Properties

Label 1375.1.bz.a.109.1
Level $1375$
Weight $1$
Character 1375.109
Analytic conductor $0.686$
Analytic rank $0$
Dimension $20$
Projective image $D_{50}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1375,1,Mod(54,1375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1375, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([11, 25]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1375.54");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1375 = 5^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1375.bz (of order \(50\), degree \(20\), 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: \(0.686214392370\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{50})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{15} + x^{10} - x^{5} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{50}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{50} - \cdots)\)

Embedding invariants

Embedding label 109.1
Root \(-0.0627905 + 0.998027i\) of defining polynomial
Character \(\chi\) \(=\) 1375.109
Dual form 1375.1.bz.a.164.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.39436 - 1.15352i) q^{3} +(-0.968583 - 0.248690i) q^{4} +(-0.309017 - 0.951057i) q^{5} +(0.426264 - 2.23455i) q^{9} +O(q^{10})\) \(q+(1.39436 - 1.15352i) q^{3} +(-0.968583 - 0.248690i) q^{4} +(-0.309017 - 0.951057i) q^{5} +(0.426264 - 2.23455i) q^{9} +(0.992115 + 0.125333i) q^{11} +(-1.63742 + 0.770513i) q^{12} +(-1.52794 - 0.969661i) q^{15} +(0.876307 + 0.481754i) q^{16} +(0.0627905 + 0.998027i) q^{20} +(-1.05491 + 1.12336i) q^{23} +(-0.809017 + 0.587785i) q^{25} +(-1.11142 - 2.02167i) q^{27} +(-1.41213 + 0.362574i) q^{31} +(1.52794 - 0.969661i) q^{33} +(-0.968583 + 2.05834i) q^{36} +(0.566335 - 1.03016i) q^{37} +(-0.929776 - 0.368125i) q^{44} +(-2.25691 + 0.285114i) q^{45} +(0.700215 + 1.76854i) q^{47} +(1.77760 - 0.339095i) q^{48} +(0.809017 - 0.587785i) q^{49} +(0.211645 + 0.134314i) q^{53} +(-0.187381 - 0.982287i) q^{55} +(-0.746226 - 1.58581i) q^{59} +(1.23879 + 1.31918i) q^{60} +(-0.728969 - 0.684547i) q^{64} +(1.68532 + 0.106032i) q^{67} +(-0.175105 + 2.78322i) q^{69} +(0.791759 - 0.313480i) q^{71} +(-0.450043 + 1.75280i) q^{75} +(0.187381 - 0.982287i) q^{80} +(-1.76665 - 0.699468i) q^{81} +(-0.791759 + 1.68257i) q^{89} +(1.30113 - 0.825723i) q^{92} +(-1.55079 + 2.13448i) q^{93} +(0.961606 - 0.0604991i) q^{97} +(0.702967 - 2.16351i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 5 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 5 q^{5} - 20 q^{12} - 5 q^{25} - 5 q^{27} + 5 q^{48} + 5 q^{49} - 5 q^{59} - 5 q^{60} + 5 q^{67} + 5 q^{69} + 5 q^{81} - 5 q^{92} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(1002\)
\(\chi(n)\) \(-1\) \(e\left(\frac{27}{50}\right)\)

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 0.125333 0.992115i \(-0.460000\pi\)
−0.125333 + 0.992115i \(0.540000\pi\)
\(3\) 1.39436 1.15352i 1.39436 1.15352i 0.425779 0.904827i \(-0.360000\pi\)
0.968583 0.248690i \(-0.0800000\pi\)
\(4\) −0.968583 0.248690i −0.968583 0.248690i
\(5\) −0.309017 0.951057i −0.309017 0.951057i
\(6\) 0 0
\(7\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(8\) 0 0
\(9\) 0.426264 2.23455i 0.426264 2.23455i
\(10\) 0 0
\(11\) 0.992115 + 0.125333i 0.992115 + 0.125333i
\(12\) −1.63742 + 0.770513i −1.63742 + 0.770513i
\(13\) 0 0 −0.982287 0.187381i \(-0.940000\pi\)
0.982287 + 0.187381i \(0.0600000\pi\)
\(14\) 0 0
\(15\) −1.52794 0.969661i −1.52794 0.969661i
\(16\) 0.876307 + 0.481754i 0.876307 + 0.481754i
\(17\) 0 0 −0.248690 0.968583i \(-0.580000\pi\)
0.248690 + 0.968583i \(0.420000\pi\)
\(18\) 0 0
\(19\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(20\) 0.0627905 + 0.998027i 0.0627905 + 0.998027i
\(21\) 0 0
\(22\) 0 0
\(23\) −1.05491 + 1.12336i −1.05491 + 1.12336i −0.0627905 + 0.998027i \(0.520000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(24\) 0 0
\(25\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(26\) 0 0
\(27\) −1.11142 2.02167i −1.11142 2.02167i
\(28\) 0 0
\(29\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(30\) 0 0
\(31\) −1.41213 + 0.362574i −1.41213 + 0.362574i −0.876307 0.481754i \(-0.840000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(32\) 0 0
