Properties

Label 1452.2.i.l.1237.1
Level $1452$
Weight $2$
Character 1452.1237
Analytic conductor $11.594$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1452,2,Mod(493,1452)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1452, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1452.493");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1452.i (of order \(5\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5942783735\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 132)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 1237.1
Root \(-0.309017 + 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 1452.1237
Dual form 1452.2.i.l.493.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.809017 + 0.587785i) q^{3} +(0.618034 - 1.90211i) q^{5} +(1.61803 - 1.17557i) q^{7} +(0.309017 + 0.951057i) q^{9} +O(q^{10})\) \(q+(0.809017 + 0.587785i) q^{3} +(0.618034 - 1.90211i) q^{5} +(1.61803 - 1.17557i) q^{7} +(0.309017 + 0.951057i) q^{9} +(-1.85410 - 5.70634i) q^{13} +(1.61803 - 1.17557i) q^{15} +(1.23607 - 3.80423i) q^{17} +(-1.61803 - 1.17557i) q^{19} +2.00000 q^{21} -8.00000 q^{23} +(0.809017 + 0.587785i) q^{25} +(-0.309017 + 0.951057i) q^{27} +(-1.23607 - 3.80423i) q^{35} +(4.85410 - 3.52671i) q^{37} +(1.85410 - 5.70634i) q^{39} -10.0000 q^{43} +2.00000 q^{45} +(-0.927051 + 2.85317i) q^{49} +(3.23607 - 2.35114i) q^{51} +(4.32624 + 13.3148i) q^{53} +(-0.618034 - 1.90211i) q^{57} +(9.70820 - 7.05342i) q^{59} +(4.32624 - 13.3148i) q^{61} +(1.61803 + 1.17557i) q^{63} -12.0000 q^{65} +4.00000 q^{67} +(-6.47214 - 4.70228i) q^{69} +(4.85410 - 3.52671i) q^{73} +(0.309017 + 0.951057i) q^{75} +(-0.618034 - 1.90211i) q^{79} +(-0.809017 + 0.587785i) q^{81} +(-4.94427 + 15.2169i) q^{83} +(-6.47214 - 4.70228i) q^{85} -14.0000 q^{89} +(-9.70820 - 7.05342i) q^{91} +(-3.23607 + 2.35114i) q^{95} +(-0.618034 - 1.90211i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} - 2 q^{5} + 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} - 2 q^{5} + 2 q^{7} - q^{9} + 6 q^{13} + 2 q^{15} - 4 q^{17} - 2 q^{19} + 8 q^{21} - 32 q^{23} + q^{25} + q^{27} + 4 q^{35} + 6 q^{37} - 6 q^{39} - 40 q^{43} + 8 q^{45} + 3 q^{49} + 4 q^{51} - 14 q^{53} + 2 q^{57} + 12 q^{59} - 14 q^{61} + 2 q^{63} - 48 q^{65} + 16 q^{67} - 8 q^{69} + 6 q^{73} - q^{75} + 2 q^{79} - q^{81} + 16 q^{83} - 8 q^{85} - 56 q^{89} - 12 q^{91} - 4 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(727\) \(1333\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{5}\right)\)

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
\(3\) 0.809017 + 0.587785i 0.467086 + 0.339358i
\(4\) 0 0
\(5\) 0.618034 1.90211i 0.276393 0.850651i −0.712454 0.701719i \(-0.752416\pi\)
0.988847 0.148932i \(-0.0475836\pi\)
\(6\) 0 0
\(7\) 1.61803 1.17557i 0.611559 0.444324i −0.238404 0.971166i \(-0.576624\pi\)
0.849963 + 0.526842i \(0.176624\pi\)
\(8\) 0 0
\(9\) 0.309017 + 0.951057i 0.103006 + 0.317019i
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −1.85410 5.70634i −0.514235 1.58265i −0.784669 0.619915i \(-0.787167\pi\)
0.270434 0.962739i \(-0.412833\pi\)
\(14\) 0 0
\(15\) 1.61803 1.17557i 0.417775 0.303531i
\(16\) 0 0
\(17\) 1.23607 3.80423i 0.299791 0.922660i −0.681780 0.731558i \(-0.738794\pi\)
0.981570 0.191103i \(-0.0612063\pi\)
\(18\) 0 0
\(19\) −1.61803 1.17557i −0.371202 0.269694i 0.386507 0.922287i \(-0.373682\pi\)
−0.757709 + 0.652592i \(0.773682\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 0.809017 + 0.587785i 0.161803 + 0.117557i
\(26\) 0 0
\(27\) −0.309017 + 0.951057i −0.0594703 + 0.183031i
\(28\) 0 0
\(29\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(30\) 0 0
\(31\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.23607 3.80423i −0.208934 0.643032i
\(36\) 0 0
\(37\) 4.85410 3.52671i 0.798009 0.579788i −0.112320 0.993672i \(-0.535828\pi\)
0.910330 + 0.413884i \(0.135828\pi\)
\(38\) 0 0
\(39\) 1.85410 5.70634i 0.296894 0.913746i
\(40\) 0 0
\(41\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(48\) 0 0
\(49\) −0.927051 + 2.85317i −0.132436 + 0.407596i
\(50\) 0 0
\(51\) 3.23607 2.35114i 0.453140 0.329226i
\(52\) 0 0
\(53\) 4.32624 + 13.3148i 0.594254 + 1.82893i 0.558403 + 0.829570i \(0.311414\pi\)
0.0358519 + 0.999357i \(0.488586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.618034 1.90211i −0.0818606 0.251941i
\(58\) 0 0
\(59\) 9.70820 7.05342i 1.26390 0.918277i 0.264958 0.964260i \(-0.414642\pi\)
0.998942 + 0.0459824i \(0.0146418\pi\)
\(60\) 0 0
\(61\) 4.32624 13.3148i 0.553918 1.70478i −0.144866 0.989451i \(-0.546275\pi\)
0.698784 0.715333i \(-0.253725\pi\)
\(62\) 0 0
\(63\) 1.61803 + 1.17557i 0.203853 + 0.148108i
\(64\) 0 0
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −6.47214 4.70228i −0.779154 0.566088i
\(70\) 0 0
