Properties

Label 84.2.k.c.5.1
Level $84$
Weight $2$
Character 84.5
Analytic conductor $0.671$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [84,2,Mod(5,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 84.k (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: \(0.670743376979\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 5.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 84.5
Dual form 84.2.k.c.17.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 - 0.866025i) q^{3} +(-2.50000 + 0.866025i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(1.50000 - 0.866025i) q^{3} +(-2.50000 + 0.866025i) q^{7} +(1.50000 - 2.59808i) q^{9} +5.19615i q^{13} +(-7.50000 - 4.33013i) q^{19} +(-3.00000 + 3.46410i) q^{21} +(2.50000 + 4.33013i) q^{25} -5.19615i q^{27} +(-1.50000 + 0.866025i) q^{31} +(5.50000 - 9.52628i) q^{37} +(4.50000 + 7.79423i) q^{39} +13.0000 q^{43} +(5.50000 - 4.33013i) q^{49} -15.0000 q^{57} +(-6.00000 - 3.46410i) q^{61} +(-1.50000 + 7.79423i) q^{63} +(-2.50000 - 4.33013i) q^{67} +(-1.50000 + 0.866025i) q^{73} +(7.50000 + 4.33013i) q^{75} +(-8.50000 + 14.7224i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-4.50000 - 12.9904i) q^{91} +(-1.50000 + 2.59808i) q^{93} +13.8564i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - 5 q^{7} + 3 q^{9} - 15 q^{19} - 6 q^{21} + 5 q^{25} - 3 q^{31} + 11 q^{37} + 9 q^{39} + 26 q^{43} + 11 q^{49} - 30 q^{57} - 12 q^{61} - 3 q^{63} - 5 q^{67} - 3 q^{73} + 15 q^{75} - 17 q^{79} - 9 q^{81} - 9 q^{91} - 3 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 1.50000 0.866025i 0.866025 0.500000i
\(4\) 0 0
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) −2.50000 + 0.866025i −0.944911 + 0.327327i
\(8\) 0 0
\(9\) 1.50000 2.59808i 0.500000 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) 5.19615i 1.44115i 0.693375 + 0.720577i \(0.256123\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −7.50000 4.33013i −1.72062 0.993399i −0.917663 0.397360i \(-0.869927\pi\)
−0.802955 0.596040i \(-0.796740\pi\)
\(20\) 0 0
\(21\) −3.00000 + 3.46410i −0.654654 + 0.755929i
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.50000 + 0.866025i −0.269408 + 0.155543i −0.628619 0.777714i \(-0.716379\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.50000 9.52628i 0.904194 1.56611i 0.0821995 0.996616i \(-0.473806\pi\)
0.821995 0.569495i \(-0.192861\pi\)
\(38\) 0 0
\(39\) 4.50000 + 7.79423i 0.720577 + 1.24808i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 13.0000 1.98248 0.991241 0.132068i \(-0.0421616\pi\)
0.991241 + 0.132068i \(0.0421616\pi\)
\(44\) 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
\(49\) 5.50000 4.33013i 0.785714 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −15.0000 −1.98680
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) −6.00000 3.46410i −0.768221 0.443533i 0.0640184 0.997949i \(-0.479608\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) −1.50000 + 7.79423i −0.188982 + 0.981981i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.50000 4.33013i −0.305424 0.529009i 0.671932 0.740613i \(-0.265465\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.50000 + 0.866025i −0.175562 + 0.101361i −0.585206 0.810885i \(-0.698986\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) 7.50000 + 4.33013i 0.866025 + 0.500000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.50000 + 14.7224i −0.956325 + 1.65640i −0.225018 + 0.974355i \(0.572244\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) −4.50000 12.9904i −0.471728 1.36176i
\(92\) 0 0