\(33\) 1.52794 0.969661i 1.52794 0.969661i
\(34\) 0 0
\(35\) 0 0
\(36\) −0.968583 + 2.05834i −0.968583 + 2.05834i
\(37\) 0.566335 1.03016i 0.566335 1.03016i −0.425779 0.904827i \(-0.640000\pi\)
0.992115 0.125333i \(-0.0400000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(42\) 0 0
\(43\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(44\) −0.929776 0.368125i −0.929776 0.368125i
\(45\) −2.25691 + 0.285114i −2.25691 + 0.285114i
\(46\) 0 0
\(47\) 0.700215 + 1.76854i 0.700215 + 1.76854i 0.637424 + 0.770513i \(0.280000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(48\) 1.77760 0.339095i 1.77760 0.339095i
\(49\) 0.809017 0.587785i 0.809017 0.587785i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.211645 + 0.134314i 0.211645 + 0.134314i 0.637424 0.770513i \(-0.280000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(54\) 0 0
\(55\) −0.187381 0.982287i −0.187381 0.982287i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.746226 1.58581i −0.746226 1.58581i −0.809017 0.587785i \(-0.800000\pi\)
0.0627905 0.998027i \(-0.480000\pi\)
\(60\) 1.23879 + 1.31918i 1.23879 + 1.31918i
\(61\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.728969 0.684547i −0.728969 0.684547i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.68532 + 0.106032i 1.68532 + 0.106032i 0.876307 0.481754i \(-0.160000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(68\) 0 0
\(69\) −0.175105 + 2.78322i −0.175105 + 2.78322i
\(70\) 0 0
\(71\) 0.791759 0.313480i 0.791759 0.313480i 0.0627905 0.998027i \(-0.480000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(72\) 0 0
\(73\) 0 0 −0.904827 0.425779i \(-0.860000\pi\)
0.904827 + 0.425779i \(0.140000\pi\)
\(74\) 0 0
\(75\) −0.450043 + 1.75280i −0.450043 + 1.75280i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(80\) 0.187381 0.982287i 0.187381 0.982287i
\(81\) −1.76665 0.699468i −1.76665 0.699468i
\(82\) 0 0
\(83\) 0 0 −0.770513 0.637424i \(-0.780000\pi\)
0.770513 + 0.637424i \(0.220000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.791759 + 1.68257i −0.791759 + 1.68257i −0.0627905 + 0.998027i \(0.520000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.30113 0.825723i 1.30113 0.825723i
\(93\) −1.55079 + 2.13448i −1.55079 + 2.13448i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.961606 0.0604991i 0.961606 0.0604991i 0.425779 0.904827i \(-0.360000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(98\) 0 0
\(99\) 0.702967 2.16351i 0.702967 2.16351i
\(100\) 0.929776 0.368125i 0.929776 0.368125i
\(101\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(102\) 0 0
\(103\) −0.419952 + 1.63560i −0.419952 + 1.63560i 0.309017 + 0.951057i \(0.400000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(108\) 0.573736 + 2.23455i 0.573736 + 2.23455i
\(109\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(110\) 0 0
\(111\) −0.398631 2.08969i −0.398631 2.08969i
\(112\) 0 0
\(113\) 0.666178 0.313480i 0.666178 0.313480i −0.0627905 0.998027i \(-0.520000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(114\) 0 0
\(115\) 1.39436 + 0.656137i 1.39436 + 0.656137i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.968583 + 0.248690i 0.968583 + 0.248690i
\(122\) 0 0
\(123\) 0 0
\(124\) 1.45794 1.45794
\(125\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(126\) 0 0
\(127\) 0 0 0.125333 0.992115i \(-0.460000\pi\)
−0.125333 + 0.992115i \(0.540000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(132\) −1.72108 + 0.559214i −1.72108 + 0.559214i
\(133\) 0 0
\(134\) 0 0
\(135\) −1.57927 + 1.68176i −1.57927 + 1.68176i
\(136\) 0 0
\(137\) −1.80608 + 0.849878i −1.80608 + 0.849878i −0.876307 + 0.481754i \(0.840000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(138\) 0 0
\(139\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(140\) 0 0
\(141\) 3.01639 + 1.65828i 3.01639 + 1.65828i
\(142\) 0 0
\(143\) 0 0
\(144\) 1.45004 1.75280i 1.45004 1.75280i
\(145\) 0 0
\(146\) 0 0
\(147\) 0.450043 1.75280i 0.450043 1.75280i
\(148\) −0.804733 + 0.856954i −0.804733 + 0.856954i
\(149\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(150\) 0 0
\(151\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.781202 + 1.23098i 0.781202 + 1.23098i
\(156\) 0 0
