Properties

Label 1334.2.a.f.1.1
Level $1334$
Weight $2$
Character 1334.1
Self dual yes
Analytic conductor $10.652$
Analytic rank $1$
Dimension $5$
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] = [1334,2,Mod(1,1334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1334, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1334.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1334 = 2 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1334.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: \(10.6520436296\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.207184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 6x^{3} + 2x^{2} + 7x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.139666\) of defining polynomial
Character \(\chi\) \(=\) 1334.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.54471 q^{3} +1.00000 q^{4} -0.504925 q^{5} +2.54471 q^{6} +2.12016 q^{7} -1.00000 q^{8} +3.47557 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.54471 q^{3} +1.00000 q^{4} -0.504925 q^{5} +2.54471 q^{6} +2.12016 q^{7} -1.00000 q^{8} +3.47557 q^{9} +0.504925 q^{10} -0.850485 q^{11} -2.54471 q^{12} -5.05101 q^{13} -2.12016 q^{14} +1.28489 q^{15} +1.00000 q^{16} +6.51398 q^{17} -3.47557 q^{18} -4.69560 q^{19} -0.504925 q^{20} -5.39520 q^{21} +0.850485 q^{22} +1.00000 q^{23} +2.54471 q^{24} -4.74505 q^{25} +5.05101 q^{26} -1.21019 q^{27} +2.12016 q^{28} +1.00000 q^{29} -1.28489 q^{30} +10.4596 q^{31} -1.00000 q^{32} +2.16424 q^{33} -6.51398 q^{34} -1.07052 q^{35} +3.47557 q^{36} +6.58588 q^{37} +4.69560 q^{38} +12.8534 q^{39} +0.504925 q^{40} -8.56343 q^{41} +5.39520 q^{42} +4.24169 q^{43} -0.850485 q^{44} -1.75490 q^{45} -1.00000 q^{46} -0.904416 q^{47} -2.54471 q^{48} -2.50492 q^{49} +4.74505 q^{50} -16.5762 q^{51} -5.05101 q^{52} +3.47576 q^{53} +1.21019 q^{54} +0.429431 q^{55} -2.12016 q^{56} +11.9490 q^{57} -1.00000 q^{58} +2.19195 q^{59} +1.28489 q^{60} -12.9842 q^{61} -10.4596 q^{62} +7.36876 q^{63} +1.00000 q^{64} +2.55038 q^{65} -2.16424 q^{66} -14.7941 q^{67} +6.51398 q^{68} -2.54471 q^{69} +1.07052 q^{70} -1.16201 q^{71} -3.47557 q^{72} -5.09617 q^{73} -6.58588 q^{74} +12.0748 q^{75} -4.69560 q^{76} -1.80316 q^{77} -12.8534 q^{78} +7.30088 q^{79} -0.504925 q^{80} -7.34713 q^{81} +8.56343 q^{82} +6.12415 q^{83} -5.39520 q^{84} -3.28907 q^{85} -4.24169 q^{86} -2.54471 q^{87} +0.850485 q^{88} +2.44271 q^{89} +1.75490 q^{90} -10.7090 q^{91} +1.00000 q^{92} -26.6166 q^{93} +0.904416 q^{94} +2.37093 q^{95} +2.54471 q^{96} -14.5252 q^{97} +2.50492 q^{98} -2.95592 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} + q^{3} + 5 q^{4} - 5 q^{5} - q^{6} - 2 q^{7} - 5 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{2} + q^{3} + 5 q^{4} - 5 q^{5} - q^{6} - 2 q^{7} - 5 q^{8} + 2 q^{9} + 5 q^{10} + q^{11} + q^{12} - 11 q^{13} + 2 q^{14} + 5 q^{15} + 5 q^{16} + 4 q^{17} - 2 q^{18} - 12 q^{19} - 5 q^{20} - 8 q^{21} - q^{22} + 5 q^{23} - q^{24} - 4 q^{25} + 11 q^{26} - 5 q^{27} - 2 q^{28} + 5 q^{29} - 5 q^{30} - 5 q^{31} - 