\(93\) −1.50000 + 2.59808i −0.155543 + 0.269408i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.8564i 1.40690i 0.710742 + 0.703452i \(0.248359\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 13.5000 + 7.79423i 1.33019 + 0.767988i 0.985329 0.170664i \(-0.0545913\pi\)
0.344865 + 0.938652i \(0.387925\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) 0 0
\(109\) −9.50000 16.4545i −0.909935 1.57605i −0.814152 0.580651i \(-0.802798\pi\)
−0.0957826 0.995402i \(-0.530535\pi\)
\(110\) 0 0
\(111\) 19.0526i 1.80839i
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 13.5000 + 7.79423i 1.24808 + 0.720577i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.50000 + 9.52628i −0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1.00000 −0.0887357 −0.0443678 0.999015i \(-0.514127\pi\)
−0.0443678 + 0.999015i \(0.514127\pi\)
\(128\) 0 0
\(129\) 19.5000 11.2583i 1.71688 0.991241i
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 22.5000 + 4.33013i 1.95100 + 0.375470i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(138\) 0 0
\(139\) 5.19615i 0.440732i 0.975417 + 0.220366i \(0.0707252\pi\)
−0.975417 + 0.220366i \(0.929275\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.50000 11.2583i 0.371154 0.928571i
\(148\) 0 0
\(149\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) −2.00000 3.46410i −0.162758 0.281905i 0.773099 0.634285i \(-0.218706\pi\)
−0.935857 + 0.352381i \(0.885372\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −18.0000 + 10.3923i −1.43656 + 0.829396i −0.997609 0.0691164i \(-0.977982\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 + 6.92820i −0.313304 + 0.542659i −0.979076 0.203497i \(-0.934769\pi\)
0.665771 + 0.746156i \(0.268103\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −14.0000 −1.07692
\(170\) 0 0
\(171\) −22.5000 + 12.9904i −1.72062 + 0.993399i
\(172\) 0 0
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) −10.0000 8.66025i −0.755929 0.654654i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0 0
\(181\) 19.0526i 1.41617i −0.706129 0.708083i \(-0.749560\pi\)
0.706129 0.708083i \(-0.250440\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 4.50000 + 12.9904i 0.327327 + 0.944911i
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 0 0
\(193\) 11.5000 + 19.9186i 0.827788 + 1.43377i 0.899770 + 0.436365i \(0.143734\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 3.00000 1.73205i 0.212664 0.122782i −0.389885 0.920864i \(-0.627485\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(200\) 0 0
\(201\) −7.50000 4.33013i −0.529009 0.305424i
\(202\) 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
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3.00000 3.46410i 0.203653 0.235159i
\(218\) 0 0
\(219\) −1.50000 + 2.59808i −0.101361 + 0.175562i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 10.3923i 0.695920i −0.937509 0.347960i \(-0.886874\pi\)
0.937509 0.347960i \(-0.113126\pi\)
\(224\) 0 0
\(225\) 15.0000 1.00000
\(226\) 0 0
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) 0 0
\(229\) −7.50000 4.33013i −0.495614 0.286143i 0.231287 0.972886i \(-0.425707\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 29.4449i 1.91265i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 24.0000 13.8564i 1.54598 0.892570i 0.547533 0.836784i \(-0.315567\pi\)
0.998443 0.0557856i \(-0.0177663\pi\)
\(242\) 0 0