\(157\) 0.292352 0.402389i 0.292352 0.402389i −0.637424 0.770513i \(-0.720000\pi\)
0.929776 + 0.368125i \(0.120000\pi\)
\(158\) 0 0
\(159\) 0.450043 0.0568536i 0.450043 0.0568536i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.937209 0.998027i −0.937209 0.998027i 0.0627905 0.998027i \(-0.480000\pi\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) −1.39436 1.15352i −1.39436 1.15352i
\(166\) 0 0
\(167\) 0 0 −0.770513 0.637424i \(-0.780000\pi\)
0.770513 + 0.637424i \(0.220000\pi\)
\(168\) 0 0
\(169\) 0.929776 + 0.368125i 0.929776 + 0.368125i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.982287 0.187381i \(-0.0600000\pi\)
−0.982287 + 0.187381i \(0.940000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(177\) −2.86977 1.35041i −2.86977 1.35041i
\(178\) 0 0
\(179\) 0.116762 0.0462295i 0.116762 0.0462295i −0.309017 0.951057i \(-0.600000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(180\) 2.25691 + 0.285114i 2.25691 + 0.285114i
\(181\) 0.101597 1.61484i 0.101597 1.61484i −0.535827 0.844328i \(-0.680000\pi\)
0.637424 0.770513i \(-0.280000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.15475 0.220280i −1.15475 0.220280i
\(186\) 0 0
\(187\) 0 0
\(188\) −0.238398 1.88711i −0.238398 1.88711i
\(189\) 0 0
\(190\) 0 0
\(191\) −0.791759 1.68257i −0.791759 1.68257i −0.728969 0.684547i \(-0.760000\pi\)
−0.0627905 0.998027i \(-0.520000\pi\)
\(192\) −1.80608 0.113629i −1.80608 0.113629i
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.929776 + 0.368125i −0.929776 + 0.368125i
\(197\) 0 0 −0.844328 0.535827i \(-0.820000\pi\)
0.844328 + 0.535827i \(0.180000\pi\)
\(198\) 0 0
\(199\) −0.101597 0.0738147i −0.101597 0.0738147i 0.535827 0.844328i \(-0.320000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(200\) 0 0
\(201\) 2.47226 1.79620i 2.47226 1.79620i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.06054 + 2.83609i 2.06054 + 2.83609i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.876307 0.481754i \(-0.160000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(212\) −0.171593 0.182728i −0.171593 0.182728i
\(213\) 0.742395 1.35041i 0.742395 1.35041i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −0.0627905 + 0.998027i −0.0627905 + 0.998027i
\(221\) 0 0
\(222\) 0 0
\(223\) 0.946441 + 1.72157i 0.946441 + 1.72157i 0.637424 + 0.770513i \(0.280000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(224\) 0 0
\(225\) 0.968583 + 2.05834i 0.968583 + 2.05834i
\(226\) 0 0
\(227\) 0 0 0.684547 0.728969i \(-0.260000\pi\)
−0.684547 + 0.728969i \(0.740000\pi\)
\(228\) 0 0
\(229\) 1.03799 1.63560i 1.03799 1.63560i 0.309017 0.951057i \(-0.400000\pi\)
0.728969 0.684547i \(-0.240000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.248690 0.968583i \(-0.580000\pi\)
0.248690 + 0.968583i \(0.420000\pi\)
\(234\) 0 0
\(235\) 1.46560 1.21245i 1.46560 1.21245i
\(236\) 0.328407 + 1.72157i 0.328407 + 1.72157i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(240\) −0.871808 1.58581i −0.871808 1.58581i
\(241\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(242\) 0 0
\(243\) −1.07609 + 0.349641i −1.07609 + 0.349641i
\(244\) 0 0
\(245\) −0.809017 0.587785i −0.809017 0.587785i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.98423 −1.98423 −0.992115 0.125333i \(-0.960000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(252\) 0 0
\(253\) −1.18738 + 0.982287i −1.18738 + 0.982287i
\(254\) 0 0
\(255\) 0 0
\(256\) 0.535827 + 0.844328i 0.535827 + 0.844328i
\(257\) −1.46560 + 0.476203i −1.46560 + 0.476203i −0.929776 0.368125i \(-0.880000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.982287 0.187381i \(-0.940000\pi\)
0.982287 + 0.187381i \(0.0600000\pi\)
\(264\) 0 0
\(265\) 0.0623382 0.242791i 0.0623382 0.242791i
\(266\) 0 0
\(267\) 0.836878 + 3.25943i 0.836878 + 3.25943i
\(268\) −1.60601 0.521823i −1.60601 0.521823i
\(269\) −1.23480 + 1.49261i −1.23480 + 1.49261i −0.425779 + 0.904827i \(0.640000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(270\) 0 0
\(271\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.876307 + 0.481754i −0.876307 + 0.481754i
\(276\) 0.861763 2.65223i 0.861763 2.65223i
\(277\) 0 0 −0.481754 0.876307i \(-0.660000\pi\)
0.481754 + 0.876307i \(0.340000\pi\)
\(278\) 0 0