5 q^{32} - 11 q^{33} - 4 q^{34} - 4 q^{35} + 2 q^{36} + 12 q^{38} - 9 q^{39} + 5 q^{40} - 6 q^{41} + 8 q^{42} - 7 q^{43} + q^{44} + 6 q^{45} - 5 q^{46} + 5 q^{47} + q^{48} - 15 q^{49} + 4 q^{50} - 42 q^{51} - 11 q^{52} - q^{53} + 5 q^{54} - 31 q^{55} + 2 q^{56} - 4 q^{57} - 5 q^{58} - 12 q^{59} + 5 q^{60} - 20 q^{61} + 5 q^{62} + 10 q^{63} + 5 q^{64} + 3 q^{65} + 11 q^{66} - 4 q^{67} + 4 q^{68} + q^{69} + 4 q^{70} - 4 q^{71} - 2 q^{72} + 4 q^{73} - 12 q^{75} - 12 q^{76} + 14 q^{77} + 9 q^{78} - q^{79} - 5 q^{80} - 23 q^{81} + 6 q^{82} + 22 q^{83} - 8 q^{84} - 16 q^{85} + 7 q^{86} + q^{87} - q^{88} + 8 q^{89} - 6 q^{90} - 18 q^{91} + 5 q^{92} - 37 q^{93} - 5 q^{94} + 18 q^{95} - q^{96} - 46 q^{97} + 15 q^{98} - 24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.54471 −1.46919 −0.734596 0.678505i \(-0.762628\pi\)
−0.734596 + 0.678505i \(0.762628\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.504925 −0.225809 −0.112905 0.993606i \(-0.536015\pi\)
−0.112905 + 0.993606i \(0.536015\pi\)
\(6\) 2.54471 1.03888
\(7\) 2.12016 0.801345 0.400672 0.916221i \(-0.368777\pi\)
0.400672 + 0.916221i \(0.368777\pi\)
\(8\) −1.00000 −0.353553
\(9\) 3.47557 1.15852
\(10\) 0.504925 0.159671
\(11\) −0.850485 −0.256431 −0.128215 0.991746i \(-0.540925\pi\)
−0.128215 + 0.991746i \(0.540925\pi\)
\(12\) −2.54471 −0.734596
\(13\) −5.05101 −1.40090 −0.700450 0.713702i \(-0.747017\pi\)
−0.700450 + 0.713702i \(0.747017\pi\)
\(14\) −2.12016 −0.566636
\(15\) 1.28489 0.331757
\(16\) 1.00000 0.250000
\(17\) 6.51398 1.57987 0.789936 0.613189i \(-0.210114\pi\)
0.789936 + 0.613189i \(0.210114\pi\)
\(18\) −3.47557 −0.819199
\(19\) −4.69560 −1.07725 −0.538623 0.842547i \(-0.681055\pi\)
−0.538623 + 0.842547i \(0.681055\pi\)
\(20\) −0.504925 −0.112905
\(21\) −5.39520 −1.17733
\(22\) 0.850485 0.181324
\(23\) 1.00000 0.208514
\(24\) 2.54471 0.519438
\(25\) −4.74505 −0.949010
\(26\) 5.05101 0.990585
\(27\) −1.21019 −0.232901
\(28\) 2.12016 0.400672
\(29\) 1.00000 0.185695
\(30\) −1.28489 −0.234588
\(31\) 10.4596 1.87859 0.939296 0.343107i \(-0.111479\pi\)
0.939296 + 0.343107i \(0.111479\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.16424 0.376746
\(34\) −6.51398 −1.11714
\(35\) −1.07052 −0.180951
\(36\) 3.47557 0.579261
\(37\) 6.58588 1.08271 0.541356 0.840794i \(-0.317911\pi\)
0.541356 + 0.840794i \(0.317911\pi\)
\(38\) 4.69560 0.761728
\(39\) 12.8534 2.05819
\(40\) 0.504925 0.0798356
\(41\) −8.56343 −1.33738 −0.668691 0.743540i \(-0.733145\pi\)
−0.668691 + 0.743540i \(0.733145\pi\)
\(42\) 5.39520 0.832497
\(43\) 4.24169 0.646852 0.323426 0.946253i \(-0.395165\pi\)
0.323426 + 0.946253i \(0.395165\pi\)
\(44\) −0.850485 −0.128215
\(45\) −1.75490 −0.261605
\(46\) −1.00000 −0.147442
\(47\) −0.904416 −0.131923 −0.0659613 0.997822i \(-0.521011\pi\)
−0.0659613 + 0.997822i \(0.521011\pi\)
\(48\) −2.54471 −0.367298
\(49\) −2.50492 −0.357846
\(50\) 4.74505 0.671052
\(51\) −16.5762 −2.32114
\(52\) −5.05101 −0.700450
\(53\) 3.47576 0.477432 0.238716 0.971089i \(-0.423273\pi\)
