Properties

Label 2008.1.w.a.1531.1
Level $2008$
Weight $1$
Character 2008.1531
Analytic conductor $1.002$
Analytic rank $0$
Dimension $20$
Projective image $D_{25}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2008,1,Mod(51,2008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2008, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([25, 25, 18]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2008.51");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2008 = 2^{3} \cdot 251 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2008.w (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: \(1.00212254537\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{25}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{25} - \cdots)\)

Embedding invariants

Embedding label 1531.1
Root \(0.929776 - 0.368125i\) of defining polynomial
Character \(\chi\) \(=\) 2008.1531
Dual form 2008.1.w.a.1747.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.309017 + 0.951057i) q^{2} +(0.791759 - 1.68257i) q^{3} +(-0.809017 + 0.587785i) q^{4} +(1.84489 + 0.233064i) q^{6} +(-0.809017 - 0.587785i) q^{8} +(-1.56675 - 1.89387i) q^{9} +O(q^{10})\) \(q+(0.309017 + 0.951057i) q^{2} +(0.791759 - 1.68257i) q^{3} +(-0.809017 + 0.587785i) q^{4} +(1.84489 + 0.233064i) q^{6} +(-0.809017 - 0.587785i) q^{8} +(-1.56675 - 1.89387i) q^{9} +(-0.328407 - 1.72157i) q^{11} +(0.348445 + 1.82662i) q^{12} +(0.309017 - 0.951057i) q^{16} +(-1.98423 + 0.250666i) q^{17} +(1.31703 - 2.07531i) q^{18} +(0.0388067 - 0.616814i) q^{19} +(1.53583 - 0.844328i) q^{22} +(-1.62954 + 0.895846i) q^{24} +(0.309017 + 0.951057i) q^{25} +(-2.62594 + 0.674226i) q^{27} +1.00000 q^{32} +(-3.15669 - 0.810500i) q^{33} +(-0.851559 - 1.80965i) q^{34} +(2.38072 + 0.611264i) q^{36} +(0.598617 - 0.153699i) q^{38} +(1.69755 + 0.933237i) q^{41} +(1.84489 - 0.730444i) q^{43} +(1.27760 + 1.19975i) q^{44} +(-1.35556 - 1.27295i) q^{48} +(0.876307 - 0.481754i) q^{49} +(-0.809017 + 0.587785i) q^{50} +(-1.14927 + 3.53708i) q^{51} +(-1.45269 - 2.28907i) q^{54} +(-1.00711 - 0.553664i) q^{57} +(0.371808 + 0.0469702i) q^{59} +(0.309017 + 0.951057i) q^{64} +(-0.204639 - 3.25265i) q^{66} +(0.110048 + 0.0604991i) q^{67} +(1.45794 - 1.36909i) q^{68} +(0.154335 + 2.45309i) q^{72} +(-0.263146 - 0.559214i) q^{73} +(1.84489 + 0.233064i) q^{75} +(0.331159 + 0.521823i) q^{76} +(-0.484103 + 2.53776i) q^{81} +(-0.362989 + 1.90285i) q^{82} +(-0.620759 + 0.582932i) q^{83} +(1.26480 + 1.52888i) q^{86} +(-0.746226 + 1.58581i) q^{88} +(-1.06320 + 0.134314i) q^{89} +(0.791759 - 1.68257i) q^{96} +(1.41213 - 1.32608i) q^{97} +(0.728969 + 0.684547i) q^{98} +(-2.74590 + 3.31923i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 5 q^{2} - 5 q^{4} - 5 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 5 q^{2} - 5 q^{4} - 5 q^{8} - 5 q^{11} - 5 q^{16} + 20 q^{22} - 5 q^{25} - 5 q^{27} + 20 q^{32} - 5 q^{33} - 5 q^{44} - 5 q^{50} - 5 q^{54} - 5 q^{59} - 5 q^{64} - 5 q^{66} - 5 q^{81} - 5 q^{83} - 5 q^{88} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2008\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(503\) \(1005\)
\(\chi(n)\) \(e\left(\frac{1}{25}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(3\) 0.791759 1.68257i 0.791759 1.68257i 0.0627905 0.998027i \(-0.480000\pi\)
0.728969 0.684547i \(-0.240000\pi\)
\(4\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(5\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(6\) 1.84489 + 0.233064i 1.84489 + 0.233064i
\(7\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(8\) −0.809017 0.587785i −0.809017 0.587785i
\(9\) −1.56675 1.89387i −1.56675 1.89387i
\(10\) 0 0
\(11\) −0.328407 1.72157i −0.328407 1.72157i −0.637424 0.770513i \(-0.720000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(12\) 0.348445 + 1.82662i 0.348445 + 1.82662i
\(13\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.309017 0.951057i 0.309017 0.951057i
\(17\) −1.98423 + 0.250666i −1.98423 + 0.250666i −0.992115 + 0.125333i \(0.960000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(18\) 1.31703 2.07531i 1.31703 2.07531i
\(19\) 0.0388067 0.616814i 0.0388067 0.616814i −0.929776 0.368125i \(-0.880000\pi\)
0.968583 0.248690i \(-0.0800000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.53583 0.844328i 1.53583 0.844328i
\(23\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(24\) −1.62954 + 0.895846i −1.62954 + 0.895846i
