Properties

Label 2667.2.a.l.1.10
Level $2667$
Weight $2$
Character 2667.1
Self dual yes
Analytic conductor $21.296$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2667,2,Mod(1,2667)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2667, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2667.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2667 = 3 \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2667.a (trivial)

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: yes
Analytic conductor: \(21.2961022191\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} - 372 x^{4} + 146 x^{3} + 116 x^{2} - 12 x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(1.45503\) of defining polynomial
Character \(\chi\) \(=\) 2667.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.45503 q^{2} -1.00000 q^{3} +0.117115 q^{4} -0.420019 q^{5} -1.45503 q^{6} +1.00000 q^{7} -2.73966 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.45503 q^{2} -1.00000 q^{3} +0.117115 q^{4} -0.420019 q^{5} -1.45503 q^{6} +1.00000 q^{7} -2.73966 q^{8} +1.00000 q^{9} -0.611141 q^{10} +4.09314 q^{11} -0.117115 q^{12} +5.95755 q^{13} +1.45503 q^{14} +0.420019 q^{15} -4.22051 q^{16} -7.99832 q^{17} +1.45503 q^{18} +1.20264 q^{19} -0.0491906 q^{20} -1.00000 q^{21} +5.95565 q^{22} -0.806334 q^{23} +2.73966 q^{24} -4.82358 q^{25} +8.66843 q^{26} -1.00000 q^{27} +0.117115 q^{28} +5.46630 q^{29} +0.611141 q^{30} +3.58774 q^{31} -0.661667 q^{32} -4.09314 q^{33} -11.6378 q^{34} -0.420019 q^{35} +0.117115 q^{36} -3.79745 q^{37} +1.74988 q^{38} -5.95755 q^{39} +1.15071 q^{40} +9.24830 q^{41} -1.45503 q^{42} -4.78854 q^{43} +0.479369 q^{44} -0.420019 q^{45} -1.17324 q^{46} +10.3421 q^{47} +4.22051 q^{48} +1.00000 q^{49} -7.01846 q^{50} +7.99832 q^{51} +0.697719 q^{52} +3.29407 q^{53} -1.45503 q^{54} -1.71920 q^{55} -2.73966 q^{56} -1.20264 q^{57} +7.95363 q^{58} +5.19779 q^{59} +0.0491906 q^{60} -4.46437 q^{61} +5.22027 q^{62} +1.00000 q^{63} +7.47828 q^{64} -2.50229 q^{65} -5.95565 q^{66} +10.1874 q^{67} -0.936724 q^{68} +0.806334 q^{69} -0.611141 q^{70} +8.24577 q^{71} -2.73966 q^{72} +8.96444 q^{73} -5.52541 q^{74} +4.82358 q^{75} +0.140847 q^{76} +4.09314 q^{77} -8.66843 q^{78} -11.1975 q^{79} +1.77270 q^{80} +1.00000 q^{81} +13.4566 q^{82} +14.0246 q^{83} -0.117115 q^{84} +3.35945 q^{85} -6.96747 q^{86} -5.46630 q^{87} -11.2138 q^{88} +12.7389 q^{89} -0.611141 q^{90} +5.95755 q^{91} -0.0944338 q^{92} -3.58774 q^{93} +15.0480 q^{94} -0.505131 q^{95} +0.661667 q^{96} -8.63018 q^{97} +1.45503 q^{98} +4.09314 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 4 q^{2} - 13 q^{3} + 10 q^{4} + 12 q^{5} - 4 q^{6} + 13 q^{7} + 9 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 4 q^{2} - 13 q^{3} + 10 q^{4} + 12 q^{5} - 4 q^{6} + 13 q^{7} + 9 q^{8} + 13 q^{9} + 6 q^{10} + 3 q^{11} - 10 q^{12} + 21 q^{13} + 4 q^{14} - 12 q^{15} + 8 q^{16} + 17 q^{17} + 4 q^{18} + 5 q^{19} + 29 q^{20} - 13 q^{21} + q^{22} + 4 q^{23} - 9 q^{24} + q^{25} + 22 q^{26} - 13 q^{27} + 10 q^{28} + 21 q^{29} - 6 q^{30} - 7 q^{31} + 12 q^{32} - 3 q^{33} + 2 q^{34} + 12 q^{35} + 10 q^{36} + 7 q^{37} - 9 q^{38} - 21 q^{39} + 29 q^{40} + 21 q^{41} - 4 q^{42} - 9 q^{43} - 2 q^{44} + 12 q^{45} - 28 q^{46} + 23 q^{47} - 8 q^{48} + 13 q^{49} + 15 q^{50} - 17 q^{51} + 15 q^{52} + 31 q^{53} - 4 q^{54} - 8 q^{55} + 9 q^{56} - 5 q^{57} - 25 q^{58} + 28 q^{59} - 29 q^{60} + 29 q^{61} - 3 q^{62} + 13 q^{63} + 9 q^{64} + 30 q^{65} - q^{66} - 18 q^{67} + 34 q^{68} - 4 q^{69} + 6 q^{70} + 10 q^{71} + 9 q^{72} + 24 q^{73} - 19 q^{74} - q^{75} + 3 q^{77} - 22 q^{78} - 28 q^{79} + 26 q^{80} + 13 q^{81} + 18 q^{82} + 26 q^{83} - 10 q^{84} + 20 q^{85} - 2 q^{86} - 21 q^{87} - 17 q^{88} + 44 q^{89} + 6 q^{90} + 21 q^{91} + 6 q^{92} + 7 q^{93} - 9 q^{94} - 2 q^{95} - 12 q^{96} + 17 q^{97} + 4 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.45503 1.02886 0.514431 0.857532i \(-0.328003\pi\)
0.514431 + 0.857532i \(0.328003\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.117115 0.0585575
\(5\) −0.420019 −0.187838 −0.0939191 0.995580i \(-0.529940\pi\)
−0.0939191 + 0.995580i \(0.529940\pi\)
\(6\) −1.45503 −0.594014
\(7\) 1.00000 0.377964
\(8\) −2.73966 −0.968615
\(9\) 1.00000 0.333333
\(10\) −0.611141 −0.193260
\(11\) 4.09314 1.23413 0.617065 0.786912i \(-0.288322\pi\)
0.617065 + 0.786912i \(0.288322\pi\)
\(12\) −0.117115 −0.0338082
\(13\) 5.95755 1.65233 0.826164 0.563430i \(-0.190518\pi\)
0.826164 + 0.563430i \(0.190518\pi\)
\(14\) 1.45503 0.388873
\(15\) 0.420019 0.108448
\(16\) −4.22051 −1.05513
\(17\) −7.99832 −1.93988 −0.969939 0.243348i \(-0.921754\pi\)
−0.969939 + 0.243348i \(0.921754\pi\)
\(18\) 1.45503 0.342954
\(19\) 1.20264 0.275904 0.137952 0.990439i \(-0.455948\pi\)
0.137952 + 0.990439i \(0.455948\pi\)
\(20\) −0.0491906 −0.0109993
\(21\) −1.00000 −0.218218
\(22\) 5.95565 1.26975
\(23\) −0.806334 −0.168132 −0.0840661 0.996460i \(-0.526791\pi\)
−0.0840661 + 0.996460i \(0.526791\pi\)
\(24\) 2.73966 0.559230
\(25\) −4.82358 −0.964717
\(26\) 8.66843 1.70002
\(27\) −1.00000 −0.192450
\(28\) 0.117115 0.0221327
\(29\) 5.46630 1.01507 0.507533 0.861632i \(-0.330558\pi\)
0.507533 + 0.861632i \(0.330558\pi\)
\(30\) 0.611141 0.111579
\(31\) 3.58774 0.644376 0.322188 0.946676i \(-0.395582\pi\)
0.322188 + 0.946676i \(0.395582\pi\)
\(32\) −0.661667 −0.116967
