Properties

Label 2100.2.bi.g.101.1
Level $2100$
Weight $2$
Character 2100.101
Analytic conductor $16.769$
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] = [2100,2,Mod(101,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.bi (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: \(16.7685844245\)
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 101.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2100.101
Dual form 2100.2.bi.g.1601.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 + 0.866025i) q^{3} +(-2.00000 - 1.73205i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\) \(q+(1.50000 + 0.866025i) q^{3} +(-2.00000 - 1.73205i) q^{7} +(1.50000 + 2.59808i) q^{9} +6.92820i q^{13} +(-3.00000 + 1.73205i) q^{19} +(-1.50000 - 4.33013i) q^{21} +5.19615i q^{27} +(1.50000 + 0.866025i) q^{31} +(0.500000 + 0.866025i) q^{37} +(-6.00000 + 10.3923i) q^{39} -13.0000 q^{43} +(1.00000 + 6.92820i) q^{49} -6.00000 q^{57} +(7.50000 - 4.33013i) q^{61} +(1.50000 - 7.79423i) q^{63} +(-8.00000 + 13.8564i) q^{67} +(13.5000 + 7.79423i) q^{73} +(6.50000 + 11.2583i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(12.0000 - 13.8564i) q^{91} +(1.50000 + 2.59808i) q^{93} +19.0526i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - 4 q^{7} + 3 q^{9} - 6 q^{19} - 3 q^{21} + 3 q^{31} + q^{37} - 12 q^{39} - 26 q^{43} + 2 q^{49} - 12 q^{57} + 15 q^{61} + 3 q^{63} - 16 q^{67} + 27 q^{73} + 13 q^{79} - 9 q^{81} + 24 q^{91} + 3 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(6\) 0 0
\(7\) −2.00000 1.73205i −0.755929 0.654654i
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) 6.92820i 1.92154i 0.277350 + 0.960769i \(0.410544\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −3.00000 + 1.73205i −0.688247 + 0.397360i −0.802955 0.596040i \(-0.796740\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) −1.50000 4.33013i −0.327327 0.944911i
\(22\) 0 0
\(23\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 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 \(-0.283621\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.500000 + 0.866025i 0.0821995 + 0.142374i 0.904194 0.427121i \(-0.140472\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) −6.00000 + 10.3923i −0.960769 + 1.66410i
\(40\) 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 \(-0.957838\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 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
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 7.50000 4.33013i 0.960277 0.554416i 0.0640184 0.997949i \(-0.479608\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 1.50000 7.79423i 0.188982 0.981981i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 + 13.8564i −0.977356 + 1.69283i −0.305424 + 0.952217i \(0.598798\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 13.5000 + 7.79423i 1.58006 + 0.912245i 0.994850 + 0.101361i \(0.0323196\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.50000 + 11.2583i 0.731307 + 1.26666i 0.956325 + 0.292306i \(0.0944227\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 12.0000 13.8564i 1.25794 1.45255i
\(92\) 0 0
\(93\) 1.50000 + 2.59808i 0.155543 + 0.269408i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 19.0526i 1.93449i 0.253837 + 0.967247i \(0.418307\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) −16.5000 + 9.52628i −1.62579 + 0.938652i −0.640464 + 0.767988i \(0.721258\pi\)
−0.985329 + 0.170664i \(0.945409\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) 0 0
\(109\) 1.00000 1.73205i 0.0957826 0.165900i −0.814152 0.580651i \(-0.802798\pi\)
0.909935 + 0.414751i \(0.136131\pi\)
\(110\) 0 0
\(111\) 1.73205i 0.164399i
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −18.0000 + 10.3923i −1.66410 + 0.960769i
\(118\) 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
\(127\) 19.0000 1.68598 0.842989 0.537931i \(-0.180794\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(128\) 0 0
\(129\) −19.5000 11.2583i −1.71688 0.991241i
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) 9.00000 + 1.73205i 0.780399 + 0.150188i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) 22.5167i 1.90984i −0.296866 0.954919i \(-0.595942\pi\)
0.296866 0.954919i \(-0.404058\pi\)
