Properties

Label 117.2.q.b.82.1
Level $117$
Weight $2$
Character 117.82
Analytic conductor $0.934$
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] = [117,2,Mod(10,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.q (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: \(0.934249703649\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 82.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 117.82
Dual form 117.2.q.b.10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{4} +(4.50000 - 2.59808i) q^{7} +O(q^{10})\) \(q+(-1.00000 - 1.73205i) q^{4} +(4.50000 - 2.59808i) q^{7} +(-2.50000 + 2.59808i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(-3.00000 + 1.73205i) q^{19} +5.00000 q^{25} +(-9.00000 - 5.19615i) q^{28} +8.66025i q^{31} +(-6.00000 - 3.46410i) q^{37} +(6.50000 + 11.2583i) q^{43} +(10.0000 - 17.3205i) q^{49} +(7.00000 + 1.73205i) q^{52} +(-6.50000 - 11.2583i) q^{61} +8.00000 q^{64} +(-10.5000 - 6.06218i) q^{67} +1.73205i q^{73} +(6.00000 + 3.46410i) q^{76} +13.0000 q^{79} +(-4.50000 + 18.1865i) q^{91} +(-16.5000 + 9.52628i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 9 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 9 q^{7} - 5 q^{13} - 4 q^{16} - 6 q^{19} + 10 q^{25} - 18 q^{28} - 12 q^{37} + 13 q^{43} + 20 q^{49} + 14 q^{52} - 13 q^{61} + 16 q^{64} - 21 q^{67} + 12 q^{76} + 26 q^{79} - 9 q^{91} - 33 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\)

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 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 0 0
\(4\) −1.00000 1.73205i −0.500000 0.866025i
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 4.50000 2.59808i 1.70084 0.981981i 0.755929 0.654654i \(-0.227186\pi\)
0.944911 0.327327i \(-0.106148\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) −2.50000 + 2.59808i −0.693375 + 0.720577i
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 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
\(21\) 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
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −9.00000 5.19615i −1.70084 0.981981i
\(29\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(30\) 0 0
\(31\) 8.66025i 1.55543i 0.628619 + 0.777714i \(0.283621\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 3.46410i −0.986394 0.569495i −0.0821995 0.996616i \(-0.526194\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) 6.50000 + 11.2583i 0.991241 + 1.71688i 0.609994 + 0.792406i \(0.291172\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 10.0000 17.3205i 1.42857 2.47436i
\(50\) 0 0
\(51\) 0 0
\(52\) 7.00000 + 1.73205i 0.970725 + 0.240192i
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) −6.50000 11.2583i −0.832240 1.44148i −0.896258 0.443533i \(-0.853725\pi\)
0.0640184 0.997949i \(-0.479608\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −10.5000 6.06218i −1.28278 0.740613i −0.305424 0.952217i \(-0.598798\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) 0 0
\(73\) 1.73205i 0.202721i 0.994850 + 0.101361i \(0.0323196\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 6.00000 + 3.46410i 0.688247 + 0.397360i
\(77\) 0 0
\(78\) 0 0
\(79\) 13.0000 1.46261 0.731307 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) −4.50000 + 18.1865i −0.471728 + 1.90647i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −16.5000 + 9.52628i −1.67532 + 0.967247i −0.710742 + 0.703452i \(0.751641\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −5.00000 8.66025i −0.500000 0.866025i
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) −13.0000 −1.28093 −0.640464 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(108\) 0 0
\(109\) 8.66025i 0.829502i −0.909935 0.414751i \(-0.863869\pi\)
0.909935 0.414751i \(-0.136131\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 20.7846i 1.96396i
\(113\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.50000 9.52628i −0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 15.0000 8.66025i 1.34704 0.777714i
\(125\) 0 0
\(126\) 0 0
\(127\) −0.500000 + 0.866025i −0.0443678 + 0.0768473i −0.887357 0.461084i \(-0.847461\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −9.00000 + 15.5885i −0.780399 + 1.35169i
