Properties

Label 2925.1.er.a.883.1
Level $2925$
Weight $1$
Character 2925.883
Analytic conductor $1.460$
Analytic rank $0$
Dimension $32$
Projective image $D_{40}$
CM discriminant -39
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2925,1,Mod(298,2925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2925, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 11, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2925.298");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2925.er (of order \(20\), degree \(8\), 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: \(1.45976516195\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{80})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{32} - x^{24} + x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{40}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{40} + \cdots)\)

Embedding invariants

Embedding label 883.1
Root \(0.649448 + 0.760406i\) of defining polynomial
Character \(\chi\) \(=\) 2925.883
Dual form 2925.1.er.a.2872.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.905182 + 1.77652i) q^{2} +(-1.74888 - 2.40713i) q^{4} +(0.233445 - 0.972370i) q^{5} +(3.89008 - 0.616129i) q^{8} +O(q^{10})\) \(q+(-0.905182 + 1.77652i) q^{2} +(-1.74888 - 2.40713i) q^{4} +(0.233445 - 0.972370i) q^{5} +(3.89008 - 0.616129i) q^{8} +(1.51612 + 1.29489i) q^{10} +(1.62182 - 0.526961i) q^{11} +(-0.891007 + 0.453990i) q^{13} +(-1.50723 + 4.63877i) q^{16} +(-2.74889 + 1.13863i) q^{20} +(-0.531885 + 3.35819i) q^{22} +(-0.891007 - 0.453990i) q^{25} -1.99383i q^{26} +(-4.09156 - 4.09156i) q^{32} +(0.309017 - 3.92643i) q^{40} +(1.23532 + 0.401381i) q^{41} +(0.642040 + 0.642040i) q^{43} +(-4.10483 - 2.98233i) q^{44} +(1.28290 + 0.203192i) q^{47} -1.00000i q^{49} +(1.61305 - 1.17195i) q^{50} +(2.65108 + 1.35079i) q^{52} +(-0.133795 - 1.70002i) q^{55} +(0.600958 - 1.84956i) q^{59} +(-0.0966818 - 0.297556i) q^{61} +(6.33356 - 2.05790i) q^{64} +(0.233445 + 0.972370i) q^{65} +(-0.614234 - 0.845420i) q^{71} +(-0.831254 - 1.14412i) q^{79} +(4.15874 + 2.54848i) q^{80} +(-1.83125 + 1.83125i) q^{82} +(-0.154986 + 0.0245474i) q^{83} +(-1.72176 + 0.559433i) q^{86} +(5.98433 - 3.04917i) q^{88} +(0.236511 + 0.727907i) q^{89} +(-1.52224 + 2.09518i) q^{94} +(1.77652 + 0.905182i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 8 q^{10} + 8 q^{16} - 8 q^{22} - 8 q^{40} + 8 q^{43} + 8 q^{52} + 40 q^{64} - 32 q^{82} + 24 q^{88} - 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(2251\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{20}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.905182 + 1.77652i −0.905182 + 1.77652i −0.382683 + 0.923880i \(0.625000\pi\)
−0.522499 + 0.852640i \(0.675000\pi\)
\(3\) 0 0
\(4\) −1.74888 2.40713i −1.74888 2.40713i
\(5\) 0.233445 0.972370i 0.233445 0.972370i
\(6\) 0 0
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 3.89008 0.616129i 3.89008 0.616129i
\(9\) 0 0
\(10\) 1.51612 + 1.29489i 1.51612 + 1.29489i
\(11\) 1.62182 0.526961i 1.62182 0.526961i 0.649448 0.760406i \(-0.275000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(12\) 0 0
\(13\) −0.891007 + 0.453990i −0.891007 + 0.453990i
\(14\) 0 0
\(15\) 0 0
\(16\) −1.50723 + 4.63877i −1.50723 + 4.63877i
\(17\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(18\) 0 0
\(19\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(20\) −2.74889 + 1.13863i −2.74889 + 1.13863i
\(21\) 0 0
\(22\) −0.531885 + 3.35819i −0.531885 + 3.35819i
\(23\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(24\) 0 0
\(25\) −0.891007 0.453990i −0.891007 0.453990i
\(26\) 1.99383i 1.99383i
