Properties

Label 441.2.e.h.226.1
Level $441$
Weight $2$
Character 441.226
Analytic conductor $3.521$
Analytic rank $0$
Dimension $4$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,2,Mod(226,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.226");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 226.1
Root \(-1.32288 + 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 441.226
Dual form 441.2.e.h.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.32288 + 2.29129i) q^{2} +(-2.50000 - 4.33013i) q^{4} +7.93725 q^{8} +O(q^{10})\) \(q+(-1.32288 + 2.29129i) q^{2} +(-2.50000 - 4.33013i) q^{4} +7.93725 q^{8} +(2.64575 + 4.58258i) q^{11} +(-5.50000 + 9.52628i) q^{16} -14.0000 q^{22} +(-2.64575 + 4.58258i) q^{23} +(2.50000 + 4.33013i) q^{25} -10.5830 q^{29} +(-6.61438 - 11.4564i) q^{32} +(-3.00000 + 5.19615i) q^{37} +12.0000 q^{43} +(13.2288 - 22.9129i) q^{44} +(-7.00000 - 12.1244i) q^{46} -13.2288 q^{50} +(5.29150 + 9.16515i) q^{53} +(14.0000 - 24.2487i) q^{58} +13.0000 q^{64} +(-2.00000 - 3.46410i) q^{67} -5.29150 q^{71} +(-7.93725 - 13.7477i) q^{74} +(-4.00000 + 6.92820i) q^{79} +(-15.8745 + 27.4955i) q^{86} +(21.0000 + 36.3731i) q^{88} +26.4575 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{4} - 22 q^{16} - 56 q^{22} + 10 q^{25} - 12 q^{37} + 48 q^{43} - 28 q^{46} + 56 q^{58} + 52 q^{64} - 8 q^{67} - 16 q^{79} + 84 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.32288 + 2.29129i −0.935414 + 1.62019i −0.161521 + 0.986869i \(0.551640\pi\)
−0.773893 + 0.633316i \(0.781693\pi\)
\(3\) 0 0
\(4\) −2.50000 4.33013i −1.25000 2.16506i
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 7.93725 2.80624
\(9\) 0 0
\(10\) 0 0
\(11\) 2.64575 + 4.58258i 0.797724 + 1.38170i 0.921095 + 0.389338i \(0.127296\pi\)
−0.123371 + 0.992361i \(0.539370\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −5.50000 + 9.52628i −1.37500 + 2.38157i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −14.0000 −2.98481
\(23\) −2.64575 + 4.58258i −0.551677 + 0.955533i 0.446476 + 0.894795i \(0.352679\pi\)
−0.998154 + 0.0607377i \(0.980655\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −10.5830 −1.96521 −0.982607 0.185695i \(-0.940546\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) −6.61438 11.4564i −1.16927 2.02523i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.00000 + 5.19615i −0.493197 + 0.854242i −0.999969 0.00783774i \(-0.997505\pi\)
0.506772 + 0.862080i \(0.330838\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 13.2288 22.9129i 1.99431 3.45425i
\(45\) 0 0
\(46\) −7.00000 12.1244i −1.03209 1.78764i
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −13.2288 −1.87083
\(51\) 0 0
\(52\) 0 0
\(53\) 5.29150 + 9.16515i 0.726844 + 1.25893i 0.958211 + 0.286064i \(0.0923469\pi\)
−0.231367 + 0.972867i \(0.574320\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 14.0000 24.2487i 1.83829 3.18401i
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 13.0000 1.62500
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 3.46410i −0.244339 0.423207i 0.717607 0.696449i \(-0.245238\pi\)
−0.961946 + 0.273241i \(0.911904\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.29150 −0.627986 −0.313993 0.949425i \(-0.601667\pi\)
−0.313993 + 0.949425i \(0.601667\pi\)
\(72\) 0 0
\(73\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(74\) −7.93725 13.7477i −0.922687 1.59814i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 + 6.92820i −0.450035 + 0.779484i −0.998388 0.0567635i \(-0.981922\pi\)
0.548352 + 0.836247i \(0.315255\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −15.8745 + 27.4955i −1.71179 + 2.96491i
\(87\) 0 0
\(88\) 21.0000 + 36.3731i 2.23861 + 3.87738i
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 26.4575 2.75839
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 12.5000 21.6506i 1.25000 2.16506i
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −28.0000 −2.71960