\(243\) −13.5000 7.79423i −0.866025 0.500000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 22.5000 38.9711i 1.43164 2.47967i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) −5.50000 + 28.5788i −0.341753 + 1.77580i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) 15.0000 + 8.66025i 0.911185 + 0.526073i 0.880812 0.473466i \(-0.156997\pi\)
0.0303728 + 0.999539i \(0.490331\pi\)
\(272\) 0 0
\(273\) −18.0000 15.5885i −1.08941 0.943456i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.50000 4.33013i −0.150210 0.260172i 0.781094 0.624413i \(-0.214662\pi\)
−0.931305 + 0.364241i \(0.881328\pi\)
\(278\) 0 0
\(279\) 5.19615i 0.311086i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 19.5000 11.2583i 1.15915 0.669238i 0.208053 0.978117i \(-0.433287\pi\)
0.951101 + 0.308879i \(0.0999539\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.50000 14.7224i 0.500000 0.866025i
\(290\) 0 0
\(291\) 12.0000 + 20.7846i 0.703452 + 1.21842i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −32.5000 + 11.2583i −1.87327 + 0.648919i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 29.4449i 1.68051i 0.542194 + 0.840254i \(0.317594\pi\)
−0.542194 + 0.840254i \(0.682406\pi\)
\(308\) 0 0
\(309\) 27.0000 1.53598
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) −28.5000 16.4545i −1.61092 0.930062i −0.989158 0.146852i \(-0.953086\pi\)
−0.621757 0.783210i \(-0.713581\pi\)
\(314\) 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
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −22.5000 + 12.9904i −1.24808 + 0.720577i
\(326\) 0 0
\(327\) −28.5000 16.4545i −1.57605 0.909935i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −15.5000 + 26.8468i −0.851957 + 1.47563i 0.0274825 + 0.999622i \(0.491251\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 0 0
\(333\) −16.5000 28.5788i −0.904194 1.56611i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −29.0000 −1.57973 −0.789865 0.613280i \(-0.789850\pi\)
−0.789865 + 0.613280i \(0.789850\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) 34.6410i 1.85429i −0.374701 0.927146i \(-0.622255\pi\)
0.374701 0.927146i \(-0.377745\pi\)
\(350\) 0 0
\(351\) 27.0000 1.44115
\(352\) 0 0
\(353\) 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
\(358\) 0 0
\(359\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) 28.0000 + 48.4974i 1.47368 + 2.55250i
\(362\) 0 0
\(363\) 19.0526i 1.00000i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 19.5000 11.2583i 1.01789 0.587680i 0.104399 0.994535i \(-0.466708\pi\)
0.913493 + 0.406855i \(0.133375\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 12.5000 21.6506i 0.647225 1.12103i −0.336557 0.941663i \(-0.609263\pi\)
0.983783 0.179364i \(-0.0574041\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −29.0000 −1.48963 −0.744815 0.667271i \(-0.767462\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) 0 0
\(381\) −1.50000 + 0.866025i −0.0768473 + 0.0443678i
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 19.5000 33.7750i 0.991241 1.71688i
\(388\) 0 0
\(389\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 34.5000 + 19.9186i 1.73151 + 0.999685i 0.878300 + 0.478110i \(0.158678\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) 0 0
\(399\) 37.5000 12.9904i 1.87735 0.650332i
\(400\) 0 0
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) −4.50000 7.79423i −0.224161 0.388258i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −22.5000 + 12.9904i −1.11255 + 0.642333i −0.939490 0.342578i \(-0.888700\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.50000 + 7.79423i 0.220366 + 0.381685i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 41.0000 1.99822 0.999109 0.0422075i \(-0.0134391\pi\)