\(71\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(72\) 0 0
\(73\) 4.85410 3.52671i 0.568130 0.412770i −0.266296 0.963891i \(-0.585800\pi\)
0.834425 + 0.551121i \(0.185800\pi\)
\(74\) 0 0
\(75\) 0.309017 + 0.951057i 0.0356822 + 0.109819i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.618034 1.90211i −0.0695343 0.214004i 0.910251 0.414057i \(-0.135889\pi\)
−0.979785 + 0.200053i \(0.935889\pi\)
\(80\) 0 0
\(81\) −0.809017 + 0.587785i −0.0898908 + 0.0653095i
\(82\) 0 0
\(83\) −4.94427 + 15.2169i −0.542704 + 1.67027i 0.183682 + 0.982986i \(0.441198\pi\)
−0.726386 + 0.687287i \(0.758802\pi\)
\(84\) 0 0
\(85\) −6.47214 4.70228i −0.702002 0.510034i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) −9.70820 7.05342i −1.01770 0.739400i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.23607 + 2.35114i −0.332014 + 0.241222i
\(96\) 0 0
\(97\) −0.618034 1.90211i −0.0627518 0.193130i 0.914766 0.403985i \(-0.132375\pi\)
−0.977517 + 0.210855i \(0.932375\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.47214 7.60845i −0.245987 0.757069i −0.995473 0.0950490i \(-0.969699\pi\)
0.749486 0.662020i \(-0.230301\pi\)
\(102\) 0 0
\(103\) 9.70820 7.05342i 0.956578 0.694994i 0.00422434 0.999991i \(-0.498655\pi\)
0.952353 + 0.304997i \(0.0986553\pi\)
\(104\) 0 0
\(105\) 1.23607 3.80423i 0.120628 0.371254i
\(106\) 0 0
\(107\) 12.9443 + 9.40456i 1.25137 + 0.909174i 0.998301 0.0582746i \(-0.0185599\pi\)
0.253069 + 0.967448i \(0.418560\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 8.09017 + 5.87785i 0.761059 + 0.552942i 0.899235 0.437466i \(-0.144124\pi\)
−0.138176 + 0.990408i \(0.544124\pi\)
\(114\) 0 0
\(115\) −4.94427 + 15.2169i −0.461056 + 1.41898i
\(116\) 0 0
\(117\) 4.85410 3.52671i 0.448762 0.326045i
\(118\) 0 0
\(119\) −2.47214 7.60845i −0.226620 0.697466i
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.70820 7.05342i 0.868328 0.630877i
\(126\) 0 0
\(127\) 1.85410 5.70634i 0.164525 0.506356i −0.834476 0.551044i \(-0.814230\pi\)
0.999001 + 0.0446885i \(0.0142295\pi\)
\(128\) 0 0
\(129\) −8.09017 5.87785i −0.712300 0.517516i
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 0 0
\(135\) 1.61803 + 1.17557i 0.139258 + 0.101177i
\(136\) 0 0
\(137\) 4.32624 13.3148i 0.369615 1.13756i −0.577425 0.816444i \(-0.695942\pi\)
0.947040 0.321115i \(-0.104058\pi\)
\(138\) 0 0
\(139\) 11.3262 8.22899i 0.960679 0.697974i 0.00737063 0.999973i \(-0.497654\pi\)
0.953308 + 0.301999i \(0.0976538\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2.42705 + 1.76336i −0.200180 + 0.145439i
\(148\) 0 0
\(149\) 1.23607 3.80423i 0.101263 0.311654i −0.887572 0.460668i \(-0.847610\pi\)
0.988835 + 0.149014i \(0.0476099\pi\)
\(150\) 0 0
\(151\) 8.09017 + 5.87785i 0.658369 + 0.478333i 0.866112 0.499851i \(-0.166612\pi\)
−0.207743 + 0.978183i \(0.566612\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.09017 + 5.87785i 0.645666 + 0.469104i 0.861792 0.507262i \(-0.169342\pi\)
−0.216126 + 0.976365i \(0.569342\pi\)
\(158\) 0 0
\(159\) −4.32624 + 13.3148i −0.343093 + 1.05593i
\(160\) 0 0
\(161\) −12.9443 + 9.40456i −1.02015 + 0.741183i
\(162\) 0 0
\(163\) 4.94427 + 15.2169i 0.387265 + 1.19188i 0.934824 + 0.355112i \(0.115557\pi\)
−0.547558 + 0.836768i \(0.684443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.70820 + 11.4127i 0.286949 + 0.883140i 0.985808 + 0.167879i \(0.0536919\pi\)
−0.698858 + 0.715260i \(0.746308\pi\)
\(168\) 0 0
\(169\) −18.6074 + 13.5191i −1.43134 + 1.03993i
\(170\) 0 0
\(171\) 0.618034 1.90211i 0.0472622 0.145458i
\(172\) 0 0
\(173\) 6.47214 + 4.70228i 0.492067 + 0.357508i 0.805979 0.591944i \(-0.201639\pi\)
−0.313911 + 0.949452i \(0.601639\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 3.23607 + 2.35114i 0.241875 + 0.175733i 0.702118 0.712060i \(-0.252238\pi\)
−0.460243 + 0.887793i \(0.652238\pi\)
\(180\) 0 0
\(181\) 6.79837 20.9232i 0.505319 1.55521i −0.294914 0.955524i \(-0.595291\pi\)
0.800233 0.599689i \(-0.204709\pi\)
\(182\) 0 0
\(183\) 11.3262 8.22899i 0.837260 0.608305i
\(184\) 0 0
\(185\) −3.70820 11.4127i −0.272633 0.839077i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.618034 + 1.90211i 0.0449554 + 0.138358i
\(190\) 0 0
\(191\) −19.4164 + 14.1068i −1.40492 + 1.02074i −0.410886 + 0.911687i \(0.634780\pi\)
−0.994036 + 0.109049i \(0.965220\pi\)
\(192\) 0 0
\(193\) −3.09017 + 9.51057i −0.222435 + 0.684585i 0.776107 + 0.630602i \(0.217192\pi\)
−0.998542 + 0.0539836i \(0.982808\pi\)
\(194\) 0 0
\(195\) −9.70820 7.05342i −0.695219 0.505106i
\(196\) 0 0
\(197\) −16.0000 −1.13995 −0.569976 0.821661i \(-0.693048\pi\)
