Properties

Label 3267.1.bf.a.112.1
Level $3267$
Weight $1$
Character 3267.112
Analytic conductor $1.630$
Analytic rank $0$
Dimension $24$
Projective image $D_{9}$
CM discriminant -11
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3267,1,Mod(40,3267)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3267, base_ring=CyclotomicField(90))
 
chi = DirichletCharacter(H, H._module([40, 63]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3267.40");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3267 = 3^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3267.bf (of order \(90\), degree \(24\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.63044539627\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{45})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 297)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.459450093735369.1

Embedding invariants

Embedding label 112.1
Root \(-0.719340 + 0.694658i\) of defining polynomial
Character \(\chi\) \(=\) 3267.112
Dual form 3267.1.bf.a.1546.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.559193 - 0.829038i) q^{3} +(-0.241922 - 0.970296i) q^{4} +(1.51718 - 0.213226i) q^{5} +(-0.374607 - 0.927184i) q^{9} +O(q^{10})\) \(q+(0.559193 - 0.829038i) q^{3} +(-0.241922 - 0.970296i) q^{4} +(1.51718 - 0.213226i) q^{5} +(-0.374607 - 0.927184i) q^{9} +(-0.939693 - 0.342020i) q^{12} +(0.671624 - 1.37703i) q^{15} +(-0.882948 + 0.469472i) q^{16} +(-0.573931 - 1.42053i) q^{20} +(-0.766044 - 0.642788i) q^{23} +(1.29510 - 0.371365i) q^{25} +(-0.978148 - 0.207912i) q^{27} +(-1.59381 + 0.995922i) q^{31} +(-0.809017 + 0.587785i) q^{36} +(1.39963 - 0.623157i) q^{37} +(-0.766044 - 1.32683i) q^{45} +(-0.0840186 + 0.336980i) q^{47} +(-0.104528 + 0.994522i) q^{48} +(0.990268 - 0.139173i) q^{49} +(-0.580762 + 1.78740i) q^{53} +(1.35192 - 1.30553i) q^{59} +(-1.49861 - 0.318539i) q^{60} +(0.669131 + 0.743145i) q^{64} +(1.76604 - 0.642788i) q^{67} +(-0.961262 + 0.275637i) q^{69} +(-0.339707 - 0.0722070i) q^{71} +(0.416337 - 1.28135i) q^{75} +(-1.23949 + 0.900539i) q^{80} +(-0.719340 + 0.694658i) q^{81} +(0.500000 + 0.866025i) q^{89} +(-0.438371 + 0.898794i) q^{92} +(-0.0655896 + 1.87824i) q^{93} +(0.343916 + 0.0483343i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 3 q^{5} + 3 q^{15} - 6 q^{20} + 3 q^{25} + 3 q^{27} + 3 q^{31} - 6 q^{36} + 3 q^{47} + 3 q^{48} + 3 q^{59} + 3 q^{64} + 24 q^{67} - 6 q^{75} + 12 q^{89} - 6 q^{93} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(3026\)
\(\chi(n)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{1}{9}\right)\)

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 0.615661 0.788011i \(-0.288889\pi\)
−0.615661 + 0.788011i \(0.711111\pi\)
\(3\) 0.559193 0.829038i 0.559193 0.829038i
\(4\) −0.241922 0.970296i −0.241922 0.970296i
\(5\) 1.51718 0.213226i 1.51718 0.213226i 0.669131 0.743145i \(-0.266667\pi\)
0.848048 + 0.529919i \(0.177778\pi\)
\(6\) 0 0
\(7\) 0 0 0.997564 0.0697565i \(-0.0222222\pi\)
−0.997564 + 0.0697565i \(0.977778\pi\)
\(8\) 0 0
\(9\) −0.374607 0.927184i −0.374607 0.927184i
\(10\) 0 0
\(11\) 0 0
\(12\) −0.939693 0.342020i −0.939693 0.342020i
\(13\) 0 0 −0.0348995 0.999391i \(-0.511111\pi\)
0.0348995 + 0.999391i \(0.488889\pi\)
\(14\) 0 0
\(15\) 0.671624 1.37703i 0.671624 1.37703i
\(16\) −0.882948 + 0.469472i −0.882948 + 0.469472i
\(17\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(18\) 0 0
\(19\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(20\) −0.573931 1.42053i −0.573931 1.42053i
\(21\) 0 0
\(22\) 0 0
\(23\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(24\) 0 0
\(25\) 1.29510 0.371365i 1.29510 0.371365i
\(26\) 0 0
\(27\) −0.978148 0.207912i −0.978148 0.207912i
\(28\) 0 0
\(29\) 0 0 −0.559193 0.829038i \(-0.688889\pi\)
0.559193 + 0.829038i \(0.311111\pi\)
\(30\) 0 0
\(31\) −1.59381 + 0.995922i −1.59381 + 0.995922i −0.615661 + 0.788011i \(0.711111\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(37\) 1.39963 0.623157i 1.39963 0.623157i 0.438371 0.898794i \(-0.355556\pi\)
0.961262 + 0.275637i \(0.0888889\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.559193 0.829038i \(-0.311111\pi\)
−0.559193 + 0.829038i \(0.688889\pi\)
\(42\) 0 0
\(43\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(44\) 0 0