0.999109 + 0.0422075i \(0.0134391\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 18.0000 + 3.46410i 0.871081 + 0.167640i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 0 0
\(433\) 19.0526i 0.915608i −0.889053 0.457804i \(-0.848636\pi\)
0.889053 0.457804i \(-0.151364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −27.0000 15.5885i −1.28864 0.743996i −0.310228 0.950662i \(-0.600405\pi\)
−0.978412 + 0.206666i \(0.933739\pi\)
\(440\) 0 0
\(441\) −3.00000 20.7846i −0.142857 0.989743i
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −6.00000 3.46410i −0.281905 0.162758i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −15.5000 + 26.8468i −0.725059 + 1.25584i 0.233890 + 0.972263i \(0.424854\pi\)
−0.958950 + 0.283577i \(0.908479\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −43.0000 −1.99838 −0.999190 0.0402476i \(-0.987185\pi\)
−0.999190 + 0.0402476i \(0.987185\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 0 0
\(469\) 10.0000 + 8.66025i 0.461757 + 0.399893i
\(470\) 0 0
\(471\) −18.0000 + 31.1769i −0.829396 + 1.43656i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 43.3013i 1.98680i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) 49.5000 + 28.5788i 2.25701 + 1.30308i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −9.50000 16.4545i −0.430486 0.745624i 0.566429 0.824110i \(-0.308325\pi\)
−0.996915 + 0.0784867i \(0.974991\pi\)
\(488\) 0 0
\(489\) 13.8564i 0.626608i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 5.50000 9.52628i 0.246214 0.426455i −0.716258 0.697835i \(-0.754147\pi\)
0.962472 + 0.271380i \(0.0874801\pi\)
\(500\) 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
\(506\) 0 0
\(507\) −21.0000 + 12.1244i −0.932643 + 0.538462i
\(508\) 0 0
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 3.00000 3.46410i 0.132712 0.153243i
\(512\) 0 0
\(513\) −22.5000 + 38.9711i −0.993399 + 1.72062i
\(514\) 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
\(523\) 13.5000 + 7.79423i 0.590314 + 0.340818i 0.765222 0.643767i \(-0.222629\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 0 0
\(525\) −22.5000 4.33013i −0.981981 0.188982i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −11.5000 19.9186i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8.50000 + 14.7224i −0.365444 + 0.632967i −0.988847 0.148933i \(-0.952416\pi\)
0.623404 + 0.781900i \(0.285749\pi\)
\(542\) 0 0
\(543\) −16.5000 28.5788i −0.708083 1.22644i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −40.0000 −1.71028 −0.855138 0.518400i \(-0.826528\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 0 0
\(549\) −18.0000 + 10.3923i −0.768221 + 0.443533i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.50000 44.1673i 0.361457 1.87818i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) 67.5500i 2.85706i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 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
\(567\) 18.0000 + 15.5885i 0.755929 + 0.654654i
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) −23.5000 40.7032i −0.983444 1.70338i −0.648655 0.761083i \(-0.724668\pi\)
−0.334790 0.942293i \(-0.608665\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 40.5000 23.3827i 1.68604 0.973434i 0.728535 0.685009i \(-0.240202\pi\)
0.957503 0.288425i \(-0.0931316\pi\)
\(578\) 0 0
\(579\) 34.5000 + 19.9186i 1.43377 + 0.827788i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 15.0000 0.618064
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3.00000 5.19615i 0.122782 0.212664i