\(25\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(26\) 0 0
\(27\) −2.62594 + 0.674226i −2.62594 + 0.674226i
\(28\) 0 0
\(29\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(32\) 1.00000 1.00000
\(33\) −3.15669 0.810500i −3.15669 0.810500i
\(34\) −0.851559 1.80965i −0.851559 1.80965i
\(35\) 0 0
\(36\) 2.38072 + 0.611264i 2.38072 + 0.611264i
\(37\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(38\) 0.598617 0.153699i 0.598617 0.153699i
\(39\) 0 0
\(40\) 0 0
\(41\) 1.69755 + 0.933237i 1.69755 + 0.933237i 0.968583 + 0.248690i \(0.0800000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(42\) 0 0
\(43\) 1.84489 0.730444i 1.84489 0.730444i 0.876307 0.481754i \(-0.160000\pi\)
0.968583 0.248690i \(-0.0800000\pi\)
\(44\) 1.27760 + 1.19975i 1.27760 + 1.19975i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(48\) −1.35556 1.27295i −1.35556 1.27295i
\(49\) 0.876307 0.481754i 0.876307 0.481754i
\(50\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(51\) −1.14927 + 3.53708i −1.14927 + 3.53708i
\(52\) 0 0
\(53\) 0 0 −0.929776 0.368125i \(-0.880000\pi\)
0.929776 + 0.368125i \(0.120000\pi\)
\(54\) −1.45269 2.28907i −1.45269 2.28907i
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00711 0.553664i −1.00711 0.553664i
\(58\) 0 0
\(59\) 0.371808 + 0.0469702i 0.371808 + 0.0469702i 0.309017 0.951057i \(-0.400000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(60\) 0 0
\(61\) 0 0 −0.929776 0.368125i \(-0.880000\pi\)
0.929776 + 0.368125i \(0.120000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(65\) 0 0
\(66\) −0.204639 3.25265i −0.204639 3.25265i
\(67\) 0.110048 + 0.0604991i 0.110048 + 0.0604991i 0.535827 0.844328i \(-0.320000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(68\) 1.45794 1.36909i 1.45794 1.36909i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(72\) 0.154335 + 2.45309i 0.154335 + 2.45309i
\(73\) −0.263146 0.559214i −0.263146 0.559214i 0.728969 0.684547i \(-0.240000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(74\) 0 0
\(75\) 1.84489 + 0.233064i 1.84489 + 0.233064i
\(76\) 0.331159 + 0.521823i 0.331159 + 0.521823i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(80\) 0 0
\(81\) −0.484103 + 2.53776i −0.484103 + 2.53776i
\(82\) −0.362989 + 1.90285i −0.362989 + 1.90285i
\(83\) −0.620759 + 0.582932i −0.620759 + 0.582932i −0.929776 0.368125i \(-0.880000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.26480 + 1.52888i 1.26480 + 1.52888i
\(87\) 0 0
\(88\) −0.746226 + 1.58581i −0.746226 + 1.58581i
\(89\) −1.06320 + 0.134314i −1.06320 + 0.134314i −0.637424 0.770513i \(-0.720000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0.791759 1.68257i 0.791759 1.68257i
\(97\) 1.41213 1.32608i 1.41213 1.32608i 0.535827 0.844328i \(-0.320000\pi\)
0.876307 0.481754i \(-0.160000\pi\)
\(98\) 0.728969 + 0.684547i 0.728969 + 0.684547i
\(99\) −2.74590 + 3.31923i −2.74590 + 3.31923i
\(100\) −0.809017 0.587785i −0.809017 0.587785i
\(101\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(102\) −3.71911 −3.71911
\(103\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.72897 0.684547i 1.72897 0.684547i 0.728969 0.684547i \(-0.240000\pi\)
1.00000 \(0\)
\(108\) 1.72813 2.08895i 1.72813 2.08895i
\(109\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.98423 −1.98423 −0.992115 0.125333i \(-0.960000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(114\) 0.215351 1.12891i 0.215351 1.12891i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0.0702235 + 0.368125i 0.0702235 + 0.368125i
\(119\) 0 0
\(120\) 0 0
\(121\) −1.92617 + 0.762627i −1.92617 + 0.762627i
\(122\) 0 0
\(123\) 2.91429 2.11736i 2.91429 2.11736i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(128\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(129\) 0.231683 3.68250i 0.231683 3.68250i
\(130\) 0 0
\(131\) 1.60528 + 0.202793i 1.60528 + 0.202793i 0.876307 0.481754i \(-0.160000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(132\) 3.03021 1.19975i 3.03021 1.19975i
\(133\) 0 0
\(134\) −0.0235315 + 0.123357i −0.0235315 + 0.123357i
\(135\) 0 0
\(136\) 1.75261 + 0.963507i 1.75261 + 0.963507i
\(137\) 0.348445 1.82662i 0.348445 1.82662i −0.187381 0.982287i \(-0.560000\pi\)
0.535827 0.844328i \(-0.320000\pi\)
\(138\) 0 0
\(139\) −0.0534698 + 0.849878i −0.0534698 + 0.849878i 0.876307 + 0.481754i \(0.160000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −2.28533 + 0.904827i −2.28533 + 0.904827i
\(145\) 0 0