\(107\) 2.64575 4.58258i 0.255774 0.443014i −0.709331 0.704875i \(-0.751003\pi\)
0.965106 + 0.261861i \(0.0843362\pi\)
\(108\) 0 0
\(109\) 9.00000 + 15.5885i 0.862044 + 1.49310i 0.869953 + 0.493135i \(0.164149\pi\)
−0.00790932 + 0.999969i \(0.502518\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 21.1660 1.99113 0.995565 0.0940721i \(-0.0299884\pi\)
0.995565 + 0.0940721i \(0.0299884\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 26.4575 + 45.8258i 2.45652 + 4.25481i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.50000 + 14.7224i −0.772727 + 1.33840i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −3.96863 + 6.87386i −0.350780 + 0.607569i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 10.5830 0.914232
\(135\) 0 0
\(136\) 0 0
\(137\) −10.5830 18.3303i −0.904167 1.56606i −0.822031 0.569442i \(-0.807159\pi\)
−0.0821359 0.996621i \(-0.526174\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.00000 12.1244i 0.587427 1.01745i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 30.0000 2.46598
\(149\) −5.29150 + 9.16515i −0.433497 + 0.750838i −0.997172 0.0751583i \(-0.976054\pi\)
0.563675 + 0.825997i \(0.309387\pi\)
\(150\) 0 0
\(151\) −12.0000 20.7846i −0.976546 1.69143i −0.674735 0.738060i \(-0.735742\pi\)
−0.301811 0.953368i \(-0.597591\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(158\) −10.5830 18.3303i −0.841939 1.45828i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.0000 17.3205i 0.783260 1.35665i −0.146772 0.989170i \(-0.546888\pi\)
0.930033 0.367477i \(-0.119778\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −30.0000 51.9615i −2.28748 3.96203i
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −58.2065 −4.38748
\(177\) 0 0
\(178\) 0 0
\(179\) −13.2288 22.9129i −0.988764 1.71259i −0.623841 0.781551i \(-0.714429\pi\)
−0.364922 0.931038i \(-0.618904\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −21.0000 + 36.3731i −1.54814 + 2.68146i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.2288 22.9129i 0.957199 1.65792i 0.227946 0.973674i \(-0.426799\pi\)
0.729253 0.684244i \(-0.239868\pi\)
\(192\) 0 0
\(193\) 9.00000 + 15.5885i 0.647834 + 1.12208i 0.983639 + 0.180150i \(0.0576584\pi\)
−0.335805 + 0.941932i \(0.609008\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.5830 0.754008 0.377004 0.926212i \(-0.376954\pi\)
0.377004 + 0.926212i \(0.376954\pi\)
\(198\) 0 0
\(199\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(200\) 19.8431 + 34.3693i 1.40312 + 2.43028i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 26.4575 45.8258i 1.81711 3.14733i
\(213\) 0 0
\(214\) 7.00000 + 12.1244i 0.478510 + 0.828804i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −47.6235 −3.22547
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −28.0000 + 48.4974i −1.86253 + 3.22600i
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) 0 0
\(229\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −84.0000 −5.51487
\(233\) 10.5830 18.3303i 0.693316 1.20086i −0.277429 0.960746i \(-0.589482\pi\)
0.970745 0.240112i \(-0.0771842\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 26.4575 1.71139 0.855697 0.517477i \(-0.173129\pi\)
0.855697 + 0.517477i \(0.173129\pi\)
\(240\) 0 0
\(241\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(242\) −22.4889 38.9519i −1.44564 2.50392i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −28.0000 −1.76034
\(254\) −21.1660 + 36.6606i −1.32807 + 2.30029i
\(255\) 0 0
\(256\) 2.50000 + 4.33013i 0.156250 + 0.270633i
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2.64575 4.58258i −0.163144 0.282574i 0.772851 0.634588i \(-0.218830\pi\)
−0.935995 + 0.352014i \(0.885497\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −10.0000 + 17.3205i −0.610847 + 1.05802i
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 56.0000 3.38308
\(275\) −13.2288 + 22.9129i −0.797724 + 1.38170i
\(276\) 0 0