\(45\) −0.766044 1.32683i −0.766044 1.32683i
\(46\) 0 0
\(47\) −0.0840186 + 0.336980i −0.0840186 + 0.336980i −0.997564 0.0697565i \(-0.977778\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(48\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(49\) 0.990268 0.139173i 0.990268 0.139173i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.580762 + 1.78740i −0.580762 + 1.78740i 0.0348995 + 0.999391i \(0.488889\pi\)
−0.615661 + 0.788011i \(0.711111\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.35192 1.30553i 1.35192 1.30553i 0.438371 0.898794i \(-0.355556\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(60\) −1.49861 0.318539i −1.49861 0.318539i
\(61\) 0 0 −0.848048 0.529919i \(-0.822222\pi\)
0.848048 + 0.529919i \(0.177778\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.76604 0.642788i 1.76604 0.642788i 0.766044 0.642788i \(-0.222222\pi\)
1.00000 \(0\)
\(68\) 0 0
\(69\) −0.961262 + 0.275637i −0.961262 + 0.275637i
\(70\) 0 0
\(71\) −0.339707 0.0722070i −0.339707 0.0722070i 0.0348995 0.999391i \(-0.488889\pi\)
−0.374607 + 0.927184i \(0.622222\pi\)
\(72\) 0 0
\(73\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(74\) 0 0
\(75\) 0.416337 1.28135i 0.416337 1.28135i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.615661 0.788011i \(-0.288889\pi\)
−0.615661 + 0.788011i \(0.711111\pi\)
\(80\) −1.23949 + 0.900539i −1.23949 + 0.900539i
\(81\) −0.719340 + 0.694658i −0.719340 + 0.694658i
\(82\) 0 0
\(83\) 0 0 0.0348995 0.999391i \(-0.488889\pi\)
−0.0348995 + 0.999391i \(0.511111\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.438371 + 0.898794i −0.438371 + 0.898794i
\(93\) −0.0655896 + 1.87824i −0.0655896 + 1.87824i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.343916 + 0.0483343i 0.343916 + 0.0483343i 0.309017 0.951057i \(-0.400000\pi\)
0.0348995 + 0.999391i \(0.488889\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −0.673648 1.16679i −0.673648 1.16679i
\(101\) 0 0 −0.374607 0.927184i \(-0.622222\pi\)
0.374607 + 0.927184i \(0.377778\pi\)
\(102\) 0 0
\(103\) −1.10209 + 1.06428i −1.10209 + 1.06428i −0.104528 + 0.994522i \(0.533333\pi\)
−0.997564 + 0.0697565i \(0.977778\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(108\) 0.0348995 + 0.999391i 0.0348995 + 0.999391i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0.266044 1.50881i 0.266044 1.50881i
\(112\) 0 0
\(113\) 0.152245 + 0.312148i 0.152245 + 0.312148i 0.961262 0.275637i \(-0.0888889\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(114\) 0 0
\(115\) −1.29929 0.811883i −1.29929 0.811883i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 1.35192 + 1.30553i 1.35192 + 1.30553i
\(125\) 0.486087 0.216420i 0.486087 0.216420i
\(126\) 0 0
\(127\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1.52836 0.106873i −1.52836 0.106873i
\(136\) 0 0
\(137\) −0.0655896 + 1.87824i −0.0655896 + 1.87824i 0.309017 + 0.951057i \(0.400000\pi\)
−0.374607 + 0.927184i \(0.622222\pi\)
\(138\) 0 0
\(139\) 0 0 −0.241922 0.970296i \(-0.577778\pi\)
0.241922 + 0.970296i \(0.422222\pi\)
\(140\) 0 0
\(141\) 0.232387 + 0.258091i 0.232387 + 0.258091i
\(142\) 0 0
\(143\) 0 0
\(144\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(145\) 0 0
\(146\) 0 0
\(147\) 0.438371 0.898794i 0.438371 0.898794i
\(148\) −0.943248 1.20730i −0.943248 1.20730i
\(149\) 0 0 −0.615661 0.788011i \(-0.711111\pi\)
0.615661 + 0.788011i \(0.288889\pi\)
\(150\) 0 0
\(151\) 0 0 −0.719340 0.694658i \(-0.755556\pi\)
0.719340 + 0.694658i \(0.244444\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.20574 + 1.85083i −2.20574 + 1.85083i
\(156\) 0 0
\(157\) −0.823868 1.68918i −0.823868 1.68918i −0.719340 0.694658i \(-0.755556\pi\)
−0.104528 0.994522i \(-0.533333\pi\)
\(158\) 0 0
\(159\) 1.15707 + 1.48098i 1.15707 + 1.48098i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.107320 0.330298i 0.107320 0.330298i −0.882948 0.469472i \(-0.844444\pi\)
0.990268 + 0.139173i \(0.0444444\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.882948 0.469472i \(-0.844444\pi\)
0.882948 + 0.469472i \(0.155556\pi\)
\(168\) 0 0
\(169\) −0.997564 + 0.0697565i −0.997564 + 0.0697565i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.719340 0.694658i \(-0.755556\pi\)
0.719340 + 0.694658i \(0.244444\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −0.326352 1.85083i −0.326352 1.85083i
