3553i 1.50210i
\(555\) 0 0
\(556\) 0 0
\(557\) − 1.41421i − 0.0599222i −0.999551 0.0299611i \(-0.990462\pi\)
0.999551 0.0299611i \(-0.00953833\pi\)
\(558\) 0 0
\(559\) 24.2487i 1.02561i
\(560\) 0 0
\(561\) 0 0
\(562\) −8.00000 −0.337460
\(563\) −22.0454 −0.929103 −0.464552 0.885546i \(-0.653784\pi\)
−0.464552 + 0.885546i \(0.653784\pi\)
\(564\) 0 0
\(565\) 6.92820i 0.291472i
\(566\) 36.7423 1.54440
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) 36.7696i 1.54146i 0.637162 + 0.770730i \(0.280108\pi\)
−0.637162 + 0.770730i \(0.719892\pi\)
\(570\) 0 0
\(571\) 41.0000 1.71580 0.857898 0.513820i \(-0.171770\pi\)
0.857898 + 0.513820i \(0.171770\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 7.07107i 0.294884i
\(576\) 0 0
\(577\) 3.46410i 0.144212i 0.997397 + 0.0721062i \(0.0229721\pi\)
−0.997397 + 0.0721062i \(0.977028\pi\)
\(578\) 15.5563i 0.647059i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) −34.2929 −1.41905
\(585\) 0 0
\(586\) 34.6410i 1.43101i
\(587\) −26.9444 −1.11211 −0.556057 0.831144i \(-0.687686\pi\)
−0.556057 + 0.831144i \(0.687686\pi\)
\(588\) 0 0
\(589\) −9.00000 −0.370839
\(590\) − 50.9117i − 2.09600i
\(591\) 0 0
\(592\) 16.0000 0.657596
\(593\) −9.79796 −0.402354 −0.201177 0.979555i \(-0.564477\pi\)
−0.201177 + 0.979555i \(0.564477\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) − 34.6410i − 1.41658i
\(599\) 7.07107i 0.288916i 0.989511 + 0.144458i \(0.0461439\pi\)
−0.989511 + 0.144458i \(0.953856\pi\)
\(600\) 0 0
\(601\) − 22.5167i − 0.918474i −0.888314 0.459237i \(-0.848123\pi\)
0.888314 0.459237i \(-0.151877\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −51.4393 −2.09130
\(606\) 0 0
\(607\) 29.4449i 1.19513i 0.801820 + 0.597565i \(0.203865\pi\)
−0.801820 + 0.597565i \(0.796135\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −18.0000 −0.728799
\(611\) − 25.4558i − 1.02983i
\(612\) 0 0
\(613\) −31.0000 −1.25208 −0.626039 0.779792i \(-0.715325\pi\)
−0.626039 + 0.779792i \(0.715325\pi\)
\(614\) −7.34847 −0.296560
\(615\) 0 0
\(616\) 0 0
\(617\) 45.2548i 1.82189i 0.412527 + 0.910946i \(0.364646\pi\)
−0.412527 + 0.910946i \(0.635354\pi\)
\(618\) 0 0
\(619\) − 34.6410i − 1.39234i −0.717877 0.696170i \(-0.754886\pi\)
0.717877 0.696170i \(-0.245114\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 7.34847 0.293704
\(627\) 0 0
\(628\) 0 0
\(629\) −9.79796 −0.390670
\(630\) 0 0
\(631\) 11.0000 0.437903 0.218952 0.975736i \(-0.429736\pi\)
0.218952 + 0.975736i \(0.429736\pi\)
\(632\) 11.3137i 0.450035i
\(633\) 0 0
\(634\) 4.00000 0.158860
\(635\) −2.44949 −0.0972050
\(636\) 0 0
\(637\) 0 0
\(638\) 11.3137i 0.447914i
\(639\) 0 0
\(640\) 27.7128i 1.09545i
\(641\) 7.07107i 0.279290i 0.990202 + 0.139645i \(0.0445962\pi\)
−0.990202 + 0.139645i \(0.955404\pi\)
\(642\) 0 0
\(643\) 22.5167i 0.887970i 0.896034 + 0.443985i \(0.146436\pi\)
−0.896034 + 0.443985i \(0.853564\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) 24.4949 0.962994 0.481497 0.876448i \(-0.340093\pi\)