\(277\) 5.00000 + 8.66025i 0.300421 + 0.520344i 0.976231 0.216731i \(-0.0695395\pi\)
−0.675810 + 0.737075i \(0.736206\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −21.1660 −1.26266 −0.631329 0.775515i \(-0.717490\pi\)
−0.631329 + 0.775515i \(0.717490\pi\)
\(282\) 0 0
\(283\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(284\) 13.2288 + 22.9129i 0.784982 + 1.35963i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.50000 14.7224i 0.500000 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −23.8118 + 41.2432i −1.38403 + 2.39721i
\(297\) 0 0
\(298\) −14.0000 24.2487i −0.810998 1.40469i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 63.4980 3.65390
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 40.0000 2.25018
\(317\) 5.29150 9.16515i 0.297200 0.514766i −0.678294 0.734791i \(-0.737280\pi\)
0.975494 + 0.220024i \(0.0706137\pi\)
\(318\) 0 0
\(319\) −28.0000 48.4974i −1.56770 2.71533i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 26.4575 + 45.8258i 1.46535 + 2.53805i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 18.0000 31.1769i 0.989369 1.71364i 0.368744 0.929531i \(-0.379788\pi\)
0.620625 0.784107i \(-0.286879\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) 17.1974 29.7867i 0.935414 1.62019i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 95.2470 5.13538
\(345\) 0 0
\(346\) 0 0
\(347\) 18.5203 + 32.0780i 0.994220 + 1.72204i 0.590091 + 0.807337i \(0.299092\pi\)
0.404128 + 0.914702i \(0.367575\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 35.0000 60.6218i 1.86551 3.23115i
\(353\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 70.0000 3.69961
\(359\) −18.5203 + 32.0780i −0.977462 + 1.69301i −0.305903 + 0.952063i \(0.598958\pi\)
−0.671559 + 0.740951i \(0.734375\pi\)
\(360\) 0 0
\(361\) 9.50000 + 16.4545i 0.500000 + 0.866025i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(368\) −29.1033 50.4083i −1.51711 2.62772i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −11.0000 + 19.0526i −0.569558 + 0.986504i 0.427051 + 0.904227i \(0.359552\pi\)
−0.996610 + 0.0822766i \(0.973781\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 35.0000 + 60.6218i 1.79076 + 3.10168i
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −47.6235 −2.42397
\(387\) 0 0
\(388\) 0 0
\(389\) −5.29150 9.16515i −0.268290 0.464692i 0.700130 0.714015i \(-0.253125\pi\)
−0.968420 + 0.249323i \(0.919792\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −14.0000 + 24.2487i −0.705310 + 1.22163i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −55.0000 −2.75000
\(401\) −10.5830 + 18.3303i −0.528490 + 0.915372i 0.470958 + 0.882156i \(0.343908\pi\)
−0.999448 + 0.0332161i \(0.989425\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −31.7490 −1.57374
\(408\) 0 0
\(409\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −15.8745 + 27.4955i −0.772759 + 1.33846i
\(423\) 0 0
\(424\) 42.0000 + 72.7461i 2.03970 + 3.53286i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −26.4575 −1.27887
\(429\) 0 0
\(430\) 0 0
\(431\) 13.2288 + 22.9129i 0.637207 + 1.10367i 0.986043 + 0.166491i \(0.0532436\pi\)
−0.348836 + 0.937184i \(0.613423\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 45.0000 77.9423i 2.15511 3.73276i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.5203 32.0780i 0.879924 1.52407i 0.0285009 0.999594i \(-0.490927\pi\)
0.851423 0.524479i \(-0.175740\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −42.3320 −1.99777 −0.998886 0.0471929i \(-0.984972\pi\)
−0.998886 + 0.0471929i \(0.984972\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −52.9150 91.6515i −2.48891 4.31092i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.00000 + 5.19615i −0.140334 + 0.243066i −0.927622 0.373519i \(-0.878151\pi\)
0.787288 + 0.616585i \(0.211484\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 58.2065 100.817i 2.70217 4.68030i
\(465\) 0 0
\(466\) 28.0000 + 48.4974i 1.29707 + 2.24660i