−0.569976 + 0.821661i \(0.693048\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 3.23607 + 2.35114i 0.228255 + 0.165837i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.47214 7.60845i −0.171825 0.528824i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 3.09017 + 9.51057i 0.212736 + 0.654734i 0.999307 + 0.0372338i \(0.0118546\pi\)
−0.786571 + 0.617500i \(0.788145\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.18034 + 19.0211i −0.421496 + 1.29723i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) −9.70820 7.05342i −0.650109 0.472332i 0.213199 0.977009i \(-0.431612\pi\)
−0.863308 + 0.504677i \(0.831612\pi\)
\(224\) 0 0
\(225\) −0.309017 + 0.951057i −0.0206011 + 0.0634038i
\(226\) 0 0
\(227\) −19.4164 + 14.1068i −1.28871 + 0.936304i −0.999779 0.0210448i \(-0.993301\pi\)
−0.288934 + 0.957349i \(0.593301\pi\)
\(228\) 0 0
\(229\) 0.618034 + 1.90211i 0.0408408 + 0.125695i 0.969398 0.245494i \(-0.0789502\pi\)
−0.928557 + 0.371189i \(0.878950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.94427 15.2169i −0.323910 0.996893i −0.971930 0.235269i \(-0.924403\pi\)
0.648020 0.761623i \(-0.275597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.618034 1.90211i 0.0401456 0.123556i
\(238\) 0 0
\(239\) −3.23607 2.35114i −0.209324 0.152083i 0.478184 0.878260i \(-0.341295\pi\)
−0.687508 + 0.726177i \(0.741295\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 4.85410 + 3.52671i 0.310117 + 0.225313i
\(246\) 0 0
\(247\) −3.70820 + 11.4127i −0.235947 + 0.726171i
\(248\) 0 0
\(249\) −12.9443 + 9.40456i −0.820310 + 0.595990i
\(250\) 0 0
\(251\) −1.23607 3.80423i −0.0780199 0.240121i 0.904438 0.426605i \(-0.140291\pi\)
−0.982458 + 0.186485i \(0.940291\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −2.47214 7.60845i −0.154811 0.476460i
\(256\) 0 0
\(257\) −1.61803 + 1.17557i −0.100930 + 0.0733301i −0.637106 0.770776i \(-0.719869\pi\)
0.536175 + 0.844107i \(0.319869\pi\)
\(258\) 0 0
\(259\) 3.70820 11.4127i 0.230417 0.709149i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 28.0000 1.72003
\(266\) 0 0
\(267\) −11.3262 8.22899i −0.693155 0.503606i
\(268\) 0 0
\(269\) 4.32624 13.3148i 0.263775 0.811817i −0.728198 0.685367i \(-0.759642\pi\)
0.991973 0.126450i \(-0.0403583\pi\)
\(270\) 0 0
\(271\) −8.09017 + 5.87785i −0.491443 + 0.357054i −0.805739 0.592271i \(-0.798232\pi\)
0.314296 + 0.949325i \(0.398232\pi\)
\(272\) 0 0
\(273\) −3.70820 11.4127i −0.224431 0.690727i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.09017 + 9.51057i 0.185670 + 0.571434i 0.999959 0.00902525i \(-0.00287287\pi\)
−0.814289 + 0.580460i \(0.802873\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −9.88854 + 30.4338i −0.589901 + 1.81553i −0.0112742 + 0.999936i \(0.503589\pi\)
−0.578627 + 0.815592i \(0.696411\pi\)
\(282\) 0 0
\(283\) −11.3262 8.22899i −0.673275 0.489163i 0.197845 0.980233i \(-0.436606\pi\)
−0.871120 + 0.491070i \(0.836606\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.809017 + 0.587785i 0.0475892 + 0.0345756i
\(290\) 0 0
\(291\) 0.618034 1.90211i 0.0362298 0.111504i
\(292\) 0 0
\(293\) −16.1803 + 11.7557i −0.945266 + 0.686776i −0.949682 0.313215i \(-0.898594\pi\)
0.00441682 + 0.999990i \(0.498594\pi\)
\(294\) 0 0
\(295\) −7.41641 22.8254i −0.431800 1.32894i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 14.8328 + 45.6507i 0.857804 + 2.64005i
\(300\) 0 0
\(301\) −16.1803 + 11.7557i −0.932619 + 0.677588i
\(302\) 0 0
\(303\) 2.47214 7.60845i 0.142020 0.437094i
\(304\) 0 0
\(305\) −22.6525 16.4580i −1.29708 0.942382i
\(306\) 0 0
\(307\) 26.0000 1.48390 0.741949 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) 6.47214 + 4.70228i 0.367001 + 0.266642i 0.755966 0.654611i \(-0.227167\pi\)
−0.388965 + 0.921252i \(0.627167\pi\)
\(312\) 0 0
\(313\) 4.32624 13.3148i 0.244533 0.752596i −0.751179 0.660098i \(-0.770515\pi\)
0.995713 0.0924984i \(-0.0294853\pi\)
\(314\) 0 0
\(315\) 3.23607 2.35114i 0.182332 0.132472i
\(316\) 0 0
\(317\) 5.56231 + 17.1190i 0.312410 + 0.961500i 0.976807 + 0.214120i \(0.0686884\pi\)
−0.664397 + 0.747380i \(0.731312\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 4.94427 + 15.2169i 0.275962 + 0.849325i
\(322\) 0 0
\(323\) −6.47214 + 4.70228i −0.360119 + 0.261642i
\(324\) 0 0
\(325\) 1.85410 5.70634i 0.102847 0.316531i
\(326\) 0 0
\(327\) 14.5623 + 10.5801i 0.805297 + 0.585083i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) 4.85410 + 3.52671i 0.266003 + 0.193263i
\(334\) 0 0
\(335\) 2.47214 7.60845i 0.135067 0.415694i
\(336\) 0 0
\(337\) −14.5623 + 10.5801i −0.793259 + 0.576337i −0.908929 0.416951i \(-0.863099\pi\)
0.115670 + 0.993288i \(0.463099\pi\)
\(338\) 0 0
\(339\) 3.09017 + 9.51057i 0.167835 + 0.516543i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 6.18034 + 19.0211i 0.333707 + 1.02704i
\(344\) 0 0
\(345\) −12.9443 + 9.40456i −0.696896 + 0.506325i
\(346\) 0 0