\(140\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) 11.5000 19.9186i 0.935857 1.62095i 0.162758 0.986666i \(-0.447961\pi\)
0.773099 0.634285i \(-0.218706\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.50000 0.866025i −0.119713 0.0691164i 0.438948 0.898513i \(-0.355351\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −12.5000 21.6506i −0.979076 1.69581i −0.665771 0.746156i \(-0.731897\pi\)
−0.313304 0.949653i \(-0.601436\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −35.0000 −2.69231
\(170\) 0 0
\(171\) −9.00000 5.19615i −0.688247 0.397360i
\(172\) 0 0
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) 0 0
\(181\) 19.0526i 1.41617i −0.706129 0.708083i \(-0.749560\pi\)
0.706129 0.708083i \(-0.250440\pi\)
\(182\) 0 0
\(183\) 15.0000 1.10883
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 9.00000 10.3923i 0.654654 0.755929i
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 0 0
\(193\) −11.5000 + 19.9186i −0.827788 + 1.43377i 0.0719816 + 0.997406i \(0.477068\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 22.5000 + 12.9904i 1.59498 + 0.920864i 0.992434 + 0.122782i \(0.0391815\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(200\) 0 0
\(201\) −24.0000 + 13.8564i −1.69283 + 0.977356i
\(202\) 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
\(211\) 13.0000 0.894957 0.447478 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.50000 4.33013i −0.101827 0.293948i
\(218\) 0 0
\(219\) 13.5000 + 23.3827i 0.912245 + 1.58006i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 29.4449i 1.97177i −0.167412 0.985887i \(-0.553541\pi\)
0.167412 0.985887i \(-0.446459\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(228\) 0 0
\(229\) 18.0000 10.3923i 1.18947 0.686743i 0.231287 0.972886i \(-0.425707\pi\)
0.958187 + 0.286143i \(0.0923732\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 22.5167i 1.46261i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 22.5000 + 12.9904i 1.44935 + 0.836784i 0.998443 0.0557856i \(-0.0177663\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) −13.5000 + 7.79423i −0.866025 + 0.500000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −12.0000 20.7846i −0.763542 1.32249i
\(248\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) 0.500000 2.59808i 0.0310685 0.161437i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) −15.0000 + 8.66025i −0.911185 + 0.526073i −0.880812 0.473466i \(-0.843003\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 0 0
\(273\) 30.0000 10.3923i 1.81568 0.628971i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 13.0000 22.5167i 0.781094 1.35290i −0.150210 0.988654i \(-0.547995\pi\)
0.931305 0.364241i \(-0.118672\pi\)
\(278\) 0 0
\(279\) 5.19615i 0.311086i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 9.00000 + 5.19615i 0.534994 + 0.308879i 0.743048 0.669238i \(-0.233379\pi\)
−0.208053 + 0.978117i \(0.566713\pi\)
\(284\) 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
\(291\) −16.5000 + 28.5788i −0.967247 + 1.67532i
\(292\) 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
\(301\) 26.0000 + 22.5167i 1.49862 + 1.29784i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 29.4449i 1.68051i −0.542194 0.840254i \(-0.682406\pi\)
0.542194 0.840254i \(-0.317594\pi\)
\(308\) 0 0
\(309\) −33.0000 −1.87730
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) −28.5000 + 16.4545i −1.61092 + 0.930062i −0.621757 + 0.783210i \(0.713581\pi\)
−0.989158 + 0.146852i \(0.953086\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 3.00000 1.73205i 0.165900 0.0957826i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.500000 0.866025i −0.0274825 0.0476011i 0.851957 0.523612i \(-0.175416\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 0 0
\(333\) −1.50000 + 2.59808i −0.0821995 + 0.142374i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 29.0000 1.57973 0.789865 0.613280i \(-0.210150\pi\)
0.789865 + 0.613280i \(0.210150\pi\)
\(338\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0 0
\(349\) 34.6410i 1.85429i −0.374701 0.927146i \(-0.622255\pi\)
0.374701 0.927146i \(-0.377745\pi\)
\(350\) 0 0
\(351\) −36.0000 −1.92154
\(352\) 0 0
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0 0