\(33\) −4.09314 −0.712525
\(34\) −11.6378 −1.99587
\(35\) −0.420019 −0.0709962
\(36\) 0.117115 0.0195192
\(37\) −3.79745 −0.624297 −0.312149 0.950033i \(-0.601049\pi\)
−0.312149 + 0.950033i \(0.601049\pi\)
\(38\) 1.74988 0.283867
\(39\) −5.95755 −0.953972
\(40\) 1.15071 0.181943
\(41\) 9.24830 1.44434 0.722171 0.691715i \(-0.243145\pi\)
0.722171 + 0.691715i \(0.243145\pi\)
\(42\) −1.45503 −0.224516
\(43\) −4.78854 −0.730245 −0.365123 0.930959i \(-0.618973\pi\)
−0.365123 + 0.930959i \(0.618973\pi\)
\(44\) 0.479369 0.0722676
\(45\) −0.420019 −0.0626128
\(46\) −1.17324 −0.172985
\(47\) 10.3421 1.50855 0.754273 0.656561i \(-0.227990\pi\)
0.754273 + 0.656561i \(0.227990\pi\)
\(48\) 4.22051 0.609179
\(49\) 1.00000 0.142857
\(50\) −7.01846 −0.992561
\(51\) 7.99832 1.11999
\(52\) 0.697719 0.0967563
\(53\) 3.29407 0.452475 0.226238 0.974072i \(-0.427357\pi\)
0.226238 + 0.974072i \(0.427357\pi\)
\(54\) −1.45503 −0.198005
\(55\) −1.71920 −0.231817
\(56\) −2.73966 −0.366102
\(57\) −1.20264 −0.159293
\(58\) 7.95363 1.04436
\(59\) 5.19779 0.676695 0.338347 0.941021i \(-0.390132\pi\)
0.338347 + 0.941021i \(0.390132\pi\)
\(60\) 0.0491906 0.00635048
\(61\) −4.46437 −0.571605 −0.285802 0.958289i \(-0.592260\pi\)
−0.285802 + 0.958289i \(0.592260\pi\)
\(62\) 5.22027 0.662974
\(63\) 1.00000 0.125988
\(64\) 7.47828 0.934785
\(65\) −2.50229 −0.310371
\(66\) −5.95565 −0.733090
\(67\) 10.1874 1.24459 0.622297 0.782781i \(-0.286200\pi\)
0.622297 + 0.782781i \(0.286200\pi\)
\(68\) −0.936724 −0.113594
\(69\) 0.806334 0.0970712
\(70\) −0.611141 −0.0730453
\(71\) 8.24577 0.978592 0.489296 0.872118i \(-0.337254\pi\)
0.489296 + 0.872118i \(0.337254\pi\)
\(72\) −2.73966 −0.322872
\(73\) 8.96444 1.04921 0.524604 0.851346i \(-0.324213\pi\)
0.524604 + 0.851346i \(0.324213\pi\)
\(74\) −5.52541 −0.642316
\(75\) 4.82358 0.556979
\(76\) 0.140847 0.0161563
\(77\) 4.09314 0.466457
\(78\) −8.66843 −0.981506
\(79\) −11.1975 −1.25982 −0.629911 0.776667i \(-0.716909\pi\)
−0.629911 + 0.776667i \(0.716909\pi\)
\(80\) 1.77270 0.198194
\(81\) 1.00000 0.111111
\(82\) 13.4566 1.48603
\(83\) 14.0246 1.53940 0.769699 0.638407i \(-0.220406\pi\)
0.769699 + 0.638407i \(0.220406\pi\)
\(84\) −0.117115 −0.0127783
\(85\) 3.35945 0.364383
\(86\) −6.96747 −0.751322
\(87\) −5.46630 −0.586049
\(88\) −11.2138 −1.19540
\(89\) 12.7389 1.35032 0.675162 0.737669i \(-0.264074\pi\)
0.675162 + 0.737669i \(0.264074\pi\)
\(90\) −0.611141 −0.0644199
\(91\) 5.95755 0.624521
\(92\) −0.0944338 −0.00984540
\(93\) −3.58774 −0.372031
\(94\) 15.0480 1.55209
\(95\) −0.505131 −0.0518254
\(96\) 0.661667 0.0675311
\(97\) −8.63018 −0.876262 −0.438131 0.898911i \(-0.644359\pi\)
−0.438131 + 0.898911i \(0.644359\pi\)
\(98\) 1.45503 0.146980
\(99\) 4.09314 0.411376
\(100\) −0.564914 −0.0564914
\(101\) −13.9999 −1.39305 −0.696523 0.717535i \(-0.745270\pi\)
−0.696523 + 0.717535i \(0.745270\pi\)
\(102\) 11.6378 1.15231
\(103\) 18.1046 1.78390 0.891950 0.452133i \(-0.149337\pi\)
0.891950 + 0.452133i \(0.149337\pi\)
\(104\) −16.3216 −1.60047
\(105\) 0.420019 0.0409897
\(106\) 4.79297 0.465534
\(107\) −8.11725 −0.784725 −0.392362 0.919811i \(-0.628342\pi\)
−0.392362 + 0.919811i \(0.628342\pi\)
\(108\) −0.117115 −0.0112694
\(109\) 5.74750 0.550511 0.275256 0.961371i \(-0.411238\pi\)
0.275256 + 0.961371i \(0.411238\pi\)
\(110\) −2.50149 −0.238507
\(111\) 3.79745 0.360438
\(112\) −4.22051 −0.398801
\(113\) 14.6228 1.37559 0.687797 0.725903i \(-0.258578\pi\)
0.687797 + 0.725903i \(0.258578\pi\)
\(114\) −1.74988 −0.163891
\(115\) 0.338676 0.0315817
\(116\) 0.640186 0.0594398
\(117\) 5.95755 0.550776
\(118\) 7.56294 0.696225
\(119\) −7.99832 −0.733205
\(120\) −1.15071 −0.105045
\(121\) 5.75383 0.523075
\(122\) −6.49580 −0.588102
\(123\) −9.24830 −0.833891
\(124\) 0.420178 0.0377331
\(125\) 4.12609 0.369049
\(126\) 1.45503 0.129624
\(127\) −1.00000 −0.0887357
\(128\) 12.2045 1.07873
\(129\) 4.78854 0.421607
\(130\) −3.64091 −0.319329
\(131\) −8.56457 −0.748290 −0.374145 0.927370i \(-0.622064\pi\)
−0.374145 + 0.927370i \(0.622064\pi\)
\(132\) −0.479369 −0.0417237
\(133\) 1.20264 0.104282
\(134\) 14.8230 1.28052
\(135\) 0.420019 0.0361495
\(136\) 21.9127 1.87899
\(137\) 11.6339 0.993952 0.496976 0.867764i \(-0.334444\pi\)
0.496976 + 0.867764i \(0.334444\pi\)
\(138\) 1.17324 0.0998728
\(139\) −5.36556 −0.455101 −0.227550 0.973766i \(-0.573072\pi\)
−0.227550 + 0.973766i \(0.573072\pi\)
\(140\) −0.0491906 −0.00415736
\(141\) −10.3421 −0.870959
\(142\) 11.9978 1.00684
\(143\) 24.3851 2.03919
\(144\) −4.22051 −0.351710
\(145\) −2.29595 −0.190668
\(146\) 13.0435 1.07949
\(147\) −1.00000 −0.0824786
\(148\) −0.444739 −0.0365573
\(149\) −2.26030 −0.185171 −0.0925856 0.995705i \(-0.529513\pi\)
−0.0925856 + 0.995705i \(0.529513\pi\)
\(150\) 7.01846 0.573055
\(151\) 5.84583 0.475727 0.237863 0.971299i \(-0.423553\pi\)
0.237863 + 0.971299i \(0.423553\pi\)
\(152\) −3.29482 −0.267245
\(153\) −7.99832 −0.646626
\(154\) 5.95565 0.479920
\(155\) −1.50692 −0.121039
\(156\) −0.697719 −0.0558622
\(157\) −2.77780 −0.221692 −0.110846 0.993838i \(-0.535356\pi\)
−0.110846 + 0.993838i \(0.535356\pi\)
\(158\) −16.2928 −1.29618
\(159\) −3.29407 −0.261237
\(160\) 0.277913 0.0219709
\(161\) −0.806334 −0.0635480
\(162\) 1.45503 0.114318
\(163\) −7.22816 −0.566153 −0.283076 0.959097i \(-0.591355\pi\)
−0.283076 + 0.959097i \(0.591355\pi\)