0.238716 + 0.971089i \(0.423273\pi\)
\(54\) 1.21019 0.164686
\(55\) 0.429431 0.0579044
\(56\) −2.12016 −0.283318
\(57\) 11.9490 1.58268
\(58\) −1.00000 −0.131306
\(59\) 2.19195 0.285367 0.142683 0.989768i \(-0.454427\pi\)
0.142683 + 0.989768i \(0.454427\pi\)
\(60\) 1.28489 0.165878
\(61\) −12.9842 −1.66245 −0.831227 0.555933i \(-0.812361\pi\)
−0.831227 + 0.555933i \(0.812361\pi\)
\(62\) −10.4596 −1.32837
\(63\) 7.36876 0.928376
\(64\) 1.00000 0.125000
\(65\) 2.55038 0.316336
\(66\) −2.16424 −0.266400
\(67\) −14.7941 −1.80739 −0.903693 0.428181i \(-0.859154\pi\)
−0.903693 + 0.428181i \(0.859154\pi\)
\(68\) 6.51398 0.789936
\(69\) −2.54471 −0.306348
\(70\) 1.07052 0.127952
\(71\) −1.16201 −0.137905 −0.0689523 0.997620i \(-0.521966\pi\)
−0.0689523 + 0.997620i \(0.521966\pi\)
\(72\) −3.47557 −0.409600
\(73\) −5.09617 −0.596461 −0.298231 0.954494i \(-0.596396\pi\)
−0.298231 + 0.954494i \(0.596396\pi\)
\(74\) −6.58588 −0.765593
\(75\) 12.0748 1.39428
\(76\) −4.69560 −0.538623
\(77\) −1.80316 −0.205490
\(78\) −12.8534 −1.45536
\(79\) 7.30088 0.821413 0.410707 0.911768i \(-0.365282\pi\)
0.410707 + 0.911768i \(0.365282\pi\)
\(80\) −0.504925 −0.0564523
\(81\) −7.34713 −0.816348
\(82\) 8.56343 0.945673
\(83\) 6.12415 0.672213 0.336106 0.941824i \(-0.390890\pi\)
0.336106 + 0.941824i \(0.390890\pi\)
\(84\) −5.39520 −0.588664
\(85\) −3.28907 −0.356750
\(86\) −4.24169 −0.457394
\(87\) −2.54471 −0.272822
\(88\) 0.850485 0.0906620
\(89\) 2.44271 0.258927 0.129463 0.991584i \(-0.458674\pi\)
0.129463 + 0.991584i \(0.458674\pi\)
\(90\) 1.75490 0.184983
\(91\) −10.7090 −1.12260
\(92\) 1.00000 0.104257
\(93\) −26.6166 −2.76001
\(94\) 0.904416 0.0932833
\(95\) 2.37093 0.243252
\(96\) 2.54471 0.259719
\(97\) −14.5252 −1.47481 −0.737406 0.675450i \(-0.763949\pi\)
−0.737406 + 0.675450i \(0.763949\pi\)
\(98\) 2.50492 0.253036
\(99\) −2.95592 −0.297081
\(100\) −4.74505 −0.474505
\(101\) −2.74730 −0.273367 −0.136683 0.990615i \(-0.543644\pi\)
−0.136683 + 0.990615i \(0.543644\pi\)
\(102\) 16.5762 1.64129
\(103\) 0.112367 0.0110718 0.00553590 0.999985i \(-0.498238\pi\)
0.00553590 + 0.999985i \(0.498238\pi\)
\(104\) 5.05101 0.495293
\(105\) 2.72417 0.265852
\(106\) −3.47576 −0.337596
\(107\) −18.0811 −1.74797 −0.873984 0.485954i \(-0.838472\pi\)
−0.873984 + 0.485954i \(0.838472\pi\)
\(108\) −1.21019 −0.116450
\(109\) −15.4803 −1.48274 −0.741370 0.671097i \(-0.765823\pi\)
−0.741370 + 0.671097i \(0.765823\pi\)
\(110\) −0.429431 −0.0409446
\(111\) −16.7592 −1.59071
\(112\) 2.12016 0.200336
\(113\) −7.30730 −0.687413 −0.343706 0.939077i \(-0.611682\pi\)
−0.343706 + 0.939077i \(0.611682\pi\)
\(114\) −11.9490 −1.11912
\(115\) −0.504925 −0.0470845
\(116\) 1.00000 0.0928477
\(117\) −17.5551 −1.62297
\(118\) −2.19195 −0.201785
\(119\) 13.8107 1.26602
\(120\) −1.28489 −0.117294
\(121\) −10.2767 −0.934243
\(122\) 12.9842 1.17553
\(123\) 21.7915 1.96487
\(124\) 10.4596 0.939296
\(125\) 4.92052 0.440104
\(126\) −7.36876 −0.656461
\(127\) 1.40232 0.124436 0.0622181 0.998063i \(-0.480183\pi\)