\(134\) 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
\(139\) 3.50000 + 6.06218i 0.296866 + 0.514187i 0.975417 0.220366i \(-0.0707252\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 13.8564i 1.13899i
\(149\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) 24.2487i 1.97333i −0.162758 0.986666i \(-0.552039\pi\)
0.162758 0.986666i \(-0.447961\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 11.0000 0.877896 0.438948 0.898513i \(-0.355351\pi\)
0.438948 + 0.898513i \(0.355351\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.50000 + 2.59808i −0.352467 + 0.203497i −0.665771 0.746156i \(-0.731897\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(168\) 0 0
\(169\) −0.500000 12.9904i −0.0384615 0.999260i
\(170\) 0 0
\(171\) 0 0
\(172\) 13.0000 22.5167i 0.991241 1.71688i
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 22.5000 12.9904i 1.70084 0.981981i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(180\) 0 0
\(181\) 26.0000 1.93256 0.966282 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) 0 0
\(193\) −13.5000 7.79423i −0.971751 0.561041i −0.0719816 0.997406i \(-0.522932\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −40.0000 −2.85714
\(197\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(198\) 0 0
\(199\) 8.50000 + 14.7224i 0.602549 + 1.04365i 0.992434 + 0.122782i \(0.0391815\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −4.00000 13.8564i −0.277350 0.960769i
\(209\) 0 0
\(210\) 0 0
\(211\) −6.50000 + 11.2583i −0.447478 + 0.775055i −0.998221 0.0596196i \(-0.981011\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 22.5000 + 38.9711i 1.52740 + 2.64553i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 9.00000 + 5.19615i 0.602685 + 0.347960i 0.770097 0.637927i \(-0.220208\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 0 0
\(229\) 20.7846i 1.37349i −0.726900 0.686743i \(-0.759040\pi\)
0.726900 0.686743i \(-0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 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 \(0.684433\pi\)
−0.998443 + 0.0557856i \(0.982234\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −13.0000 + 22.5167i −0.832240 + 1.44148i
\(245\) 0 0
\(246\) 0 0
\(247\) 3.00000 12.1244i 0.190885 0.771454i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 13.8564i −0.500000 0.866025i
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) −36.0000 −2.23693
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 24.2487i 1.48123i
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) 13.5000 + 7.79423i 0.820067 + 0.473466i 0.850439 0.526073i \(-0.176336\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −13.0000 22.5167i −0.781094 1.35290i −0.931305 0.364241i \(-0.881328\pi\)
0.150210 0.988654i \(-0.452005\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −12.5000 + 21.6506i −0.743048 + 1.28700i 0.208053 + 0.978117i \(0.433287\pi\)
−0.951101 + 0.308879i \(0.900046\pi\)
\(284\) 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
\(291\) 0 0
\(292\) 3.00000 1.73205i 0.175562 0.101361i
\(293\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 58.5000 + 33.7750i 3.37188 + 1.94676i
\(302\) 0 0
\(303\) 0 0
\(304\) 13.8564i 0.794719i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.73205i 0.0988534i −0.998778 0.0494267i \(-0.984261\pi\)
0.998778 0.0494267i \(-0.0157394\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 13.0000 0.734803 0.367402 0.930062i \(-0.380247\pi\)
0.367402 + 0.930062i \(0.380247\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −13.0000 22.5167i −0.731307 1.26666i
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −12.5000 + 12.9904i −0.693375 + 0.720577i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 16.5000 9.52628i 0.906922 0.523612i 0.0274825 0.999622i \(-0.491251\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 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
\(343\) 67.5500i 3.64736i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(348\) 0 0