\(164\) 1.08311 0.0845770
\(165\) 1.71920 0.133839
\(166\) 20.4062 1.58383
\(167\) 19.3352 1.49620 0.748102 0.663584i \(-0.230965\pi\)
0.748102 + 0.663584i \(0.230965\pi\)
\(168\) 2.73966 0.211369
\(169\) 22.4925 1.73019
\(170\) 4.88810 0.374900
\(171\) 1.20264 0.0919681
\(172\) −0.560810 −0.0427614
\(173\) 10.9047 0.829067 0.414533 0.910034i \(-0.363945\pi\)
0.414533 + 0.910034i \(0.363945\pi\)
\(174\) −7.95363 −0.602963
\(175\) −4.82358 −0.364629
\(176\) −17.2752 −1.30216
\(177\) −5.19779 −0.390690
\(178\) 18.5355 1.38930
\(179\) −24.2428 −1.81199 −0.905995 0.423289i \(-0.860876\pi\)
−0.905995 + 0.423289i \(0.860876\pi\)
\(180\) −0.0491906 −0.00366645
\(181\) −2.20670 −0.164023 −0.0820114 0.996631i \(-0.526134\pi\)
−0.0820114 + 0.996631i \(0.526134\pi\)
\(182\) 8.66843 0.642546
\(183\) 4.46437 0.330016
\(184\) 2.20908 0.162855
\(185\) 1.59500 0.117267
\(186\) −5.22027 −0.382768
\(187\) −32.7383 −2.39406
\(188\) 1.21121 0.0883367
\(189\) −1.00000 −0.0727393
\(190\) −0.734982 −0.0533212
\(191\) −21.7204 −1.57164 −0.785818 0.618458i \(-0.787758\pi\)
−0.785818 + 0.618458i \(0.787758\pi\)
\(192\) −7.47828 −0.539699
\(193\) −6.27244 −0.451500 −0.225750 0.974185i \(-0.572483\pi\)
−0.225750 + 0.974185i \(0.572483\pi\)
\(194\) −12.5572 −0.901553
\(195\) 2.50229 0.179193
\(196\) 0.117115 0.00836536
\(197\) 20.0593 1.42917 0.714584 0.699550i \(-0.246616\pi\)
0.714584 + 0.699550i \(0.246616\pi\)
\(198\) 5.95565 0.423250
\(199\) −0.345721 −0.0245075 −0.0122538 0.999925i \(-0.503901\pi\)
−0.0122538 + 0.999925i \(0.503901\pi\)
\(200\) 13.2150 0.934439
\(201\) −10.1874 −0.718567
\(202\) −20.3703 −1.43325
\(203\) 5.46630 0.383659
\(204\) 0.936724 0.0655838
\(205\) −3.88446 −0.271303
\(206\) 26.3428 1.83539
\(207\) −0.806334 −0.0560441
\(208\) −25.1439 −1.74342
\(209\) 4.92257 0.340501
\(210\) 0.611141 0.0421727
\(211\) −17.4577 −1.20184 −0.600919 0.799310i \(-0.705199\pi\)
−0.600919 + 0.799310i \(0.705199\pi\)
\(212\) 0.385785 0.0264958
\(213\) −8.24577 −0.564991
\(214\) −11.8109 −0.807374
\(215\) 2.01128 0.137168
\(216\) 2.73966 0.186410
\(217\) 3.58774 0.243551
\(218\) 8.36280 0.566400
\(219\) −8.96444 −0.605761
\(220\) −0.201344 −0.0135746
\(221\) −47.6504 −3.20532
\(222\) 5.52541 0.370841
\(223\) −8.39861 −0.562412 −0.281206 0.959647i \(-0.590734\pi\)
−0.281206 + 0.959647i \(0.590734\pi\)
\(224\) −0.661667 −0.0442095
\(225\) −4.82358 −0.321572
\(226\) 21.2766 1.41530
\(227\) 7.77384 0.515968 0.257984 0.966149i \(-0.416942\pi\)
0.257984 + 0.966149i \(0.416942\pi\)
\(228\) −0.140847 −0.00932782
\(229\) −25.0564 −1.65577 −0.827885 0.560897i \(-0.810456\pi\)
−0.827885 + 0.560897i \(0.810456\pi\)
\(230\) 0.492783 0.0324932
\(231\) −4.09314 −0.269309
\(232\) −14.9758 −0.983208
\(233\) −30.1760 −1.97689 −0.988446 0.151571i \(-0.951567\pi\)
−0.988446 + 0.151571i \(0.951567\pi\)
\(234\) 8.66843 0.566673
\(235\) −4.34387 −0.283363
\(236\) 0.608739 0.0396256
\(237\) 11.1975 0.727359
\(238\) −11.6378 −0.754367
\(239\) −5.76988 −0.373222 −0.186611 0.982434i \(-0.559750\pi\)
−0.186611 + 0.982434i \(0.559750\pi\)
\(240\) −1.77270 −0.114427
\(241\) −11.9312 −0.768559 −0.384279 0.923217i \(-0.625550\pi\)
−0.384279 + 0.923217i \(0.625550\pi\)
\(242\) 8.37199 0.538172
\(243\) −1.00000 −0.0641500
\(244\) −0.522845 −0.0334717
\(245\) −0.420019 −0.0268340
\(246\) −13.4566 −0.857959
\(247\) 7.16478 0.455884
\(248\) −9.82916 −0.624152
\(249\) −14.0246 −0.888772
\(250\) 6.00359 0.379701
\(251\) 27.8947 1.76070 0.880350 0.474325i \(-0.157308\pi\)
0.880350 + 0.474325i \(0.157308\pi\)
\(252\) 0.117115 0.00737755
\(253\) −3.30044 −0.207497
\(254\) −1.45503 −0.0912968
\(255\) −3.35945 −0.210377
\(256\) 2.80131 0.175082
\(257\) −1.37008 −0.0854634 −0.0427317 0.999087i \(-0.513606\pi\)
−0.0427317 + 0.999087i \(0.513606\pi\)
\(258\) 6.96747 0.433776
\(259\) −3.79745 −0.235962
\(260\) −0.293055 −0.0181745
\(261\) 5.46630 0.338355
\(262\) −12.4617 −0.769888
\(263\) 6.11351 0.376975 0.188488 0.982076i \(-0.439642\pi\)
0.188488 + 0.982076i \(0.439642\pi\)
\(264\) 11.2138 0.690162
\(265\) −1.38357 −0.0849921
\(266\) 1.74988 0.107292
\(267\) −12.7389 −0.779610
\(268\) 1.19310 0.0728803
\(269\) −14.3108 −0.872547 −0.436274 0.899814i \(-0.643702\pi\)
−0.436274 + 0.899814i \(0.643702\pi\)
\(270\) 0.611141 0.0371929
\(271\) −19.0355 −1.15632 −0.578162 0.815922i \(-0.696230\pi\)
−0.578162 + 0.815922i \(0.696230\pi\)
\(272\) 33.7570 2.04682
\(273\) −5.95755 −0.360568
\(274\) 16.9277 1.02264
\(275\) −19.7436 −1.19059
\(276\) 0.0944338 0.00568425
\(277\) 17.7110 1.06415 0.532076 0.846697i \(-0.321412\pi\)
0.532076 + 0.846697i \(0.321412\pi\)
\(278\) −7.80705 −0.468236
\(279\) 3.58774 0.214792
\(280\) 1.15071 0.0687680
\(281\) −27.6078 −1.64694 −0.823472 0.567357i \(-0.807966\pi\)
−0.823472 + 0.567357i \(0.807966\pi\)
\(282\) −15.0480 −0.896097
\(283\) 11.8324 0.703360 0.351680 0.936120i \(-0.385610\pi\)
0.351680 + 0.936120i \(0.385610\pi\)
\(284\) 0.965703 0.0573039
\(285\) 0.505131 0.0299214
\(286\) 35.4811 2.09804
\(287\) 9.24830 0.545910
\(288\) −0.661667 −0.0389891
\(289\) 46.9732 2.76313
\(290\) −3.34068 −0.196171
\(291\) 8.63018 0.505910
\(292\) 1.04987 0.0614390
\(293\) 17.9876 1.05085 0.525423 0.850841i \(-0.323907\pi\)
0.525423 + 0.850841i \(0.323907\pi\)
\(294\) −1.45503 −0.0848591
\(295\) −2.18317 −0.127109
\(296\) 10.4037 0.604703
\(297\) −4.09314 −0.237508