\(146\) 0.450527 0.423073i 0.450527 0.423073i
\(147\) −0.116762 1.85588i −0.116762 1.85588i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0.348445 + 1.82662i 0.348445 + 1.82662i
\(151\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(152\) −0.393950 + 0.476203i −0.393950 + 0.476203i
\(153\) 3.58352 + 3.36515i 3.58352 + 3.36515i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −2.56315 + 0.323801i −2.56315 + 0.323801i
\(163\) −0.456288 + 0.969661i −0.456288 + 0.969661i 0.535827 + 0.844328i \(0.320000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(164\) −1.92189 + 0.242791i −1.92189 + 0.242791i
\(165\) 0 0
\(166\) −0.746226 0.410241i −0.746226 0.410241i
\(167\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(168\) 0 0
\(169\) −0.187381 + 0.982287i −0.187381 + 0.982287i
\(170\) 0 0
\(171\) −1.22897 + 0.892898i −1.22897 + 0.892898i
\(172\) −1.06320 + 1.67534i −1.06320 + 1.67534i
\(173\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.73879 0.219661i −1.73879 0.219661i
\(177\) 0.373413 0.588404i 0.373413 0.588404i
\(178\) −0.456288 0.969661i −0.456288 0.969661i
\(179\) −0.0800484 1.27233i −0.0800484 1.27233i −0.809017 0.587785i \(-0.800000\pi\)
0.728969 0.684547i \(-0.240000\pi\)
\(180\) 0 0
\(181\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.08317 + 3.33367i 1.08317 + 3.33367i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.425779 0.904827i \(-0.360000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(192\) 1.84489 + 0.233064i 1.84489 + 0.233064i
\(193\) 0.238883 0.288760i 0.238883 0.288760i −0.637424 0.770513i \(-0.720000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(194\) 1.69755 + 0.933237i 1.69755 + 0.933237i
\(195\) 0 0
\(196\) −0.425779 + 0.904827i −0.425779 + 0.904827i
\(197\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(198\) −4.00530 1.58581i −4.00530 1.58581i
\(199\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(200\) 0.309017 0.951057i 0.309017 0.951057i
\(201\) 0.188925 0.137262i 0.188925 0.137262i
\(202\) 0 0
\(203\) 0 0
\(204\) −1.14927 3.53708i −1.14927 3.53708i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.07463 + 0.135758i −1.07463 + 0.135758i
\(210\) 0 0
\(211\) 0.331159 + 1.01920i 0.331159 + 1.01920i 0.968583 + 0.248690i \(0.0800000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 1.18532 + 1.43281i 1.18532 + 1.43281i
\(215\) 0 0
\(216\) 2.52073 + 0.998027i 2.52073 + 0.998027i
\(217\) 0 0
\(218\) 0 0
\(219\) −1.14927 −1.14927
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(224\) 0 0
\(225\) 1.31703 2.07531i 1.31703 2.07531i
\(226\) −0.613161 1.88711i −0.613161 1.88711i
\(227\) −1.41789 + 0.779494i −1.41789 + 0.779494i −0.992115 0.125333i \(-0.960000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(228\) 1.14020 0.144041i 1.14020 0.144041i
\(229\) 0 0 0.876307 0.481754i \(-0.160000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.06320 + 1.67534i −1.06320 + 1.67534i −0.425779 + 0.904827i \(0.640000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.328407 + 0.180543i −0.328407 + 0.180543i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(240\) 0 0
\(241\) −0.393950 + 1.21245i −0.393950 + 1.21245i 0.535827 + 0.844328i \(0.320000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(242\) −1.32052 1.59624i −1.32052 1.59624i
\(243\) 1.69334 + 1.23028i 1.69334 + 1.23028i
\(244\) 0 0
\(245\) 0 0
\(246\) 2.91429 + 2.11736i 2.91429 + 2.11736i
\(247\) 0 0
\(248\) 0 0
\(249\) 0.489334 + 1.50602i 0.489334 + 1.50602i
\(250\) 0 0
\(251\) 0.968583 + 0.248690i 0.968583 + 0.248690i
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.809017 0.587785i −0.809017 0.587785i
\(257\) 0.371808 + 0.0469702i 0.371808 + 0.0469702i 0.309017 0.951057i \(-0.400000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(258\) 3.57386 0.917611i 3.57386 0.917611i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0.303189 + 1.58937i 0.303189 + 1.58937i
\(263\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(264\) 2.07741 + 2.51116i 2.07741 + 2.51116i
\(265\) 0 0
\(266\) 0 0
\(267\) −0.615808 + 1.89526i −0.615808 + 1.89526i
\(268\) −0.124591 + 0.0157395i −0.124591 + 0.0157395i
\(269\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −0.374763 + 1.96457i −0.374763 + 1.96457i
\(273\) 0 0
\(274\) 1.84489 0.233064i 1.84489 0.233064i
\(275\) 1.53583 0.844328i 1.53583 0.844328i
\(276\) 0 0
\(277\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(278\) −0.824805 + 0.211774i −0.824805 + 0.211774i
