Properties

Label 1216.2.t.a.1185.1
Level $1216$
Weight $2$
Character 1216.1185
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(353,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.353");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.t (of order \(6\), degree \(2\), 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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 1185.1
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1216.1185
Dual form 1216.2.t.a.353.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.98735 + 1.72474i) q^{3} +(4.44949 - 7.70674i) q^{9} +O(q^{10})\) \(q+(-2.98735 + 1.72474i) q^{3} +(4.44949 - 7.70674i) q^{9} +0.550510i q^{11} +(3.00000 + 5.19615i) q^{17} +(-2.98735 + 3.17423i) q^{19} +(-2.50000 + 4.33013i) q^{25} +20.3485i q^{27} +(-0.949490 - 1.64456i) q^{33} +(-6.39898 - 11.0834i) q^{41} +(8.66025 - 5.00000i) q^{43} -7.00000 q^{49} +(-17.9241 - 10.3485i) q^{51} +(3.44949 - 14.6349i) q^{57} +(-8.00853 + 4.62372i) q^{59} +(-12.4261 - 7.17423i) q^{67} +(7.84847 + 13.5939i) q^{73} -17.2474i q^{75} +(-21.7474 - 37.6677i) q^{81} -6.55051i q^{83} +(-9.00000 + 15.5885i) q^{89} +(-4.84847 - 8.39780i) q^{97} +(4.24264 + 2.44949i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{9} + 24 q^{17} - 20 q^{25} + 12 q^{33} - 12 q^{41} - 56 q^{49} + 8 q^{57} + 4 q^{73} - 76 q^{81} - 72 q^{89} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.98735 + 1.72474i −1.72474 + 0.995782i −0.816497 + 0.577350i \(0.804087\pi\)
−0.908248 + 0.418432i \(0.862580\pi\)
\(4\) 0 0
\(5\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 4.44949 7.70674i 1.48316 2.56891i
\(10\) 0 0
\(11\) 0.550510i 0.165985i 0.996550 + 0.0829925i \(0.0264478\pi\)
−0.996550 + 0.0829925i \(0.973552\pi\)
\(12\) 0 0
\(13\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 + 5.19615i 0.727607 + 1.26025i 0.957892 + 0.287129i \(0.0927008\pi\)
−0.230285 + 0.973123i \(0.573966\pi\)
\(18\) 0 0
\(19\) −2.98735 + 3.17423i −0.685344 + 0.728219i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) −2.50000 + 4.33013i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 20.3485i 3.91606i
\(28\) 0 0
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −0.949490 1.64456i −0.165285 0.286282i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.39898 11.0834i −0.999353 1.73093i −0.530831 0.847477i \(-0.678120\pi\)
−0.468521 0.883452i \(-0.655213\pi\)
\(42\) 0 0
\(43\) 8.66025 5.00000i 1.32068 0.762493i 0.336840 0.941562i \(-0.390642\pi\)
0.983836 + 0.179069i \(0.0573086\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −17.9241 10.3485i −2.50987 1.44908i
\(52\) 0 0
\(53\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.44949 14.6349i 0.456896 1.93845i
\(58\) 0 0
\(59\) −8.00853 + 4.62372i −1.04262 + 0.601958i −0.920575 0.390567i \(-0.872279\pi\)
−0.122047 + 0.992524i \(0.538946\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.4261 7.17423i −1.51809 0.876472i −0.999773 0.0212861i \(-0.993224\pi\)
−0.518321 0.855186i \(-0.673443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) 0 0
\(73\) 7.84847 + 13.5939i 0.918594 + 1.59105i 0.801553 + 0.597924i \(0.204008\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 17.2474i 1.99156i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) 0 0
\(81\) −21.7474 37.6677i −2.41638 4.18530i
\(82\) 0 0
\(83\) 6.55051i 0.719012i −0.933143 0.359506i \(-0.882945\pi\)
0.933143 0.359506i \(-0.117055\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.00000 + 15.5885i −0.953998 + 1.65237i −0.217354 + 0.976093i \(0.569742\pi\)
−0.736644 + 0.676280i \(0.763591\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.84847 8.39780i −0.492287 0.852667i 0.507673 0.861550i \(-0.330506\pi\)
−0.999961 + 0.00888289i \(0.997172\pi\)
\(98\) 0 0
\(99\) 4.24264 + 2.44949i 0.426401 + 0.246183i
\(100\) 0 0
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000i 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 0 0
\(109\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −18.7980 −1.76836 −0.884182 0.467143i \(-0.845283\pi\)
−0.884182 + 0.467143i \(0.845283\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.6969 0.972449
\(122\) 0 0
\(123\) 38.2319 + 22.0732i 3.44726 + 1.99027i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(128\) 0 0