\(349\) 25.5000 + 14.7224i 1.36498 + 0.788074i 0.990282 0.139072i \(-0.0444119\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −3.50000 + 6.06218i −0.184211 + 0.319062i
\(362\) 0 0
\(363\) 0 0
\(364\) 36.0000 10.3923i 1.88691 0.544705i
\(365\) 0 0
\(366\) 0 0
\(367\) 15.5000 26.8468i 0.809093 1.40139i −0.104399 0.994535i \(-0.533292\pi\)
0.913493 0.406855i \(-0.133375\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6.50000 11.2583i −0.336557 0.582934i 0.647225 0.762299i \(-0.275929\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 10.5000 + 6.06218i 0.539349 + 0.311393i 0.744815 0.667271i \(-0.232538\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 33.0000 + 19.0526i 1.67532 + 0.967247i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 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 \(0.825345\pi\)
−0.878300 + 0.478110i \(0.841322\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −10.0000 + 17.3205i −0.500000 + 0.866025i
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) −22.5000 21.6506i −1.12080 1.07849i
\(404\) 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
\(412\) 13.0000 + 22.5167i 0.640464 + 1.10932i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(420\) 0 0
\(421\) 36.3731i 1.77271i 0.463002 + 0.886357i \(0.346772\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −58.5000 33.7750i −2.83101 1.63449i
\(428\) 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
\(433\) 17.5000 + 30.3109i 0.840996 + 1.45665i 0.889053 + 0.457804i \(0.151364\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −15.0000 + 8.66025i −0.718370 + 0.414751i
\(437\) 0 0
\(438\) 0 0
\(439\) 6.50000 11.2583i 0.310228 0.537331i −0.668184 0.743996i \(-0.732928\pi\)
0.978412 + 0.206666i \(0.0662612\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 36.0000 20.7846i 1.70084 0.981981i
\(449\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.5000 + 6.06218i 0.491169 + 0.283577i 0.725059 0.688686i \(-0.241812\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(462\) 0 0
\(463\) 1.73205i 0.0804952i −0.999190 0.0402476i \(-0.987185\pi\)
0.999190 0.0402476i \(-0.0128147\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −63.0000 −2.90907
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −15.0000 + 8.66025i −0.688247 + 0.397360i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 24.0000 6.92820i 1.09431 0.315899i
\(482\) 0 0
\(483\) 0 0
\(484\) −11.0000 + 19.0526i −0.500000 + 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) 3.00000 1.73205i 0.135943 0.0784867i −0.430486 0.902597i \(-0.641658\pi\)
0.566429 + 0.824110i \(0.308325\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −30.0000 17.3205i −1.34704 0.777714i
\(497\) 0 0
\(498\) 0 0
\(499\) 31.1769i 1.39567i 0.716258 + 0.697835i \(0.245853\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 4.50000 + 7.79423i 0.199068 + 0.344796i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −4.00000 + 6.92820i −0.174908 + 0.302949i −0.940129 0.340818i \(-0.889296\pi\)
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
\(532\) 36.0000 1.56080
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 43.3013i 1.86167i −0.365444 0.930834i \(-0.619083\pi\)
0.365444 0.930834i \(-0.380917\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 41.0000 1.75303 0.876517 0.481371i \(-0.159861\pi\)
0.876517 + 0.481371i \(0.159861\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 58.5000 33.7750i 2.48767 1.43626i
\(554\) 0 0
\(555\) 0 0
\(556\) 7.00000 12.1244i 0.296866 0.514187i
\(557\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) −45.5000 11.2583i −1.92444 0.476177i
\(560\) 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
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 13.8564i 0.576850i −0.957503 0.288425i \(-0.906868\pi\)
0.957503 0.288425i \(-0.0931316\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(588\) 0 0
\(589\) −15.0000 25.9808i −0.618064 1.07052i
\(590\) 0 0
\(591\) 0 0
\(592\) 24.0000 13.8564i 0.986394 0.569495i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 13.0000 22.5167i 0.530281 0.918474i −0.469095 0.883148i \(-0.655420\pi\)