\(361\) −3.50000 + 6.06218i −0.184211 + 0.319062i
\(362\) 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 0.913493 0.406855i \(-0.133375\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 19.0000 + 32.9090i 0.983783 + 1.70396i 0.647225 + 0.762299i \(0.275929\pi\)
0.336557 + 0.941663i \(0.390737\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 28.5000 + 16.4545i 1.46010 + 0.842989i
\(382\) 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
\(387\) −19.5000 33.7750i −0.991241 1.71688i
\(388\) 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
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 34.5000 19.9186i 1.73151 0.999685i 0.853206 0.521575i \(-0.174655\pi\)
0.878300 0.478110i \(-0.158678\pi\)
\(398\) 0 0
\(399\) 12.0000 + 10.3923i 0.600751 + 0.520266i
\(400\) 0 0
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) −6.00000 + 10.3923i −0.298881 + 0.517678i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 12.0000 + 6.92820i 0.593362 + 0.342578i 0.766426 0.642333i \(-0.222033\pi\)
−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
\(417\) 19.5000 33.7750i 0.954919 1.65397i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −22.5000 4.33013i −1.08885 0.209550i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 0 0
\(433\) 19.0526i 0.915608i 0.889053 + 0.457804i \(0.151364\pi\)
−0.889053 + 0.457804i \(0.848636\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 7.50000 4.33013i 0.357955 0.206666i −0.310228 0.950662i \(-0.600405\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) −16.5000 + 12.9904i −0.785714 + 0.618590i
\(442\) 0 0
\(443\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(444\) 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
\(453\) 34.5000 19.9186i 1.62095 0.935857i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −20.5000 35.5070i −0.958950 1.66095i −0.725059 0.688686i \(-0.758188\pi\)
−0.233890 0.972263i \(-0.575146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −23.0000 −1.06890 −0.534450 0.845200i \(-0.679481\pi\)
−0.534450 + 0.845200i \(0.679481\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) 0 0
\(469\) 40.0000 13.8564i 1.84703 0.639829i
\(470\) 0 0
\(471\) −1.50000 2.59808i −0.0691164 0.119713i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 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
\(481\) −6.00000 + 3.46410i −0.273576 + 0.157949i
\(482\) 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 \(-0.691675\pi\)
0.996915 + 0.0784867i \(0.0250088\pi\)
\(488\) 0 0
\(489\) 43.3013i 1.95815i
\(490\) 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
\(499\) 16.0000 + 27.7128i 0.716258 + 1.24060i 0.962472 + 0.271380i \(0.0874801\pi\)
−0.246214 + 0.969216i \(0.579187\pi\)
\(500\) 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
\(507\) −52.5000 30.3109i −2.33161 1.34615i
\(508\) 0 0
\(509\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) −13.5000 38.9711i −0.597205 1.72398i
\(512\) 0 0
\(513\) −9.00000 15.5885i −0.397360 0.688247i
\(514\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(522\) 0 0
\(523\) 13.5000 7.79423i 0.590314 0.340818i −0.174908 0.984585i \(-0.555963\pi\)
0.765222 + 0.643767i \(0.222629\pi\)
\(524\) 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
\(531\) 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
\(541\) 23.0000 + 39.8372i 0.988847 + 1.71273i 0.623404 + 0.781900i \(0.285749\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) 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 \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) 0 0
\(549\) 22.5000 + 12.9904i 0.960277 + 0.554416i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 6.50000 33.7750i 0.276408 1.43626i
\(554\) 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
\(559\) 90.0666i 3.80941i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 22.5000 7.79423i 0.944911 0.327327i
\(568\) 0 0
\(569\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(570\) 0 0
\(571\) 8.00000 13.8564i 0.334790 0.579873i −0.648655 0.761083i \(-0.724668\pi\)
0.983444 + 0.181210i \(0.0580014\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 40.5000 + 23.3827i 1.68604 + 0.973434i 0.957503 + 0.288425i \(0.0931316\pi\)
0.728535 + 0.685009i \(0.240202\pi\)