\(178\) 0 0
\(179\) 0.196449 + 1.86909i 0.196449 + 1.86909i 0.438371 + 0.898794i \(0.355556\pi\)
−0.241922 + 0.970296i \(0.577778\pi\)
\(180\) −1.10209 + 1.06428i −1.10209 + 1.06428i
\(181\) 1.02517 1.13856i 1.02517 1.13856i 0.0348995 0.999391i \(-0.488889\pi\)
0.990268 0.139173i \(-0.0444444\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.99062 1.24388i 1.99062 1.24388i
\(186\) 0 0
\(187\) 0 0
\(188\) 0.347296 0.347296
\(189\) 0 0
\(190\) 0 0
\(191\) 0.194206 + 0.287922i 0.194206 + 0.287922i 0.913545 0.406737i \(-0.133333\pi\)
−0.719340 + 0.694658i \(0.755556\pi\)
\(192\) 0.990268 0.139173i 0.990268 0.139173i
\(193\) 0 0 0.374607 0.927184i \(-0.377778\pi\)
−0.374607 + 0.927184i \(0.622222\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.374607 0.927184i −0.374607 0.927184i
\(197\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(198\) 0 0
\(199\) −0.173648 + 0.300767i −0.173648 + 0.300767i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(200\) 0 0
\(201\) 0.454664 1.82356i 0.454664 1.82356i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.882948 0.469472i \(-0.844444\pi\)
0.882948 + 0.469472i \(0.155556\pi\)
\(212\) 1.87481 + 0.131099i 1.87481 + 0.131099i
\(213\) −0.249824 + 0.241252i −0.249824 + 0.241252i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0.241922 0.970296i 0.241922 0.970296i −0.719340 0.694658i \(-0.755556\pi\)
0.961262 0.275637i \(-0.0888889\pi\)
\(224\) 0 0
\(225\) −0.829478 1.06168i −0.829478 1.06168i
\(226\) 0 0
\(227\) 0 0 0.438371 0.898794i \(-0.355556\pi\)
−0.438371 + 0.898794i \(0.644444\pi\)
\(228\) 0 0
\(229\) −0.0348995 0.999391i −0.0348995 0.999391i −0.882948 0.469472i \(-0.844444\pi\)
0.848048 0.529919i \(-0.177778\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(234\) 0 0
\(235\) −0.0556184 + 0.529174i −0.0556184 + 0.529174i
\(236\) −1.59381 0.995922i −1.59381 0.995922i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.997564 0.0697565i \(-0.977778\pi\)
0.997564 + 0.0697565i \(0.0222222\pi\)
\(240\) 0.0534691 + 1.53116i 0.0534691 + 1.53116i
\(241\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(242\) 0 0
\(243\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(244\) 0 0
\(245\) 1.47274 0.422301i 1.47274 0.422301i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.95630 + 0.415823i −1.95630 + 0.415823i −0.978148 + 0.207912i \(0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.559193 0.829038i 0.559193 0.829038i
\(257\) −0.961262 0.275637i −0.961262 0.275637i −0.241922 0.970296i \(-0.577778\pi\)
−0.719340 + 0.694658i \(0.755556\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(264\) 0 0
\(265\) −0.500000 + 2.83564i −0.500000 + 2.83564i
\(266\) 0 0
\(267\) 0.997564 + 0.0697565i 0.997564 + 0.0697565i
\(268\) −1.05094 1.55808i −1.05094 1.55808i
\(269\) 0.107320 + 0.330298i 0.107320 + 0.330298i 0.990268 0.139173i \(-0.0444444\pi\)
−0.882948 + 0.469472i \(0.844444\pi\)
\(270\) 0 0
\(271\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(277\) 0 0 −0.374607 0.927184i \(-0.622222\pi\)
0.374607 + 0.927184i \(0.377778\pi\)
\(278\) 0 0
\(279\) 1.52045 + 1.10467i 1.52045 + 1.10467i
\(280\) 0 0
\(281\) 0 0 0.882948 0.469472i \(-0.155556\pi\)
−0.882948 + 0.469472i \(0.844444\pi\)
\(282\) 0 0
\(283\) 0 0 0.559193 0.829038i \(-0.311111\pi\)
−0.559193 + 0.829038i \(0.688889\pi\)
\(284\) 0.0121205 + 0.347085i 0.0121205 + 0.347085i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(290\) 0 0
\(291\) 0.232387 0.258091i 0.232387 0.258091i
\(292\) 0 0
\(293\) 0 0 −0.241922 0.970296i \(-0.577778\pi\)
0.241922 + 0.970296i \(0.422222\pi\)
\(294\) 0 0
\(295\) 1.77273 2.26899i 1.77273 2.26899i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −1.34401 0.0939826i −1.34401 0.0939826i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(308\) 0 0
\(309\) 0.266044 + 1.50881i 0.266044 + 1.50881i
\(310\) 0 0
\(311\) −1.05094 + 1.55808i −1.05094 + 1.55808i −0.241922 + 0.970296i \(0.577778\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(312\) 0 0
\(313\) 0.882948 0.469472i 0.882948 0.469472i 0.0348995 0.999391i \(-0.488889\pi\)
0.848048 + 0.529919i \(0.177778\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.749213 1.85437i −0.749213 1.85437i −0.374607 0.927184i \(-0.622222\pi\)