\(298\) −3.28881 −0.190516
\(299\) −4.80378 −0.277810
\(300\) 0.564914 0.0326153
\(301\) −4.78854 −0.276007
\(302\) 8.50586 0.489457
\(303\) 13.9999 0.804275
\(304\) −5.07575 −0.291114
\(305\) 1.87512 0.107369
\(306\) −11.6378 −0.665289
\(307\) 13.1148 0.748500 0.374250 0.927328i \(-0.377900\pi\)
0.374250 + 0.927328i \(0.377900\pi\)
\(308\) 0.479369 0.0273146
\(309\) −18.1046 −1.02994
\(310\) −2.19261 −0.124532
\(311\) −9.74688 −0.552695 −0.276347 0.961058i \(-0.589124\pi\)
−0.276347 + 0.961058i \(0.589124\pi\)
\(312\) 16.3216 0.924031
\(313\) 20.0884 1.13547 0.567733 0.823213i \(-0.307821\pi\)
0.567733 + 0.823213i \(0.307821\pi\)
\(314\) −4.04178 −0.228091
\(315\) −0.420019 −0.0236654
\(316\) −1.31140 −0.0737721
\(317\) −23.1449 −1.29995 −0.649974 0.759956i \(-0.725220\pi\)
−0.649974 + 0.759956i \(0.725220\pi\)
\(318\) −4.79297 −0.268776
\(319\) 22.3743 1.25272
\(320\) −3.14102 −0.175588
\(321\) 8.11725 0.453061
\(322\) −1.17324 −0.0653821
\(323\) −9.61909 −0.535220
\(324\) 0.117115 0.00650639
\(325\) −28.7368 −1.59403
\(326\) −10.5172 −0.582493
\(327\) −5.74750 −0.317838
\(328\) −25.3371 −1.39901
\(329\) 10.3421 0.570177
\(330\) 2.50149 0.137702
\(331\) −12.8235 −0.704842 −0.352421 0.935841i \(-0.614642\pi\)
−0.352421 + 0.935841i \(0.614642\pi\)
\(332\) 1.64249 0.0901434
\(333\) −3.79745 −0.208099
\(334\) 28.1333 1.53939
\(335\) −4.27892 −0.233782
\(336\) 4.22051 0.230248
\(337\) 10.3028 0.561228 0.280614 0.959821i \(-0.409462\pi\)
0.280614 + 0.959821i \(0.409462\pi\)
\(338\) 32.7272 1.78013
\(339\) −14.6228 −0.794200
\(340\) 0.393442 0.0213374
\(341\) 14.6851 0.795244
\(342\) 1.74988 0.0946225
\(343\) 1.00000 0.0539949
\(344\) 13.1189 0.707326
\(345\) −0.338676 −0.0182337
\(346\) 15.8666 0.852996
\(347\) 19.3807 1.04041 0.520206 0.854041i \(-0.325855\pi\)
0.520206 + 0.854041i \(0.325855\pi\)
\(348\) −0.640186 −0.0343176
\(349\) 26.2455 1.40489 0.702446 0.711737i \(-0.252091\pi\)
0.702446 + 0.711737i \(0.252091\pi\)
\(350\) −7.01846 −0.375153
\(351\) −5.95755 −0.317991
\(352\) −2.70830 −0.144353
\(353\) 22.3694 1.19060 0.595300 0.803503i \(-0.297033\pi\)
0.595300 + 0.803503i \(0.297033\pi\)
\(354\) −7.56294 −0.401966
\(355\) −3.46338 −0.183817
\(356\) 1.49192 0.0790716
\(357\) 7.99832 0.423316
\(358\) −35.2740 −1.86429
\(359\) 26.1003 1.37752 0.688759 0.724990i \(-0.258156\pi\)
0.688759 + 0.724990i \(0.258156\pi\)
\(360\) 1.15071 0.0606476
\(361\) −17.5537 −0.923877
\(362\) −3.21082 −0.168757
\(363\) −5.75383 −0.301997
\(364\) 0.697719 0.0365704
\(365\) −3.76524 −0.197081
\(366\) 6.49580 0.339541
\(367\) −34.4353 −1.79751 −0.898754 0.438454i \(-0.855526\pi\)
−0.898754 + 0.438454i \(0.855526\pi\)
\(368\) 3.40314 0.177401
\(369\) 9.24830 0.481447
\(370\) 2.32078 0.120652
\(371\) 3.29407 0.171019
\(372\) −0.420178 −0.0217852
\(373\) 1.95485 0.101218 0.0506091 0.998719i \(-0.483884\pi\)
0.0506091 + 0.998719i \(0.483884\pi\)
\(374\) −47.6352 −2.46316
\(375\) −4.12609 −0.213071
\(376\) −28.3337 −1.46120
\(377\) 32.5658 1.67722
\(378\) −1.45503 −0.0748387
\(379\) −29.7272 −1.52698 −0.763492 0.645817i \(-0.776517\pi\)
−0.763492 + 0.645817i \(0.776517\pi\)
\(380\) −0.0591585 −0.00303477
\(381\) 1.00000 0.0512316
\(382\) −31.6039 −1.61700
\(383\) −35.6298 −1.82060 −0.910300 0.413949i \(-0.864149\pi\)
−0.910300 + 0.413949i \(0.864149\pi\)
\(384\) −12.2045 −0.622807
\(385\) −1.71920 −0.0876185
\(386\) −9.12660 −0.464532
\(387\) −4.78854 −0.243415
\(388\) −1.01072 −0.0513118
\(389\) −28.9904 −1.46987 −0.734935 0.678138i \(-0.762787\pi\)
−0.734935 + 0.678138i \(0.762787\pi\)
\(390\) 3.64091 0.184364
\(391\) 6.44932 0.326156
\(392\) −2.73966 −0.138374
\(393\) 8.56457 0.432026
\(394\) 29.1869 1.47042
\(395\) 4.70318 0.236643
\(396\) 0.479369 0.0240892
\(397\) −3.59161 −0.180258 −0.0901289 0.995930i \(-0.528728\pi\)
−0.0901289 + 0.995930i \(0.528728\pi\)
\(398\) −0.503035 −0.0252149
\(399\) −1.20264 −0.0602072
\(400\) 20.3580 1.01790
\(401\) 9.61960 0.480380 0.240190 0.970726i \(-0.422790\pi\)
0.240190 + 0.970726i \(0.422790\pi\)
\(402\) −14.8230 −0.739306
\(403\) 21.3741 1.06472
\(404\) −1.63960 −0.0815733
\(405\) −0.420019 −0.0208709
\(406\) 7.95363 0.394732
\(407\) −15.5435 −0.770464
\(408\) −21.9127 −1.08484
\(409\) 31.7875 1.57179 0.785894 0.618361i \(-0.212203\pi\)
0.785894 + 0.618361i \(0.212203\pi\)
\(410\) −5.65201 −0.279133
\(411\) −11.6339 −0.573858
\(412\) 2.12032 0.104461
\(413\) 5.19779 0.255766
\(414\) −1.17324 −0.0576616
\(415\) −5.89060 −0.289158
\(416\) −3.94192 −0.193268
\(417\) 5.36556 0.262753
\(418\) 7.16249 0.350329
\(419\) 21.3043 1.04078 0.520392 0.853928i \(-0.325786\pi\)
0.520392 + 0.853928i \(0.325786\pi\)
\(420\) 0.0491906 0.00240025
\(421\) −25.4806 −1.24185 −0.620924 0.783871i \(-0.713242\pi\)
−0.620924 + 0.783871i \(0.713242\pi\)
\(422\) −25.4015 −1.23653
\(423\) 10.3421 0.502849
\(424\) −9.02461 −0.438274
\(425\) 38.5806 1.87143
\(426\) −11.9978 −0.581297
\(427\) −4.46437 −0.216046
\(428\) −0.950653 −0.0459515
\(429\) −24.3851 −1.17732
\(430\) 2.92647 0.141127
\(431\) −27.6736 −1.33299 −0.666495 0.745510i \(-0.732206\pi\)
−0.666495 + 0.745510i \(0.732206\pi\)
\(432\) 4.22051 0.203060
\(433\) 28.1288 1.35178 0.675892 0.737000i \(-0.263758\pi\)
0.675892 + 0.737000i \(0.263758\pi\)
\(434\) 5.22027 0.250581
\(435\) 2.29595 0.110082
\(436\) 0.673119 0.0322366