\(129\) −17.2474 + 29.8735i −1.51855 + 2.63021i
\(130\) 0 0
\(131\) −18.4008 + 10.6237i −1.60769 + 0.928199i −0.617802 + 0.786334i \(0.711977\pi\)
−0.989886 + 0.141865i \(0.954690\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −11.2980 + 19.5686i −0.965250 + 1.67186i −0.256307 + 0.966595i \(0.582506\pi\)
−0.708942 + 0.705266i \(0.750827\pi\)
\(138\) 0 0
\(139\) −15.8902 9.17423i −1.34779 0.778148i −0.359856 0.933008i \(-0.617174\pi\)
−0.987937 + 0.154859i \(0.950508\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 20.9114 12.0732i 1.72474 0.995782i
\(148\) 0 0
\(149\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 53.3939 4.31664
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 23.0454i 1.80506i −0.430632 0.902528i \(-0.641709\pi\)
0.430632 0.902528i \(-0.358291\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(168\) 0 0
\(169\) −6.50000 11.2583i −0.500000 0.866025i
\(170\) 0 0
\(171\) 11.1708 + 37.1464i 0.854256 + 2.84066i
\(172\) 0 0
\(173\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 15.9495 27.6253i 1.19884 2.07645i
\(178\) 0 0
\(179\) 8.14643i 0.608893i 0.952529 + 0.304446i \(0.0984714\pi\)
−0.952529 + 0.304446i \(0.901529\pi\)
\(180\) 0 0
\(181\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.86054 + 1.65153i −0.209183 + 0.120772i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 11.0000 + 19.0526i 0.791797 + 1.37143i 0.924853 + 0.380325i \(0.124188\pi\)
−0.133056 + 0.991109i \(0.542479\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(200\) 0 0
\(201\) 49.4949 3.49110
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.74745 1.64456i −0.120874 0.113757i
\(210\) 0 0
\(211\) −12.1244 + 7.00000i −0.834675 + 0.481900i −0.855451 0.517884i \(-0.826720\pi\)
0.0207756 + 0.999784i \(0.493386\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −46.8922 27.0732i −3.16868 1.82944i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(224\) 0 0
\(225\) 22.2474 + 38.5337i 1.48316 + 2.56891i
\(226\) 0 0
\(227\) 17.4495i 1.15816i 0.815270 + 0.579082i \(0.196589\pi\)
−0.815270 + 0.579082i \(0.803411\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.94949 17.2330i −0.651813 1.12897i −0.982683 0.185296i \(-0.940675\pi\)
0.330870 0.943676i \(-0.392658\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −0.848469 + 1.46959i −0.0546547 + 0.0946647i −0.892058 0.451920i \(-0.850739\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) 77.0674 + 44.4949i 4.94388 + 2.85435i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 11.2980 + 19.5686i 0.715979 + 1.24011i
\(250\) 0 0
\(251\) 20.7364 + 11.9722i 1.30887 + 0.755678i 0.981908 0.189358i \(-0.0606408\pi\)
0.326965 + 0.945036i \(0.393974\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.60102 + 4.50510i −0.162247 + 0.281020i −0.935674 0.352865i \(-0.885208\pi\)
0.773427 + 0.633885i \(0.218541\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 62.0908i 3.79990i
\(268\) 0 0
\(269\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.38378 1.37628i −0.143747 0.0829925i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.74745 + 13.4190i −0.462174 + 0.800509i −0.999069 0.0431402i \(-0.986264\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 28.6182 16.5227i 1.70117 0.982173i 0.756596 0.653882i \(-0.226861\pi\)
0.944577 0.328291i \(-0.106473\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −9.50000 + 16.4545i −0.558824 + 0.967911i
\(290\) 0 0
\(291\) 28.9681 + 16.7247i 1.69814 + 0.980422i
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −11.2020 −0.650008
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −8.35847 + 4.82577i −0.477043 + 0.275421i −0.719183 0.694820i \(-0.755484\pi\)
0.242140 + 0.970241i \(0.422151\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −12.1969 + 21.1257i −0.689412 + 1.19410i 0.282617 + 0.959233i \(0.408798\pi\)
−0.972028 + 0.234863i \(0.924536\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 10.3485 + 17.9241i 0.577595 + 1.00042i
\(322\) 0 0
\(323\) −25.4558 6.00000i −1.41640 0.333849i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9.04541i 0.497181i 0.968609 + 0.248590i \(0.0799673\pi\)