−0.374607 0.927184i \(-0.622222\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.17365 + 0.984808i 1.17365 + 0.984808i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.848048 + 0.529919i 0.848048 + 0.529919i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.266044 0.223238i 0.266044 0.223238i −0.500000 0.866025i \(-0.666667\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(332\) 0 0
\(333\) −1.10209 1.06428i −1.10209 1.06428i
\(334\) 0 0
\(335\) 2.54235 1.35179i 2.54235 1.35179i
\(336\) 0 0
\(337\) 0 0 −0.961262 0.275637i \(-0.911111\pi\)
0.961262 + 0.275637i \(0.0888889\pi\)
\(338\) 0 0
\(339\) 0.343916 + 0.0483343i 0.343916 + 0.0483343i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1.39963 + 0.623157i −1.39963 + 0.623157i
\(346\) 0 0
\(347\) 0 0 0.374607 0.927184i \(-0.377778\pi\)
−0.374607 + 0.927184i \(0.622222\pi\)
\(348\) 0 0
\(349\) 0 0 0.961262 0.275637i \(-0.0888889\pi\)
−0.961262 + 0.275637i \(0.911111\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(354\) 0 0
\(355\) −0.530793 0.0371166i −0.530793 0.0371166i
\(356\) 0.719340 0.694658i 0.719340 0.694658i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(360\) 0 0
\(361\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0.152245 0.312148i 0.152245 0.312148i −0.809017 0.587785i \(-0.800000\pi\)
0.961262 + 0.275637i \(0.0888889\pi\)
\(368\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 1.83832 0.390746i 1.83832 0.390746i
\(373\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(374\) 0 0
\(375\) 0.0923963 0.524005i 0.0923963 0.524005i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.65940 + 0.882318i 1.65940 + 0.882318i 0.990268 + 0.139173i \(0.0444444\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −0.0363024 0.345394i −0.0363024 0.345394i
\(389\) 0.152245 0.312148i 0.152245 0.312148i −0.809017 0.587785i \(-0.800000\pi\)
0.961262 + 0.275637i \(0.0888889\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.939693 + 1.62760i 0.939693 + 1.62760i 0.766044 + 0.642788i \(0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.969164 + 0.935911i −0.969164 + 0.935911i
\(401\) −0.573931 + 1.42053i −0.573931 + 1.42053i 0.309017 + 0.951057i \(0.400000\pi\)
−0.882948 + 0.469472i \(0.844444\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.943248 + 1.20730i −0.943248 + 1.20730i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.848048 0.529919i \(-0.177778\pi\)
−0.848048 + 0.529919i \(0.822222\pi\)
\(410\) 0 0
\(411\) 1.52045 + 1.10467i 1.52045 + 1.10467i
\(412\) 1.29929 + 0.811883i 1.29929 + 0.811883i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.43969 + 0.524005i −1.43969 + 0.524005i −0.939693 0.342020i \(-0.888889\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) −1.10209 1.06428i −1.10209 1.06428i −0.997564 0.0697565i \(-0.977778\pi\)
−0.104528 0.994522i \(-0.533333\pi\)
\(422\) 0 0
\(423\) 0.343916 0.0483343i 0.343916 0.0483343i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(432\) 0.961262 0.275637i 0.961262 0.275637i
\(433\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(440\) 0 0
\(441\) −0.500000 0.866025i −0.500000 0.866025i
\(442\) 0 0
\(443\) −1.10209 1.06428i −1.10209 1.06428i −0.997564 0.0697565i \(-0.977778\pi\)
−0.104528 0.994522i \(-0.533333\pi\)
\(444\) −1.52836 + 0.106873i −1.52836 + 0.106873i
\(445\) 0.943248 + 1.20730i 0.943248 + 1.20730i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.02517 + 1.13856i 1.02517 + 1.13856i 0.990268 + 0.139173i \(0.0444444\pi\)
0.0348995 + 0.999391i \(0.488889\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0.266044 0.223238i 0.266044 0.223238i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.0348995 0.999391i \(-0.488889\pi\)
−0.0348995 + 0.999391i \(0.511111\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −0.473442 + 1.45710i −0.473442 + 1.45710i
\(461\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(462\) 0 0
\(463\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(464\) 0 0
\(465\) 0.300978 + 2.86361i 0.300978 + 2.86361i
\(466\) 0 0
\(467\) 0.232387 0.258091i 0.232387 0.258091i −0.615661 0.788011i \(-0.711111\pi\)
0.848048 + 0.529919i \(0.177778\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −1.86110 0.261560i −1.86110 0.261560i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.87481 0.131099i 1.87481 0.131099i