\(279\) 0 0
\(280\) 0 0
\(281\) 1.41213 + 1.32608i 1.41213 + 1.32608i 0.876307 + 0.481754i \(0.160000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(282\) 0 0
\(283\) −1.27485 −1.27485 −0.637424 0.770513i \(-0.720000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.56675 1.89387i −1.56675 1.89387i
\(289\) 2.90575 0.746070i 2.90575 0.746070i
\(290\) 0 0
\(291\) −1.11316 3.42596i −1.11316 3.42596i
\(292\) 0.541587 + 0.297740i 0.541587 + 0.297740i
\(293\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(294\) 1.72897 0.684547i 1.72897 0.684547i
\(295\) 0 0
\(296\) 0 0
\(297\) 2.02310 + 4.29931i 2.02310 + 4.29931i
\(298\) 0 0
\(299\) 0 0
\(300\) −1.62954 + 0.895846i −1.62954 + 0.895846i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −0.574633 0.227513i −0.574633 0.227513i
\(305\) 0 0
\(306\) −2.09308 + 4.44802i −2.09308 + 4.44802i
\(307\) 1.18532 + 0.469303i 1.18532 + 0.469303i 0.876307 0.481754i \(-0.160000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.425779 0.904827i \(-0.360000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(312\) 0 0
\(313\) 0.939097 0.516273i 0.939097 0.516273i 0.0627905 0.998027i \(-0.480000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0.217126 3.45111i 0.217126 3.45111i
\(322\) 0 0
\(323\) 0.0776134 + 1.23363i 0.0776134 + 1.23363i
\(324\) −1.10001 2.33764i −1.10001 2.33764i
\(325\) 0 0
\(326\) −1.06320 0.134314i −1.06320 0.134314i
\(327\) 0 0
\(328\) −0.824805 1.75280i −0.824805 1.75280i
\(329\) 0 0
\(330\) 0 0
\(331\) 0.303189 0.220280i 0.303189 0.220280i −0.425779 0.904827i \(-0.640000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(332\) 0.159566 0.836475i 0.159566 0.836475i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.11716 1.35041i −1.11716 1.35041i −0.929776 0.368125i \(-0.880000\pi\)
−0.187381 0.982287i \(-0.560000\pi\)
\(338\) −0.992115 + 0.125333i −0.992115 + 0.125333i
\(339\) −1.57103 + 3.33861i −1.57103 + 3.33861i
\(340\) 0 0
\(341\) 0 0
\(342\) −1.22897 0.892898i −1.22897 0.892898i
\(343\) 0 0
\(344\) −1.92189 0.493458i −1.92189 0.493458i
\(345\) 0 0
\(346\) 0 0
\(347\) 0.159566 0.339095i 0.159566 0.339095i −0.809017 0.587785i \(-0.800000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(348\) 0 0
\(349\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.328407 1.72157i −0.328407 1.72157i
\(353\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(354\) 0.674997 + 0.173310i 0.674997 + 0.173310i
\(355\) 0 0
\(356\) 0.781202 0.733597i 0.781202 0.733597i
\(357\) 0 0
\(358\) 1.18532 0.469303i 1.18532 0.469303i
\(359\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(360\) 0 0
\(361\) 0.613161 + 0.0774602i 0.613161 + 0.0774602i
\(362\) 0 0
\(363\) −0.241891 + 3.84475i −0.241891 + 3.84475i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.929776 0.368125i \(-0.880000\pi\)
0.929776 + 0.368125i \(0.120000\pi\)
\(368\) 0 0
\(369\) −0.892204 4.67710i −0.892204 4.67710i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(374\) −2.83579 + 2.06032i −2.83579 + 2.06032i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.101597 + 0.0738147i −0.101597 + 0.0738147i −0.637424 0.770513i \(-0.720000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.929776 0.368125i \(-0.120000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(384\) 0.348445 + 1.82662i 0.348445 + 1.82662i
\(385\) 0 0
\(386\) 0.348445 + 0.137959i 0.348445 + 0.137959i
\(387\) −4.27385 2.34957i −4.27385 2.34957i
\(388\) −0.362989 + 1.90285i −0.362989 + 1.90285i
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.992115 0.125333i −0.992115 0.125333i
\(393\) 1.61221 2.54043i 1.61221 2.54043i
\(394\) 0 0
\(395\) 0 0
\(396\) 0.270490 4.29931i 0.270490 4.29931i
\(397\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 1.00000
\(401\) −0.273190 1.43211i −0.273190 1.43211i −0.809017 0.587785i \(-0.800000\pi\)
0.535827 0.844328i \(-0.320000\pi\)
\(402\) 0.188925 + 0.137262i 0.188925 + 0.137262i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 3.00882 2.18604i 3.00882 2.18604i
\(409\) −0.824805 0.211774i −0.824805 0.211774i −0.187381 0.982287i \(-0.560000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(410\) 0 0
\(411\) −2.79753 2.03252i −2.79753 2.03252i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.38765 + 0.762866i 1.38765 + 0.762866i
\(418\) −0.461193 0.980086i −0.461193 0.980086i