0.481497 + 0.876448i \(0.340093\pi\)
\(648\) 0 0
\(649\) − 83.1384i − 3.26347i
\(650\) −4.89898 −0.192154
\(651\) 0 0
\(652\) 0 0
\(653\) 15.5563i 0.608767i 0.952550 + 0.304383i \(0.0984504\pi\)
−0.952550 + 0.304383i \(0.901550\pi\)
\(654\) 0 0
\(655\) 42.0000 1.64108
\(656\) 9.79796 0.382546
\(657\) 0 0
\(658\) 0 0
\(659\) 7.07107i 0.275450i 0.990471 + 0.137725i \(0.0439790\pi\)
−0.990471 + 0.137725i \(0.956021\pi\)
\(660\) 0 0
\(661\) 46.7654i 1.81896i 0.415745 + 0.909481i \(0.363521\pi\)
−0.415745 + 0.909481i \(0.636479\pi\)
\(662\) − 24.0416i − 0.934405i
\(663\) 0 0
\(664\) − 27.7128i − 1.07547i
\(665\) 0 0
\(666\) 0 0
\(667\) 10.0000 0.387202
\(668\) 0 0
\(669\) 0 0
\(670\) − 6.92820i − 0.267660i
\(671\) −29.3939 −1.13474
\(672\) 0 0
\(673\) −25.0000 −0.963679 −0.481840 0.876259i \(-0.660031\pi\)
−0.481840 + 0.876259i \(0.660031\pi\)
\(674\) 1.41421i 0.0544735i
\(675\) 0 0
\(676\) 0 0
\(677\) 14.6969 0.564849 0.282425 0.959289i \(-0.408861\pi\)
0.282425 + 0.959289i \(0.408861\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) − 16.9706i − 0.650791i
\(681\) 0 0
\(682\) − 41.5692i − 1.59177i
\(683\) 36.7696i 1.40695i 0.710721 + 0.703474i \(0.248369\pi\)
−0.710721 + 0.703474i \(0.751631\pi\)
\(684\) 0 0
\(685\) 17.3205i 0.661783i
\(686\) 0 0
\(687\) 0 0
\(688\) 28.0000 1.06749
\(689\) −4.89898 −0.186636
\(690\) 0 0
\(691\) − 5.19615i − 0.197671i −0.995104 0.0988355i \(-0.968488\pi\)
0.995104 0.0988355i \(-0.0315118\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −44.0000 −1.67022
\(695\) 16.9706i 0.643730i
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) −2.44949 −0.0927146
\(699\) 0 0
\(700\) 0 0
\(701\) 15.5563i 0.587555i 0.955874 + 0.293778i \(0.0949125\pi\)
−0.955874 + 0.293778i \(0.905087\pi\)
\(702\) 0 0
\(703\) − 6.92820i − 0.261302i
\(704\) 45.2548i 1.70561i
\(705\) 0 0
\(706\) − 48.4974i − 1.82522i
\(707\) 0 0
\(708\) 0 0
\(709\) 5.00000 0.187779 0.0938895 0.995583i \(-0.470070\pi\)
0.0938895 + 0.995583i \(0.470070\pi\)
\(710\) 9.79796 0.367711
\(711\) 0 0
\(712\) 6.92820i 0.259645i
\(713\) −36.7423 −1.37601
\(714\) 0 0
\(715\) −48.0000 −1.79510
\(716\) 0 0
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) −24.4949 −0.913506 −0.456753 0.889594i \(-0.650988\pi\)
−0.456753 + 0.889594i \(0.650988\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 22.6274i − 0.842105i
\(723\) 0 0
\(724\) 0 0
\(725\) − 1.41421i − 0.0525226i
\(726\) 0 0
\(727\) − 32.9090i − 1.22053i −0.792199 0.610263i \(-0.791064\pi\)
0.792199 0.610263i \(-0.208936\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −42.0000 −1.55449
\(731\) −17.1464 −0.634184
\(732\) 0 0
\(733\) − 43.3013i − 1.59937i −0.600420 0.799684i \(-0.705000\pi\)
0.600420 0.799684i \(-0.295000\pi\)
\(734\) −12.2474 −0.452062
\(735\) 0 0
\(736\) 0 0
\(737\) − 11.3137i − 0.416746i
\(738\) 0 0
\(739\) 5.00000 0.183928 0.0919640 0.995762i \(-0.470686\pi\)