\(347\) −9.88854 + 30.4338i −0.530845 + 1.63377i 0.221615 + 0.975134i \(0.428867\pi\)
−0.752460 + 0.658638i \(0.771133\pi\)
\(348\) 0 0
\(349\) −4.85410 3.52671i −0.259834 0.188781i 0.450240 0.892908i \(-0.351338\pi\)
−0.710074 + 0.704127i \(0.751338\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 2.47214 7.60845i 0.130839 0.402682i
\(358\) 0 0
\(359\) 19.4164 14.1068i 1.02476 0.744531i 0.0575058 0.998345i \(-0.481685\pi\)
0.967253 + 0.253814i \(0.0816852\pi\)
\(360\) 0 0
\(361\) −4.63525 14.2658i −0.243961 0.750834i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.70820 11.4127i −0.194096 0.597367i
\(366\) 0 0
\(367\) 3.23607 2.35114i 0.168921 0.122729i −0.500113 0.865960i \(-0.666708\pi\)
0.669034 + 0.743232i \(0.266708\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 22.6525 + 16.4580i 1.17606 + 0.854456i
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −4.94427 + 15.2169i −0.253970 + 0.781640i 0.740061 + 0.672540i \(0.234797\pi\)
−0.994031 + 0.109100i \(0.965203\pi\)
\(380\) 0 0
\(381\) 4.85410 3.52671i 0.248683 0.180679i
\(382\) 0 0
\(383\) 2.47214 + 7.60845i 0.126320 + 0.388774i 0.994139 0.108107i \(-0.0344788\pi\)
−0.867819 + 0.496880i \(0.834479\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.09017 9.51057i −0.157082 0.483449i
\(388\) 0 0
\(389\) −14.5623 + 10.5801i −0.738338 + 0.536434i −0.892190 0.451660i \(-0.850832\pi\)
0.153852 + 0.988094i \(0.450832\pi\)
\(390\) 0 0
\(391\) −9.88854 + 30.4338i −0.500085 + 1.53910i
\(392\) 0 0
\(393\) −3.23607 2.35114i −0.163238 0.118599i
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) −3.23607 2.35114i −0.162006 0.117704i
\(400\) 0 0
\(401\) −0.618034 + 1.90211i −0.0308631 + 0.0949870i −0.965301 0.261138i \(-0.915902\pi\)
0.934438 + 0.356125i \(0.115902\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0.618034 + 1.90211i 0.0307104 + 0.0945168i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −6.79837 20.9232i −0.336158 1.03459i −0.966149 0.257985i \(-0.916941\pi\)
0.629991 0.776603i \(-0.283059\pi\)
\(410\) 0 0
\(411\) 11.3262 8.22899i 0.558682 0.405906i
\(412\) 0 0
\(413\) 7.41641 22.8254i 0.364938 1.12316i
\(414\) 0 0
\(415\) 25.8885 + 18.8091i 1.27082 + 0.923304i
\(416\) 0 0
\(417\) 14.0000 0.685583
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 8.09017 + 5.87785i 0.394291 + 0.286469i 0.767211 0.641394i \(-0.221644\pi\)
−0.372921 + 0.927863i \(0.621644\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.23607 2.35114i 0.156972 0.114047i
\(426\) 0 0
\(427\) −8.65248 26.6296i −0.418723 1.28870i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.70820 + 11.4127i 0.178618 + 0.549729i 0.999780 0.0209654i \(-0.00667397\pi\)
−0.821162 + 0.570695i \(0.806674\pi\)
\(432\) 0 0
\(433\) −11.3262 + 8.22899i −0.544304 + 0.395460i −0.825681 0.564137i \(-0.809209\pi\)
0.281377 + 0.959597i \(0.409209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.9443 + 9.40456i 0.619208 + 0.449881i
\(438\) 0 0
\(439\) 2.00000 0.0954548 0.0477274 0.998860i \(-0.484802\pi\)
0.0477274 + 0.998860i \(0.484802\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −3.23607 2.35114i −0.153750 0.111706i 0.508250 0.861209i \(-0.330292\pi\)
−0.662001 + 0.749503i \(0.730292\pi\)
\(444\) 0 0
\(445\) −8.65248 + 26.6296i −0.410167 + 1.26236i
\(446\) 0 0
\(447\) 3.23607 2.35114i 0.153061 0.111205i
\(448\) 0 0
\(449\) −6.79837 20.9232i −0.320835 0.987429i −0.973286 0.229598i \(-0.926259\pi\)
0.652451 0.757831i \(-0.273741\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 3.09017 + 9.51057i 0.145189 + 0.446845i
\(454\) 0 0
\(455\) −19.4164 + 14.1068i −0.910255 + 0.661339i
\(456\) 0 0
\(457\) −6.79837 + 20.9232i −0.318015 + 0.978748i 0.656481 + 0.754342i \(0.272044\pi\)
−0.974496 + 0.224406i \(0.927956\pi\)
\(458\) 0 0
\(459\) 3.23607 + 2.35114i 0.151047 + 0.109742i
\(460\) 0 0
\(461\) 28.0000 1.30409 0.652045 0.758180i \(-0.273911\pi\)
0.652045 + 0.758180i \(0.273911\pi\)
\(462\) 0 0
\(463\) −12.0000 −0.557687 −0.278844 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.70820 + 11.4127i −0.171595 + 0.528116i −0.999462 0.0328096i \(-0.989555\pi\)
0.827866 + 0.560925i \(0.189555\pi\)
\(468\) 0 0
\(469\) 6.47214 4.70228i 0.298855 0.217131i
\(470\) 0 0
\(471\) 3.09017 + 9.51057i 0.142388 + 0.438224i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −0.618034 1.90211i −0.0283573 0.0872749i
\(476\) 0 0
\(477\) −11.3262 + 8.22899i −0.518593 + 0.376780i
\(478\) 0 0
\(479\) 3.70820 11.4127i 0.169432 0.521459i −0.829903 0.557907i \(-0.811605\pi\)
0.999336 + 0.0364486i \(0.0116045\pi\)
\(480\) 0 0
\(481\) −29.1246 21.1603i −1.32797 0.964825i
\(482\) 0 0
\(483\) −16.0000 −0.728025
\(484\) 0 0
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) 12.9443 + 9.40456i 0.586561 + 0.426161i 0.841083 0.540905i \(-0.181918\pi\)