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(32\) −4.09156 4.09156i −4.09156 4.09156i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.309017 3.92643i 0.309017 3.92643i
\(41\) 1.23532 + 0.401381i 1.23532 + 0.401381i 0.852640 0.522499i \(-0.175000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(42\) 0 0
\(43\) 0.642040 + 0.642040i 0.642040 + 0.642040i 0.951057 0.309017i \(-0.100000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(44\) −4.10483 2.98233i −4.10483 2.98233i
\(45\) 0 0
\(46\) 0 0
\(47\) 1.28290 + 0.203192i 1.28290 + 0.203192i 0.760406 0.649448i \(-0.225000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(48\) 0 0
\(49\) 1.00000i 1.00000i
\(50\) 1.61305 1.17195i 1.61305 1.17195i
\(51\) 0 0
\(52\) 2.65108 + 1.35079i 2.65108 + 1.35079i
\(53\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(54\) 0 0
\(55\) −0.133795 1.70002i −0.133795 1.70002i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.600958 1.84956i 0.600958 1.84956i 0.0784591 0.996917i \(-0.475000\pi\)
0.522499 0.852640i \(-0.325000\pi\)
\(60\) 0 0
\(61\) −0.0966818 0.297556i −0.0966818 0.297556i 0.891007 0.453990i \(-0.150000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 6.33356 2.05790i 6.33356 2.05790i
\(65\) 0.233445 + 0.972370i 0.233445 + 0.972370i
\(66\) 0 0
\(67\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.614234 0.845420i −0.614234 0.845420i 0.382683 0.923880i \(-0.375000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(72\) 0 0
\(73\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.831254 1.14412i −0.831254 1.14412i −0.987688 0.156434i \(-0.950000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(80\) 4.15874 + 2.54848i 4.15874 + 2.54848i
\(81\) 0 0
\(82\) −1.83125 + 1.83125i −1.83125 + 1.83125i
\(83\) −0.154986 + 0.0245474i −0.154986 + 0.0245474i −0.233445 0.972370i \(-0.575000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.72176 + 0.559433i −1.72176 + 0.559433i
\(87\) 0 0
\(88\) 5.98433 3.04917i 5.98433 3.04917i
\(89\) 0.236511 + 0.727907i 0.236511 + 0.727907i 0.996917 + 0.0784591i \(0.0250000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −1.52224 + 2.09518i −1.52224 + 2.09518i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(98\) 1.77652 + 0.905182i 1.77652 + 0.905182i
\(99\) 0 0
\(100\) 0.465451 + 2.93874i 0.465451 + 2.93874i
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −1.39680 0.221232i −1.39680 0.221232i −0.587785 0.809017i \(-0.700000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(104\) −3.18637 + 2.31504i −3.18637 + 2.31504i
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) 0 0
\(109\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(110\) 3.14123 + 1.30114i 3.14123 + 1.30114i
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 2.74180 + 2.74180i 2.74180 + 2.74180i
\(119\) 0 0
\(120\) 0 0
\(121\) 1.54359 1.12148i 1.54359 1.12148i
\(122\) 0.616129 + 0.0975852i 0.616129 + 0.0975852i
\(123\) 0 0
\(124\) 0 0
\(125\) −0.649448 + 0.760406i −0.649448 + 0.760406i
\(126\) 0 0
\(127\) 0.550672 + 0.280582i 0.550672 + 0.280582i 0.707107 0.707107i \(-0.250000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(128\) −1.17195 + 7.39938i −1.17195 + 7.39938i
\(129\) 0 0
\(130\) −1.93874 0.465451i −1.93874 0.465451i
\(131\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.681947 0.347469i 0.681947 0.347469i −0.0784591 0.996917i \(-0.525000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(138\) 0 0
\(139\) −1.11803 + 0.363271i −1.11803 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.05790 0.325939i 2.05790 0.325939i