\(279\) 0.208250 + 3.31004i 0.208250 + 3.31004i
\(280\) 0 0
\(281\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(282\) 0 0
\(283\) 0 0 0.844328 0.535827i \(-0.180000\pi\)
−0.844328 + 0.535827i \(0.820000\pi\)
\(284\) −0.844844 + 0.106729i −0.844844 + 0.106729i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.876307 + 0.481754i −0.876307 + 0.481754i
\(290\) 0 0
\(291\) 1.27104 1.19359i 1.27104 1.19359i
\(292\) 0 0
\(293\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(294\) 0 0
\(295\) −1.27760 + 1.19975i −1.27760 + 1.19975i
\(296\) 0 0
\(297\) −0.849276 2.14503i −0.849276 2.14503i
\(298\) 0 0
\(299\) 0 0
\(300\) 0.871808 1.58581i 0.871808 1.58581i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 1.30113 + 2.76505i 1.30113 + 2.76505i
\(310\) 0 0
\(311\) 0.781202 + 0.733597i 0.781202 + 0.733597i 0.968583 0.248690i \(-0.0800000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(312\) 0 0
\(313\) 0.250172 + 1.98031i 0.250172 + 1.98031i 0.187381 + 0.982287i \(0.440000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.96070 0.123357i −1.96070 0.123357i −0.968583 0.248690i \(-0.920000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.425779 + 0.904827i −0.425779 + 0.904827i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.53720 + 1.11684i 1.53720 + 1.11684i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.996398 + 0.394502i 0.996398 + 0.394502i 0.809017 0.587785i \(-0.200000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(332\) 0 0
\(333\) −2.06054 1.70463i −2.06054 1.70463i
\(334\) 0 0
\(335\) −0.419952 1.63560i −0.419952 1.63560i
\(336\) 0 0
\(337\) 0 0 −0.684547 0.728969i \(-0.740000\pi\)
0.684547 + 0.728969i \(0.260000\pi\)
\(338\) 0 0
\(339\) 0.567290 1.20555i 0.567290 1.20555i
\(340\) 0 0
\(341\) −1.44644 + 0.182728i −1.44644 + 0.182728i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2.70111 0.693528i 2.70111 0.693528i
\(346\) 0 0
\(347\) 0 0 0.998027 0.0627905i \(-0.0200000\pi\)
−0.998027 + 0.0627905i \(0.980000\pi\)
\(348\) 0 0
\(349\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(354\) 0 0
\(355\) −0.542804 0.656137i −0.542804 0.656137i
\(356\) 1.18532 1.43281i 1.18532 1.43281i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(360\) 0 0
\(361\) −0.187381 0.982287i −0.187381 0.982287i
\(362\) 0 0
\(363\) 1.63742 0.770513i 1.63742 0.770513i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.734796 + 1.85588i −0.734796 + 1.85588i −0.309017 + 0.951057i \(0.600000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(368\) −1.46560 + 0.476203i −1.46560 + 0.476203i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 2.03289 1.68176i 2.03289 1.68176i
\(373\) 0 0 0.125333 0.992115i \(-0.460000\pi\)
−0.125333 + 0.992115i \(0.540000\pi\)
\(374\) 0 0
\(375\) 1.80608 0.113629i 1.80608 0.113629i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.598617 + 0.153699i 0.598617 + 0.153699i 0.535827 0.844328i \(-0.320000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.503997 + 1.27295i −0.503997 + 1.27295i 0.425779 + 0.904827i \(0.360000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −0.946441 0.180543i −0.946441 0.180543i
\(389\) 0.0702235 + 0.368125i 0.0702235 + 0.368125i 1.00000 \(0\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −1.21892 + 1.92072i −1.21892 + 1.92072i
\(397\) 0.183098 0.713118i 0.183098 0.713118i −0.809017 0.587785i \(-0.800000\pi\)
0.992115 0.125333i \(-0.0400000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.992115 + 0.125333i −0.992115 + 0.125333i
\(401\) 0.613161 1.88711i 0.613161 1.88711i 0.187381 0.982287i \(-0.440000\pi\)
0.425779 0.904827i \(-0.360000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.119307 + 1.89634i −0.119307 + 1.89634i
\(406\) 0 0
\(407\) 0.690983 0.951057i 0.690983 0.951057i
\(408\) 0 0
\(409\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(410\) 0 0
\(411\) −1.53799 + 3.26839i −1.53799 + 3.26839i
\(412\) 0.813516 1.47978i 0.813516 1.47978i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.62954 + 0.645180i 1.62954 + 0.645180i 0.992115 0.125333i \(-0.0400000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(420\) 0 0