−0.254523 + 0.967067i \(0.581918\pi\)
\(488\) 0 0
\(489\) −4.94427 + 15.2169i −0.223588 + 0.688132i
\(490\) 0 0
\(491\) −12.9443 + 9.40456i −0.584167 + 0.424422i −0.840224 0.542240i \(-0.817576\pi\)
0.256057 + 0.966662i \(0.417576\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 3.23607 2.35114i 0.144866 0.105252i −0.512992 0.858394i \(-0.671463\pi\)
0.657858 + 0.753142i \(0.271463\pi\)
\(500\) 0 0
\(501\) −3.70820 + 11.4127i −0.165670 + 0.509881i
\(502\) 0 0
\(503\) 16.1803 + 11.7557i 0.721446 + 0.524161i 0.886846 0.462066i \(-0.152892\pi\)
−0.165400 + 0.986227i \(0.552892\pi\)
\(504\) 0 0
\(505\) −16.0000 −0.711991
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) 0 0
\(509\) −11.3262 8.22899i −0.502027 0.364744i 0.307764 0.951463i \(-0.400419\pi\)
−0.809791 + 0.586719i \(0.800419\pi\)
\(510\) 0 0
\(511\) 3.70820 11.4127i 0.164041 0.504867i
\(512\) 0 0
\(513\) 1.61803 1.17557i 0.0714379 0.0519027i
\(514\) 0 0
\(515\) −7.41641 22.8254i −0.326806 1.00581i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 2.47214 + 7.60845i 0.108515 + 0.333974i
\(520\) 0 0
\(521\) 33.9787 24.6870i 1.48863 1.08156i 0.513990 0.857796i \(-0.328167\pi\)
0.974644 0.223760i \(-0.0718333\pi\)
\(522\) 0 0
\(523\) 6.79837 20.9232i 0.297272 0.914910i −0.685177 0.728377i \(-0.740275\pi\)
0.982449 0.186533i \(-0.0597250\pi\)
\(524\) 0 0
\(525\) 1.61803 + 1.17557i 0.0706168 + 0.0513061i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 9.70820 + 7.05342i 0.421300 + 0.306092i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 25.8885 18.8091i 1.11926 0.813190i
\(536\) 0 0
\(537\) 1.23607 + 3.80423i 0.0533403 + 0.164164i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.03444 + 24.7275i 0.345428 + 1.06312i 0.961354 + 0.275314i \(0.0887817\pi\)
−0.615927 + 0.787803i \(0.711218\pi\)
\(542\) 0 0
\(543\) 17.7984 12.9313i 0.763801 0.554934i
\(544\) 0 0
\(545\) 11.1246 34.2380i 0.476526 1.46660i
\(546\) 0 0
\(547\) −17.7984 12.9313i −0.761004 0.552901i 0.138214 0.990402i \(-0.455864\pi\)
−0.899218 + 0.437501i \(0.855864\pi\)
\(548\) 0 0
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −3.23607 2.35114i −0.137612 0.0999807i
\(554\) 0 0
\(555\) 3.70820 11.4127i 0.157404 0.484441i
\(556\) 0 0
\(557\) 9.70820 7.05342i 0.411350 0.298863i −0.362798 0.931868i \(-0.618179\pi\)
0.774148 + 0.633005i \(0.218179\pi\)
\(558\) 0 0
\(559\) 18.5410 + 57.0634i 0.784202 + 2.41352i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.23607 3.80423i −0.0520941 0.160329i 0.921625 0.388082i \(-0.126862\pi\)
−0.973719 + 0.227753i \(0.926862\pi\)
\(564\) 0 0
\(565\) 16.1803 11.7557i 0.680712 0.494566i
\(566\) 0 0
\(567\) −0.618034 + 1.90211i −0.0259550 + 0.0798812i
\(568\) 0 0
\(569\) −16.1803 11.7557i −0.678315 0.492825i 0.194483 0.980906i \(-0.437697\pi\)
−0.872798 + 0.488081i \(0.837697\pi\)
\(570\) 0 0
\(571\) −34.0000 −1.42286 −0.711428 0.702759i \(-0.751951\pi\)
−0.711428 + 0.702759i \(0.751951\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) −6.47214 4.70228i −0.269907 0.196099i
\(576\) 0 0
\(577\) 0.618034 1.90211i 0.0257291 0.0791860i −0.937367 0.348342i \(-0.886745\pi\)
0.963097 + 0.269156i \(0.0867448\pi\)
\(578\) 0 0
\(579\) −8.09017 + 5.87785i −0.336216 + 0.244275i
\(580\) 0 0
\(581\) 9.88854 + 30.4338i 0.410246 + 1.26261i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −3.70820 11.4127i −0.153315 0.471856i
\(586\) 0 0
\(587\) 9.70820 7.05342i 0.400700 0.291126i −0.369126 0.929379i \(-0.620343\pi\)
0.769826 + 0.638254i \(0.220343\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −12.9443 9.40456i −0.532456 0.386852i
\(592\) 0 0
\(593\) −24.0000 −0.985562 −0.492781 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) 0 0
\(595\) −16.0000 −0.655936
\(596\) 0 0
\(597\) 6.47214 + 4.70228i 0.264887 + 0.192452i
\(598\) 0 0
\(599\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(600\) 0 0
\(601\) −11.3262 + 8.22899i −0.462007 + 0.335668i −0.794318 0.607502i \(-0.792172\pi\)
0.332311 + 0.943170i \(0.392172\pi\)
\(602\) 0 0
\(603\) 1.23607 + 3.80423i 0.0503366 + 0.154920i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −10.5066 32.3359i −0.426449 1.31247i −0.901600 0.432570i \(-0.857607\pi\)
0.475151 0.879904i \(-0.342393\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 33.9787 + 24.6870i 1.37239 + 0.997098i 0.997546 + 0.0700086i \(0.0223027\pi\)
0.374841 + 0.927089i \(0.377697\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 0 0
\(619\) −25.8885 18.8091i −1.04055 0.756003i −0.0701559 0.997536i \(-0.522350\pi\)
−0.970393 + 0.241533i \(0.922350\pi\)
\(620\) 0 0
\(621\) 2.47214 7.60845i 0.0992034 0.305317i
\(622\) 0 0
\(623\) −22.6525 + 16.4580i −0.907552 + 0.659375i
\(624\) 0 0