−0.968609 + 0.248590i \(0.920033\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 11.1969 + 19.3937i 0.609936 + 1.05644i 0.991250 + 0.131995i \(0.0421382\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 56.1560 32.4217i 3.04998 1.76090i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −24.9791 + 14.4217i −1.34095 + 0.774197i −0.986947 0.161048i \(-0.948512\pi\)
−0.354001 + 0.935245i \(0.615179\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4.59592 −0.244616 −0.122308 0.992492i \(-0.539030\pi\)
−0.122308 + 0.992492i \(0.539030\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(360\) 0 0
\(361\) −1.15153 18.9651i −0.0606069 0.998162i
\(362\) 0 0
\(363\) −31.9555 + 18.4495i −1.67723 + 0.968347i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(368\) 0 0
\(369\) −113.889 −5.92881
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 38.0000i 1.95193i −0.217930 0.975964i \(-0.569930\pi\)
0.217930 0.975964i \(-0.430070\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 88.9898i 4.52361i
\(388\) 0 0
\(389\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 36.6464 63.4735i 1.84857 3.20181i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.6464 32.2966i −0.931158 1.61281i −0.781345 0.624099i \(-0.785466\pi\)
−0.149813 0.988714i \(-0.547867\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 9.19694 15.9296i 0.454759 0.787666i −0.543915 0.839140i \(-0.683059\pi\)
0.998674 + 0.0514740i \(0.0163919\pi\)
\(410\) 0 0
\(411\) 77.9444i 3.84471i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 63.2929 3.09946
\(418\) 0 0
\(419\) 18.0000i 0.879358i 0.898155 + 0.439679i \(0.144908\pi\)
−0.898155 + 0.439679i \(0.855092\pi\)
\(420\) 0 0
\(421\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −30.0000 −1.45521
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(432\) 0 0
\(433\) 19.0000 32.9090i 0.913082 1.58150i 0.103396 0.994640i \(-0.467029\pi\)
0.809686 0.586864i \(-0.199638\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(440\) 0 0
\(441\) −31.1464 + 53.9472i −1.48316 + 2.56891i
\(442\) 0 0
\(443\) 16.0652 + 9.27526i 0.763281 + 0.440681i 0.830473 0.557059i \(-0.188070\pi\)
−0.0671913 + 0.997740i \(0.521404\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −25.8990 −1.22225 −0.611124 0.791535i \(-0.709282\pi\)
−0.611124 + 0.791535i \(0.709282\pi\)
\(450\) 0 0
\(451\) 6.10150 3.52270i 0.287309 0.165878i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16.3939 0.766873 0.383437 0.923567i \(-0.374740\pi\)
0.383437 + 0.923567i \(0.374740\pi\)
\(458\) 0 0
\(459\) −105.734 + 61.0454i −4.93523 + 2.84936i
\(460\) 0 0
\(461\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 41.9444i 1.94095i 0.241192 + 0.970477i \(0.422462\pi\)
−0.241192 + 0.970477i \(0.577538\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.75255 + 4.76756i 0.126562 + 0.219213i
\(474\) 0 0
\(475\) −6.27647 20.8712i −0.287984 0.957635i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 39.7474 + 68.8446i 1.79744 + 3.11326i
\(490\) 0 0
\(491\) 36.3731 21.0000i 1.64149 0.947717i 0.661190 0.750218i \(-0.270052\pi\)
0.980303 0.197499i \(-0.0632818\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −25.7576 + 14.8712i −1.15307 + 0.665725i −0.949633 0.313363i \(-0.898544\pi\)
−0.203436 + 0.979088i \(0.565211\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 38.8355 + 22.4217i 1.72474 + 0.995782i
\(508\) 0 0
\(509\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −64.5908 60.7879i −2.85175 2.68385i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −36.1918 −1.58559 −0.792797 0.609486i \(-0.791376\pi\)
−0.792797 + 0.609486i \(0.791376\pi\)
\(522\) 0 0
\(523\) 32.9090 + 19.0000i 1.43901 + 0.830812i 0.997781 0.0665832i \(-0.0212098\pi\)
0.441228 + 0.897395i \(0.354543\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 82.2929i 3.57121i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −14.0505 24.3362i −0.606324 1.05018i
\(538\) 0 0
\(539\) 3.85357i 0.165985i
\(540\) 0 0
\(541\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 39.8372 + 23.0000i 1.70331 + 0.983409i 0.942358 + 0.334606i \(0.108603\pi\)
0.760956 + 0.648803i \(0.224730\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 5.69694 9.86739i 0.240525 0.416601i
\(562\) 0 0