\(143\) −1.20582 + 1.20582i −1.20582 + 1.20582i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.39680 1.39680i 1.39680 1.39680i 0.587785 0.809017i \(-0.300000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(158\) 2.78499 0.441100i 2.78499 0.441100i
\(159\) 0 0
\(160\) −4.93366 + 3.02335i −4.93366 + 3.02335i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(164\) −1.19426 3.67555i −1.19426 3.67555i
\(165\) 0 0
\(166\) 0.0966818 0.297556i 0.0966818 0.297556i
\(167\) −0.0730378 0.461143i −0.0730378 0.461143i −0.996917 0.0784591i \(-0.975000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(168\) 0 0
\(169\) 0.587785 0.809017i 0.587785 0.809017i
\(170\) 0 0
\(171\) 0 0
\(172\) 0.422621 2.66832i 0.422621 2.66832i
\(173\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 8.31749i 8.31749i
\(177\) 0 0
\(178\) −1.50723 0.238721i −1.50723 0.238721i
\(179\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(180\) 0 0
\(181\) −0.951057 0.690983i −0.951057 0.690983i 1.00000i \(-0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −1.75454 3.44348i −1.75454 3.44348i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(192\) 0 0
\(193\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.40713 + 1.74888i −2.40713 + 1.74888i
\(197\) −1.82501 0.289053i −1.82501 0.289053i −0.852640 0.522499i \(-0.825000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(198\) 0 0
\(199\) 1.78201i 1.78201i 0.453990 + 0.891007i \(0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(200\) −3.74581 1.21709i −3.74581 1.21709i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0.678671 1.10749i 0.678671 1.10749i
\(206\) 1.65738 2.28119i 1.65738 2.28119i
\(207\) 0 0
\(208\) −0.763007 4.81744i −0.763007 4.81744i
\(209\) 0 0
\(210\) 0 0
\(211\) 0.280582 + 0.863541i 0.280582 + 0.863541i 0.987688 + 0.156434i \(0.0500000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.774181 0.474419i 0.774181 0.474419i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −3.85819 + 3.29520i −3.85819 + 3.29520i
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.347469 0.681947i 0.347469 0.681947i −0.649448 0.760406i \(-0.725000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(228\) 0 0
\(229\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(234\) 0 0
\(235\) 0.497066 1.20002i 0.497066 1.20002i
\(236\) −5.50313 + 1.78808i −5.50313 + 1.78808i
\(237\) 0 0
\(238\) 0 0
\(239\) 0.570989 + 1.75732i 0.570989 + 1.75732i 0.649448 + 0.760406i \(0.275000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(240\) 0 0
\(241\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(242\) 0.595108 + 3.75736i 0.595108 + 3.75736i
\(243\) 0 0
\(244\) −0.547171 + 0.753116i −0.547171 + 0.753116i
\(245\) −0.972370 0.233445i −0.972370 0.233445i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −0.763007 1.84206i −0.763007 1.84206i
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −0.996917 + 0.724303i −0.996917 + 0.724303i
\(255\) 0 0
\(256\) −6.69667 4.86541i −6.69667 4.86541i
\(257\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1.93235 2.26249i 1.93235 2.26249i
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(270\) 0 0
\(271\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 1.52601i 1.52601i
\(275\) −1.68429 0.266765i −1.68429 0.266765i
\(276\) 0 0
\(277\) 1.58779 + 0.809017i 1.58779 + 0.809017i 1.00000 \(0\)
0.587785 + 0.809017i \(0.300000\pi\)
\(278\) 0.366666 2.31504i 0.366666 2.31504i
\(279\) 0 0
\(280\) 0 0
\(281\) −0.274431 + 0.377723i −0.274431 + 0.377723i −0.923880 0.382683i \(-0.875000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(282\) 0 0