\(625\) −5.87132 18.0701i −0.234853 0.722803i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −7.41641 22.8254i −0.295712 0.910107i
\(630\) 0 0
\(631\) 6.47214 4.70228i 0.257652 0.187195i −0.451459 0.892292i \(-0.649096\pi\)
0.709111 + 0.705097i \(0.249096\pi\)
\(632\) 0 0
\(633\) −3.09017 + 9.51057i −0.122823 + 0.378011i
\(634\) 0 0
\(635\) −9.70820 7.05342i −0.385258 0.279907i
\(636\) 0 0
\(637\) 18.0000 0.713186
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.85410 + 3.52671i 0.191726 + 0.139297i 0.679507 0.733669i \(-0.262194\pi\)
−0.487781 + 0.872966i \(0.662194\pi\)
\(642\) 0 0
\(643\) 9.88854 30.4338i 0.389966 1.20019i −0.542847 0.839832i \(-0.682654\pi\)
0.932813 0.360361i \(-0.117346\pi\)
\(644\) 0 0
\(645\) −16.1803 + 11.7557i −0.637100 + 0.462880i
\(646\) 0 0
\(647\) 2.47214 + 7.60845i 0.0971897 + 0.299119i 0.987818 0.155613i \(-0.0497351\pi\)
−0.890628 + 0.454732i \(0.849735\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 33.9787 24.6870i 1.32969 0.966076i 0.329933 0.944004i \(-0.392974\pi\)
0.999756 0.0220720i \(-0.00702631\pi\)
\(654\) 0 0
\(655\) −2.47214 + 7.60845i −0.0965943 + 0.297287i
\(656\) 0 0
\(657\) 4.85410 + 3.52671i 0.189377 + 0.137590i
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 0 0
\(663\) −19.4164 14.1068i −0.754071 0.547865i
\(664\) 0 0
\(665\) −2.47214 + 7.60845i −0.0958653 + 0.295043i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −3.70820 11.4127i −0.143367 0.441240i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −8.03444 24.7275i −0.309705 0.953174i −0.977879 0.209169i \(-0.932924\pi\)
0.668174 0.744005i \(-0.267076\pi\)
\(674\) 0 0
\(675\) −0.809017 + 0.587785i −0.0311391 + 0.0226239i
\(676\) 0 0
\(677\) 9.88854 30.4338i 0.380048 1.16967i −0.559962 0.828518i \(-0.689184\pi\)
0.940010 0.341148i \(-0.110816\pi\)
\(678\) 0 0
\(679\) −3.23607 2.35114i −0.124189 0.0902285i
\(680\) 0 0
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −22.6525 16.4580i −0.865507 0.628827i
\(686\) 0 0
\(687\) −0.618034 + 1.90211i −0.0235795 + 0.0725701i
\(688\) 0 0
\(689\) 67.9574 49.3740i 2.58897 1.88100i
\(690\) 0 0
\(691\) 7.41641 + 22.8254i 0.282133 + 0.868317i 0.987243 + 0.159219i \(0.0508977\pi\)
−0.705110 + 0.709098i \(0.749102\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.65248 26.6296i −0.328207 1.01012i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 4.94427 15.2169i 0.187010 0.575556i
\(700\) 0 0
\(701\) −9.70820 7.05342i −0.366674 0.266404i 0.389157 0.921172i \(-0.372767\pi\)
−0.755830 + 0.654768i \(0.772767\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.9443 9.40456i −0.486819 0.353695i
\(708\) 0 0
\(709\) 5.56231 17.1190i 0.208897 0.642918i −0.790634 0.612289i \(-0.790249\pi\)
0.999531 0.0306292i \(-0.00975110\pi\)
\(710\) 0 0
\(711\) 1.61803 1.17557i 0.0606810 0.0440873i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1.23607 3.80423i −0.0461618 0.142071i
\(718\) 0 0
\(719\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(720\) 0 0
\(721\) 7.41641 22.8254i 0.276201 0.850061i
\(722\) 0 0
\(723\) 1.61803 + 1.17557i 0.0601753 + 0.0437199i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −48.0000 −1.78022 −0.890111 0.455744i \(-0.849373\pi\)
−0.890111 + 0.455744i \(0.849373\pi\)
\(728\) 0 0
\(729\) −0.809017 0.587785i −0.0299636 0.0217698i
\(730\) 0 0
\(731\) −12.3607 + 38.0423i −0.457176 + 1.40704i
\(732\) 0 0
\(733\) −27.5066 + 19.9847i −1.01598 + 0.738152i −0.965455 0.260571i \(-0.916089\pi\)
−0.0505240 + 0.998723i \(0.516089\pi\)
\(734\) 0 0
\(735\) 1.85410 + 5.70634i 0.0683896 + 0.210481i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −8.03444 24.7275i −0.295552 0.909615i −0.983036 0.183415i \(-0.941285\pi\)
0.687484 0.726200i \(-0.258715\pi\)
\(740\) 0 0
\(741\) −9.70820 + 7.05342i −0.356640 + 0.259114i
\(742\) 0 0
\(743\) 2.47214 7.60845i 0.0906939 0.279127i −0.895414 0.445235i \(-0.853120\pi\)
0.986108 + 0.166108i \(0.0531201\pi\)
\(744\) 0 0
\(745\) −6.47214 4.70228i −0.237121 0.172278i
\(746\) 0 0
\(747\) −16.0000 −0.585409
\(748\) 0 0
\(749\) 32.0000 1.16925
\(750\) 0 0
\(751\) 32.3607 + 23.5114i 1.18086 + 0.857944i 0.992268 0.124112i \(-0.0396083\pi\)
0.188590 + 0.982056i \(0.439608\pi\)
\(752\) 0 0
\(753\) 1.23607 3.80423i 0.0450448 0.138634i
\(754\) 0 0
\(755\) 16.1803 11.7557i 0.588863 0.427834i
\(756\) 0 0
\(757\) 11.7426 + 36.1401i 0.426794 + 1.31354i 0.901266 + 0.433266i \(0.142639\pi\)
−0.474473 + 0.880270i \(0.657361\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −11.1246 34.2380i −0.403267 1.24113i −0.922334 0.386394i \(-0.873720\pi\)
0.519067 0.854734i \(-0.326280\pi\)
\(762\) 0 0
\(763\) 29.1246 21.1603i 1.05438 0.766053i
\(764\) 0 0
\(765\) 2.47214 7.60845i 0.0893803 0.275084i
\(766\) 0 0