\(437\) −0.969728 −0.0463884
\(438\) −13.0435 −0.623244
\(439\) 22.3769 1.06799 0.533995 0.845488i \(-0.320690\pi\)
0.533995 + 0.845488i \(0.320690\pi\)
\(440\) 4.71001 0.224541
\(441\) 1.00000 0.0476190
\(442\) −69.3329 −3.29783
\(443\) 27.7545 1.31866 0.659328 0.751856i \(-0.270841\pi\)
0.659328 + 0.751856i \(0.270841\pi\)
\(444\) 0.444739 0.0211064
\(445\) −5.35060 −0.253643
\(446\) −12.2202 −0.578645
\(447\) 2.26030 0.106909
\(448\) 7.47828 0.353316
\(449\) 0.741298 0.0349840 0.0174920 0.999847i \(-0.494432\pi\)
0.0174920 + 0.999847i \(0.494432\pi\)
\(450\) −7.01846 −0.330854
\(451\) 37.8546 1.78250
\(452\) 1.71255 0.0805514
\(453\) −5.84583 −0.274661
\(454\) 11.3112 0.530860
\(455\) −2.50229 −0.117309
\(456\) 3.29482 0.154294
\(457\) 18.5086 0.865798 0.432899 0.901442i \(-0.357491\pi\)
0.432899 + 0.901442i \(0.357491\pi\)
\(458\) −36.4578 −1.70356
\(459\) 7.99832 0.373330
\(460\) 0.0396640 0.00184934
\(461\) −40.2184 −1.87316 −0.936579 0.350456i \(-0.886027\pi\)
−0.936579 + 0.350456i \(0.886027\pi\)
\(462\) −5.95565 −0.277082
\(463\) −33.6466 −1.56369 −0.781844 0.623474i \(-0.785721\pi\)
−0.781844 + 0.623474i \(0.785721\pi\)
\(464\) −23.0706 −1.07103
\(465\) 1.50692 0.0698816
\(466\) −43.9070 −2.03395
\(467\) 14.6853 0.679554 0.339777 0.940506i \(-0.389648\pi\)
0.339777 + 0.940506i \(0.389648\pi\)
\(468\) 0.697719 0.0322521
\(469\) 10.1874 0.470412
\(470\) −6.32046 −0.291541
\(471\) 2.77780 0.127994
\(472\) −14.2402 −0.655456
\(473\) −19.6002 −0.901217
\(474\) 16.2928 0.748352
\(475\) −5.80103 −0.266169
\(476\) −0.936724 −0.0429347
\(477\) 3.29407 0.150825
\(478\) −8.39535 −0.383994
\(479\) 10.4227 0.476227 0.238113 0.971237i \(-0.423471\pi\)
0.238113 + 0.971237i \(0.423471\pi\)
\(480\) −0.277913 −0.0126849
\(481\) −22.6235 −1.03154
\(482\) −17.3603 −0.790741
\(483\) 0.806334 0.0366894
\(484\) 0.673859 0.0306300
\(485\) 3.62484 0.164596
\(486\) −1.45503 −0.0660015
\(487\) −33.8424 −1.53355 −0.766773 0.641918i \(-0.778139\pi\)
−0.766773 + 0.641918i \(0.778139\pi\)
\(488\) 12.2308 0.553665
\(489\) 7.22816 0.326869
\(490\) −0.611141 −0.0276085
\(491\) 23.3096 1.05195 0.525975 0.850500i \(-0.323701\pi\)
0.525975 + 0.850500i \(0.323701\pi\)
\(492\) −1.08311 −0.0488306
\(493\) −43.7212 −1.96910
\(494\) 10.4250 0.469042
\(495\) −1.71920 −0.0772722
\(496\) −15.1421 −0.679900
\(497\) 8.24577 0.369873
\(498\) −20.4062 −0.914424
\(499\) −0.990923 −0.0443598 −0.0221799 0.999754i \(-0.507061\pi\)
−0.0221799 + 0.999754i \(0.507061\pi\)
\(500\) 0.483228 0.0216106
\(501\) −19.3352 −0.863834
\(502\) 40.5877 1.81152
\(503\) 12.6071 0.562124 0.281062 0.959690i \(-0.409313\pi\)
0.281062 + 0.959690i \(0.409313\pi\)
\(504\) −2.73966 −0.122034
\(505\) 5.88024 0.261667
\(506\) −4.80224 −0.213486
\(507\) −22.4925 −0.998925
\(508\) −0.117115 −0.00519614
\(509\) 32.4113 1.43661 0.718303 0.695730i \(-0.244919\pi\)
0.718303 + 0.695730i \(0.244919\pi\)
\(510\) −4.88810 −0.216449
\(511\) 8.96444 0.396563
\(512\) −20.3329 −0.898597
\(513\) −1.20264 −0.0530978
\(514\) −1.99351 −0.0879300
\(515\) −7.60429 −0.335085
\(516\) 0.560810 0.0246883
\(517\) 42.3316 1.86174
\(518\) −5.52541 −0.242773
\(519\) −10.9047 −0.478662
\(520\) 6.85541 0.300629
\(521\) 7.88673 0.345524 0.172762 0.984964i \(-0.444731\pi\)
0.172762 + 0.984964i \(0.444731\pi\)
\(522\) 7.95363 0.348121
\(523\) −5.81467 −0.254258 −0.127129 0.991886i \(-0.540576\pi\)
−0.127129 + 0.991886i \(0.540576\pi\)
\(524\) −1.00304 −0.0438180
\(525\) 4.82358 0.210518
\(526\) 8.89534 0.387855
\(527\) −28.6959 −1.25001
\(528\) 17.2752 0.751805
\(529\) −22.3498 −0.971732
\(530\) −2.01314 −0.0874452
\(531\) 5.19779 0.225565
\(532\) 0.140847 0.00610649
\(533\) 55.0972 2.38653
\(534\) −18.5355 −0.802111
\(535\) 3.40940 0.147401
\(536\) −27.9101 −1.20553
\(537\) 24.2428 1.04615
\(538\) −20.8227 −0.897731
\(539\) 4.09314 0.176304
\(540\) 0.0491906 0.00211683
\(541\) 11.4384 0.491777 0.245889 0.969298i \(-0.420920\pi\)
0.245889 + 0.969298i \(0.420920\pi\)
\(542\) −27.6972 −1.18970
\(543\) 2.20670 0.0946986
\(544\) 5.29223 0.226902
\(545\) −2.41406 −0.103407
\(546\) −8.66843 −0.370974
\(547\) −11.5776 −0.495021 −0.247510 0.968885i \(-0.579612\pi\)
−0.247510 + 0.968885i \(0.579612\pi\)
\(548\) 1.36251 0.0582034
\(549\) −4.46437 −0.190535
\(550\) −28.7276 −1.22495
\(551\) 6.57398 0.280061
\(552\) −2.20908 −0.0940245
\(553\) −11.1975 −0.476168
\(554\) 25.7701 1.09486
\(555\) −1.59500 −0.0677041
\(556\) −0.628388 −0.0266496
\(557\) −4.77210 −0.202201 −0.101100 0.994876i \(-0.532236\pi\)
−0.101100 + 0.994876i \(0.532236\pi\)
\(558\) 5.22027 0.220991
\(559\) −28.5280 −1.20660
\(560\) 1.77270 0.0749101
\(561\) 32.7383 1.38221
\(562\) −40.1702 −1.69448
\(563\) −24.0710 −1.01447 −0.507236 0.861807i \(-0.669333\pi\)
−0.507236 + 0.861807i \(0.669333\pi\)
\(564\) −1.21121 −0.0510012
\(565\) −6.14184 −0.258389
\(566\) 17.2164 0.723661
\(567\) 1.00000 0.0419961
\(568\) −22.5906 −0.947879
\(569\) −29.5305 −1.23798 −0.618991 0.785398i \(-0.712458\pi\)
−0.618991 + 0.785398i \(0.712458\pi\)
\(570\) 0.734982 0.0307850
\(571\) −10.5476 −0.441402 −0.220701 0.975342i \(-0.570834\pi\)
−0.220701 + 0.975342i \(0.570834\pi\)
\(572\) 2.85587 0.119410
\(573\) 21.7204 0.907384
\(574\) 13.4566 0.561666
\(575\) 3.88942 0.162200
\(576\) 7.47828 0.311595
\(577\) −23.4100 −0.974573 −0.487286 0.873242i \(-0.662013\pi\)