\(478\) 0 0
\(479\) 0 0 −0.848048 0.529919i \(-0.822222\pi\)
0.848048 + 0.529919i \(0.177778\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.532089 0.532089
\(486\) 0 0
\(487\) −1.23949 + 0.900539i −1.23949 + 0.900539i −0.997564 0.0697565i \(-0.977778\pi\)
−0.241922 + 0.970296i \(0.577778\pi\)
\(488\) 0 0
\(489\) −0.213817 0.273673i −0.213817 0.273673i
\(490\) 0 0
\(491\) 0 0 0.719340 0.694658i \(-0.244444\pi\)
−0.719340 + 0.694658i \(0.755556\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0.939693 1.62760i 0.939693 1.62760i
\(497\) 0 0
\(498\) 0 0
\(499\) 1.47274 + 0.422301i 1.47274 + 0.422301i 0.913545 0.406737i \(-0.133333\pi\)
0.559193 + 0.829038i \(0.311111\pi\)
\(500\) −0.327587 0.419292i −0.327587 0.419292i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(508\) 0 0
\(509\) 0.997564 + 0.0697565i 0.997564 + 0.0697565i 0.559193 0.829038i \(-0.311111\pi\)
0.438371 + 0.898794i \(0.355556\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.44514 + 1.84969i −1.44514 + 1.84969i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −0.160147 + 1.52370i −0.160147 + 1.52370i 0.559193 + 0.829038i \(0.311111\pi\)
−0.719340 + 0.694658i \(0.755556\pi\)
\(522\) 0 0
\(523\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0 0
\(530\) 0 0
\(531\) −1.71690 0.764415i −1.71690 0.764415i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.65940 + 0.882318i 1.65940 + 0.882318i
\(538\) 0 0
\(539\) 0 0
\(540\) 0.266044 + 1.50881i 0.266044 + 1.50881i
\(541\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(542\) 0 0
\(543\) −0.370646 1.48658i −0.370646 1.48658i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.241922 0.970296i \(-0.422222\pi\)
−0.241922 + 0.970296i \(0.577778\pi\)
\(548\) 1.83832 0.390746i 1.83832 0.390746i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0.0819195 2.34587i 0.0819195 2.34587i
\(556\) 0 0
\(557\) 0 0 0.913545 0.406737i \(-0.133333\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.848048 0.529919i \(-0.177778\pi\)
−0.848048 + 0.529919i \(0.822222\pi\)
\(564\) 0.194206 0.287922i 0.194206 0.287922i
\(565\) 0.297540 + 0.441122i 0.297540 + 0.441122i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.961262 0.275637i \(-0.0888889\pi\)
−0.961262 + 0.275637i \(0.911111\pi\)
\(570\) 0 0
\(571\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(572\) 0 0
\(573\) 0.347296 0.347296
\(574\) 0 0
\(575\) −1.23082 0.547995i −1.23082 0.547995i
\(576\) 0.438371 0.898794i 0.438371 0.898794i
\(577\) −1.49861 0.318539i −1.49861 0.318539i −0.615661 0.788011i \(-0.711111\pi\)
−0.882948 + 0.469472i \(0.844444\pi\)
\(578\) 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
\(587\) −1.52836 + 0.106873i −1.52836 + 0.106873i −0.809017 0.587785i \(-0.800000\pi\)
−0.719340 + 0.694658i \(0.755556\pi\)
\(588\) −0.978148 0.207912i −0.978148 0.207912i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.943248 + 1.20730i −0.943248 + 1.20730i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.152245 + 0.312148i 0.152245 + 0.312148i
\(598\) 0 0
\(599\) −0.990268 + 0.139173i −0.990268 + 0.139173i −0.615661 0.788011i \(-0.711111\pi\)
−0.374607 + 0.927184i \(0.622222\pi\)
\(600\) 0 0
\(601\) 0 0 0.997564 0.0697565i \(-0.0222222\pi\)
−0.997564 + 0.0697565i \(0.977778\pi\)
\(602\) 0 0
\(603\) −1.25755 1.39666i −1.25755 1.39666i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.0348995 0.999391i \(-0.511111\pi\)
0.0348995 + 0.999391i \(0.488889\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.43969 1.20805i −1.43969 1.20805i −0.939693 0.342020i \(-0.888889\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(618\) 0 0
\(619\) 1.47274 0.422301i 1.47274 0.422301i 0.559193 0.829038i \(-0.311111\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(620\) 2.32947 + 1.69246i 2.32947 + 1.69246i
\(621\) 0.615661 + 0.788011i 0.615661 + 0.788011i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.451237 + 0.281964i −0.451237 + 0.281964i
\(626\) 0 0
\(627\) 0 0
\(628\) −1.43969 + 1.20805i −1.43969 + 1.20805i
\(629\) 0 0
\(630\) 0 0
\(631\) −1.71690 + 0.764415i −1.71690 + 0.764415i −0.719340 + 0.694658i \(0.755556\pi\)