\(767\) −58.2492 42.3205i −2.10326 1.52811i
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 0 0
\(773\) 14.5623 + 10.5801i 0.523770 + 0.380541i 0.818022 0.575187i \(-0.195071\pi\)
−0.294252 + 0.955728i \(0.595071\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 9.70820 7.05342i 0.348280 0.253040i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 16.1803 11.7557i 0.577501 0.419579i
\(786\) 0 0
\(787\) 11.7426 36.1401i 0.418580 1.28826i −0.490429 0.871481i \(-0.663160\pi\)
0.909009 0.416776i \(-0.136840\pi\)
\(788\) 0 0
\(789\) 9.70820 + 7.05342i 0.345621 + 0.251109i
\(790\) 0 0
\(791\) 20.0000 0.711118
\(792\) 0 0
\(793\) −84.0000 −2.98293
\(794\) 0 0
\(795\) 22.6525 + 16.4580i 0.803401 + 0.583705i
\(796\) 0 0
\(797\) −1.85410 + 5.70634i −0.0656757 + 0.202129i −0.978509 0.206203i \(-0.933889\pi\)
0.912834 + 0.408332i \(0.133889\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −4.32624 13.3148i −0.152860 0.470455i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 9.88854 + 30.4338i 0.348525 + 1.07265i
\(806\) 0 0
\(807\) 11.3262 8.22899i 0.398702 0.289674i
\(808\) 0 0
\(809\) 13.5967 41.8465i 0.478036 1.47124i −0.363783 0.931484i \(-0.618515\pi\)
0.841819 0.539760i \(-0.181485\pi\)
\(810\) 0 0
\(811\) 30.7426 + 22.3358i 1.07952 + 0.784317i 0.977599 0.210475i \(-0.0675011\pi\)
0.101921 + 0.994792i \(0.467501\pi\)
\(812\) 0 0
\(813\) −10.0000 −0.350715
\(814\) 0 0
\(815\) 32.0000 1.12091
\(816\) 0 0
\(817\) 16.1803 + 11.7557i 0.566078 + 0.411280i
\(818\) 0 0
\(819\) 3.70820 11.4127i 0.129575 0.398791i
\(820\) 0 0
\(821\) 32.3607 23.5114i 1.12940 0.820554i 0.143789 0.989608i \(-0.454071\pi\)
0.985607 + 0.169055i \(0.0540714\pi\)
\(822\) 0 0
\(823\) −8.65248 26.6296i −0.301606 0.928249i −0.980922 0.194403i \(-0.937723\pi\)
0.679315 0.733846i \(-0.262277\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 4.94427 + 15.2169i 0.171929 + 0.529144i 0.999480 0.0322473i \(-0.0102664\pi\)
−0.827551 + 0.561391i \(0.810266\pi\)
\(828\) 0 0
\(829\) 11.3262 8.22899i 0.393377 0.285805i −0.373461 0.927646i \(-0.621829\pi\)
0.766838 + 0.641841i \(0.221829\pi\)
\(830\) 0 0
\(831\) −3.09017 + 9.51057i −0.107197 + 0.329918i
\(832\) 0 0
\(833\) 9.70820 + 7.05342i 0.336369 + 0.244387i
\(834\) 0 0
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 19.4164 + 14.1068i 0.670329 + 0.487022i 0.870135 0.492813i \(-0.164031\pi\)
−0.199806 + 0.979835i \(0.564031\pi\)
\(840\) 0 0
\(841\) −8.96149 + 27.5806i −0.309017 + 0.951057i
\(842\) 0 0
\(843\) −25.8885 + 18.8091i −0.891649 + 0.647821i
\(844\) 0 0
\(845\) 14.2148 + 43.7486i 0.489003 + 1.50500i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −4.32624 13.3148i −0.148476 0.456962i
\(850\) 0 0
\(851\) −38.8328 + 28.2137i −1.33117 + 0.967153i
\(852\) 0 0
\(853\) −5.56231 + 17.1190i −0.190450 + 0.586144i −1.00000 0.000907135i \(-0.999711\pi\)
0.809550 + 0.587051i \(0.199711\pi\)
\(854\) 0 0
\(855\) −3.23607 2.35114i −0.110671 0.0804073i
\(856\) 0 0
\(857\) −44.0000 −1.50301 −0.751506 0.659727i \(-0.770672\pi\)
−0.751506 + 0.659727i \(0.770672\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.47214 7.60845i 0.0841525 0.258995i −0.900123 0.435636i \(-0.856523\pi\)
0.984275 + 0.176642i \(0.0565234\pi\)
\(864\) 0 0
\(865\) 12.9443 9.40456i 0.440118 0.319765i
\(866\) 0 0
\(867\) 0.309017 + 0.951057i 0.0104948 + 0.0322996i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −7.41641 22.8254i −0.251295 0.773408i
\(872\) 0 0
\(873\) 1.61803 1.17557i 0.0547622 0.0397870i
\(874\) 0 0
\(875\) 7.41641 22.8254i 0.250720 0.771638i
\(876\) 0 0
\(877\) 14.5623 + 10.5801i 0.491734 + 0.357266i 0.805851 0.592119i \(-0.201708\pi\)
−0.314117 + 0.949384i \(0.601708\pi\)
\(878\) 0 0
\(879\) −20.0000 −0.674583
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 35.5967 + 25.8626i 1.19793 + 0.870344i 0.994079 0.108659i \(-0.0346555\pi\)
0.203847 + 0.979003i \(0.434656\pi\)
\(884\) 0 0
\(885\) 7.41641 22.8254i 0.249300 0.767266i
\(886\) 0 0
\(887\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(888\) 0 0
\(889\) −3.70820 11.4127i −0.124369 0.382769i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 6.47214 4.70228i 0.216340 0.157180i
\(896\) 0 0
\(897\) −14.8328 + 45.6507i −0.495253 + 1.52423i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 56.0000 1.86563
\(902\) 0 0
\(903\) −20.0000 −0.665558
\(904\) 0 0
\(905\) −35.5967 25.8626i −1.18328 0.859700i
\(906\) 0 0
\(907\) 4.94427 15.2169i 0.164172 0.505269i −0.834802 0.550550i \(-0.814418\pi\)
0.998974 + 0.0452806i \(0.0144182\pi\)
\(908\) 0 0
\(909\) 6.47214 4.70228i 0.214667 0.155965i
\(910\) 0 0
\(911\) −14.8328 45.6507i −0.491433 1.51248i −0.822443 0.568848i \(-0.807389\pi\)
0.331009 0.943627i \(-0.392611\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −8.65248 26.6296i −0.286042 0.880347i