0.0919640 + 0.995762i \(0.470686\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 35.3553i − 1.29706i −0.761188 0.648531i \(-0.775384\pi\)
0.761188 0.648531i \(-0.224616\pi\)
\(744\) 0 0
\(745\) − 24.2487i − 0.888404i
\(746\) 1.41421i 0.0517780i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −37.0000 −1.35015 −0.675075 0.737749i \(-0.735889\pi\)
−0.675075 + 0.737749i \(0.735889\pi\)
\(752\) −29.3939 −1.07188
\(753\) 0 0
\(754\) 6.92820i 0.252310i
\(755\) 26.9444 0.980607
\(756\) 0 0
\(757\) −7.00000 −0.254419 −0.127210 0.991876i \(-0.540602\pi\)
−0.127210 + 0.991876i \(0.540602\pi\)
\(758\) 14.1421i 0.513665i
\(759\) 0 0
\(760\) 12.0000 0.435286
\(761\) 9.79796 0.355176 0.177588 0.984105i \(-0.443171\pi\)
0.177588 + 0.984105i \(0.443171\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 48.4974i 1.75228i
\(767\) − 50.9117i − 1.83831i
\(768\) 0 0
\(769\) − 50.2295i − 1.81132i −0.424003 0.905661i \(-0.639376\pi\)
0.424003 0.905661i \(-0.360624\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −41.6413 −1.49773 −0.748867 0.662720i \(-0.769402\pi\)
−0.748867 + 0.662720i \(0.769402\pi\)
\(774\) 0 0
\(775\) 5.19615i 0.186651i
\(776\) 24.4949 0.879316
\(777\) 0 0
\(778\) −38.0000 −1.36237
\(779\) − 4.24264i − 0.152008i
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) 24.4949 0.875936
\(783\) 0 0
\(784\) 0 0
\(785\) 25.4558i 0.908558i
\(786\) 0 0
\(787\) 25.9808i 0.926114i 0.886328 + 0.463057i \(0.153248\pi\)
−0.886328 + 0.463057i \(0.846752\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 13.8564i 0.492989i
\(791\) 0 0
\(792\) 0 0
\(793\) −18.0000 −0.639199
\(794\) −26.9444 −0.956221
\(795\) 0 0
\(796\) 0 0
\(797\) −39.1918 −1.38825 −0.694123 0.719856i \(-0.744208\pi\)
−0.694123 + 0.719856i \(0.744208\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) 0 0
\(802\) −32.0000 −1.12996
\(803\) −68.5857 −2.42034
\(804\) 0 0
\(805\) 0 0
\(806\) − 25.4558i − 0.896644i
\(807\) 0 0
\(808\) 27.7128i 0.974933i
\(809\) 11.3137i 0.397769i 0.980023 + 0.198884i \(0.0637318\pi\)
−0.980023 + 0.198884i \(0.936268\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 32.0000 1.12160
\(815\) 56.3383 1.97344
\(816\) 0 0
\(817\) − 12.1244i − 0.424178i
\(818\) 14.6969 0.513866
\(819\) 0 0
\(820\) 0 0
\(821\) − 39.5980i − 1.38198i −0.722865 0.690990i \(-0.757175\pi\)
0.722865 0.690990i \(-0.242825\pi\)
\(822\) 0 0
\(823\) 23.0000 0.801730 0.400865 0.916137i \(-0.368710\pi\)
0.400865 + 0.916137i \(0.368710\pi\)
\(824\) 39.1918 1.36531
\(825\) 0 0
\(826\) 0 0
\(827\) − 22.6274i − 0.786832i −0.919360 0.393416i \(-0.871293\pi\)
0.919360 0.393416i \(-0.128707\pi\)
\(828\) 0 0
\(829\) − 32.9090i − 1.14298i −0.820611 0.571488i \(-0.806366\pi\)
0.820611 0.571488i \(-0.193634\pi\)
\(830\) − 33.9411i − 1.17811i
\(831\) 0 0
\(832\) 27.7128i 0.960769i
\(833\) 0 0
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) 0 0
\(838\) 3.46410i 0.119665i
\(839\) 24.4949 0.845658 0.422829 0.906210i \(-0.361037\pi\)