\(283\) 0.253116 + 1.59811i 0.253116 + 1.59811i 0.707107 + 0.707107i \(0.250000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(284\) −0.960814 + 2.95708i −0.960814 + 2.95708i
\(285\) 0 0
\(286\) −1.05067 3.23364i −1.05067 3.23364i
\(287\) 0 0
\(288\) 0 0
\(289\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.07538 1.07538i 1.07538 1.07538i 0.0784591 0.996917i \(-0.475000\pi\)
0.996917 0.0784591i \(-0.0250000\pi\)
\(294\) 0 0
\(295\) −1.65816 1.01612i −1.65816 1.01612i
\(296\) 0 0
\(297\) 0 0
\(298\) −1.67256 + 3.28258i −1.67256 + 3.28258i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.311904 + 0.0245474i −0.311904 + 0.0245474i
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(312\) 0 0
\(313\) −0.550672 + 0.280582i −0.550672 + 0.280582i −0.707107 0.707107i \(-0.750000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(314\) 1.21709 + 3.74581i 1.21709 + 3.74581i
\(315\) 0 0
\(316\) −1.30029 + 4.00187i −1.30029 + 4.00187i
\(317\) −0.0245474 0.154986i −0.0245474 0.154986i 0.972370 0.233445i \(-0.0750000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.522499 6.63897i −0.522499 6.63897i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1.00000 1.00000
\(326\) 0 0
\(327\) 0 0
\(328\) 5.05282 + 0.800287i 5.05282 + 0.800287i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(332\) 0.330142 + 0.330142i 0.330142 + 0.330142i
\(333\) 0 0
\(334\) 0.885341 + 0.287665i 0.885341 + 0.287665i
\(335\) 0 0
\(336\) 0 0
\(337\) 0.863541 + 1.69480i 0.863541 + 1.69480i 0.707107 + 0.707107i \(0.250000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(338\) 0.905182 + 1.77652i 0.905182 + 1.77652i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 2.89317 + 2.10201i 2.89317 + 2.10201i
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −8.79185 4.47967i −8.79185 4.47967i
\(353\) 0.237907 1.50209i 0.237907 1.50209i −0.522499 0.852640i \(-0.675000\pi\)
0.760406 0.649448i \(-0.225000\pi\)
\(354\) 0 0
\(355\) −0.965451 + 0.399903i −0.965451 + 0.399903i
\(356\) 1.33854 1.84234i 1.33854 1.84234i
\(357\) 0 0
\(358\) 0 0
\(359\) 0.469957 1.44638i 0.469957 1.44638i −0.382683 0.923880i \(-0.625000\pi\)
0.852640 0.522499i \(-0.175000\pi\)
\(360\) 0 0
\(361\) −0.309017 0.951057i −0.309017 0.951057i
\(362\) 2.08842 1.06411i 2.08842 1.06411i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.76007 + 0.278768i −1.76007 + 0.278768i −0.951057 0.309017i \(-0.900000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.533698 1.04744i 0.533698 1.04744i −0.453990 0.891007i \(-0.650000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 5.11580 5.11580
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.82501 + 0.289053i −1.82501 + 0.289053i −0.972370 0.233445i \(-0.925000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.616129 3.89008i −0.616129 3.89008i
\(393\) 0 0
\(394\) 2.16547 2.98052i 2.16547 2.98052i
\(395\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(396\) 0 0
\(397\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(398\) −3.16578 1.61305i −3.16578 1.61305i
\(399\) 0 0
\(400\) 3.44891 3.44891i 3.44891 3.44891i
\(401\) 1.04500i 1.04500i 0.852640 + 0.522499i \(0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(410\) 1.35316 + 2.20815i 1.35316 + 2.20815i
\(411\) 0 0
\(412\) 1.91031 + 3.74919i 1.91031 + 3.74919i
\(413\) 0 0
\(414\) 0 0
\(415\) −0.0123117 + 0.156434i −0.0123117 + 0.156434i
\(416\) 5.50313 + 1.78808i 5.50313 + 1.78808i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(420\) 0 0
\(421\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(422\) −1.78808 0.283203i −1.78808 0.283203i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0.142040 + 1.80478i 0.142040 + 1.80478i