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 43.3013i 1.76630i −0.469095 0.883148i \(-0.655420\pi\)
0.469095 0.883148i \(-0.344580\pi\)
\(602\) 0 0
\(603\) −15.0000 −0.610847
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 34.5000 + 19.9186i 1.40031 + 0.808470i 0.994424 0.105453i \(-0.0336291\pi\)
0.405887 + 0.913923i \(0.366962\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 5.00000 + 8.66025i 0.201948 + 0.349784i 0.949156 0.314806i \(-0.101939\pi\)
−0.747208 + 0.664590i \(0.768606\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 40.5000 23.3827i 1.62783 0.939829i 0.643094 0.765787i \(-0.277650\pi\)
0.984738 0.174042i \(-0.0556830\pi\)
\(620\) 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
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) 0 0
\(633\) 24.0000 13.8564i 0.953914 0.550743i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 22.5000 + 28.5788i 0.891482 + 1.13233i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0 0
\(643\) 19.0526i 0.751360i −0.926750 0.375680i \(-0.877409\pi\)
0.926750 0.375680i \(-0.122591\pi\)
\(644\) 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
\(651\) 1.50000 7.79423i 0.0587896 0.305480i
\(652\) 0 0
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 5.19615i 0.202721i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −43.5000 + 25.1147i −1.69195 + 0.976850i −0.739014 + 0.673690i \(0.764708\pi\)
−0.952940 + 0.303160i \(0.901958\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −9.00000 15.5885i −0.347960 0.602685i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 13.0000 0.501113 0.250557 0.968102i \(-0.419386\pi\)
0.250557 + 0.968102i \(0.419386\pi\)
\(674\) 0 0
\(675\) 22.5000 12.9904i 0.866025 0.500000i
\(676\) 0 0
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) 0 0
\(679\) −12.0000 34.6410i −0.460518 1.32940i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −15.0000 −0.572286
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −28.5000 16.4545i −1.08419 0.625958i −0.152167 0.988355i \(-0.548625\pi\)
−0.932024 + 0.362397i \(0.881959\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −82.5000 + 47.6314i −3.11155 + 1.79645i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −11.0000 + 19.0526i −0.413114 + 0.715534i −0.995228 0.0975728i \(-0.968892\pi\)
0.582115 + 0.813107i \(0.302225\pi\)
\(710\) 0 0
\(711\) 25.5000 + 44.1673i 0.956325 + 1.65640i
\(712\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) −40.5000 7.79423i −1.50830 0.290272i
\(722\) 0 0
\(723\) 24.0000 41.5692i 0.892570 1.54598i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 53.6936i 1.99138i 0.0927199 + 0.995692i \(0.470444\pi\)
−0.0927199 + 0.995692i \(0.529556\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −28.5000 16.4545i −1.05267 0.607760i −0.129275 0.991609i \(-0.541265\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 18.5000 + 32.0429i 0.680534 + 1.17872i 0.974818 + 0.223001i \(0.0715853\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 77.9423i 2.86328i
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 5.50000 9.52628i 0.200698 0.347619i −0.748056 0.663636i \(-0.769012\pi\)
0.948753 + 0.316017i \(0.102346\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 0 0
\(759\) 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
\(763\) 38.0000 + 32.9090i 1.37569 + 1.19138i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 29.4449i 1.06181i 0.847432 + 0.530904i \(0.178148\pi\)
−0.847432 + 0.530904i \(0.821852\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) −7.50000 4.33013i −0.269408 0.155543i