\(419\) −0.620759 + 0.582932i −0.620759 + 0.582932i −0.929776 0.368125i \(-0.880000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(420\) 0 0
\(421\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(422\) −0.866986 + 0.629902i −0.866986 + 0.629902i
\(423\) 0 0
\(424\) 0 0
\(425\) −0.851559 1.80965i −0.851559 1.80965i
\(426\) 0 0
\(427\) 0 0
\(428\) −0.996398 + 1.57007i −0.996398 + 1.57007i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(432\) −0.170232 + 2.70576i −0.170232 + 2.70576i
\(433\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −0.355143 1.09302i −0.355143 1.09302i
\(439\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(440\) 0 0
\(441\) −2.28533 0.904827i −2.28533 0.904827i
\(442\) 0 0
\(443\) −1.98423 0.250666i −1.98423 0.250666i −0.992115 0.125333i \(-0.960000\pi\)
−0.992115 0.125333i \(-0.960000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.791759 + 0.313480i 0.791759 + 0.313480i 0.728969 0.684547i \(-0.240000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(450\) 2.38072 + 0.611264i 2.38072 + 0.611264i
\(451\) 1.04914 3.22894i 1.04914 3.22894i
\(452\) 1.60528 1.16630i 1.60528 1.16630i
\(453\) 0 0
\(454\) −1.17950 1.10762i −1.17950 1.10762i
\(455\) 0 0
\(456\) 0.489334 + 1.03989i 0.489334 + 1.03989i
\(457\) 0.0388067 + 0.616814i 0.0388067 + 0.616814i 0.968583 + 0.248690i \(0.0800000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(458\) 0 0
\(459\) 5.04146 1.99605i 5.04146 1.99605i
\(460\) 0 0
\(461\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(462\) 0 0
\(463\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −1.92189 0.493458i −1.92189 0.493458i
\(467\) −1.35556 0.536702i −1.35556 0.536702i −0.425779 0.904827i \(-0.640000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −0.273190 0.256543i −0.273190 0.256543i
\(473\) −1.86338 2.93622i −1.86338 2.93622i
\(474\) 0 0
\(475\) 0.598617 0.153699i 0.598617 0.153699i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −1.27485 −1.27485
\(483\) 0 0
\(484\) 1.11005 1.74915i 1.11005 1.74915i
\(485\) 0 0
\(486\) −0.646797 + 1.99064i −0.646797 + 1.99064i
\(487\) 0 0 0.876307 0.481754i \(-0.160000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(488\) 0 0
\(489\) 1.27026 + 1.53548i 1.27026 + 1.53548i
\(490\) 0 0
\(491\) −0.362989 1.90285i −0.362989 1.90285i −0.425779 0.904827i \(-0.640000\pi\)
0.0627905 0.998027i \(-0.480000\pi\)
\(492\) −1.11316 + 3.42596i −1.11316 + 3.42596i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −1.28109 + 0.930769i −1.28109 + 0.930769i
\(499\) −0.0534698 + 0.113629i −0.0534698 + 0.113629i −0.929776 0.368125i \(-0.880000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0.0627905 + 0.998027i 0.0627905 + 0.998027i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.50441 + 1.09302i 1.50441 + 1.09302i
\(508\) 0 0
\(509\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.309017 0.951057i 0.309017 0.951057i
\(513\) 0.313968 + 1.64588i 0.313968 + 1.64588i
\(514\) 0.0702235 + 0.368125i 0.0702235 + 0.368125i
\(515\) 0 0
\(516\) 1.97708 + 3.11538i 1.97708 + 3.11538i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −0.0800484 + 1.27233i −0.0800484 + 1.27233i 0.728969 + 0.684547i \(0.240000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(522\) 0 0
\(523\) −0.273190 + 1.43211i −0.273190 + 1.43211i 0.535827 + 0.844328i \(0.320000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(524\) −1.41789 + 0.779494i −1.41789 + 0.779494i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −1.74630 + 2.75173i −1.74630 + 2.75173i
\(529\) 0.968583 0.248690i 0.968583 0.248690i
\(530\) 0 0
\(531\) −0.493573 0.777747i −0.493573 0.777747i
\(532\) 0 0
\(533\) 0 0
\(534\) −1.99280 −1.99280
\(535\) 0 0
\(536\) −0.0534698 0.113629i −0.0534698 0.113629i
\(537\) −2.20417 0.872693i −2.20417 0.872693i
\(538\) 0 0
\(539\) −1.11716 1.35041i −1.11716 1.35041i
\(540\) 0 0
\(541\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −1.98423 + 0.250666i −1.98423 + 0.250666i
\(545\) 0 0
\(546\) 0 0
\(547\) 0.00788530 + 0.125333i 0.00788530 + 0.125333i 1.00000 \(0\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(548\) 0.791759 + 1.68257i 0.791759 + 1.68257i
\(549\) 0 0
\(550\) 1.27760 + 1.19975i 1.27760 + 1.19975i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −0.456288 0.718995i −0.456288 0.718995i
\(557\) 0 0 0.425779 0.904827i \(-0.360000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 6.46676 + 0.816941i 6.46676 + 0.816941i