−0.997564 + 0.0697565i \(0.977778\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 1.15707 1.48098i 1.15707 1.48098i
\(637\) 0 0
\(638\) 0 0
\(639\) 0.0603074 + 0.342020i 0.0603074 + 0.342020i
\(640\) 0 0
\(641\) 0.241922 0.970296i 0.241922 0.970296i −0.719340 0.694658i \(-0.755556\pi\)
0.961262 0.275637i \(-0.0888889\pi\)
\(642\) 0 0
\(643\) −0.990268 + 0.139173i −0.990268 + 0.139173i −0.615661 0.788011i \(-0.711111\pi\)
−0.374607 + 0.927184i \(0.622222\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.618034 1.90211i 0.618034 1.90211i 0.309017 0.951057i \(-0.400000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −0.346450 0.0242262i −0.346450 0.0242262i
\(653\) −1.10209 + 1.06428i −1.10209 + 1.06428i −0.104528 + 0.994522i \(0.533333\pi\)
−0.997564 + 0.0697565i \(0.977778\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(660\) 0 0
\(661\) −0.326352 + 0.118782i −0.326352 + 0.118782i −0.500000 0.866025i \(-0.666667\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −0.669131 0.743145i −0.669131 0.743145i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.615661 0.788011i \(-0.288889\pi\)
−0.615661 + 0.788011i \(0.711111\pi\)
\(674\) 0 0
\(675\) −1.34401 + 0.0939826i −1.34401 + 0.0939826i
\(676\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(677\) 0 0 0.0348995 0.999391i \(-0.488889\pi\)
−0.0348995 + 0.999391i \(0.511111\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −0.173648 0.300767i −0.173648 0.300767i 0.766044 0.642788i \(-0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(684\) 0 0
\(685\) 0.300978 + 2.86361i 0.300978 + 2.86361i
\(686\) 0 0
\(687\) −0.848048 0.529919i −0.848048 0.529919i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1.51718 + 0.213226i 1.51718 + 0.213226i 0.848048 0.529919i \(-0.177778\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(692\) 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 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0.407604 + 0.342020i 0.407604 + 0.342020i
\(706\) 0 0
\(707\) 0 0
\(708\) −1.71690 + 0.764415i −1.71690 + 0.764415i
\(709\) 1.29929 + 0.811883i 1.29929 + 0.811883i 0.990268 0.139173i \(-0.0444444\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.86110 + 0.261560i 1.86110 + 0.261560i
\(714\) 0 0
\(715\) 0 0
\(716\) 1.76604 0.642788i 1.76604 0.642788i
\(717\) 0 0
\(718\) 0 0
\(719\) 0.317271 0.141258i 0.317271 0.141258i −0.241922 0.970296i \(-0.577778\pi\)
0.559193 + 0.829038i \(0.311111\pi\)
\(720\) 1.29929 + 0.811883i 1.29929 + 0.811883i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −1.35275 0.719272i −1.35275 0.719272i
\(725\) 0 0
\(726\) 0 0
\(727\) −0.326352 0.118782i −0.326352 0.118782i 0.173648 0.984808i \(-0.444444\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(728\) 0 0
\(729\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.241922 0.970296i \(-0.577778\pi\)
0.241922 + 0.970296i \(0.422222\pi\)
\(734\) 0 0
\(735\) 0.473442 1.45710i 0.473442 1.45710i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(740\) −1.68850 1.63057i −1.68850 1.63057i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.615661 0.788011i \(-0.711111\pi\)
0.615661 + 0.788011i \(0.288889\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0.152245 + 0.312148i 0.152245 + 0.312148i 0.961262 0.275637i \(-0.0888889\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(752\) −0.0840186 0.336980i −0.0840186 0.336980i
\(753\) −0.749213 + 1.85437i −0.749213 + 1.85437i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.580762 + 1.78740i −0.580762 + 1.78740i 0.0348995 + 0.999391i \(0.488889\pi\)
−0.615661 + 0.788011i \(0.711111\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.882948 0.469472i \(-0.844444\pi\)
0.882948 + 0.469472i \(0.155556\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0.232387 0.258091i 0.232387 0.258091i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.374607 0.927184i −0.374607 0.927184i
\(769\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(770\) 0 0
\(771\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(772\) 0 0
\(773\) −0.209057 1.98904i −0.209057 1.98904i −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 0.994522i \(-0.533333\pi\)
\(774\) 0 0
\(775\) −1.69430 + 1.88171i −1.69430 + 1.88171i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(785\) −1.61013 2.38712i −1.61013 2.38712i