0.422829 + 0.906210i \(0.361037\pi\)
\(840\) 0 0
\(841\) 27.0000 0.931034
\(842\) − 15.5563i − 0.536107i
\(843\) 0 0
\(844\) 0 0
\(845\) 2.44949 0.0842650
\(846\) 0 0
\(847\) 0 0
\(848\) 5.65685i 0.194257i
\(849\) 0 0
\(850\) − 3.46410i − 0.118818i
\(851\) − 28.2843i − 0.969572i
\(852\) 0 0
\(853\) 8.66025i 0.296521i 0.988948 + 0.148261i \(0.0473675\pi\)
−0.988948 + 0.148261i \(0.952633\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −40.0000 −1.36717
\(857\) 4.89898 0.167346 0.0836730 0.996493i \(-0.473335\pi\)
0.0836730 + 0.996493i \(0.473335\pi\)
\(858\) 0 0
\(859\) 50.2295i 1.71381i 0.515476 + 0.856904i \(0.327615\pi\)
−0.515476 + 0.856904i \(0.672385\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 40.0000 1.36241
\(863\) − 1.41421i − 0.0481404i −0.999710 0.0240702i \(-0.992337\pi\)
0.999710 0.0240702i \(-0.00766252\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) −7.34847 −0.249711
\(867\) 0 0
\(868\) 0 0
\(869\) 22.6274i 0.767583i
\(870\) 0 0
\(871\) − 6.92820i − 0.234753i
\(872\) − 48.0833i − 1.62830i
\(873\) 0 0
\(874\) 17.3205i 0.585875i
\(875\) 0 0
\(876\) 0 0
\(877\) 35.0000 1.18187 0.590933 0.806721i \(-0.298760\pi\)
0.590933 + 0.806721i \(0.298760\pi\)
\(878\) 29.3939 0.991995
\(879\) 0 0
\(880\) 55.4256i 1.86840i
\(881\) 36.7423 1.23788 0.618941 0.785438i \(-0.287562\pi\)
0.618941 + 0.785438i \(0.287562\pi\)
\(882\) 0 0
\(883\) −37.0000 −1.24515 −0.622575 0.782560i \(-0.713913\pi\)
−0.622575 + 0.782560i \(0.713913\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 22.0000 0.739104
\(887\) 53.8888 1.80941 0.904704 0.426041i \(-0.140092\pi\)
0.904704 + 0.426041i \(0.140092\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 8.48528i 0.284427i
\(891\) 0 0
\(892\) 0 0
\(893\) 12.7279i 0.425924i
\(894\) 0 0
\(895\) − 24.2487i − 0.810545i
\(896\) 0 0
\(897\) 0 0
\(898\) −50.0000 −1.66852
\(899\) 7.34847 0.245085
\(900\) 0 0
\(901\) − 3.46410i − 0.115406i
\(902\) 19.5959 0.652473
\(903\) 0 0
\(904\) 8.00000 0.266076
\(905\) 38.1838i 1.26927i
\(906\) 0 0
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 52.3259i − 1.73363i −0.498626 0.866817i \(-0.666162\pi\)
0.498626 0.866817i \(-0.333838\pi\)
\(912\) 0 0
\(913\) − 55.4256i − 1.83432i
\(914\) − 7.07107i − 0.233890i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 5.00000 0.164935 0.0824674 0.996594i \(-0.473720\pi\)
0.0824674 + 0.996594i \(0.473720\pi\)
\(920\) 48.9898 1.61515
\(921\) 0 0
\(922\) − 34.6410i − 1.14084i
\(923\) 9.79796 0.322504
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 1.41421i 0.0464739i
\(927\) 0 0
\(928\) 0 0
\(929\) 22.0454 0.723286 0.361643 0.932317i \(-0.382216\pi\)
0.361643 + 0.932317i \(0.382216\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) − 6.92820i − 0.226698i
\(935\) − 33.9411i − 1.10999i
\(936\) 0 0
\(937\) − 10.3923i − 0.339502i −0.985487 0.169751i \(-0.945704\pi\)
0.985487 0.169751i \(-0.0542963\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 14.6969 0.479107 0.239553 0.970883i \(-0.422999\pi\)