\(431\) −1.17195 + 1.61305i −1.17195 + 1.61305i −0.522499 + 0.852640i \(0.675000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(432\) 0 0
\(433\) 0.183900 + 1.16110i 0.183900 + 1.16110i 0.891007 + 0.453990i \(0.150000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −1.80902 + 0.587785i −1.80902 + 0.587785i −0.809017 + 0.587785i \(0.800000\pi\)
−1.00000 \(\pi\)
\(440\) −1.56791 6.53080i −1.56791 6.53080i
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) 0 0
\(445\) 0.763007 0.0600500i 0.763007 0.0600500i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.94474 1.94474 0.972370 0.233445i \(-0.0750000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(450\) 0 0
\(451\) 2.21498 2.21498
\(452\) 0 0
\(453\) 0 0
\(454\) 0.896969 + 1.23457i 0.896969 + 1.23457i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.444039 + 0.144277i −0.444039 + 0.144277i −0.522499 0.852640i \(-0.675000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(462\) 0 0
\(463\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1.68193 + 1.96929i 1.68193 + 1.96929i
\(471\) 0 0
\(472\) 1.19821 7.56520i 1.19821 7.56520i
\(473\) 1.37960 + 0.702942i 1.37960 + 0.702942i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −3.63877 0.576324i −3.63877 0.576324i
\(479\) 0.377723 0.274431i 0.377723 0.274431i −0.382683 0.923880i \(-0.625000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −5.39911 1.75428i −5.39911 1.75428i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(488\) −0.559433 1.09795i −0.559433 1.09795i
\(489\) 0 0
\(490\) 1.29489 1.51612i 1.29489 1.51612i
\(491\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 2.96620 + 0.233445i 2.96620 + 0.233445i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −0.287665 1.81624i −0.287665 1.81624i
\(509\) −0.616129 + 1.89625i −0.616129 + 1.89625i −0.233445 + 0.972370i \(0.575000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 8.03013 4.09156i 8.03013 4.09156i
\(513\) 0 0
\(514\) 0 0
\(515\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(516\) 0 0
\(517\) 2.18771 0.346500i 2.18771 0.346500i
\(518\) 0 0
\(519\) 0 0
\(520\) 1.50723 + 3.63877i 1.50723 + 3.63877i
\(521\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(522\) 0 0
\(523\) 0.142040 0.278768i 0.142040 0.278768i −0.809017 0.587785i \(-0.800000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.28290 + 0.203192i −1.28290 + 0.203192i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.526961 1.62182i −0.526961 1.62182i
\(540\) 0 0
\(541\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.309017 + 1.95106i −0.309017 + 1.95106i 1.00000i \(0.5\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(548\) −2.02905 1.03385i −2.02905 1.03385i
\(549\) 0 0
\(550\) 1.99850 2.75070i 1.99850 2.75070i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −2.87447 + 2.08842i −2.87447 + 2.08842i
\(555\) 0 0
\(556\) 2.82975 + 2.05593i 2.82975 + 2.05593i
\(557\) −0.330142 0.330142i −0.330142 0.330142i 0.522499 0.852640i \(-0.325000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(558\) 0 0
\(559\) −0.863541 0.280582i −0.863541 0.280582i
\(560\) 0 0
\(561\) 0 0
\(562\) −0.422621 0.829441i −0.422621 0.829441i
\(563\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −3.06820 0.996917i −3.06820 0.996917i
\(567\) 0 0
\(568\) −2.91031 2.91031i −2.91031 2.91031i
\(569\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(570\) 0 0