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 31.7490 + 54.9909i 1.45982 + 2.52848i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −35.0000 + 60.6218i −1.60086 + 2.77278i
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 85.0000 3.86364
\(485\) 0 0
\(486\) 0 0
\(487\) −12.0000 20.7846i −0.543772 0.941841i −0.998683 0.0513038i \(-0.983662\pi\)
0.454911 0.890537i \(-0.349671\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5.29150 0.238802 0.119401 0.992846i \(-0.461903\pi\)
0.119401 + 0.992846i \(0.461903\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 18.0000 31.1769i 0.805791 1.39567i −0.109965 0.993935i \(-0.535074\pi\)
0.915756 0.401735i \(-0.131593\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 37.0405 64.1561i 1.64665 2.85208i
\(507\) 0 0
\(508\) −40.0000 69.2820i −1.77471 3.07389i
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −29.1033 −1.28619
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(522\) 0 0
\(523\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 14.0000 0.610429
\(527\) 0 0
\(528\) 0 0
\(529\) −2.50000 4.33013i −0.108696 0.188266i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −15.8745 27.4955i −0.685674 1.18762i
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 17.0000 29.4449i 0.730887 1.26593i −0.225617 0.974216i \(-0.572440\pi\)
0.956504 0.291718i \(-0.0942267\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) −52.9150 + 91.6515i −2.26042 + 3.91516i
\(549\) 0 0
\(550\) −35.0000 60.6218i −1.49241 2.58492i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −26.4575 −1.12407
\(555\) 0 0
\(556\) 0 0
\(557\) −5.29150 9.16515i −0.224208 0.388340i 0.731873 0.681441i \(-0.238646\pi\)
−0.956082 + 0.293101i \(0.905313\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 28.0000 48.4974i 1.18111 2.04574i
\(563\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −42.0000 −1.76228
\(569\) −21.1660 + 36.6606i −0.887325 + 1.53689i −0.0443003 + 0.999018i \(0.514106\pi\)
−0.843025 + 0.537874i \(0.819228\pi\)
\(570\) 0 0
\(571\) −2.00000 3.46410i −0.0836974 0.144968i 0.821138 0.570730i \(-0.193340\pi\)
−0.904835 + 0.425762i \(0.860006\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −26.4575 −1.10335
\(576\) 0 0
\(577\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(578\) 22.4889 + 38.9519i 0.935414 + 1.62019i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −28.0000 + 48.4974i −1.15964 + 2.00856i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −33.0000 57.1577i −1.35629 2.34917i
\(593\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 52.9150 2.16748
\(597\) 0 0
\(598\) 0 0
\(599\) −18.5203 32.0780i −0.756717 1.31067i −0.944516 0.328465i \(-0.893469\pi\)
0.187799 0.982208i \(-0.439865\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −60.0000 + 103.923i −2.44137 + 4.22857i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 19.0000 + 32.9090i 0.767403 + 1.32918i 0.938967 + 0.344008i \(0.111785\pi\)
−0.171564 + 0.985173i \(0.554882\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 42.3320 1.70422 0.852111 0.523360i \(-0.175322\pi\)
0.852111 + 0.523360i \(0.175322\pi\)
\(618\) 0 0
\(619\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.5000 + 21.6506i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −31.7490 + 54.9909i −1.26291 + 2.18742i
\(633\) 0 0
\(634\) 14.0000 + 24.2487i 0.556011 + 0.963039i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 148.162 5.86579
\(639\) 0 0
\(640\) 0 0
\(641\) 10.5830 + 18.3303i 0.418004 + 0.724003i 0.995739 0.0922210i \(-0.0293966\pi\)
−0.577735 + 0.816224i \(0.696063\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −100.000 −3.91630
\(653\) 5.29150 9.16515i 0.207072 0.358660i −0.743719 0.668493i \(-0.766940\pi\)
0.950791 + 0.309833i \(0.100273\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −26.4575 −1.03064 −0.515319 0.856998i \(-0.672327\pi\)
−0.515319 + 0.856998i \(0.672327\pi\)