0.239553 + 0.970883i \(0.422999\pi\)
\(942\) 0 0
\(943\) − 17.3205i − 0.564033i
\(944\) −58.7878 −1.91338
\(945\) 0 0
\(946\) 56.0000 1.82072
\(947\) − 9.89949i − 0.321690i −0.986980 0.160845i \(-0.948578\pi\)
0.986980 0.160845i \(-0.0514220\pi\)
\(948\) 0 0
\(949\) −42.0000 −1.36338
\(950\) 2.44949 0.0794719
\(951\) 0 0
\(952\) 0 0
\(953\) − 18.3848i − 0.595541i −0.954637 0.297771i \(-0.903757\pi\)
0.954637 0.297771i \(-0.0962431\pi\)
\(954\) 0 0
\(955\) − 3.46410i − 0.112096i
\(956\) 0 0
\(957\) 0 0
\(958\) 3.46410i 0.111920i
\(959\) 0 0
\(960\) 0 0
\(961\) 4.00000 0.129032
\(962\) 19.5959 0.631798
\(963\) 0 0
\(964\) 0 0
\(965\) −9.79796 −0.315407
\(966\) 0 0
\(967\) −34.0000 −1.09337 −0.546683 0.837340i \(-0.684110\pi\)
−0.546683 + 0.837340i \(0.684110\pi\)
\(968\) 59.3970i 1.90909i
\(969\) 0 0
\(970\) 30.0000 0.963242
\(971\) 12.2474 0.393039 0.196520 0.980500i \(-0.437036\pi\)
0.196520 + 0.980500i \(0.437036\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 18.3848i 0.589086i
\(975\) 0 0
\(976\) 20.7846i 0.665299i
\(977\) 36.7696i 1.17636i 0.808729 + 0.588181i \(0.200156\pi\)
−0.808729 + 0.588181i \(0.799844\pi\)
\(978\) 0 0
\(979\) 13.8564i 0.442853i
\(980\) 0 0
\(981\) 0 0
\(982\) −8.00000 −0.255290
\(983\) 4.89898 0.156253 0.0781266 0.996943i \(-0.475106\pi\)
0.0781266 + 0.996943i \(0.475106\pi\)
\(984\) 0 0
\(985\) 6.92820i 0.220751i
\(986\) −4.89898 −0.156015
\(987\) 0 0
\(988\) 0 0
\(989\) − 49.4975i − 1.57393i
\(990\) 0 0
\(991\) −58.0000 −1.84243 −0.921215 0.389053i \(-0.872802\pi\)
−0.921215 + 0.389053i \(0.872802\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 4.24264i − 0.134501i
\(996\) 0 0
\(997\) − 39.8372i − 1.26166i −0.775923 0.630828i \(-0.782715\pi\)
0.775923 0.630828i \(-0.217285\pi\)
\(998\) 35.3553i 1.11915i
\(999\) 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 1323.2.c.c.1322.2 4
3.2 odd 2 inner 1323.2.c.c.1322.3 4
7.2 even 3 189.2.p.b.80.1 yes 4
7.3 odd 6 189.2.p.b.26.2 yes 4
7.6 odd 2 inner 1323.2.c.c.1322.1 4
21.2 odd 6 189.2.p.b.80.2 yes 4
21.17 even 6 189.2.p.b.26.1 4
21.20 even 2 inner 1323.2.c.c.1322.4 4
63.2 odd 6 567.2.s.c.458.1 4
63.16 even 3 567.2.s.c.458.2 4
63.23 odd 6 567.2.i.e.269.2 4
63.31 odd 6 567.2.s.c.26.1 4
63.38 even 6 567.2.i.e.215.2 4
63.52 odd 6 567.2.i.e.215.1 4
63.58 even 3 567.2.i.e.269.1 4
63.59 even 6 567.2.s.c.26.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.p.b.26.1 4 21.17 even 6
189.2.p.b.26.2 yes 4 7.3 odd 6
189.2.p.b.80.1 yes 4 7.2 even 3
189.2.p.b.80.2 yes 4 21.2 odd 6
567.2.i.e.215.1 4 63.52 odd 6
567.2.i.e.215.2 4 63.38 even 6
567.2.i.e.269.1 4 63.58 even 3
567.2.i.e.269.2 4 63.23 odd 6
567.2.s.c.26.1 4 63.31 odd 6
567.2.s.c.26.2 4 63.59 even 6
567.2.s.c.458.1 4 63.2 odd 6
567.2.s.c.458.2 4 63.16 even 3
1323.2.c.c.1322.1 4 7.6 odd 2 inner
1323.2.c.c.1322.2 4 1.1 even 1 trivial
1323.2.c.c.1322.3 4 3.2 odd 2 inner
1323.2.c.c.1322.4 4 21.20 even 2 inner