\(562\) −0.824805 + 1.75280i −0.824805 + 1.75280i
\(563\) 1.50441 + 0.595638i 1.50441 + 0.595638i 0.968583 0.248690i \(-0.0800000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −0.393950 1.21245i −0.393950 1.21245i
\(567\) 0 0
\(568\) 0 0
\(569\) −0.328407 0.180543i −0.328407 0.180543i 0.309017 0.951057i \(-0.400000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(570\) 0 0
\(571\) 0.541587 1.66683i 0.541587 1.66683i −0.187381 0.982287i \(-0.560000\pi\)
0.728969 0.684547i \(-0.240000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.31703 2.07531i 1.31703 2.07531i
\(577\) −0.124591 0.0157395i −0.124591 0.0157395i 0.0627905 0.998027i \(-0.480000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(578\) 1.60748 + 2.53298i 1.60748 + 2.53298i
\(579\) −0.296722 0.630566i −0.296722 0.630566i
\(580\) 0 0
\(581\) 0 0
\(582\) 2.91429 2.11736i 2.91429 2.11736i
\(583\) 0 0
\(584\) −0.115808 + 0.607087i −0.115808 + 0.607087i
\(585\) 0 0
\(586\) 0 0
\(587\) 0.939097 + 0.516273i 0.939097 + 0.516273i 0.876307 0.481754i \(-0.160000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(588\) 1.18532 + 1.43281i 1.18532 + 1.43281i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.50441 + 1.09302i 1.50441 + 1.09302i 0.968583 + 0.248690i \(0.0800000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(594\) −3.46372 + 3.25265i −3.46372 + 3.25265i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(600\) −1.35556 1.27295i −1.35556 1.27295i
\(601\) −0.393950 + 0.476203i −0.393950 + 0.476203i −0.929776 0.368125i \(-0.880000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(602\) 0 0
\(603\) −0.0578390 0.303203i −0.0578390 0.303203i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(608\) 0.0388067 0.616814i 0.0388067 0.616814i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −4.87711 0.616122i −4.87711 0.616122i
\(613\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(614\) −0.0800484 + 1.27233i −0.0800484 + 1.27233i
\(615\) 0 0
\(616\) 0 0
\(617\) −0.328407 0.180543i −0.328407 0.180543i 0.309017 0.951057i \(-0.400000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(618\) 0 0
\(619\) 0.371808 1.94908i 0.371808 1.94908i 0.0627905 0.998027i \(-0.480000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(626\) 0.781202 + 0.733597i 0.781202 + 0.733597i
\(627\) −0.622428 + 1.91564i −0.622428 + 1.91564i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(632\) 0 0
\(633\) 1.97708 + 0.249764i 1.97708 + 0.249764i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.101597 + 1.61484i −0.101597 + 1.61484i 0.535827 + 0.844328i \(0.320000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(642\) 3.34930 0.859954i 3.34930 0.859954i
\(643\) −1.44644 0.182728i −1.44644 0.182728i −0.637424 0.770513i \(-0.720000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.14927 + 0.455027i −1.14927 + 0.455027i
\(647\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(648\) 1.88330 1.76854i 1.88330 1.76854i
\(649\) −0.0412417 0.655518i −0.0412417 0.655518i
\(650\) 0 0
\(651\) 0 0
\(652\) −0.200808 1.05267i −0.200808 1.05267i
\(653\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.41213 1.32608i 1.41213 1.32608i
\(657\) −0.646797 + 1.37451i −0.646797 + 1.37451i
\(658\) 0 0
\(659\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(660\) 0 0
\(661\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(662\) 0.303189 + 0.220280i 0.303189 + 0.220280i
\(663\) 0 0
\(664\) 0.844844 0.106729i 0.844844 0.106729i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.866986 + 0.629902i −0.866986 + 0.629902i −0.929776 0.368125i \(-0.880000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(674\) 0.939097 1.47978i 0.939097 1.47978i
\(675\) −1.45269 2.28907i −1.45269 2.28907i
\(676\) −0.425779 0.904827i −0.425779 0.904827i
\(677\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(678\) −3.66068 0.462452i −3.66068 0.462452i
\(679\) 0 0
\(680\) 0 0
\(681\) 0.188925 + 3.00288i 0.188925 + 3.00288i
\(682\) 0 0
\(683\) 0.110048 1.74915i 0.110048 1.74915i −0.425779 0.904827i \(-0.640000\pi\)
0.535827 0.844328i \(-0.320000\pi\)
\(684\) 0.469424 1.44474i 0.469424 1.44474i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −0.124591 1.98031i −0.124591 1.98031i
\(689\) 0 0
\(690\) 0 0
\(691\) 1.69755 0.933237i 1.69755 0.933237i 0.728969 0.684547i \(-0.240000\pi\)
0.968583 0.248690i \(-0.0800000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0.371808 + 0.0469702i 0.371808 + 0.0469702i