\(578\) 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
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 22.5000 + 38.9711i 0.920864 + 1.59498i
\(598\) 0 0
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) 41.5692i 1.69564i 0.530281 + 0.847822i \(0.322086\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) −48.0000 −1.95471
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4.50000 2.59808i 0.182649 0.105453i −0.405887 0.913923i \(-0.633038\pi\)
0.588537 + 0.808470i \(0.299704\pi\)
\(608\) 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 \(-0.898061\pi\)
0.747208 + 0.664590i \(0.231394\pi\)
\(614\) 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 −0.984738 0.174042i \(-0.944317\pi\)
−0.643094 0.765787i \(-0.722350\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −43.0000 −1.71180 −0.855901 0.517139i \(-0.826997\pi\)
−0.855901 + 0.517139i \(0.826997\pi\)
\(632\) 0 0
\(633\) 19.5000 + 11.2583i 0.775055 + 0.447478i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −48.0000 + 6.92820i −1.90183 + 0.274505i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) 50.2295i 1.98086i −0.138027 0.990429i \(-0.544076\pi\)
0.138027 0.990429i \(-0.455924\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 46.7654i 1.82449i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −13.5000 7.79423i −0.525089 0.303160i 0.213925 0.976850i \(-0.431375\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 25.5000 44.1673i 0.985887 1.70761i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −37.0000 −1.42625 −0.713123 0.701039i \(-0.752720\pi\)
−0.713123 + 0.701039i \(0.752720\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(678\) 0 0
\(679\) 33.0000 38.1051i 1.26642 1.46234i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 36.0000 1.37349
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 16.5000 9.52628i 0.627690 0.362397i −0.152167 0.988355i \(-0.548625\pi\)
0.779857 + 0.625958i \(0.215292\pi\)
\(692\) 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
\(703\) −3.00000 1.73205i −0.113147 0.0653255i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −15.5000 26.8468i −0.582115 1.00825i −0.995228 0.0975728i \(-0.968892\pi\)
0.413114 0.910679i \(-0.364441\pi\)
\(710\) 0 0
\(711\) −19.5000 + 33.7750i −0.731307 + 1.26666i
\(712\) 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
\(721\) 49.5000 + 9.52628i 1.84348 + 0.354777i
\(722\) 0 0
\(723\) 22.5000 + 38.9711i 0.836784 + 1.44935i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 31.1769i 1.15629i 0.815935 + 0.578144i \(0.196223\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 46.5000 26.8468i 1.71752 0.991609i 0.794121 0.607760i \(-0.207932\pi\)
0.923396 0.383849i \(-0.125402\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −26.5000 + 45.8993i −0.974818 + 1.68843i −0.294285 + 0.955718i \(0.595081\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(740\) 0 0
\(741\) 41.5692i 1.52708i
\(742\) 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
\(751\) 20.5000 + 35.5070i 0.748056 + 1.29567i 0.948753 + 0.316017i \(0.102346\pi\)
−0.200698 + 0.979653i \(0.564321\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −55.0000 −1.99901 −0.999505 0.0314762i \(-0.989979\pi\)
−0.999505 + 0.0314762i \(0.989979\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) −5.00000 + 1.73205i −0.181012 + 0.0627044i
\(764\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 3.00000 3.46410i 0.107624 0.124274i
\(778\) 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
\(787\) −43.5000 25.1147i −1.55061 0.895244i −0.998092 0.0617409i \(-0.980335\pi\)
−0.552515 0.833503i \(-0.686332\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 30.0000 + 51.9615i 1.06533 + 1.84521i
\(794\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 0 0
\(811\) 43.3013i 1.52051i −0.649623 0.760257i \(-0.725073\pi\)
0.649623 0.760257i \(-0.274927\pi\)
\(812\) 0 0
\(813\) −30.0000 −1.05215
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 39.0000 22.5167i 1.36444 0.787758i
\(818\) 0 0
\(819\) 54.0000 + 10.3923i 1.88691 + 0.363137i
\(820\) 0 0
\(821\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(822\) 0 0