\(571\) 1.44168 1.04744i 1.44168 1.04744i 0.453990 0.891007i \(-0.350000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(572\) 5.01138 + 0.793725i 5.01138 + 0.793725i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(578\) 0.311904 1.96929i 0.311904 1.96929i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0.937016 + 2.88384i 0.937016 + 2.88384i
\(587\) −0.416003 + 0.211964i −0.416003 + 0.211964i −0.649448 0.760406i \(-0.725000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 3.30610 2.02598i 3.30610 2.02598i
\(591\) 0 0
\(592\) 0 0
\(593\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.23151 4.44780i −3.23151 4.44780i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.730153 1.76274i −0.730153 1.76274i
\(606\) 0 0
\(607\) 0.221232 0.221232i 0.221232 0.221232i −0.587785 0.809017i \(-0.700000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0.238721 0.576324i 0.238721 0.576324i
\(611\) −1.23532 + 0.401381i −1.23532 + 0.401381i
\(612\) 0 0
\(613\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.203192 1.28290i −0.203192 1.28290i −0.852640 0.522499i \(-0.825000\pi\)
0.649448 0.760406i \(-0.275000\pi\)
\(618\) 0 0
\(619\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(626\) 1.23226i 1.23226i
\(627\) 0 0
\(628\) −5.80513 0.919442i −5.80513 0.919442i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(632\) −3.93857 3.93857i −3.93857 3.93857i
\(633\) 0 0
\(634\) 0.297556 + 0.0966818i 0.297556 + 0.0966818i
\(635\) 0.401381 0.469957i 0.401381 0.469957i
\(636\) 0 0
\(637\) 0.453990 + 0.891007i 0.453990 + 0.891007i
\(638\) 0 0
\(639\) 0 0
\(640\) 6.92135 + 2.86692i 6.92135 + 2.86692i
\(641\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(642\) 0 0
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(648\) 0 0
\(649\) 3.31633i 3.31633i
\(650\) −0.905182 + 1.77652i −0.905182 + 1.77652i
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.72383 + 5.12541i −3.72383 + 5.12541i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(660\) 0 0
\(661\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −0.982296 + 0.982296i −0.982296 + 0.982296i
\(669\) 0 0
\(670\) 0 0
\(671\) −0.313601 0.431634i −0.313601 0.431634i
\(672\) 0 0
\(673\) −0.642040 + 1.26007i −0.642040 + 1.26007i 0.309017 + 0.951057i \(0.400000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(674\) −3.79250 −3.79250
\(675\) 0 0
\(676\) −2.97538 −2.97538
\(677\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.68429 + 0.266765i −1.68429 + 0.266765i −0.923880 0.382683i \(-0.875000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(684\) 0 0
\(685\) −0.178671 0.744220i −0.178671 0.744220i
\(686\) 0 0
\(687\) 0 0
\(688\) −3.94597 + 2.01057i −3.94597 + 2.01057i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.0922342 + 1.17195i 0.0922342 + 1.17195i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 9.18746 6.67508i 9.18746 6.67508i
\(705\) 0 0
\(706\) 2.45314 + 1.78231i 2.45314 + 1.78231i
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(710\) 0.163474 2.07713i 0.163474 2.07713i
\(711\) 0 0
\(712\) 1.36853 + 2.68590i 1.36853 + 2.68590i
\(713\) 0 0
\(714\) 0 0
\(715\) 0.891007 + 1.45399i 0.891007 + 1.45399i
\(716\) 0 0
\(717\) 0 0
\(718\) 2.14412 + 2.14412i 2.14412 + 2.14412i
\(719\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.96929 + 0.311904i 1.96929 + 0.311904i
\(723\) 0 0
\(724\) 3.49777i 3.49777i
\(725\) 0 0
\(726\) 0 0
\(727\) −1.44168 0.734572i −1.44168 0.734572i −0.453990 0.891007i \(-0.650000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(734\) 1.09795 3.37914i 1.09795 3.37914i
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −0.738925 + 0.738925i −0.738925 + 0.738925i −0.972370 0.233445i \(-0.925000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(744\) 0 0