−0.487286 + 0.873242i \(0.662013\pi\)
\(578\) 68.3474 2.84288
\(579\) 6.27244 0.260674
\(580\) −0.268890 −0.0111651
\(581\) 14.0246 0.581838
\(582\) 12.5572 0.520512
\(583\) 13.4831 0.558413
\(584\) −24.5595 −1.01628
\(585\) −2.50229 −0.103457
\(586\) 26.1725 1.08118
\(587\) −37.0926 −1.53098 −0.765488 0.643451i \(-0.777502\pi\)
−0.765488 + 0.643451i \(0.777502\pi\)
\(588\) −0.117115 −0.00482974
\(589\) 4.31475 0.177786
\(590\) −3.17658 −0.130778
\(591\) −20.0593 −0.825130
\(592\) 16.0272 0.658714
\(593\) −13.9145 −0.571400 −0.285700 0.958319i \(-0.592226\pi\)
−0.285700 + 0.958319i \(0.592226\pi\)
\(594\) −5.95565 −0.244363
\(595\) 3.35945 0.137724
\(596\) −0.264715 −0.0108432
\(597\) 0.345721 0.0141494
\(598\) −6.98964 −0.285828
\(599\) 21.8080 0.891051 0.445526 0.895269i \(-0.353017\pi\)
0.445526 + 0.895269i \(0.353017\pi\)
\(600\) −13.2150 −0.539498
\(601\) 17.0329 0.694785 0.347393 0.937720i \(-0.387067\pi\)
0.347393 + 0.937720i \(0.387067\pi\)
\(602\) −6.96747 −0.283973
\(603\) 10.1874 0.414865
\(604\) 0.684635 0.0278574
\(605\) −2.41672 −0.0982535
\(606\) 20.3703 0.827488
\(607\) −38.0454 −1.54422 −0.772108 0.635491i \(-0.780798\pi\)
−0.772108 + 0.635491i \(0.780798\pi\)
\(608\) −0.795746 −0.0322718
\(609\) −5.46630 −0.221506
\(610\) 2.72836 0.110468
\(611\) 61.6134 2.49261
\(612\) −0.936724 −0.0378648
\(613\) 16.0498 0.648245 0.324123 0.946015i \(-0.394931\pi\)
0.324123 + 0.946015i \(0.394931\pi\)
\(614\) 19.0824 0.770103
\(615\) 3.88446 0.156637
\(616\) −11.2138 −0.451817
\(617\) 28.5918 1.15106 0.575532 0.817779i \(-0.304795\pi\)
0.575532 + 0.817779i \(0.304795\pi\)
\(618\) −26.3428 −1.05966
\(619\) −0.179381 −0.00720994 −0.00360497 0.999994i \(-0.501148\pi\)
−0.00360497 + 0.999994i \(0.501148\pi\)
\(620\) −0.176483 −0.00708772
\(621\) 0.806334 0.0323571
\(622\) −14.1820 −0.568647
\(623\) 12.7389 0.510375
\(624\) 25.1439 1.00656
\(625\) 22.3849 0.895395
\(626\) 29.2293 1.16824
\(627\) −4.92257 −0.196589
\(628\) −0.325322 −0.0129818
\(629\) 30.3732 1.21106
\(630\) −0.611141 −0.0243484
\(631\) −29.5538 −1.17652 −0.588258 0.808673i \(-0.700186\pi\)
−0.588258 + 0.808673i \(0.700186\pi\)
\(632\) 30.6774 1.22028
\(633\) 17.4577 0.693882
\(634\) −33.6766 −1.33747
\(635\) 0.420019 0.0166680
\(636\) −0.385785 −0.0152974
\(637\) 5.95755 0.236047
\(638\) 32.5554 1.28888
\(639\) 8.24577 0.326197
\(640\) −5.12611 −0.202627
\(641\) −4.17299 −0.164823 −0.0824116 0.996598i \(-0.526262\pi\)
−0.0824116 + 0.996598i \(0.526262\pi\)
\(642\) 11.8109 0.466137
\(643\) −0.715991 −0.0282359 −0.0141180 0.999900i \(-0.504494\pi\)
−0.0141180 + 0.999900i \(0.504494\pi\)
\(644\) −0.0944338 −0.00372121
\(645\) −2.01128 −0.0791940
\(646\) −13.9961 −0.550668
\(647\) 18.4983 0.727245 0.363622 0.931546i \(-0.381540\pi\)
0.363622 + 0.931546i \(0.381540\pi\)
\(648\) −2.73966 −0.107624
\(649\) 21.2753 0.835129
\(650\) −41.8129 −1.64004
\(651\) −3.58774 −0.140614
\(652\) −0.846526 −0.0331525
\(653\) 24.6148 0.963251 0.481626 0.876377i \(-0.340046\pi\)
0.481626 + 0.876377i \(0.340046\pi\)
\(654\) −8.36280 −0.327011
\(655\) 3.59728 0.140558
\(656\) −39.0326 −1.52397
\(657\) 8.96444 0.349736
\(658\) 15.0480 0.586633
\(659\) −29.6799 −1.15616 −0.578082 0.815979i \(-0.696199\pi\)
−0.578082 + 0.815979i \(0.696199\pi\)
\(660\) 0.201344 0.00783731
\(661\) 0.127223 0.00494841 0.00247420 0.999997i \(-0.499212\pi\)
0.00247420 + 0.999997i \(0.499212\pi\)
\(662\) −18.6586 −0.725186
\(663\) 47.6504 1.85059
\(664\) −38.4225 −1.49108
\(665\) −0.505131 −0.0195881
\(666\) −5.52541 −0.214105
\(667\) −4.40766 −0.170665
\(668\) 2.26445 0.0876140
\(669\) 8.39861 0.324709
\(670\) −6.22596 −0.240530
\(671\) −18.2733 −0.705434
\(672\) 0.661667 0.0255244
\(673\) 18.7015 0.720891 0.360445 0.932780i \(-0.382625\pi\)
0.360445 + 0.932780i \(0.382625\pi\)
\(674\) 14.9908 0.577426
\(675\) 4.82358 0.185660
\(676\) 2.63420 0.101316
\(677\) 37.6012 1.44513 0.722567 0.691301i \(-0.242962\pi\)
0.722567 + 0.691301i \(0.242962\pi\)
\(678\) −21.2766 −0.817122
\(679\) −8.63018 −0.331196
\(680\) −9.20373 −0.352947
\(681\) −7.77384 −0.297894
\(682\) 21.3673 0.818196
\(683\) 1.76839 0.0676657 0.0338328 0.999428i \(-0.489229\pi\)
0.0338328 + 0.999428i \(0.489229\pi\)
\(684\) 0.140847 0.00538542
\(685\) −4.88647 −0.186702
\(686\) 1.45503 0.0555533
\(687\) 25.0564 0.955960
\(688\) 20.2101 0.770503
\(689\) 19.6246 0.747637
\(690\) −0.492783 −0.0187599
\(691\) −6.84776 −0.260501 −0.130250 0.991481i \(-0.541578\pi\)
−0.130250 + 0.991481i \(0.541578\pi\)
\(692\) 1.27710 0.0485481
\(693\) 4.09314 0.155486
\(694\) 28.1996 1.07044
\(695\) 2.25364 0.0854854
\(696\) 14.9758 0.567655
\(697\) −73.9708 −2.80185
\(698\) 38.1881 1.44544
\(699\) 30.1760 1.14136
\(700\) −0.564914 −0.0213518
\(701\) −27.6613 −1.04475 −0.522376 0.852715i \(-0.674954\pi\)
−0.522376 + 0.852715i \(0.674954\pi\)
\(702\) −8.66843 −0.327169
\(703\) −4.56696 −0.172246
\(704\) 30.6097 1.15365
\(705\) 4.34387 0.163600
\(706\) 32.5481 1.22496
\(707\) −13.9999 −0.526522
\(708\) −0.608739 −0.0228778
\(709\) 45.0037 1.69015 0.845075 0.534648i \(-0.179556\pi\)
0.845075 + 0.534648i \(0.179556\pi\)
\(710\) −5.03932 −0.189122
\(711\) −11.1975 −0.419941
\(712\) −34.9003 −1.30794
\(713\) −2.89291 −0.108340
\(714\) 11.6378 0.435534
\(715\) −10.2422 −0.383037
\(716\) −2.83919 −0.106106
\(717\) 5.76988 0.215480
\(718\) 37.9767 1.41728