\(823\) 2.50000 4.33013i 0.0871445 0.150939i −0.819159 0.573567i \(-0.805559\pi\)
0.906303 + 0.422628i \(0.138892\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 49.5000 + 28.5788i 1.71921 + 0.992584i 0.920383 + 0.391018i \(0.127877\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) 39.0000 22.5167i 1.35290 0.781094i
\(832\) 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
\(849\) 9.00000 + 15.5885i 0.308879 + 0.534994i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 53.6936i 1.83843i −0.393753 0.919216i \(-0.628823\pi\)
0.393753 0.919216i \(-0.371177\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(858\) 0 0
\(859\) −15.0000 + 8.66025i −0.511793 + 0.295484i −0.733571 0.679613i \(-0.762148\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 29.4449i 1.00000i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −96.0000 55.4256i −3.25284 1.87803i
\(872\) 0 0
\(873\) −49.5000 + 28.5788i −1.67532 + 0.967247i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −12.5000 21.6506i −0.422095 0.731090i 0.574049 0.818821i \(-0.305372\pi\)
−0.996144 + 0.0877308i \(0.972038\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(888\) 0 0
\(889\) −38.0000 32.9090i −1.27448 1.10373i
\(890\) 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
\(903\) 19.5000 + 56.2917i 0.648919 + 1.87327i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −29.5000 + 51.0955i −0.979531 + 1.69660i −0.315442 + 0.948945i \(0.602153\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) 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 0.857661 + 0.514216i \(0.171917\pi\)
0.0164935 + 0.999864i \(0.494750\pi\)
\(920\) 0 0
\(921\) 25.5000 44.1673i 0.840254 1.45536i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −49.5000 28.5788i −1.62579 0.938652i
\(928\) 0 0
\(929\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(930\) 0 0
\(931\) −15.0000 19.0526i −0.491605 0.624422i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 5.19615i 0.169751i −0.996392 0.0848755i \(-0.972951\pi\)
0.996392 0.0848755i \(-0.0270492\pi\)
\(938\) 0 0
\(939\) −57.0000 −1.86012
\(940\) 0 0
\(941\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(948\) 0 0
\(949\) −54.0000 + 93.5307i −1.75291 + 3.03614i
\(950\) 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
\(967\) 61.0000 1.96163 0.980814 0.194946i \(-0.0624533\pi\)
0.980814 + 0.194946i \(0.0624533\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(972\) 0 0
\(973\) −39.0000 + 45.0333i −1.25028 + 1.44370i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 0 0
\(983\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(984\) 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 −0.270011 + 0.962857i \(0.587027\pi\)
−0.698853 + 0.715265i \(0.746306\pi\)
\(992\) 0 0
\(993\) 1.73205i 0.0549650i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 34.5000 + 19.9186i 1.09263 + 0.630828i 0.934274 0.356555i \(-0.116049\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 0 0
\(999\) −4.50000 + 2.59808i −0.142374 + 0.0821995i
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 2100.2.bi.g.101.1 yes 2
3.2 odd 2 CM 2100.2.bi.g.101.1 yes 2
5.2 odd 4 2100.2.bo.d.1949.2 4
5.3 odd 4 2100.2.bo.d.1949.1 4
5.4 even 2 2100.2.bi.c.101.1 2
7.5 odd 6 inner 2100.2.bi.g.1601.1 yes 2
15.2 even 4 2100.2.bo.d.1949.2 4
15.8 even 4 2100.2.bo.d.1949.1 4
15.14 odd 2 2100.2.bi.c.101.1 2
21.5 even 6 inner 2100.2.bi.g.1601.1 yes 2
35.12 even 12 2100.2.bo.d.1349.1 4
35.19 odd 6 2100.2.bi.c.1601.1 yes 2
35.33 even 12 2100.2.bo.d.1349.2 4
105.47 odd 12 2100.2.bo.d.1349.1 4
105.68 odd 12 2100.2.bo.d.1349.2 4
105.89 even 6 2100.2.bi.c.1601.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.bi.c.101.1 2 5.4 even 2
2100.2.bi.c.101.1 2 15.14 odd 2
2100.2.bi.c.1601.1 yes 2 35.19 odd 6
2100.2.bi.c.1601.1 yes 2 105.89 even 6
2100.2.bi.g.101.1 yes 2 1.1 even 1 trivial
2100.2.bi.g.101.1 yes 2 3.2 odd 2 CM
2100.2.bi.g.1601.1 yes 2 7.5 odd 6 inner
2100.2.bi.g.1601.1 yes 2 21.5 even 6 inner
2100.2.bo.d.1349.1 4 35.12 even 12
2100.2.bo.d.1349.1 4 105.47 odd 12
2100.2.bo.d.1349.2 4 35.33 even 12
2100.2.bo.d.1349.2 4 105.68 odd 12
2100.2.bo.d.1949.1 4 5.3 odd 4
2100.2.bo.d.1949.1 4 15.8 even 4
2100.2.bo.d.1949.2 4 5.2 odd 4
2100.2.bo.d.1949.2 4 15.2 even 4