\(776\) 0 0
\(777\) 16.5000 + 47.6314i 0.591934 + 1.70877i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 3.00000 1.73205i 0.106938 0.0617409i −0.445577 0.895244i \(-0.647001\pi\)
0.552515 + 0.833503i \(0.313668\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 18.0000 31.1769i 0.639199 1.10712i
\(794\) 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
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(810\) 0 0
\(811\) 10.3923i 0.364923i −0.983213 0.182462i \(-0.941593\pi\)
0.983213 0.182462i \(-0.0584065\pi\)
\(812\) 0 0
\(813\) 30.0000 1.05215
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −97.5000 56.2917i −3.41109 1.96940i
\(818\) 0 0
\(819\) −40.5000 7.79423i −1.41518 0.272352i
\(820\) 0 0
\(821\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) 26.0000 + 45.0333i 0.906303 + 1.56976i 0.819159 + 0.573567i \(0.194441\pi\)
0.0871445 + 0.996196i \(0.472226\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 19.5000 11.2583i 0.677263 0.391018i −0.121560 0.992584i \(-0.538790\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) −7.50000 4.33013i −0.260172 0.150210i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 4.50000 + 7.79423i 0.155543 + 0.269408i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 5.50000 28.5788i 0.188982 0.981981i
\(848\) 0 0
\(849\) 19.5000 33.7750i 0.669238 1.15915i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 53.6936i 1.83843i 0.393753 + 0.919216i \(0.371177\pi\)
−0.393753 + 0.919216i \(0.628823\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(858\) 0 0
\(859\) 15.0000 + 8.66025i 0.511793 + 0.295484i 0.733571 0.679613i \(-0.237852\pi\)
−0.221777 + 0.975097i \(0.571186\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 29.4449i 1.00000i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 22.5000 12.9904i 0.762383 0.440162i
\(872\) 0 0
\(873\) 36.0000 + 20.7846i 1.21842 + 0.703452i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 17.0000 29.4449i 0.574049 0.994282i −0.422095 0.906552i \(-0.638705\pi\)
0.996144 0.0877308i \(-0.0279615\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 55.0000 1.85090 0.925449 0.378873i \(-0.123688\pi\)
0.925449 + 0.378873i \(0.123688\pi\)
\(884\) 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
\(889\) 2.50000 0.866025i 0.0838473 0.0290456i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −39.0000 + 45.0333i −1.29784 + 1.49862i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −9.50000 16.4545i −0.315442 0.546362i 0.664089 0.747653i \(-0.268820\pi\)
−0.979531 + 0.201291i \(0.935486\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 26.5000 45.8993i 0.874154 1.51408i 0.0164935 0.999864i \(-0.494750\pi\)
0.857661 0.514216i \(-0.171917\pi\)
\(920\) 0 0
\(921\) 25.5000 + 44.1673i 0.840254 + 1.45536i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 55.0000 1.80839
\(926\) 0 0
\(927\) 40.5000 23.3827i 1.33019 0.767988i
\(928\) 0 0
\(929\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(930\) 0 0
\(931\) −60.0000 + 8.66025i −1.96642 + 0.283828i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 5.19615i 0.169751i 0.996392 + 0.0848755i \(0.0270492\pi\)
−0.996392 + 0.0848755i \(0.972951\pi\)
\(938\) 0 0
\(939\) −57.0000 −1.86012
\(940\) 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
\(946\) 0 0
\(947\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(948\) 0 0