\(916\) 0 0
\(917\) −6.47214 + 4.70228i −0.213729 + 0.155283i
\(918\) 0 0
\(919\) 10.5066 32.3359i 0.346580 1.06666i −0.614152 0.789187i \(-0.710502\pi\)
0.960732 0.277476i \(-0.0894980\pi\)
\(920\) 0 0
\(921\) 21.0344 + 15.2824i 0.693108 + 0.503573i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 0 0
\(927\) 9.70820 + 7.05342i 0.318859 + 0.231665i
\(928\) 0 0
\(929\) −12.9787 + 39.9444i −0.425818 + 1.31053i 0.476391 + 0.879233i \(0.341945\pi\)
−0.902209 + 0.431299i \(0.858055\pi\)
\(930\) 0 0
\(931\) 4.85410 3.52671i 0.159087 0.115583i
\(932\) 0 0
\(933\) 2.47214 + 7.60845i 0.0809341 + 0.249090i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 11.7426 + 36.1401i 0.383616 + 1.18065i 0.937480 + 0.348040i \(0.113153\pi\)
−0.553864 + 0.832607i \(0.686847\pi\)
\(938\) 0 0
\(939\) 11.3262 8.22899i 0.369618 0.268543i
\(940\) 0 0
\(941\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) −29.1246 21.1603i −0.945425 0.686891i
\(950\) 0 0
\(951\) −5.56231 + 17.1190i −0.180370 + 0.555122i
\(952\) 0 0
\(953\) 9.70820 7.05342i 0.314480 0.228483i −0.419337 0.907831i \(-0.637737\pi\)
0.733816 + 0.679348i \(0.237737\pi\)
\(954\) 0 0
\(955\) 14.8328 + 45.6507i 0.479979 + 1.47722i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.65248 26.6296i −0.279403 0.859914i
\(960\) 0 0
\(961\) 25.0795 18.2213i 0.809017 0.587785i
\(962\) 0 0
\(963\) −4.94427 + 15.2169i −0.159327 + 0.490358i
\(964\) 0 0
\(965\) 16.1803 + 11.7557i 0.520864 + 0.378430i
\(966\) 0 0
\(967\) −34.0000 −1.09337 −0.546683 0.837340i \(-0.684110\pi\)
−0.546683 + 0.837340i \(0.684110\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) −48.5410 35.2671i −1.55776 1.13178i −0.937820 0.347123i \(-0.887159\pi\)
−0.619936 0.784652i \(-0.712841\pi\)
\(972\) 0 0
\(973\) 8.65248 26.6296i 0.277386 0.853705i
\(974\) 0 0
\(975\) 4.85410 3.52671i 0.155456 0.112945i
\(976\) 0 0
\(977\) −3.09017 9.51057i −0.0988633 0.304270i 0.889378 0.457173i \(-0.151138\pi\)
−0.988241 + 0.152903i \(0.951138\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 5.56231 + 17.1190i 0.177591 + 0.546568i
\(982\) 0 0
\(983\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(984\) 0 0
\(985\) −9.88854 + 30.4338i −0.315075 + 0.969702i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 80.0000 2.54385
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 6.47214 + 4.70228i 0.205387 + 0.149222i
\(994\) 0 0
\(995\) 4.94427 15.2169i 0.156744 0.482408i
\(996\) 0 0
\(997\) 8.09017 5.87785i 0.256218 0.186153i −0.452260 0.891886i \(-0.649382\pi\)
0.708478 + 0.705733i \(0.249382\pi\)
\(998\) 0 0
\(999\) 1.85410 + 5.70634i 0.0586612 + 0.180541i
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 1452.2.i.l.1237.1 4
11.2 odd 10 1452.2.i.k.493.1 4
11.3 even 5 1452.2.a.c.1.1 1
11.4 even 5 inner 1452.2.i.l.1213.1 4
11.5 even 5 inner 1452.2.i.l.565.1 4
11.6 odd 10 1452.2.i.k.565.1 4
11.7 odd 10 1452.2.i.k.1213.1 4
11.8 odd 10 132.2.a.a.1.1 1
11.9 even 5 inner 1452.2.i.l.493.1 4
11.10 odd 2 1452.2.i.k.1237.1 4
33.8 even 10 396.2.a.b.1.1 1
33.14 odd 10 4356.2.a.c.1.1 1
44.3 odd 10 5808.2.a.bf.1.1 1
44.19 even 10 528.2.a.h.1.1 1
55.8 even 20 3300.2.c.c.1849.1 2
55.19 odd 10 3300.2.a.k.1.1 1
55.52 even 20 3300.2.c.c.1849.2 2
77.41 even 10 6468.2.a.l.1.1 1
88.19 even 10 2112.2.a.b.1.1 1
88.85 odd 10 2112.2.a.t.1.1 1
99.41 even 30 3564.2.i.g.1189.1 2
99.52 odd 30 3564.2.i.b.2377.1 2
99.74 even 30 3564.2.i.g.2377.1 2
99.85 odd 30 3564.2.i.b.1189.1 2
132.107 odd 10 1584.2.a.c.1.1 1
165.8 odd 20 9900.2.c.k.5149.1 2
165.74 even 10 9900.2.a.g.1.1 1
165.107 odd 20 9900.2.c.k.5149.2 2
264.107 odd 10 6336.2.a.bz.1.1 1
264.173 even 10 6336.2.a.cf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
132.2.a.a.1.1 1 11.8 odd 10
396.2.a.b.1.1 1 33.8 even 10
528.2.a.h.1.1 1 44.19 even 10
1452.2.a.c.1.1 1 11.3 even 5
1452.2.i.k.493.1 4 11.2 odd 10
1452.2.i.k.565.1 4 11.6 odd 10
1452.2.i.k.1213.1 4 11.7 odd 10
1452.2.i.k.1237.1 4 11.10 odd 2
1452.2.i.l.493.1 4 11.9 even 5 inner
1452.2.i.l.565.1 4 11.5 even 5 inner
1452.2.i.l.1213.1 4 11.4 even 5 inner
1452.2.i.l.1237.1 4 1.1 even 1 trivial
1584.2.a.c.1.1 1 132.107 odd 10
2112.2.a.b.1.1 1 88.19 even 10
2112.2.a.t.1.1 1 88.85 odd 10
3300.2.a.k.1.1 1 55.19 odd 10
3300.2.c.c.1849.1 2 55.8 even 20
3300.2.c.c.1849.2 2 55.52 even 20
3564.2.i.b.1189.1 2 99.85 odd 30
3564.2.i.b.2377.1 2 99.52 odd 30
3564.2.i.g.1189.1 2 99.41 even 30
3564.2.i.g.2377.1 2 99.74 even 30
4356.2.a.c.1.1 1 33.14 odd 10
5808.2.a.bf.1.1 1 44.3 odd 10
6336.2.a.bz.1.1 1 264.107 odd 10
6336.2.a.cf.1.1 1 264.173 even 10
6468.2.a.l.1.1 1 77.41 even 10
9900.2.a.g.1.1 1 165.74 even 10
9900.2.c.k.5149.1 2 165.8 odd 20
9900.2.c.k.5149.2 2 165.107 odd 20