0.999376 0.0353259i \(-0.0112469\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −42.0000 + 24.2487i −1.70896 + 0.986666i
\(605\) 0 0
\(606\) 0 0
\(607\) −10.0000 17.3205i −0.405887 0.703018i 0.588537 0.808470i \(-0.299704\pi\)
−0.994424 + 0.105453i \(0.966371\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −28.5000 16.4545i −1.15110 0.664590i −0.201948 0.979396i \(-0.564727\pi\)
−0.949156 + 0.314806i \(0.898061\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(618\) 0 0
\(619\) 46.7654i 1.87966i 0.341644 + 0.939829i \(0.389016\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −11.0000 19.0526i −0.438948 0.760280i
\(629\) 0 0
\(630\) 0 0
\(631\) 43.5000 25.1147i 1.73171 0.999802i 0.855901 0.517139i \(-0.173003\pi\)
0.875806 0.482663i \(-0.160330\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 20.0000 + 69.2820i 0.792429 + 2.74505i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(642\) 0 0
\(643\) 16.5000 9.52628i 0.650696 0.375680i −0.138027 0.990429i \(-0.544076\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) 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
\(651\) 0 0
\(652\) 9.00000 + 5.19615i 0.352467 + 0.203497i
\(653\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(660\) 0 0
\(661\) 13.5000 + 7.79423i 0.525089 + 0.303160i 0.739014 0.673690i \(-0.235292\pi\)
−0.213925 + 0.976850i \(0.568625\pi\)
\(662\) 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
\(673\) 6.50000 11.2583i 0.250557 0.433977i −0.713123 0.701039i \(-0.752720\pi\)
0.963679 + 0.267063i \(0.0860531\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −22.0000 + 13.8564i −0.846154 + 0.532939i
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) −49.5000 + 85.7365i −1.89964 + 3.29027i
\(680\) 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
\(687\) 0 0
\(688\) −52.0000 −1.98248
\(689\) 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
\(700\) −45.0000 25.9808i −1.70084 0.981981i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −4.50000 + 2.59808i −0.169001 + 0.0975728i −0.582115 0.813107i \(-0.697775\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 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
\(721\) −58.5000 + 33.7750i −2.17865 + 1.25785i
\(722\) 0 0
\(723\) 0 0
\(724\) −26.0000 45.0333i −0.966282 1.67365i
\(725\) 0 0
\(726\) 0 0
\(727\) 49.0000 1.81731 0.908655 0.417548i \(-0.137111\pi\)
0.908655 + 0.417548i \(0.137111\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 53.6936i 1.98322i 0.129275 + 0.991609i \(0.458735\pi\)
−0.129275 + 0.991609i \(0.541265\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 45.0000 + 25.9808i 1.65535 + 0.955718i 0.974818 + 0.223001i \(0.0715853\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 26.0000 45.0333i 0.948753 1.64329i 0.200698 0.979653i \(-0.435679\pi\)
0.748056 0.663636i \(-0.230988\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 13.0000 22.5167i 0.472493 0.818382i −0.527011 0.849858i \(-0.676688\pi\)
0.999505 + 0.0314762i \(0.0100208\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(762\) 0 0
\(763\) −22.5000 38.9711i −0.814555 1.41085i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −48.0000 27.7128i −1.73092 0.999350i −0.883493 0.468445i \(-0.844814\pi\)
−0.847432 0.530904i \(-0.821852\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 31.1769i 1.12208i
\(773\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(774\) 0 0
\(775\) 43.3013i 1.55543i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 40.0000 + 69.2820i 1.42857 + 2.47436i
\(785\) 0 0
\(786\) 0 0
\(787\) −43.5000 + 25.1147i −1.55061 + 0.895244i −0.552515 + 0.833503i \(0.686332\pi\)
−0.998092 + 0.0617409i \(0.980335\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 45.5000 + 11.2583i 1.61575 + 0.399795i
\(794\) 0 0
\(795\) 0 0
\(796\) 17.0000 29.4449i 0.602549 1.04365i
\(797\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(798\) 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 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(810\) 0 0
\(811\) 53.6936i 1.88544i −0.333590 0.942718i \(-0.608260\pi\)
0.333590 0.942718i \(-0.391740\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −39.0000 22.5167i −1.36444 0.787758i