\(563\) 46.8434i 1.97421i 0.160066 + 0.987106i \(0.448829\pi\)
−0.160066 + 0.987106i \(0.551171\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) 25.7423i 1.07728i −0.842535 0.538642i \(-0.818938\pi\)
0.842535 0.538642i \(-0.181062\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 12.3939 0.515964 0.257982 0.966150i \(-0.416942\pi\)
0.257982 + 0.966150i \(0.416942\pi\)
\(578\) 0 0
\(579\) −65.7216 37.9444i −2.73130 1.57691i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.19615 + 3.00000i −0.214468 + 0.123823i −0.603386 0.797449i \(-0.706182\pi\)
0.388918 + 0.921272i \(0.372849\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 24.0959 41.7354i 0.989501 1.71387i 0.369586 0.929197i \(-0.379500\pi\)
0.619915 0.784669i \(-0.287167\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(600\) 0 0
\(601\) 37.6969 1.53769 0.768845 0.639435i \(-0.220832\pi\)
0.768845 + 0.639435i \(0.220832\pi\)
\(602\) 0 0
\(603\) −110.580 + 63.8434i −4.50316 + 2.59990i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −24.6464 + 42.6889i −0.992228 + 1.71859i −0.388351 + 0.921512i \(0.626955\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 26.0000i 1.04503i 0.852631 + 0.522514i \(0.175006\pi\)
−0.852631 + 0.522514i \(0.824994\pi\)
\(620\) 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
\(627\) 8.05669 + 1.89898i 0.321753 + 0.0758379i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(632\) 0 0
\(633\) 24.1464 41.8228i 0.959734 1.66231i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.74745 3.02667i −0.0690201 0.119546i 0.829450 0.558581i \(-0.188654\pi\)
−0.898470 + 0.439034i \(0.855321\pi\)
\(642\) 0 0
\(643\) −15.2867 + 8.82577i −0.602848 + 0.348054i −0.770161 0.637850i \(-0.779824\pi\)
0.167313 + 0.985904i \(0.446491\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −2.54541 4.40878i −0.0999160 0.173060i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 139.687 5.44970
\(658\) 0 0
\(659\) −15.5885 9.00000i −0.607240 0.350590i 0.164644 0.986353i \(-0.447352\pi\)
−0.771885 + 0.635763i \(0.780686\pi\)
\(660\) 0 0
\(661\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 0 0
\(675\) −88.1115 50.8712i −3.39141 1.95803i
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −30.0959 52.1277i −1.15328 1.99754i
\(682\) 0 0
\(683\) 42.0000i 1.60709i 0.595247 + 0.803543i \(0.297054\pi\)
−0.595247 + 0.803543i \(0.702946\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 46.0000i 1.74992i −0.484193 0.874961i \(-0.660887\pi\)
0.484193 0.874961i \(-0.339113\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 38.3939 66.5001i 1.45427 2.51887i
\(698\) 0 0
\(699\) 59.4451 + 34.3207i 2.24842 + 1.29813i
\(700\) 0 0
\(701\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 5.85357i 0.217697i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(728\) 0 0
\(729\) −176.485 −6.53647
\(730\) 0 0
\(731\) 51.9615 + 30.0000i 1.92187 + 1.10959i
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.94949 6.84072i 0.145481 0.251981i
\(738\) 0 0
\(739\) −46.5422 + 26.8712i −1.71208 + 0.988472i −0.780340 + 0.625355i \(0.784954\pi\)
−0.931744 + 0.363117i \(0.881713\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −50.4831 29.1464i −1.84708 1.06641i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(752\) 0 0
\(753\) −82.5959 −3.00996
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 17.2020 0.623574 0.311787 0.950152i \(-0.399073\pi\)
0.311787 + 0.950152i \(0.399073\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 11.0000 19.0526i 0.396670 0.687053i −0.596643 0.802507i \(-0.703499\pi\)
0.993313 + 0.115454i \(0.0368323\pi\)
\(770\) 0 0
\(771\) 17.9444i 0.646251i
\(772\) 0 0
\(773\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 54.2971 + 12.7980i 1.94540 + 0.458534i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 47.0454i 1.67699i −0.544911 0.838494i \(-0.683437\pi\)
0.544911 0.838494i \(-0.316563\pi\)
\(788\) 0 0
\(789\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 80.0908 + 138.721i 2.82987 + 4.90148i
\(802\) 0 0
\(803\) −7.48361 + 4.32066i −0.264091 + 0.152473i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −45.9898 −1.61692 −0.808458 0.588555i \(-0.799697\pi\)
−0.808458 + 0.588555i \(0.799697\pi\)