\(745\) 0.431351 1.79671i 0.431351 1.79671i
\(746\) 1.37771 + 1.89625i 1.37771 + 1.89625i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.17557 −1.17557 −0.587785 0.809017i \(-0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(752\) −2.87619 + 5.64484i −2.87619 + 5.64484i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.34500 + 1.34500i −1.34500 + 1.34500i −0.453990 + 0.891007i \(0.650000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.44638 0.469957i 1.44638 0.469957i 0.522499 0.852640i \(-0.325000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 1.13846 3.50381i 1.13846 3.50381i
\(767\) 0.304224 + 1.92080i 0.304224 + 1.92080i
\(768\) 0 0
\(769\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.73278 + 0.882893i 1.73278 + 0.882893i 0.972370 + 0.233445i \(0.0750000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −1.44168 1.04744i −1.44168 1.04744i
\(782\) 0 0
\(783\) 0 0
\(784\) 4.63877 + 1.50723i 4.63877 + 1.50723i
\(785\) −1.03213 1.68429i −1.03213 1.68429i
\(786\) 0 0
\(787\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(788\) 2.49594 + 4.89856i 2.49594 + 4.89856i
\(789\) 0 0
\(790\) 0.221232 2.81102i 0.221232 2.81102i
\(791\) 0 0
\(792\) 0 0
\(793\) 0.221232 + 0.221232i 0.221232 + 0.221232i
\(794\) 0 0
\(795\) 0 0
\(796\) 4.28954 3.11653i 4.28954 3.11653i
\(797\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.78808 + 5.50313i 1.78808 + 5.50313i
\(801\) 0 0
\(802\) −1.85646 0.945913i −1.85646 0.945913i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(810\) 0 0
\(811\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) −3.85279 + 0.303221i −3.85279 + 0.303221i
\(821\) 1.17195 + 1.61305i 1.17195 + 1.61305i 0.649448 + 0.760406i \(0.275000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(822\) 0 0
\(823\) −0.142040 + 0.278768i −0.142040 + 0.278768i −0.951057 0.309017i \(-0.900000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(824\) −5.56999 −5.56999
\(825\) 0 0
\(826\) 0 0
\(827\) 0.474419 0.931099i 0.474419 0.931099i −0.522499 0.852640i \(-0.675000\pi\)
0.996917 0.0784591i \(-0.0250000\pi\)
\(828\) 0 0
\(829\) −0.533698 0.734572i −0.533698 0.734572i 0.453990 0.891007i \(-0.350000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(830\) −0.266765 0.163474i −0.266765 0.163474i
\(831\) 0 0
\(832\) −4.70898 + 4.70898i −4.70898 + 4.70898i
\(833\) 0 0
\(834\) 0 0
\(835\) −0.465451 0.0366318i −0.465451 0.0366318i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0.144277 + 0.444039i 0.144277 + 0.444039i 0.996917 0.0784591i \(-0.0250000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(840\) 0 0
\(841\) 0.309017 0.951057i 0.309017 0.951057i
\(842\) 0 0
\(843\) 0 0
\(844\) 1.58795 2.18563i 1.58795 2.18563i
\(845\) −0.649448 0.760406i −0.649448 0.760406i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 0 0
\(859\) −1.69480 0.550672i −1.69480 0.550672i −0.707107 0.707107i \(-0.750000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(860\) −2.49594 1.03385i −2.49594 1.03385i
\(861\) 0 0
\(862\) −1.80478 3.54209i −1.80478 3.54209i
\(863\) 0.690434 + 1.35505i 0.690434 + 1.35505i 0.923880 + 0.382683i \(0.125000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −2.22917 0.724303i −2.22917 0.724303i
\(867\) 0 0
\(868\) 0 0
\(869\) −1.95105 1.41752i −1.95105 1.41752i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(878\) 0.593278 3.74581i 0.593278 3.74581i
\(879\) 0 0
\(880\) 8.08767 + 1.94168i 8.08767 + 1.94168i
\(881\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(882\) 0 0