\(786\) 0 0
\(787\) 0 0 0.374607 0.927184i \(-0.377778\pi\)
−0.374607 + 0.927184i \(0.622222\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 2.07126 + 2.00019i 2.07126 + 2.00019i
\(796\) 0.333843 + 0.0957278i 0.333843 + 0.0957278i
\(797\) −0.943248 1.20730i −0.943248 1.20730i −0.978148 0.207912i \(-0.933333\pi\)
0.0348995 0.999391i \(-0.488889\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0.615661 0.788011i 0.615661 0.788011i
\(802\) 0 0
\(803\) 0 0
\(804\) −1.87939 −1.87939
\(805\) 0 0
\(806\) 0 0
\(807\) 0.333843 + 0.0957278i 0.333843 + 0.0957278i
\(808\) 0 0
\(809\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(810\) 0 0
\(811\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0.0923963 0.524005i 0.0923963 0.524005i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.438371 0.898794i \(-0.355556\pi\)
−0.438371 + 0.898794i \(0.644444\pi\)
\(822\) 0 0
\(823\) −0.0348995 0.999391i −0.0348995 0.999391i −0.882948 0.469472i \(-0.844444\pi\)
0.848048 0.529919i \(-0.177778\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(828\) 0.997564 + 0.0697565i 0.997564 + 0.0697565i
\(829\) 0.196449 1.86909i 0.196449 1.86909i −0.241922 0.970296i \(-0.577778\pi\)
0.438371 0.898794i \(-0.355556\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.76604 0.642788i 1.76604 0.642788i
\(838\) 0 0
\(839\) −0.961262 + 0.275637i −0.961262 + 0.275637i −0.719340 0.694658i \(-0.755556\pi\)
−0.241922 + 0.970296i \(0.577778\pi\)
\(840\) 0 0
\(841\) −0.374607 + 0.927184i −0.374607 + 0.927184i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.49861 + 0.318539i −1.49861 + 0.318539i
\(846\) 0 0
\(847\) 0 0
\(848\) −0.326352 1.85083i −0.326352 1.85083i
\(849\) 0 0
\(850\) 0 0
\(851\) −1.47274 0.422301i −1.47274 0.422301i
\(852\) 0.294524 + 0.184039i 0.294524 + 0.184039i
\(853\) 0 0 0.882948 0.469472i \(-0.155556\pi\)
−0.882948 + 0.469472i \(0.844444\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(858\) 0 0
\(859\) 0.0603074 0.342020i 0.0603074 0.342020i −0.939693 0.342020i \(-0.888889\pi\)
1.00000 \(0\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.848048 0.529919i 0.848048 0.529919i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −0.0840186 0.336980i −0.0840186 0.336980i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.559193 0.829038i \(-0.311111\pi\)
−0.559193 + 0.829038i \(0.688889\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.173648 + 0.300767i −0.173648 + 0.300767i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(882\) 0 0
\(883\) 0.317271 + 0.141258i 0.317271 + 0.141258i 0.559193 0.829038i \(-0.311111\pi\)
−0.241922 + 0.970296i \(0.577778\pi\)
\(884\) 0 0
\(885\) −0.889779 2.73846i −0.889779 2.73846i
\(886\) 0 0
\(887\) 0 0 −0.241922 0.970296i \(-0.577778\pi\)
0.241922 + 0.970296i \(0.422222\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −1.00000 −1.00000
\(893\) 0 0
\(894\) 0 0
\(895\) 0.696586 + 2.79386i 0.696586 + 2.79386i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −0.829478 + 1.06168i −0.829478 + 1.06168i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.31259 1.94600i 1.31259 1.94600i
\(906\) 0 0
\(907\) 0.882948 0.469472i 0.882948 0.469472i 0.0348995 0.999391i \(-0.488889\pi\)
0.848048 + 0.529919i \(0.177778\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −0.573931 1.42053i −0.573931 1.42053i −0.882948 0.469472i \(-0.844444\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −0.961262 + 0.275637i −0.961262 + 0.275637i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.58125 1.32683i 1.58125 1.32683i
\(926\) 0 0
\(927\) 1.39963 + 0.623157i 1.39963 + 0.623157i
\(928\) 0 0
\(929\) 1.65940 0.882318i 1.65940 0.882318i 0.669131 0.743145i \(-0.266667\pi\)
0.990268 0.139173i \(-0.0444444\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0.704030 + 1.74254i 0.704030 + 1.74254i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(938\) 0 0
\(939\) 0.104528 0.994522i 0.104528 0.994522i
\(940\) 0.526911 0.0740525i 0.526911 0.0740525i
\(941\) 0 0 0.374607 0.927184i \(-0.377778\pi\)
−0.374607 + 0.927184i \(0.622222\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −0.580762 + 1.78740i −0.580762 + 1.78740i
\(945\) 0 0
\(946\) 0 0
\(947\) 1.76604 + 0.642788i 1.76604 + 0.642788i 1.00000 \(0\)