\(660\) 0 0
\(661\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(662\) 47.6235 + 82.4864i 1.85094 + 3.20592i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 28.0000 48.4974i 1.08416 1.87783i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 39.6863 68.7386i 1.52866 2.64771i
\(675\) 0 0
\(676\) 32.5000 + 56.2917i 1.25000 + 2.16506i
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −2.64575 4.58258i −0.101237 0.175347i 0.810958 0.585105i \(-0.198947\pi\)
−0.912194 + 0.409757i \(0.865613\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −66.0000 + 114.315i −2.51623 + 4.35823i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −98.0000 −3.72003
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 52.9150 1.99857 0.999286 0.0377695i \(-0.0120253\pi\)
0.999286 + 0.0377695i \(0.0120253\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 34.3948 + 59.5735i 1.29630 + 2.24526i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.00000 + 5.19615i −0.112667 + 0.195146i −0.916845 0.399244i \(-0.869273\pi\)
0.804178 + 0.594389i \(0.202606\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −66.1438 + 114.564i −2.47191 + 4.28147i
\(717\) 0 0
\(718\) −49.0000 84.8705i −1.82866 3.16734i
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −50.2693 −1.87083
\(723\) 0 0
\(724\) 0 0
\(725\) −26.4575 45.8258i −0.982607 1.70193i
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 70.0000 2.58023
\(737\) 10.5830 18.3303i 0.389830 0.675205i
\(738\) 0 0
\(739\) 26.0000 + 45.0333i 0.956425 + 1.65658i 0.731072 + 0.682300i \(0.239020\pi\)
0.225354 + 0.974277i \(0.427646\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 37.0405 1.35888 0.679442 0.733729i \(-0.262222\pi\)
0.679442 + 0.733729i \(0.262222\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −29.1033 50.4083i −1.06555 1.84558i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −24.0000 + 41.5692i −0.875772 + 1.51688i −0.0198348 + 0.999803i \(0.506314\pi\)
−0.855938 + 0.517079i \(0.827019\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 54.0000 1.96266 0.981332 0.192323i \(-0.0616021\pi\)
0.981332 + 0.192323i \(0.0616021\pi\)
\(758\) −15.8745 + 27.4955i −0.576588 + 0.998680i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −132.288 −4.78600
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 45.0000 77.9423i 1.61959 2.80520i
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 28.0000 1.00385
\(779\) 0 0
\(780\) 0 0
\(781\) −14.0000 24.2487i −0.500959 0.867687i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(788\) −26.4575 45.8258i −0.942510 1.63247i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 33.0719 57.2822i 1.16927 2.02523i
\(801\) 0 0
\(802\) −28.0000 48.4974i −0.988714 1.71250i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −21.1660 36.6606i −0.744157 1.28892i −0.950587 0.310457i \(-0.899518\pi\)
0.206430 0.978461i \(-0.433815\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 42.0000 72.7461i 1.47210 2.54975i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 26.4575 45.8258i 0.923374 1.59933i 0.129217 0.991616i \(-0.458754\pi\)
0.794156 0.607714i \(-0.207913\pi\)
\(822\) 0 0
\(823\) −16.0000 27.7128i −0.557725 0.966008i −0.997686 0.0679910i \(-0.978341\pi\)
0.439961 0.898017i \(-0.354992\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37.0405 1.28803 0.644013 0.765015i \(-0.277268\pi\)
0.644013 + 0.765015i \(0.277268\pi\)
\(828\) 0 0
\(829\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 83.0000 2.86207
\(842\) 34.3948 59.5735i 1.18532 2.05304i
\(843\) 0 0
\(844\) −30.0000 51.9615i −1.03264 1.78859i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −116.413 −3.99764
\(849\) 0 0
\(850\) 0 0
\(851\) −15.8745 27.4955i −0.544171 0.942532i
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 21.0000 36.3731i 0.717765 1.24321i
\(857\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(858\) 0 0