\(695\) 0 0
\(696\) 0 0
\(697\) −3.60226 1.42624i −3.60226 1.42624i
\(698\) 0 0
\(699\) 1.97708 + 3.11538i 1.97708 + 3.11538i
\(700\) 0 0
\(701\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.53583 0.844328i 1.53583 0.844328i
\(705\) 0 0
\(706\) −0.500000 1.53884i −0.500000 1.53884i
\(707\) 0 0
\(708\) 0.0437581 + 0.695516i 0.0437581 + 0.695516i
\(709\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.939097 + 0.516273i 0.939097 + 0.516273i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0.812619 + 0.982287i 0.812619 + 0.982287i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.115808 + 0.607087i 0.115808 + 0.607087i
\(723\) 1.72813 + 1.62282i 1.72813 + 1.62282i
\(724\) 0 0
\(725\) 0 0
\(726\) −3.73132 + 0.958040i −3.73132 + 0.958040i
\(727\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(728\) 0 0
\(729\) 1.14680 0.630457i 1.14680 0.630457i
\(730\) 0 0
\(731\) −3.47759 + 1.91182i −3.47759 + 1.91182i
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.0680131 0.209323i 0.0680131 0.209323i
\(738\) 4.17248 2.29384i 4.17248 2.29384i
\(739\) 0.939097 + 1.47978i 0.939097 + 1.47978i 0.876307 + 0.481754i \(0.160000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 2.07657 + 0.262332i 2.07657 + 0.262332i
\(748\) −2.83579 2.06032i −2.83579 2.06032i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(752\) 0 0
\(753\) 1.18532 1.43281i 1.18532 1.43281i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(758\) −0.101597 0.0738147i −0.101597 0.0738147i
\(759\) 0 0
\(760\) 0 0
\(761\) 0.688925 + 0.500534i 0.688925 + 0.500534i 0.876307 0.481754i \(-0.160000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.62954 + 0.895846i −1.62954 + 0.895846i
\(769\) 0.541587 1.66683i 0.541587 1.66683i −0.187381 0.982287i \(-0.560000\pi\)
0.728969 0.684547i \(-0.240000\pi\)
\(770\) 0 0
\(771\) 0.373413 0.588404i 0.373413 0.588404i
\(772\) −0.0235315 + 0.374023i −0.0235315 + 0.374023i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0.913880 4.79072i 0.913880 4.79072i
\(775\) 0 0
\(776\) −1.92189 + 0.242791i −1.92189 + 0.242791i
\(777\) 0 0
\(778\) 0 0
\(779\) 0.641510 1.01086i 0.641510 1.01086i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.187381 0.982287i −0.187381 0.982287i
\(785\) 0 0
\(786\) 2.91429 + 0.748263i 2.91429 + 0.748263i
\(787\) 0.844844 + 1.79538i 0.844844 + 1.79538i 0.535827 + 0.844328i \(0.320000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 4.17248 1.07131i 4.17248 1.07131i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(801\) 1.92015 + 1.80314i 1.92015 + 1.80314i
\(802\) 1.27760 0.702367i 1.27760 0.702367i
\(803\) −0.876307 + 0.636674i −0.876307 + 0.636674i
\(804\) −0.0721631 + 0.222095i −0.0721631 + 0.222095i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.62954 0.645180i −1.62954 0.645180i −0.637424 0.770513i \(-0.720000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(810\) 0 0
\(811\) −0.683098 + 0.825723i −0.683098 + 0.825723i −0.992115 0.125333i \(-0.960000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 3.00882 + 2.18604i 3.00882 + 2.18604i
\(817\) −0.378954 1.16630i −0.378954 1.16630i
\(818\) −0.0534698 0.849878i −0.0534698 0.849878i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(822\) 1.06856 3.28869i 1.06856 3.28869i
\(823\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(824\) 0 0
\(825\) −0.204639 3.25265i −0.204639 3.25265i
\(826\) 0 0
\(827\) 0.781202 1.23098i 0.781202 1.23098i −0.187381 0.982287i \(-0.560000\pi\)
0.968583 0.248690i \(-0.0800000\pi\)
\(828\) 0 0
\(829\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.61803 + 1.17557i −1.61803 + 1.17557i
\(834\) −0.296722 + 1.55547i −0.296722 + 1.55547i
\(835\) 0 0
\(836\) 0.789600 0.741484i 0.789600 0.741484i
\(837\) 0 0
\(838\) −0.746226 0.410241i −0.746226 0.410241i
\(839\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(840\) 0 0
\(841\) −0.425779 + 0.904827i −0.425779 + 0.904827i
\(842\) 0 0
\(843\) 3.34930 1.32608i 3.34930 1.32608i
\(844\) −0.866986 0.629902i −0.866986 0.629902i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −1.00937 + 2.14503i −1.00937 + 2.14503i
\(850\) 1.45794 1.36909i 1.45794 1.36909i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.80113 0.462452i −1.80113 0.462452i
\(857\) 0.0672897 + 1.06954i 0.0672897 + 1.06954i 0.876307 + 0.481754i \(0.160000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(858\) 0 0