0.766044 + 0.642788i \(0.222222\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −1.95630 0.415823i −1.95630 0.415823i
\(952\) 0 0
\(953\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(954\) 0 0
\(955\) 0.356037 + 0.395419i 0.356037 + 0.395419i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 1.47274 0.422301i 1.47274 0.422301i
\(961\) 1.10999 2.27583i 1.10999 2.27583i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(972\) 0.913545 0.406737i 0.913545 0.406737i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.76590 0.938943i −1.76590 0.938943i −0.882948 0.469472i \(-0.844444\pi\)
−0.882948 0.469472i \(-0.844444\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.766044 1.32683i −0.766044 1.32683i
\(981\) 0 0
\(982\) 0 0
\(983\) −0.823868 + 1.68918i −0.823868 + 1.68918i −0.104528 + 0.994522i \(0.533333\pi\)
−0.719340 + 0.694658i \(0.755556\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) 0 0
\(993\) −0.0363024 0.345394i −0.0363024 0.345394i
\(994\) 0 0
\(995\) −0.199324 + 0.493344i −0.199324 + 0.493344i
\(996\) 0 0
\(997\) 0 0 −0.559193 0.829038i \(-0.688889\pi\)
0.559193 + 0.829038i \(0.311111\pi\)
\(998\) 0 0
\(999\) −1.49861 + 0.318539i −1.49861 + 0.318539i
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 3267.1.bf.a.112.1 24
11.2 odd 10 inner 3267.1.bf.a.1570.1 24
11.3 even 5 inner 3267.1.bf.a.2218.1 24
11.4 even 5 297.1.q.a.274.1 yes 6
11.5 even 5 inner 3267.1.bf.a.2272.1 24
11.6 odd 10 inner 3267.1.bf.a.2272.1 24
11.7 odd 10 297.1.q.a.274.1 yes 6
11.8 odd 10 inner 3267.1.bf.a.2218.1 24
11.9 even 5 inner 3267.1.bf.a.1570.1 24
11.10 odd 2 CM 3267.1.bf.a.112.1 24
27.7 even 9 inner 3267.1.bf.a.2653.1 24
33.26 odd 10 891.1.q.a.307.1 6
33.29 even 10 891.1.q.a.307.1 6
99.4 even 15 2673.1.q.b.109.1 6
99.7 odd 30 2673.1.q.d.1000.1 6
99.29 even 30 2673.1.q.a.1000.1 6
99.40 odd 30 2673.1.q.b.109.1 6
99.59 odd 30 2673.1.q.c.109.1 6
99.70 even 15 2673.1.q.d.1000.1 6
99.92 odd 30 2673.1.q.a.1000.1 6
99.95 even 30 2673.1.q.c.109.1 6
297.7 odd 90 297.1.q.a.142.1 6
297.29 even 90 2673.1.q.c.1594.1 6
297.61 odd 90 inner 3267.1.bf.a.1546.1 24
297.70 even 45 2673.1.q.d.703.1 6
297.92 odd 90 2673.1.q.a.703.1 6
297.106 odd 90 2673.1.q.b.1594.1 6
297.115 even 45 inner 3267.1.bf.a.1546.1 24
297.128 even 90 891.1.q.a.505.1 6
297.142 odd 18 inner 3267.1.bf.a.2653.1 24
297.169 even 45 297.1.q.a.142.1 6
297.191 odd 90 2673.1.q.c.1594.1 6
297.196 even 45 inner 3267.1.bf.a.844.1 24
297.205 odd 90 2673.1.q.d.703.1 6
297.223 even 45 inner 3267.1.bf.a.1492.1 24
297.227 even 90 2673.1.q.a.703.1 6
297.250 odd 90 inner 3267.1.bf.a.1492.1 24
297.268 even 45 2673.1.q.b.1594.1 6
297.277 odd 90 inner 3267.1.bf.a.844.1 24
297.290 odd 90 891.1.q.a.505.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
297.1.q.a.142.1 6 297.7 odd 90
297.1.q.a.142.1 6 297.169 even 45
297.1.q.a.274.1 yes 6 11.4 even 5
297.1.q.a.274.1 yes 6 11.7 odd 10
891.1.q.a.307.1 6 33.26 odd 10
891.1.q.a.307.1 6 33.29 even 10
891.1.q.a.505.1 6 297.128 even 90
891.1.q.a.505.1 6 297.290 odd 90
2673.1.q.a.703.1 6 297.92 odd 90
2673.1.q.a.703.1 6 297.227 even 90
2673.1.q.a.1000.1 6 99.29 even 30
2673.1.q.a.1000.1 6 99.92 odd 30
2673.1.q.b.109.1 6 99.4 even 15
2673.1.q.b.109.1 6 99.40 odd 30
2673.1.q.b.1594.1 6 297.106 odd 90
2673.1.q.b.1594.1 6 297.268 even 45
2673.1.q.c.109.1 6 99.59 odd 30
2673.1.q.c.109.1 6 99.95 even 30
2673.1.q.c.1594.1 6 297.29 even 90
2673.1.q.c.1594.1 6 297.191 odd 90
2673.1.q.d.703.1 6 297.70 even 45
2673.1.q.d.703.1 6 297.205 odd 90
2673.1.q.d.1000.1 6 99.7 odd 30
2673.1.q.d.1000.1 6 99.70 even 15
3267.1.bf.a.112.1 24 1.1 even 1 trivial
3267.1.bf.a.112.1 24 11.10 odd 2 CM
3267.1.bf.a.844.1 24 297.196 even 45 inner
3267.1.bf.a.844.1 24 297.277 odd 90 inner
3267.1.bf.a.1492.1 24 297.223 even 45 inner
3267.1.bf.a.1492.1 24 297.250 odd 90 inner
3267.1.bf.a.1546.1 24 297.61 odd 90 inner
3267.1.bf.a.1546.1 24 297.115 even 45 inner
3267.1.bf.a.1570.1 24 11.2 odd 10 inner
3267.1.bf.a.1570.1 24 11.9 even 5 inner
3267.1.bf.a.2218.1 24 11.3 even 5 inner
3267.1.bf.a.2218.1 24 11.8 odd 10 inner
3267.1.bf.a.2272.1 24 11.5 even 5 inner
3267.1.bf.a.2272.1 24 11.6 odd 10 inner
3267.1.bf.a.2653.1 24 27.7 even 9 inner
3267.1.bf.a.2653.1 24 297.142 odd 18 inner