\(818\) 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
\(823\) 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
\(829\) 26.5000 45.8993i 0.920383 1.59415i 0.121560 0.992584i \(-0.461210\pi\)
0.798823 0.601566i \(-0.205456\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −20.0000 + 20.7846i −0.693375 + 0.720577i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(840\) 0 0
\(841\) 14.5000 + 25.1147i 0.500000 + 0.866025i
\(842\) 0 0
\(843\) 0 0
\(844\) 26.0000 0.894957
\(845\) 0 0
\(846\) 0 0
\(847\) −49.5000 28.5788i −1.70084 0.981981i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 46.7654i 1.60122i −0.599189 0.800608i \(-0.704510\pi\)
0.599189 0.800608i \(-0.295490\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 13.0000 0.443554 0.221777 0.975097i \(-0.428814\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 45.0000 77.9423i 1.52740 2.64553i
\(869\) 0 0
\(870\) 0 0
\(871\) 42.0000 12.1244i 1.42312 0.410818i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 42.0000 24.2487i 1.41824 0.818821i 0.422095 0.906552i \(-0.361295\pi\)
0.996144 + 0.0877308i \(0.0279615\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(882\) 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
\(889\) 5.19615i 0.174273i
\(890\) 0 0
\(891\) 0 0
\(892\) 20.7846i 0.695920i
\(893\) 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
\(907\) −20.0000 + 34.6410i −0.664089 + 1.15024i 0.315442 + 0.948945i \(0.397847\pi\)
−0.979531 + 0.201291i \(0.935486\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −36.0000 + 20.7846i −1.18947 + 0.686743i
\(917\) 0 0
\(918\) 0 0
\(919\) −26.0000 45.0333i −0.857661 1.48551i −0.874154 0.485648i \(-0.838584\pi\)
0.0164935 0.999864i \(-0.494750\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −30.0000 17.3205i −0.986394 0.569495i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(930\) 0 0
\(931\) 69.2820i 2.27063i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 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 4.33013i −0.146076 0.140562i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −44.0000 −1.41935
\(962\) 0 0
\(963\) 0 0
\(964\) 48.0000 + 27.7128i 1.54598 + 0.892570i
\(965\) 0 0
\(966\) 0 0
\(967\) 58.8897i 1.89377i 0.321578 + 0.946883i \(0.395787\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) 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
\(973\) 31.5000 + 18.1865i 1.00984 + 0.583033i
\(974\) 0 0
\(975\) 0 0
\(976\) 52.0000 1.66448
\(977\) 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
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −24.0000 + 6.92820i −0.763542 + 0.220416i
\(989\) 0 0
\(990\) 0 0
\(991\) −22.0000 + 38.1051i −0.698853 + 1.21045i 0.270011 + 0.962857i \(0.412973\pi\)
−0.968864 + 0.247592i \(0.920361\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −29.5000 51.0955i −0.934274 1.61821i −0.775923 0.630828i \(-0.782715\pi\)
−0.158352 0.987383i \(-0.550618\pi\)
\(998\) 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 117.2.q.b.82.1 yes 2
3.2 odd 2 CM 117.2.q.b.82.1 yes 2
4.3 odd 2 1872.2.by.a.433.1 2
12.11 even 2 1872.2.by.a.433.1 2
13.4 even 6 1521.2.b.d.1351.1 2
13.6 odd 12 1521.2.a.i.1.1 2
13.7 odd 12 1521.2.a.i.1.2 2
13.9 even 3 1521.2.b.d.1351.2 2
13.10 even 6 inner 117.2.q.b.10.1 2
39.17 odd 6 1521.2.b.d.1351.1 2
39.20 even 12 1521.2.a.i.1.2 2
39.23 odd 6 inner 117.2.q.b.10.1 2
39.32 even 12 1521.2.a.i.1.1 2
39.35 odd 6 1521.2.b.d.1351.2 2
52.23 odd 6 1872.2.by.a.1297.1 2
156.23 even 6 1872.2.by.a.1297.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.q.b.10.1 2 13.10 even 6 inner
117.2.q.b.10.1 2 39.23 odd 6 inner
117.2.q.b.82.1 yes 2 1.1 even 1 trivial
117.2.q.b.82.1 yes 2 3.2 odd 2 CM
1521.2.a.i.1.1 2 13.6 odd 12
1521.2.a.i.1.1 2 39.32 even 12
1521.2.a.i.1.2 2 13.7 odd 12
1521.2.a.i.1.2 2 39.20 even 12
1521.2.b.d.1351.1 2 13.4 even 6
1521.2.b.d.1351.1 2 39.17 odd 6
1521.2.b.d.1351.2 2 13.9 even 3
1521.2.b.d.1351.2 2 39.35 odd 6
1872.2.by.a.433.1 2 4.3 odd 2
1872.2.by.a.433.1 2 12.11 even 2
1872.2.by.a.1297.1 2 52.23 odd 6
1872.2.by.a.1297.1 2 156.23 even 6