\(859\) 0.0672897 1.06954i 0.0672897 1.06954i −0.809017 0.587785i \(-0.800000\pi\)
0.876307 0.481754i \(-0.160000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(864\) −2.62594 + 0.674226i −2.62594 + 0.674226i
\(865\) 0 0
\(866\) −1.61803 −1.61803
\(867\) 1.04534 5.47985i 1.04534 5.47985i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −4.72389 0.596766i −4.72389 0.596766i
\(874\) 0 0
\(875\) 0 0
\(876\) 0.929776 0.675522i 0.929776 0.675522i
\(877\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.41789 + 1.03016i −1.41789 + 1.03016i −0.425779 + 0.904827i \(0.640000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(882\) 0.154335 2.45309i 0.154335 2.45309i
\(883\) −0.574633 + 0.227513i −0.574633 + 0.227513i −0.637424 0.770513i \(-0.720000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −0.374763 1.96457i −0.374763 1.96457i
\(887\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4.52791 4.52791
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −0.0534698 + 0.849878i −0.0534698 + 0.849878i
\(899\) 0 0
\(900\) 0.154335 + 2.45309i 0.154335 + 2.45309i
\(901\) 0 0
\(902\) 3.39510 3.39510
\(903\) 0 0
\(904\) 1.60528 + 1.16630i 1.60528 + 1.16630i
\(905\) 0 0
\(906\) 0 0
\(907\) 1.06279 0.998027i 1.06279 0.998027i 0.0627905 0.998027i \(-0.480000\pi\)
1.00000 \(0\)
\(908\) 0.688925 1.46404i 0.688925 1.46404i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(912\) −0.837780 + 0.786727i −0.837780 + 0.786727i
\(913\) 1.20742 + 0.877242i 1.20742 + 0.877242i
\(914\) −0.574633 + 0.227513i −0.574633 + 0.227513i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 3.45626 + 4.17789i 3.45626 + 4.17789i
\(919\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(920\) 0 0
\(921\) 1.72813 1.62282i 1.72813 1.62282i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.26480 + 0.159781i 1.26480 + 0.159781i 0.728969 0.684547i \(-0.240000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(930\) 0 0
\(931\) −0.263146 0.559214i −0.263146 0.559214i
\(932\) −0.124591 1.98031i −0.124591 1.98031i
\(933\) 0 0
\(934\) 0.0915446 1.45506i 0.0915446 1.45506i
\(935\) 0 0
\(936\) 0 0
\(937\) 1.53583 + 0.844328i 1.53583 + 0.844328i 1.00000 \(0\)
0.535827 + 0.844328i \(0.320000\pi\)
\(938\) 0 0
\(939\) −0.125129 1.98886i −0.125129 1.98886i
\(940\) 0 0
\(941\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0.159566 0.339095i 0.159566 0.339095i
\(945\) 0 0
\(946\) 2.21670 2.67953i 2.21670 2.67953i
\(947\) 0.541587 + 0.297740i 0.541587 + 0.297740i 0.728969 0.684547i \(-0.240000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0.331159 + 0.521823i 0.331159 + 0.521823i
\(951\) 0 0
\(952\) 0 0
\(953\) −0.613161 + 1.88711i −0.613161 + 1.88711i −0.187381 + 0.982287i \(0.560000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.929776 + 0.368125i −0.929776 + 0.368125i
\(962\) 0 0
\(963\) −4.00530 2.20193i −4.00530 2.20193i
\(964\) −0.393950 1.21245i −0.393950 1.21245i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(968\) 2.00657 + 0.515199i 2.00657 + 0.515199i
\(969\) 2.13712 + 0.846147i 2.13712 + 0.846147i
\(970\) 0 0
\(971\) 0.598617 + 0.153699i 0.598617 + 0.153699i 0.535827 0.844328i \(-0.320000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(972\) −2.09308 −2.09308
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −0.824805 + 0.211774i −0.824805 + 0.211774i −0.637424 0.770513i \(-0.720000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(978\) −1.06779 + 1.68257i −1.06779 + 1.68257i
\(979\) 0.580394 + 1.78627i 0.580394 + 1.78627i
\(980\) 0 0
\(981\) 0 0
\(982\) 1.69755 0.933237i 1.69755 0.933237i
\(983\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(984\) −3.60226 −3.60226
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(992\) 0 0
\(993\) −0.130584 0.684547i −0.130584 0.684547i
\(994\) 0 0
\(995\) 0 0
\(996\) −1.28109 0.930769i −1.28109 0.930769i
\(997\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(998\) −0.124591 0.0157395i −0.124591 0.0157395i
\(999\) 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 2008.1.w.a.1531.1 20
8.3 odd 2 CM 2008.1.w.a.1531.1 20
251.241 even 25 inner 2008.1.w.a.1747.1 yes 20
2008.1747 odd 50 inner 2008.1.w.a.1747.1 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2008.1.w.a.1531.1 20 1.1 even 1 trivial
2008.1.w.a.1531.1 20 8.3 odd 2 CM
2008.1.w.a.1747.1 yes 20 251.241 even 25 inner
2008.1.w.a.1747.1 yes 20 2008.1747 odd 50 inner