\(421\) −1.18532 1.43281i −1.18532 1.43281i −0.876307 0.481754i \(-0.840000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(422\) 0 0
\(423\) 4.25037 0.810802i 4.25037 0.810802i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(432\) 2.30703i 2.30703i
\(433\) −1.17325 0.0738147i −1.17325 0.0738147i −0.535827 0.844328i \(-0.680000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(440\) 0 0
\(441\) −0.968583 2.05834i −0.968583 2.05834i
\(442\) 0 0
\(443\) 0.963507i 0.963507i 0.876307 + 0.481754i \(0.160000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(444\) −0.133579 + 2.12318i −0.133579 + 2.12318i
\(445\) 1.84489 + 0.233064i 1.84489 + 0.233064i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.03137 0.749337i −1.03137 0.749337i −0.0627905 0.998027i \(-0.520000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −0.723208 + 0.137959i −0.723208 + 0.137959i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −1.18738 0.982287i −1.18738 0.982287i
\(461\) 0 0 0.876307 0.481754i \(-0.160000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(462\) 0 0
\(463\) 0.354691 0.645180i 0.354691 0.645180i −0.637424 0.770513i \(-0.720000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(464\) 0 0
\(465\) 2.50923 + 0.815299i 2.50923 + 0.815299i
\(466\) 0 0
\(467\) −1.60601 + 1.01920i −1.60601 + 1.01920i −0.637424 + 0.770513i \(0.720000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −0.0565168 0.898309i −0.0565168 0.898309i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.390348 0.415678i 0.390348 0.415678i
\(478\) 0 0
\(479\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −0.876307 0.481754i −0.876307 0.481754i
\(485\) −0.354691 0.895846i −0.354691 0.895846i
\(486\) 0 0
\(487\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(488\) 0 0
\(489\) −2.45805 0.310524i −2.45805 0.310524i
\(490\) 0 0
\(491\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −2.27485 −2.27485
\(496\) −1.41213 0.362574i −1.41213 0.362574i
\(497\) 0 0
\(498\) 0 0
\(499\) −1.98423 −1.98423 −0.992115 0.125333i \(-0.960000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(500\) −0.637424 0.770513i −0.637424 0.770513i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.770513 0.637424i \(-0.220000\pi\)
−0.770513 + 0.637424i \(0.780000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.72108 0.559214i 1.72108 0.559214i
\(508\) 0 0
\(509\) −0.0235315 + 0.123357i −0.0235315 + 0.123357i −0.992115 0.125333i \(-0.960000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.68532 0.106032i 1.68532 0.106032i
\(516\) 0 0
\(517\) 0.473036 + 1.84235i 0.473036 + 1.84235i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −0.574221 + 0.904827i −0.574221 + 0.904827i 0.425779 + 0.904827i \(0.360000\pi\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 0.684547 0.728969i \(-0.260000\pi\)
−0.684547 + 0.728969i \(0.740000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 1.80608 0.113629i 1.80608 0.113629i
\(529\) −0.0863221 1.37205i −0.0863221 1.37205i
\(530\) 0 0
\(531\) −3.86167 + 0.991509i −3.86167 + 0.991509i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0.109482 0.199148i 0.109482 0.199148i
\(538\) 0 0
\(539\) 0.876307 0.481754i 0.876307 0.481754i
\(540\) 1.94789 1.23617i 1.94789 1.23617i
\(541\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(542\) 0 0
\(543\) −1.72108 2.36887i −1.72108 2.36887i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.368125 0.929776i \(-0.620000\pi\)
0.368125 + 0.929776i \(0.380000\pi\)
\(548\) 1.96070 0.374023i 1.96070 0.374023i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1.86423 + 1.02487i −1.86423 + 1.02487i
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.125333 0.992115i \(-0.540000\pi\)
0.125333 + 0.992115i \(0.460000\pi\)
\(564\) −2.50923 2.35633i −2.50923 2.35633i
\(565\) −0.503997 0.536702i −0.503997 0.536702i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(570\) 0 0
\(571\) 0 0 0.929776 0.368125i \(-0.120000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(572\) 0 0
\(573\) −3.04488 1.43281i −3.04488 1.43281i
\(574\) 0 0
\(575\) 0.193142 1.52888i 0.193142 1.52888i
\(576\) −1.84039 + 1.33712i −1.84039 + 1.33712i
\(577\) 1.92978 0.368125i 1.92978 0.368125i 0.929776 0.368125i \(-0.120000\pi\)