\(859\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −70.0000 −2.38421
\(863\) 29.1033 50.4083i 0.990687 1.71592i 0.377424 0.926041i \(-0.376810\pi\)
0.613263 0.789879i \(-0.289857\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −42.3320 −1.43602
\(870\) 0 0
\(871\) 0 0
\(872\) 71.4353 + 123.730i 2.41910 + 4.19001i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −25.0000 + 43.3013i −0.844190 + 1.46218i 0.0421327 + 0.999112i \(0.486585\pi\)
−0.886323 + 0.463068i \(0.846749\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 49.0000 + 84.8705i 1.64619 + 2.85128i
\(887\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 56.0000 96.9948i 1.86874 3.23676i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 168.000 5.58760
\(905\) 0 0
\(906\) 0 0
\(907\) 30.0000 + 51.9615i 0.996134 + 1.72535i 0.574148 + 0.818752i \(0.305333\pi\)
0.421986 + 0.906602i \(0.361333\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 58.2065 1.92847 0.964234 0.265052i \(-0.0853891\pi\)
0.964234 + 0.265052i \(0.0853891\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −7.93725 13.7477i −0.262541 0.454734i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −24.0000 + 41.5692i −0.791687 + 1.37124i 0.133235 + 0.991084i \(0.457464\pi\)
−0.924922 + 0.380158i \(0.875870\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −30.0000 −0.986394
\(926\) 52.9150 91.6515i 1.73890 3.01186i
\(927\) 0 0
\(928\) 70.0000 + 121.244i 2.29786 + 3.98001i
\(929\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −105.830 −3.46658
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −168.000 −5.46215
\(947\) −29.1033 + 50.4083i −0.945729 + 1.63805i −0.191444 + 0.981504i \(0.561317\pi\)
−0.754285 + 0.656547i \(0.772016\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −21.1660 −0.685634 −0.342817 0.939402i \(-0.611381\pi\)
−0.342817 + 0.939402i \(0.611381\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −66.1438 114.564i −2.13924 3.70528i
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 15.5000 26.8468i 0.500000 0.866025i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) −67.4667 + 116.856i −2.16846 + 3.75588i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 63.4980 2.03461
\(975\) 0 0
\(976\) 0 0
\(977\) −21.1660 36.6606i −0.677161 1.17288i −0.975832 0.218521i \(-0.929877\pi\)
0.298672 0.954356i \(-0.403456\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −7.00000 + 12.1244i −0.223379 + 0.386904i
\(983\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −31.7490 + 54.9909i −1.00956 + 1.74861i
\(990\) 0 0
\(991\) −12.0000 20.7846i −0.381193 0.660245i 0.610040 0.792370i \(-0.291153\pi\)
−0.991233 + 0.132125i \(0.957820\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(998\) 47.6235 + 82.4864i 1.50750 + 2.61106i
\(999\) 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 441.2.e.h.226.1 4
3.2 odd 2 inner 441.2.e.h.226.2 4
7.2 even 3 441.2.a.h.1.2 yes 2
7.3 odd 6 inner 441.2.e.h.361.1 4
7.4 even 3 inner 441.2.e.h.361.1 4
7.5 odd 6 441.2.a.h.1.2 yes 2
7.6 odd 2 CM 441.2.e.h.226.1 4
21.2 odd 6 441.2.a.h.1.1 2
21.5 even 6 441.2.a.h.1.1 2
21.11 odd 6 inner 441.2.e.h.361.2 4
21.17 even 6 inner 441.2.e.h.361.2 4
21.20 even 2 inner 441.2.e.h.226.2 4
28.19 even 6 7056.2.a.co.1.2 2
28.23 odd 6 7056.2.a.co.1.2 2
84.23 even 6 7056.2.a.co.1.1 2
84.47 odd 6 7056.2.a.co.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.a.h.1.1 2 21.2 odd 6
441.2.a.h.1.1 2 21.5 even 6
441.2.a.h.1.2 yes 2 7.2 even 3
441.2.a.h.1.2 yes 2 7.5 odd 6
441.2.e.h.226.1 4 1.1 even 1 trivial
441.2.e.h.226.1 4 7.6 odd 2 CM
441.2.e.h.226.2 4 3.2 odd 2 inner
441.2.e.h.226.2 4 21.20 even 2 inner
441.2.e.h.361.1 4 7.3 odd 6 inner
441.2.e.h.361.1 4 7.4 even 3 inner
441.2.e.h.361.2 4 21.11 odd 6 inner
441.2.e.h.361.2 4 21.17 even 6 inner
7056.2.a.co.1.1 2 84.23 even 6
7056.2.a.co.1.1 2 84.47 odd 6
7056.2.a.co.1.2 2 28.19 even 6
7056.2.a.co.1.2 2 28.23 odd 6