\(719\) 0.0406017 0.00151419 0.000757094 1.00000i \(-0.499759\pi\)
0.000757094 1.00000i \(0.499759\pi\)
\(720\) 1.77270 0.0660645
\(721\) 18.1046 0.674251
\(722\) −25.5411 −0.950542
\(723\) 11.9312 0.443728
\(724\) −0.258438 −0.00960477
\(725\) −26.3671 −0.979251
\(726\) −8.37199 −0.310714
\(727\) −29.7498 −1.10336 −0.551679 0.834056i \(-0.686013\pi\)
−0.551679 + 0.834056i \(0.686013\pi\)
\(728\) −16.3216 −0.604921
\(729\) 1.00000 0.0370370
\(730\) −5.47853 −0.202770
\(731\) 38.3003 1.41659
\(732\) 0.522845 0.0193249
\(733\) −3.84361 −0.141967 −0.0709835 0.997477i \(-0.522614\pi\)
−0.0709835 + 0.997477i \(0.522614\pi\)
\(734\) −50.1044 −1.84939
\(735\) 0.420019 0.0154926
\(736\) 0.533524 0.0196660
\(737\) 41.6987 1.53599
\(738\) 13.4566 0.495343
\(739\) −13.5757 −0.499391 −0.249695 0.968324i \(-0.580331\pi\)
−0.249695 + 0.968324i \(0.580331\pi\)
\(740\) 0.186799 0.00686686
\(741\) −7.16478 −0.263205
\(742\) 4.79297 0.175955
\(743\) 7.62422 0.279706 0.139853 0.990172i \(-0.455337\pi\)
0.139853 + 0.990172i \(0.455337\pi\)
\(744\) 9.82916 0.360354
\(745\) 0.949370 0.0347822
\(746\) 2.84437 0.104140
\(747\) 14.0246 0.513133
\(748\) −3.83415 −0.140190
\(749\) −8.11725 −0.296598
\(750\) −6.00359 −0.219220
\(751\) −38.7369 −1.41353 −0.706765 0.707449i \(-0.749846\pi\)
−0.706765 + 0.707449i \(0.749846\pi\)
\(752\) −43.6488 −1.59171
\(753\) −27.8947 −1.01654
\(754\) 47.3842 1.72563
\(755\) −2.45536 −0.0893597
\(756\) −0.117115 −0.00425943
\(757\) 13.7525 0.499841 0.249921 0.968266i \(-0.419595\pi\)
0.249921 + 0.968266i \(0.419595\pi\)
\(758\) −43.2540 −1.57106
\(759\) 3.30044 0.119798
\(760\) 1.38389 0.0501988
\(761\) 29.3366 1.06345 0.531725 0.846917i \(-0.321544\pi\)
0.531725 + 0.846917i \(0.321544\pi\)
\(762\) 1.45503 0.0527102
\(763\) 5.74750 0.208074
\(764\) −2.54379 −0.0920311
\(765\) 3.35945 0.121461
\(766\) −51.8425 −1.87315
\(767\) 30.9661 1.11812
\(768\) −2.80131 −0.101084
\(769\) −29.4417 −1.06170 −0.530848 0.847467i \(-0.678127\pi\)
−0.530848 + 0.847467i \(0.678127\pi\)
\(770\) −2.50149 −0.0901474
\(771\) 1.37008 0.0493423
\(772\) −0.734598 −0.0264387
\(773\) −14.9074 −0.536180 −0.268090 0.963394i \(-0.586392\pi\)
−0.268090 + 0.963394i \(0.586392\pi\)
\(774\) −6.96747 −0.250441
\(775\) −17.3057 −0.621641
\(776\) 23.6437 0.848760
\(777\) 3.79745 0.136233
\(778\) −42.1819 −1.51229
\(779\) 11.1224 0.398500
\(780\) 0.293055 0.0104931
\(781\) 33.7511 1.20771
\(782\) 9.38395 0.335570
\(783\) −5.46630 −0.195350
\(784\) −4.22051 −0.150733
\(785\) 1.16673 0.0416423
\(786\) 12.4617 0.444495
\(787\) −52.7973 −1.88202 −0.941010 0.338379i \(-0.890121\pi\)
−0.941010 + 0.338379i \(0.890121\pi\)
\(788\) 2.34925 0.0836885
\(789\) −6.11351 −0.217647
\(790\) 6.84328 0.243473
\(791\) 14.6228 0.519926
\(792\) −11.2138 −0.398465
\(793\) −26.5968 −0.944478
\(794\) −5.22590 −0.185460
\(795\) 1.38357 0.0490702
\(796\) −0.0404891 −0.00143510
\(797\) −19.3955 −0.687023 −0.343511 0.939148i \(-0.611616\pi\)
−0.343511 + 0.939148i \(0.611616\pi\)
\(798\) −1.74988 −0.0619449
\(799\) −82.7192 −2.92639
\(800\) 3.19161 0.112840
\(801\) 12.7389 0.450108
\(802\) 13.9968 0.494245
\(803\) 36.6927 1.29486
\(804\) −1.19310 −0.0420775
\(805\) 0.338676 0.0119367
\(806\) 31.1000 1.09545
\(807\) 14.3108 0.503765
\(808\) 38.3550 1.34932
\(809\) 16.1411 0.567490 0.283745 0.958900i \(-0.408423\pi\)
0.283745 + 0.958900i \(0.408423\pi\)
\(810\) −0.611141 −0.0214733
\(811\) −29.7370 −1.04421 −0.522104 0.852882i \(-0.674853\pi\)
−0.522104 + 0.852882i \(0.674853\pi\)
\(812\) 0.640186 0.0224661
\(813\) 19.0355 0.667604
\(814\) −22.6163 −0.792701
\(815\) 3.03596 0.106345
\(816\) −33.7570 −1.18173
\(817\) −5.75888 −0.201478
\(818\) 46.2517 1.61715
\(819\) 5.95755 0.208174
\(820\) −0.454929 −0.0158868
\(821\) 13.5406 0.472570 0.236285 0.971684i \(-0.424070\pi\)
0.236285 + 0.971684i \(0.424070\pi\)
\(822\) −16.9277 −0.590421
\(823\) −20.0304 −0.698216 −0.349108 0.937083i \(-0.613515\pi\)
−0.349108 + 0.937083i \(0.613515\pi\)
\(824\) −49.6004 −1.72791
\(825\) 19.7436 0.687385
\(826\) 7.56294 0.263148
\(827\) −40.3235 −1.40219 −0.701094 0.713069i \(-0.747304\pi\)
−0.701094 + 0.713069i \(0.747304\pi\)
\(828\) −0.0944338 −0.00328180
\(829\) −49.5541 −1.72108 −0.860542 0.509379i \(-0.829875\pi\)
−0.860542 + 0.509379i \(0.829875\pi\)
\(830\) −8.57100 −0.297504
\(831\) −17.7110 −0.614388
\(832\) 44.5523 1.54457
\(833\) −7.99832 −0.277125
\(834\) 7.80705 0.270336
\(835\) −8.12116 −0.281044
\(836\) 0.576507 0.0199389
\(837\) −3.58774 −0.124010
\(838\) 30.9984 1.07082
\(839\) −34.0265 −1.17473 −0.587363 0.809324i \(-0.699834\pi\)
−0.587363 + 0.809324i \(0.699834\pi\)
\(840\) −1.15071 −0.0397032
\(841\) 0.880417 0.0303592
\(842\) −37.0750 −1.27769
\(843\) 27.6078 0.950863
\(844\) −2.04456 −0.0703767
\(845\) −9.44726 −0.324996
\(846\) 15.0480 0.517362
\(847\) 5.75383 0.197704
\(848\) −13.9027 −0.477419
\(849\) −11.8324 −0.406085
\(850\) 56.1359 1.92545
\(851\) 3.06201 0.104964
\(852\) −0.965703 −0.0330844
\(853\) 27.8105 0.952213 0.476107 0.879388i \(-0.342048\pi\)
0.476107 + 0.879388i \(0.342048\pi\)
\(854\) −6.49580 −0.222282
\(855\) −0.505131 −0.0172751
\(856\) 22.2385 0.760096
\(857\) −55.1687 −1.88453 −0.942263 0.334874i \(-0.891306\pi\)
−0.942263 + 0.334874i \(0.891306\pi\)
\(858\) −35.4811 −1.21131
\(859\) 14.0090 0.477980 0.238990 0.971022i \(-0.423184\pi\)