1.00000 \(0\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0.193142 + 0.159781i 0.193142 + 0.159781i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.659566 + 0.702367i 0.659566 + 0.702367i 0.968583 0.248690i \(-0.0800000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(588\) −0.871808 + 1.58581i −0.871808 + 1.58581i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0.992567 0.629902i 0.992567 0.629902i
\(593\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.226810 + 0.0142697i −0.226810 + 0.0142697i
\(598\) 0 0
\(599\) 0.115808 0.356420i 0.115808 0.356420i −0.876307 0.481754i \(-0.840000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(600\) 0 0
\(601\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(602\) 0 0
\(603\) 0.955326 3.72075i 0.955326 3.72075i
\(604\) 0 0
\(605\) −0.0627905 0.998027i −0.0627905 0.998027i
\(606\) 0 0
\(607\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.904827 0.425779i \(-0.140000\pi\)
−0.904827 + 0.425779i \(0.860000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.723208 1.82662i 0.723208 1.82662i 0.187381 0.982287i \(-0.440000\pi\)
0.535827 0.844328i \(-0.320000\pi\)
\(618\) 0 0
\(619\) −0.781202 1.23098i −0.781202 1.23098i −0.968583 0.248690i \(-0.920000\pi\)
0.187381 0.982287i \(-0.440000\pi\)
\(620\) −0.450527 1.38658i −0.450527 1.38658i
\(621\) 3.44351 + 0.884142i 3.44351 + 0.884142i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.309017 0.951057i 0.309017 0.951057i
\(626\) 0 0
\(627\) 0 0
\(628\) −0.383238 + 0.317042i −0.383238 + 0.317042i
\(629\) 0 0
\(630\) 0 0
\(631\) 0.574221 + 0.904827i 0.574221 + 0.904827i 1.00000 \(0\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −0.450043 0.0568536i −0.450043 0.0568536i
\(637\) 0 0
\(638\) 0 0
\(639\) −0.362989 1.90285i −0.362989 1.90285i
\(640\) 0 0
\(641\) 0.746226 + 0.410241i 0.746226 + 0.410241i 0.809017 0.587785i \(-0.200000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(642\) 0 0
\(643\) 1.30209 + 0.423073i 1.30209 + 0.423073i 0.876307 0.481754i \(-0.160000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.419952 + 1.63560i −0.419952 + 1.63560i 0.309017 + 0.951057i \(0.400000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(648\) 0 0
\(649\) −0.541587 1.66683i −0.541587 1.66683i
\(650\) 0 0
\(651\) 0 0
\(652\) 0.659566 + 1.19975i 0.659566 + 1.19975i
\(653\) −1.36639 + 0.0859661i −1.36639 + 0.0859661i −0.728969 0.684547i \(-0.760000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(660\) 1.06369 + 1.46404i 1.06369 + 1.46404i
\(661\) −0.0534698 + 0.113629i −0.0534698 + 0.113629i −0.929776 0.368125i \(-0.880000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 3.30554 + 1.30876i 3.30554 + 1.30876i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.982287 0.187381i \(-0.0600000\pi\)
−0.982287 + 0.187381i \(0.940000\pi\)
\(674\) 0 0
\(675\) 2.08747 + 0.982287i 2.08747 + 0.982287i
\(676\) −0.809017 0.587785i −0.809017 0.587785i
\(677\) 0 0 −0.904827 0.425779i \(-0.860000\pi\)
0.904827 + 0.425779i \(0.140000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −0.496398 0.0312307i −0.496398 0.0312307i −0.187381 0.982287i \(-0.560000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(684\) 0 0
\(685\) 1.36639 + 1.45506i 1.36639 + 1.45506i
\(686\) 0 0
\(687\) −0.439368 3.47796i −0.439368 3.47796i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0.542804 + 1.15352i 0.542804 + 1.15352i 0.968583 + 0.248690i \(0.0800000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(692\) 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
\(704\) −0.637424 0.770513i −0.637424 0.770513i
\(705\) 0.644998 3.38120i 0.644998 3.38120i
\(706\) 0 0
\(707\) 0 0
\(708\) 2.44378 + 2.02167i 2.44378 + 2.02167i
\(709\) 1.44644 1.35830i 1.44644 1.35830i 0.637424 0.770513i \(-0.280000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.08237 1.96882i 1.08237 1.96882i
\(714\) 0 0
\(715\) 0 0
\(716\) −0.124591 + 0.0157395i −0.124591 + 0.0157395i
\(717\) 0 0
\(718\) 0 0
\(719\) −1.87631 + 0.481754i −1.87631 + 0.481754i −0.876307 + 0.481754i \(0.840000\pi\)
−1.00000 \(\pi\)
\(720\) −2.11510 0.837427i −2.11510 0.837427i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(725\) 0 0
\(726\) 0 0