\(883\) −0.0966818 0.610425i −0.0966818 0.610425i −0.987688 0.156434i \(-0.950000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −0.583981 + 1.40985i −0.583981 + 1.40985i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −1.76034 + 3.45487i −1.76034 + 3.45487i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −2.00496 + 3.93496i −2.00496 + 3.93496i
\(903\) 0 0
\(904\) 0 0
\(905\) −0.893911 + 0.763472i −0.893911 + 0.763472i
\(906\) 0 0
\(907\) −0.437016 + 0.437016i −0.437016 + 0.437016i −0.891007 0.453990i \(-0.850000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(908\) −2.24922 + 0.356241i −2.24922 + 0.356241i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(912\) 0 0
\(913\) −0.238424 + 0.121483i −0.238424 + 0.121483i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −0.183900 + 0.253116i −0.183900 + 0.253116i −0.891007 0.453990i \(-0.850000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0.145625 0.919442i 0.145625 0.919442i
\(923\) 0.931099 + 0.474419i 0.931099 + 0.474419i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.23036 + 0.893911i −1.23036 + 0.893911i −0.996917 0.0784591i \(-0.975000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −0.734572 1.44168i −0.734572 1.44168i −0.891007 0.453990i \(-0.850000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −3.75792 + 0.902198i −3.75792 + 0.902198i
\(941\) 0.993851 + 0.322922i 0.993851 + 0.322922i 0.760406 0.649448i \(-0.225000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 7.67389 + 5.57541i 7.67389 + 5.57541i
\(945\) 0 0
\(946\) −2.49758 + 1.81460i −2.49758 + 1.81460i
\(947\) 1.92080 + 0.304224i 1.92080 + 0.304224i 0.996917 0.0784591i \(-0.0250000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 3.23151 4.44780i 3.23151 4.44780i
\(957\) 0 0
\(958\) 0.145625 + 0.919442i 0.145625 + 0.919442i
\(959\) 0 0
\(960\) 0 0
\(961\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(968\) 5.31371 5.31371i 5.31371 5.31371i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1.52601 1.52601
\(977\) −0.882893 + 1.73278i −0.882893 + 1.73278i −0.233445 + 0.972370i \(0.575000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(978\) 0 0
\(979\) 0.767157 + 1.05590i 0.767157 + 1.05590i
\(980\) 1.13863 + 2.74889i 1.13863 + 2.74889i
\(981\) 0 0
\(982\) 0 0
\(983\) −1.50209 + 0.237907i −1.50209 + 0.237907i −0.852640 0.522499i \(-0.825000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(984\) 0 0
\(985\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −0.190983 + 0.587785i −0.190983 + 0.587785i 0.809017 + 0.587785i \(0.200000\pi\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.73278 + 0.416003i 1.73278 + 0.416003i
\(996\) 0 0
\(997\) −0.142040 + 0.896802i −0.142040 + 0.896802i 0.809017 + 0.587785i \(0.200000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(998\) 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 2925.1.er.a.883.1 32
3.2 odd 2 inner 2925.1.er.a.883.4 yes 32
13.12 even 2 inner 2925.1.er.a.883.4 yes 32
25.22 odd 20 inner 2925.1.er.a.2872.4 yes 32
39.38 odd 2 CM 2925.1.er.a.883.1 32
75.47 even 20 inner 2925.1.er.a.2872.1 yes 32
325.272 odd 20 inner 2925.1.er.a.2872.1 yes 32
975.272 even 20 inner 2925.1.er.a.2872.4 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2925.1.er.a.883.1 32 1.1 even 1 trivial
2925.1.er.a.883.1 32 39.38 odd 2 CM
2925.1.er.a.883.4 yes 32 3.2 odd 2 inner
2925.1.er.a.883.4 yes 32 13.12 even 2 inner
2925.1.er.a.2872.1 yes 32 75.47 even 20 inner
2925.1.er.a.2872.1 yes 32 325.272 odd 20 inner
2925.1.er.a.2872.4 yes 32 25.22 odd 20 inner
2925.1.er.a.2872.4 yes 32 975.272 even 20 inner