0.238990 + 0.971022i \(0.423184\pi\)
\(860\) 0.235551 0.00803222
\(861\) −9.24830 −0.315181
\(862\) −40.2659 −1.37146
\(863\) −7.64204 −0.260138 −0.130069 0.991505i \(-0.541520\pi\)
−0.130069 + 0.991505i \(0.541520\pi\)
\(864\) 0.661667 0.0225104
\(865\) −4.58017 −0.155731
\(866\) 40.9283 1.39080
\(867\) −46.9732 −1.59529
\(868\) 0.420178 0.0142618
\(869\) −45.8332 −1.55478
\(870\) 3.34068 0.113260
\(871\) 60.6922 2.05648
\(872\) −15.7462 −0.533233
\(873\) −8.63018 −0.292087
\(874\) −1.41098 −0.0477272
\(875\) 4.12609 0.139487
\(876\) −1.04987 −0.0354718
\(877\) 0.769283 0.0259768 0.0129884 0.999916i \(-0.495866\pi\)
0.0129884 + 0.999916i \(0.495866\pi\)
\(878\) 32.5590 1.09881
\(879\) −17.9876 −0.606706
\(880\) 7.25590 0.244596
\(881\) −3.94890 −0.133042 −0.0665209 0.997785i \(-0.521190\pi\)
−0.0665209 + 0.997785i \(0.521190\pi\)
\(882\) 1.45503 0.0489934
\(883\) 37.5238 1.26278 0.631388 0.775467i \(-0.282486\pi\)
0.631388 + 0.775467i \(0.282486\pi\)
\(884\) −5.58058 −0.187695
\(885\) 2.18317 0.0733865
\(886\) 40.3836 1.35671
\(887\) 23.7485 0.797397 0.398698 0.917082i \(-0.369462\pi\)
0.398698 + 0.917082i \(0.369462\pi\)
\(888\) −10.4037 −0.349126
\(889\) −1.00000 −0.0335389
\(890\) −7.78528 −0.260963
\(891\) 4.09314 0.137125
\(892\) −0.983604 −0.0329335
\(893\) 12.4378 0.416214
\(894\) 3.28881 0.109994
\(895\) 10.1824 0.340361
\(896\) 12.2045 0.407723
\(897\) 4.80378 0.160393
\(898\) 1.07861 0.0359937
\(899\) 19.6116 0.654085
\(900\) −0.564914 −0.0188305
\(901\) −26.3470 −0.877746
\(902\) 55.0796 1.83395
\(903\) 4.78854 0.159353
\(904\) −40.0614 −1.33242
\(905\) 0.926857 0.0308098
\(906\) −8.50586 −0.282588
\(907\) −25.2818 −0.839468 −0.419734 0.907647i \(-0.637877\pi\)
−0.419734 + 0.907647i \(0.637877\pi\)
\(908\) 0.910434 0.0302138
\(909\) −13.9999 −0.464348
\(910\) −3.64091 −0.120695
\(911\) 3.77037 0.124918 0.0624589 0.998048i \(-0.480106\pi\)
0.0624589 + 0.998048i \(0.480106\pi\)
\(912\) 5.07575 0.168075
\(913\) 57.4047 1.89982
\(914\) 26.9306 0.890787
\(915\) −1.87512 −0.0619897
\(916\) −2.93448 −0.0969578
\(917\) −8.56457 −0.282827
\(918\) 11.6378 0.384105
\(919\) 31.0902 1.02557 0.512785 0.858517i \(-0.328614\pi\)
0.512785 + 0.858517i \(0.328614\pi\)
\(920\) −0.927855 −0.0305905
\(921\) −13.1148 −0.432146
\(922\) −58.5190 −1.92722
\(923\) 49.1246 1.61696
\(924\) −0.479369 −0.0157701
\(925\) 18.3173 0.602270
\(926\) −48.9568 −1.60882
\(927\) 18.1046 0.594634
\(928\) −3.61687 −0.118730
\(929\) 29.7655 0.976574 0.488287 0.872683i \(-0.337622\pi\)
0.488287 + 0.872683i \(0.337622\pi\)
\(930\) 2.19261 0.0718986
\(931\) 1.20264 0.0394149
\(932\) −3.53406 −0.115762
\(933\) 9.74688 0.319098
\(934\) 21.3676 0.699168
\(935\) 13.7507 0.449696
\(936\) −16.3216 −0.533490
\(937\) 31.5815 1.03172 0.515862 0.856672i \(-0.327472\pi\)
0.515862 + 0.856672i \(0.327472\pi\)
\(938\) 14.8230 0.483989
\(939\) −20.0884 −0.655562
\(940\) −0.508732 −0.0165930
\(941\) −7.00125 −0.228234 −0.114117 0.993467i \(-0.536404\pi\)
−0.114117 + 0.993467i \(0.536404\pi\)
\(942\) 4.04178 0.131688
\(943\) −7.45721 −0.242840
\(944\) −21.9373 −0.714000
\(945\) 0.420019 0.0136632
\(946\) −28.5189 −0.927228
\(947\) 34.3629 1.11664 0.558322 0.829624i \(-0.311445\pi\)
0.558322 + 0.829624i \(0.311445\pi\)
\(948\) 1.31140 0.0425923
\(949\) 53.4061 1.73364
\(950\) −8.44067 −0.273852
\(951\) 23.1449 0.750526
\(952\) 21.9127 0.710193
\(953\) 1.56016 0.0505386 0.0252693 0.999681i \(-0.491956\pi\)
0.0252693 + 0.999681i \(0.491956\pi\)
\(954\) 4.79297 0.155178
\(955\) 9.12300 0.295213
\(956\) −0.675739 −0.0218550
\(957\) −22.3743 −0.723260
\(958\) 15.1654 0.489972
\(959\) 11.6339 0.375679
\(960\) 3.14102 0.101376
\(961\) −18.1282 −0.584779
\(962\) −32.9179 −1.06132
\(963\) −8.11725 −0.261575
\(964\) −1.39733 −0.0450049
\(965\) 2.63455 0.0848091
\(966\) 1.17324 0.0377484
\(967\) 30.5017 0.980868 0.490434 0.871478i \(-0.336838\pi\)
0.490434 + 0.871478i \(0.336838\pi\)
\(968\) −15.7635 −0.506658
\(969\) 9.61909 0.309010
\(970\) 5.27426 0.169346
\(971\) −25.2755 −0.811131 −0.405565 0.914066i \(-0.632925\pi\)
−0.405565 + 0.914066i \(0.632925\pi\)
\(972\) −0.117115 −0.00375647
\(973\) −5.36556 −0.172012
\(974\) −49.2418 −1.57781
\(975\) 28.7368 0.920313
\(976\) 18.8420 0.603116
\(977\) 32.0198 1.02440 0.512202 0.858865i \(-0.328830\pi\)
0.512202 + 0.858865i \(0.328830\pi\)
\(978\) 10.5172 0.336303
\(979\) 52.1423 1.66647
\(980\) −0.0491906 −0.00157134
\(981\) 5.74750 0.183504
\(982\) 33.9162 1.08231
\(983\) 44.6458 1.42398 0.711989 0.702190i \(-0.247794\pi\)
0.711989 + 0.702190i \(0.247794\pi\)
\(984\) 25.3371 0.807719
\(985\) −8.42530 −0.268452
\(986\) −63.6157 −2.02594
\(987\) −10.3421 −0.329192
\(988\) 0.839104 0.0266955
\(989\) 3.86116 0.122778
\(990\) −2.50149 −0.0795025
\(991\) 23.0161 0.731129 0.365565 0.930786i \(-0.380876\pi\)
0.365565 + 0.930786i \(0.380876\pi\)
\(992\) −2.37389 −0.0753710
\(993\) 12.8235 0.406941
\(994\) 11.9978 0.380548
\(995\) 0.145209 0.00460345
\(996\) −1.64249 −0.0520443
\(997\) 31.6803 1.00333 0.501663 0.865063i \(-0.332722\pi\)
0.501663 + 0.865063i \(0.332722\pi\)
\(998\) −1.44182 −0.0456401
\(999\) 3.79745 0.120146
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 2667.2.a.l.1.10 13
3.2 odd 2 8001.2.a.o.1.4 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.l.1.10 13 1.1 even 1 trivial
8001.2.a.o.1.4 13 3.2 odd 2