\(727\) −0.659566 + 0.702367i −0.659566 + 0.702367i −0.968583 0.248690i \(-0.920000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(728\) 0 0
\(729\) −0.0790194 + 0.124515i −0.0790194 + 0.124515i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.248690 0.968583i \(-0.580000\pi\)
0.248690 + 0.968583i \(0.420000\pi\)
\(734\) 0 0
\(735\) −1.80608 + 0.113629i −1.80608 + 0.113629i
\(736\) 0 0
\(737\) 1.65875 + 0.316423i 1.65875 + 0.316423i
\(738\) 0 0
\(739\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(740\) 1.06369 + 0.500534i 1.06369 + 0.500534i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.85955 −1.85955 −0.929776 0.368125i \(-0.880000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(752\) −0.238398 + 1.88711i −0.238398 + 1.88711i
\(753\) −2.76673 + 2.28884i −2.76673 + 2.28884i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.916350 + 0.297740i −0.916350 + 0.297740i −0.728969 0.684547i \(-0.760000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(758\) 0 0
\(759\) −0.522555 + 2.73933i −0.522555 + 2.73933i
\(760\) 0 0
\(761\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0.348445 + 1.82662i 0.348445 + 1.82662i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 1.72108 + 0.559214i 1.72108 + 0.559214i
\(769\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(770\) 0 0
\(771\) −1.49427 + 2.35460i −1.49427 + 2.35460i
\(772\) 0 0
\(773\) 0.171593 0.182728i 0.171593 0.182728i −0.637424 0.770513i \(-0.720000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(774\) 0 0
\(775\) 0.929324 1.12336i 0.929324 1.12336i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0.824805 0.211774i 0.824805 0.211774i
\(782\) 0 0
\(783\) 0 0
\(784\) 0.992115 0.125333i 0.992115 0.125333i
\(785\) −0.473036 0.153699i −0.473036 0.153699i
\(786\) 0 0
\(787\) 0 0 0.481754 0.876307i \(-0.340000\pi\)
−0.481754 + 0.876307i \(0.660000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −0.193142 0.410447i −0.193142 0.410447i
\(796\) 0.0800484 + 0.0967619i 0.0800484 + 0.0967619i
\(797\) −0.271031 0.684547i −0.271031 0.684547i 0.728969 0.684547i \(-0.240000\pi\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 3.42230 + 2.48645i 3.42230 + 2.48645i
\(802\) 0 0
\(803\) 0 0
\(804\) −2.84129 + 1.12495i −2.84129 + 1.12495i
\(805\) 0 0
\(806\) 0 0
\(807\) 3.50560i 3.50560i
\(808\) 0 0
\(809\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(810\) 0 0
\(811\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −0.659566 + 1.19975i −0.659566 + 1.19975i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.929776 0.368125i \(-0.120000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(822\) 0 0
\(823\) −0.226810 0.106729i −0.226810 0.106729i 0.309017 0.951057i \(-0.400000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(824\) 0 0
\(825\) −0.666178 + 1.68257i −0.666178 + 1.68257i
\(826\) 0 0
\(827\) 0 0 0.982287 0.187381i \(-0.0600000\pi\)
−0.982287 + 0.187381i \(0.940000\pi\)
\(828\) −1.29050 3.25943i −1.29050 3.25943i
\(829\) −0.238883 0.288760i −0.238883 0.288760i 0.637424 0.770513i \(-0.280000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2.30248 + 2.45189i 2.30248 + 2.45189i
\(838\) 0 0
\(839\) −0.456288 + 0.969661i −0.456288 + 0.969661i 0.535827 + 0.844328i \(0.320000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(840\) 0 0
\(841\) −0.992115 + 0.125333i −0.992115 + 0.125333i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.0627905 0.998027i 0.0627905 0.998027i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.120759 + 0.219661i 0.120759 + 0.219661i
\(849\) 0 0
\(850\) 0 0
\(851\) 0.559811 + 1.72292i 0.559811 + 1.72292i
\(852\) −1.05491 + 1.12336i −1.05491 + 1.12336i
\(853\) 0 0 0.248690 0.968583i \(-0.420000\pi\)
−0.248690 + 0.968583i \(0.580000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(858\) 0 0
\(859\) 1.73879 + 0.955910i 1.73879 + 0.955910i 0.929776 + 0.368125i \(0.120000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.23879 0.582932i 1.23879 0.582932i 0.309017 0.951057i \(-0.400000\pi\)
0.929776 + 0.368125i \(0.120000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.666178 + 1.68257i −0.666178 + 1.68257i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0.274709 2.17455i 0.274709 2.17455i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.125333 0.992115i \(-0.460000\pi\)