\(810\) 0 0
\(811\) 32.9090 + 19.0000i 1.15559 + 0.667180i 0.950243 0.311509i \(-0.100834\pi\)
0.205347 + 0.978689i \(0.434168\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −10.0000 + 42.4264i −0.349856 + 1.48431i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(824\) 0 0
\(825\) 9.49490 0.330570
\(826\) 0 0
\(827\) −8.53344 4.92679i −0.296737 0.171321i 0.344239 0.938882i \(-0.388137\pi\)
−0.640976 + 0.767561i \(0.721470\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −21.0000 36.3731i −0.727607 1.26025i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(840\) 0 0
\(841\) −14.5000 25.1147i −0.500000 0.866025i
\(842\) 0 0
\(843\) 53.4495i 1.84090i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −56.9949 + 98.7181i −1.95606 + 3.38799i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −23.2980 40.3532i −0.795843 1.37844i −0.922302 0.386469i \(-0.873695\pi\)
0.126459 0.991972i \(-0.459639\pi\)
\(858\) 0 0
\(859\) 31.4787 + 18.1742i 1.07404 + 0.620097i 0.929282 0.369370i \(-0.120427\pi\)
0.144757 + 0.989467i \(0.453760\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 65.5403i 2.22587i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −86.2929 −2.92057
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 39.9898 1.34729 0.673645 0.739055i \(-0.264728\pi\)
0.673645 + 0.739055i \(0.264728\pi\)
\(882\) 0 0
\(883\) 45.4138 + 26.2196i 1.52829 + 0.882361i 0.999434 + 0.0336527i \(0.0107140\pi\)
0.528861 + 0.848709i \(0.322619\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 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
\(891\) 20.7364 11.9722i 0.694697 0.401084i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 40.2176 23.2196i 1.33540 0.770996i 0.349281 0.937018i \(-0.386426\pi\)
0.986122 + 0.166022i \(0.0530924\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 3.60612 0.119345
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 16.6464 28.8325i 0.548518 0.950062i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −25.7474 44.5959i −0.844746 1.46314i −0.885841 0.463988i \(-0.846418\pi\)
0.0410949 0.999155i \(-0.486915\pi\)
\(930\) 0 0
\(931\) 20.9114 22.2196i 0.685344 0.728219i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −13.5454 + 23.4613i −0.442509 + 0.766448i −0.997875 0.0651578i \(-0.979245\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) 0 0
\(939\) 84.1464i 2.74601i
\(940\) 0 0
\(941\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −25.9808 + 15.0000i −0.844261 + 0.487435i −0.858710 0.512461i \(-0.828734\pi\)
0.0144491 + 0.999896i \(0.495401\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −9.09592 15.7546i −0.294646 0.510341i 0.680257 0.732974i \(-0.261868\pi\)
−0.974902 + 0.222633i \(0.928535\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −46.2405 26.6969i −1.49008 0.860297i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(968\) 0 0
\(969\) 86.3939 25.9808i 2.77537 0.834622i
\(970\) 0 0
\(971\) 0.0481621 0.0278064i 0.00154560 0.000892350i −0.499227 0.866471i \(-0.666383\pi\)
0.500773 + 0.865579i \(0.333049\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 56.8888 1.82003 0.910017 0.414572i \(-0.136069\pi\)
0.910017 + 0.414572i \(0.136069\pi\)
\(978\) 0 0
\(979\) −8.58161 4.95459i −0.274269 0.158349i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 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
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(992\) 0 0
\(993\) −15.6010 27.0218i −0.495083 0.857510i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1216.2.t.a.1185.1 yes 8
4.3 odd 2 inner 1216.2.t.a.1185.4 yes 8
8.3 odd 2 CM 1216.2.t.a.1185.1 yes 8
8.5 even 2 inner 1216.2.t.a.1185.4 yes 8
19.11 even 3 inner 1216.2.t.a.353.4 yes 8
76.11 odd 6 inner 1216.2.t.a.353.1 8
152.11 odd 6 inner 1216.2.t.a.353.4 yes 8
152.125 even 6 inner 1216.2.t.a.353.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.t.a.353.1 8 76.11 odd 6 inner
1216.2.t.a.353.1 8 152.125 even 6 inner
1216.2.t.a.353.4 yes 8 19.11 even 3 inner
1216.2.t.a.353.4 yes 8 152.11 odd 6 inner
1216.2.t.a.1185.1 yes 8 1.1 even 1 trivial
1216.2.t.a.1185.1 yes 8 8.3 odd 2 CM
1216.2.t.a.1185.4 yes 8 4.3 odd 2 inner
1216.2.t.a.1185.4 yes 8 8.5 even 2 inner