\(949\) −4.50000 7.79423i −0.146076 0.253011i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −14.0000 + 24.2487i −0.451613 + 0.782216i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 41.0000 1.31847 0.659236 0.751936i \(-0.270880\pi\)
0.659236 + 0.751936i \(0.270880\pi\)
\(968\) 0 0
\(969\) 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
\(973\) −4.50000 12.9904i −0.144263 0.416452i
\(974\) 0 0
\(975\) −22.5000 + 38.9711i −0.720577 + 1.24808i
\(976\) 0 0
\(977\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −57.0000 −1.81987
\(982\) 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
\(991\) −30.5000 52.8275i −0.968864 1.67812i −0.698853 0.715265i \(-0.746306\pi\)
−0.270011 0.962857i \(-0.587027\pi\)
\(992\) 0 0
\(993\) 53.6936i 1.70391i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 19.5000 11.2583i 0.617571 0.356555i −0.158352 0.987383i \(-0.550618\pi\)
0.775923 + 0.630828i \(0.217285\pi\)
\(998\) 0 0
\(999\) −49.5000 28.5788i −1.56611 0.904194i
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 84.2.k.c.5.1 2
3.2 odd 2 CM 84.2.k.c.5.1 2
4.3 odd 2 336.2.bc.a.257.1 2
5.2 odd 4 2100.2.bo.e.1349.1 4
5.3 odd 4 2100.2.bo.e.1349.2 4
5.4 even 2 2100.2.bi.d.1601.1 2
7.2 even 3 588.2.f.b.293.2 2
7.3 odd 6 inner 84.2.k.c.17.1 yes 2
7.4 even 3 588.2.k.b.521.1 2
7.5 odd 6 588.2.f.b.293.1 2
7.6 odd 2 588.2.k.b.509.1 2
9.2 odd 6 2268.2.w.e.1349.1 2
9.4 even 3 2268.2.bm.d.593.1 2
9.5 odd 6 2268.2.bm.d.593.1 2
9.7 even 3 2268.2.w.e.1349.1 2
12.11 even 2 336.2.bc.a.257.1 2
15.2 even 4 2100.2.bo.e.1349.1 4
15.8 even 4 2100.2.bo.e.1349.2 4
15.14 odd 2 2100.2.bi.d.1601.1 2
21.2 odd 6 588.2.f.b.293.2 2
21.5 even 6 588.2.f.b.293.1 2
21.11 odd 6 588.2.k.b.521.1 2
21.17 even 6 inner 84.2.k.c.17.1 yes 2
21.20 even 2 588.2.k.b.509.1 2
28.3 even 6 336.2.bc.a.17.1 2
28.19 even 6 2352.2.k.b.881.2 2
28.23 odd 6 2352.2.k.b.881.1 2
35.3 even 12 2100.2.bo.e.1949.1 4
35.17 even 12 2100.2.bo.e.1949.2 4
35.24 odd 6 2100.2.bi.d.101.1 2
63.31 odd 6 2268.2.w.e.269.1 2
63.38 even 6 2268.2.bm.d.1025.1 2
63.52 odd 6 2268.2.bm.d.1025.1 2
63.59 even 6 2268.2.w.e.269.1 2
84.23 even 6 2352.2.k.b.881.1 2
84.47 odd 6 2352.2.k.b.881.2 2
84.59 odd 6 336.2.bc.a.17.1 2
105.17 odd 12 2100.2.bo.e.1949.2 4
105.38 odd 12 2100.2.bo.e.1949.1 4
105.59 even 6 2100.2.bi.d.101.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.2.k.c.5.1 2 1.1 even 1 trivial
84.2.k.c.5.1 2 3.2 odd 2 CM
84.2.k.c.17.1 yes 2 7.3 odd 6 inner
84.2.k.c.17.1 yes 2 21.17 even 6 inner
336.2.bc.a.17.1 2 28.3 even 6
336.2.bc.a.17.1 2 84.59 odd 6
336.2.bc.a.257.1 2 4.3 odd 2
336.2.bc.a.257.1 2 12.11 even 2
588.2.f.b.293.1 2 7.5 odd 6
588.2.f.b.293.1 2 21.5 even 6
588.2.f.b.293.2 2 7.2 even 3
588.2.f.b.293.2 2 21.2 odd 6
588.2.k.b.509.1 2 7.6 odd 2
588.2.k.b.509.1 2 21.20 even 2
588.2.k.b.521.1 2 7.4 even 3
588.2.k.b.521.1 2 21.11 odd 6
2100.2.bi.d.101.1 2 35.24 odd 6
2100.2.bi.d.101.1 2 105.59 even 6
2100.2.bi.d.1601.1 2 5.4 even 2
2100.2.bi.d.1601.1 2 15.14 odd 2
2100.2.bo.e.1349.1 4 5.2 odd 4
2100.2.bo.e.1349.1 4 15.2 even 4
2100.2.bo.e.1349.2 4 5.3 odd 4
2100.2.bo.e.1349.2 4 15.8 even 4
2100.2.bo.e.1949.1 4 35.3 even 12
2100.2.bo.e.1949.1 4 105.38 odd 12
2100.2.bo.e.1949.2 4 35.17 even 12
2100.2.bo.e.1949.2 4 105.17 odd 12
2268.2.w.e.269.1 2 63.31 odd 6
2268.2.w.e.269.1 2 63.59 even 6
2268.2.w.e.1349.1 2 9.2 odd 6
2268.2.w.e.1349.1 2 9.7 even 3
2268.2.bm.d.593.1 2 9.4 even 3
2268.2.bm.d.593.1 2 9.5 odd 6
2268.2.bm.d.1025.1 2 63.38 even 6
2268.2.bm.d.1025.1 2 63.52 odd 6
2352.2.k.b.881.1 2 28.23 odd 6
2352.2.k.b.881.1 2 84.23 even 6
2352.2.k.b.881.2 2 28.19 even 6
2352.2.k.b.881.2 2 84.47 odd 6