−0.125333 + 0.992115i \(0.540000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0.309017 0.951057i 0.309017 0.951057i
\(881\) −0.200808 0.316423i −0.200808 0.316423i 0.728969 0.684547i \(-0.240000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(882\) 0 0
\(883\) 0.503997 1.27295i 0.503997 1.27295i −0.425779 0.904827i \(-0.640000\pi\)
0.929776 0.368125i \(-0.120000\pi\)
\(884\) 0 0
\(885\) −0.397510 + 3.14661i −0.397510 + 3.14661i
\(886\) 0 0
\(887\) 0 0 0.904827 0.425779i \(-0.140000\pi\)
−0.904827 + 0.425779i \(0.860000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.66506 0.915373i −1.66506 0.915373i
\(892\) −0.488570 1.90285i −0.488570 1.90285i
\(893\) 0 0
\(894\) 0 0
\(895\) −0.0800484 0.0967619i −0.0800484 0.0967619i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −0.426264 2.23455i −0.426264 2.23455i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.56720 + 0.402389i −1.56720 + 0.402389i
\(906\) 0 0
\(907\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0.456288 0.969661i 0.456288 0.969661i −0.535827 0.844328i \(-0.680000\pi\)
0.992115 0.125333i \(-0.0400000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −1.41213 + 1.32608i −1.41213 + 1.32608i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.929776 0.368125i \(-0.880000\pi\)
0.929776 + 0.368125i \(0.120000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0.147338 + 1.16630i 0.147338 + 1.16630i
\(926\) 0 0
\(927\) 3.47583 + 1.63560i 3.47583 + 1.63560i
\(928\) 0 0
\(929\) −0.116762 + 0.0462295i −0.116762 + 0.0462295i −0.425779 0.904827i \(-0.640000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.93550 + 0.121771i 1.93550 + 0.121771i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.125333 0.992115i \(-0.540000\pi\)
0.125333 + 0.992115i \(0.460000\pi\)
\(938\) 0 0
\(939\) 2.63316 + 2.47270i 2.63316 + 2.47270i
\(940\) −1.72108 + 0.809880i −1.72108 + 0.809880i
\(941\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0.110048 1.74915i 0.110048 1.74915i
\(945\) 0 0
\(946\) 0 0
\(947\) 1.65875 + 1.05267i 1.65875 + 1.05267i 0.929776 + 0.368125i \(0.120000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −2.87622 + 2.08969i −2.87622 + 2.08969i
\(952\) 0 0
\(953\) 0 0 −0.368125 0.929776i \(-0.620000\pi\)
0.368125 + 0.929776i \(0.380000\pi\)
\(954\) 0 0
\(955\) −1.35556 + 1.27295i −1.35556 + 1.27295i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0.450043 + 1.75280i 0.450043 + 1.75280i
\(961\) 0.986354 0.542253i 0.986354 0.542253i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.844328 0.535827i \(-0.180000\pi\)
−0.844328 + 0.535827i \(0.820000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0.124591 + 1.98031i 0.124591 + 1.98031i 0.187381 + 0.982287i \(0.440000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(972\) 1.12923 0.0710452i 1.12923 0.0710452i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −0.804733 + 0.856954i −0.804733 + 0.856954i −0.992115 0.125333i \(-0.960000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(978\) 0 0
\(979\) −0.996398 + 1.57007i −0.996398 + 1.57007i
\(980\) 0.637424 + 0.770513i 0.637424 + 0.770513i
\(981\) 0 0
\(982\) 0 0
\(983\) −0.496398 1.93334i −0.496398 1.93334i −0.309017 0.951057i \(-0.600000\pi\)
−0.187381 0.982287i \(-0.560000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0.328407 1.72157i 0.328407 1.72157i −0.309017 0.951057i \(-0.600000\pi\)
0.637424 0.770513i \(-0.280000\pi\)
\(992\) 0 0
\(993\) 1.84441 0.599284i 1.84441 0.599284i
\(994\) 0 0
\(995\) −0.0388067 + 0.119435i −0.0388067 + 0.119435i
\(996\) 0 0
\(997\) 0 0 0.770513 0.637424i \(-0.220000\pi\)
−0.770513 + 0.637424i \(0.780000\pi\)
\(998\) 0 0
\(999\) −2.71208 −2.71208
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 1375.1.bz.a.109.1 20
11.10 odd 2 CM 1375.1.bz.a.109.1 20
125.39 even 50 inner 1375.1.bz.a.164.1 yes 20
1375.164 odd 50 inner 1375.1.bz.a.164.1 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1375.1.bz.a.109.1 20 1.1 even 1 trivial
1375.1.bz.a.109.1 20 11.10 odd 2 CM
1375.1.bz.a.164.1 yes 20 125.39 even 50 inner
1375.1.bz.a.164.1 yes 20 1375.164 odd 50 inner