0.0622181 + 0.998063i \(0.480183\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.7939 −0.950350
\(130\) −2.55038 −0.223683
\(131\) −17.9044 −1.56431 −0.782157 0.623081i \(-0.785881\pi\)
−0.782157 + 0.623081i \(0.785881\pi\)
\(132\) 2.16424 0.188373
\(133\) −9.95543 −0.863245
\(134\) 14.7941 1.27801
\(135\) 0.611053 0.0525911
\(136\) −6.51398 −0.558569
\(137\) 21.6999 1.85395 0.926973 0.375128i \(-0.122401\pi\)
0.926973 + 0.375128i \(0.122401\pi\)
\(138\) 2.54471 0.216620
\(139\) −14.3121 −1.21394 −0.606968 0.794726i \(-0.707615\pi\)
−0.606968 + 0.794726i \(0.707615\pi\)
\(140\) −1.07052 −0.0904755
\(141\) 2.30148 0.193819
\(142\) 1.16201 0.0975133
\(143\) 4.29581 0.359234
\(144\) 3.47557 0.289631
\(145\) −0.504925 −0.0419317
\(146\) 5.09617 0.421762
\(147\) 6.37432 0.525745
\(148\) 6.58588 0.541356
\(149\) 2.45800 0.201367 0.100683 0.994919i \(-0.467897\pi\)
0.100683 + 0.994919i \(0.467897\pi\)
\(150\) −12.0748 −0.985903
\(151\) −19.9665 −1.62485 −0.812427 0.583063i \(-0.801854\pi\)
−0.812427 + 0.583063i \(0.801854\pi\)
\(152\) 4.69560 0.380864
\(153\) 22.6398 1.83032
\(154\) 1.80316 0.145303
\(155\) −5.28129 −0.424204
\(156\) 12.8534 1.02909
\(157\) −3.40867 −0.272042 −0.136021 0.990706i \(-0.543431\pi\)
−0.136021 + 0.990706i \(0.543431\pi\)
\(158\) −7.30088 −0.580827
\(159\) −8.84482 −0.701440
\(160\) 0.504925 0.0399178
\(161\) 2.12016 0.167092
\(162\) 7.34713 0.577245
\(163\) 14.5175 1.13710 0.568549 0.822650i \(-0.307505\pi\)
0.568549 + 0.822650i \(0.307505\pi\)
\(164\) −8.56343 −0.668691
\(165\) −1.09278 −0.0850727
\(166\) −6.12415 −0.475326
\(167\) 14.6764 1.13569 0.567846 0.823135i \(-0.307777\pi\)
0.567846 + 0.823135i \(0.307777\pi\)
\(168\) 5.39520 0.416249
\(169\) 12.5127 0.962519
\(170\) 3.28907 0.252260
\(171\) −16.3199 −1.24801
\(172\) 4.24169 0.323426
\(173\) −4.89401 −0.372085 −0.186042 0.982542i \(-0.559566\pi\)
−0.186042 + 0.982542i \(0.559566\pi\)
\(174\) 2.54471 0.192914
\(175\) −10.0603 −0.760484
\(176\) −0.850485 −0.0641077
\(177\) −5.57787 −0.419259
\(178\) −2.44271 −0.183089
\(179\) 23.9294 1.78857 0.894283 0.447502i \(-0.147686\pi\)
0.894283 + 0.447502i \(0.147686\pi\)
\(180\) −1.75490 −0.130803
\(181\) −20.2876 −1.50797 −0.753983 0.656894i \(-0.771870\pi\)
−0.753983 + 0.656894i \(0.771870\pi\)
\(182\) 10.7090 0.793801
\(183\) 33.0410 2.44246
\(184\) −1.00000 −0.0737210
\(185\) −3.32537 −0.244486
\(186\) 26.6166 1.95162
\(187\) −5.54004 −0.405128
\(188\) −0.904416 −0.0659613
\(189\) −2.56579 −0.186634
\(190\) −2.37093 −0.172005
\(191\) 3.15447 0.228250 0.114125 0.993466i \(-0.463594\pi\)
0.114125 + 0.993466i \(0.463594\pi\)
\(192\) −2.54471 −0.183649
\(193\) −12.8415 −0.924353 −0.462176 0.886788i \(-0.652931\pi\)
−0.462176 + 0.886788i \(0.652931\pi\)
\(194\) 14.5252 1.04285
\(195\) −6.48999 −0.464758
\(196\) −2.50492 −0.178923
\(197\) 19.3191 1.37643 0.688216 0.725506i \(-0.258394\pi\)
0.688216 + 0.725506i \(0.258394\pi\)
\(198\) 2.95592 0.210068
\(199\) −3.71208 −0.263143 −0.131571 0.991307i \(-0.542002\pi\)
−0.131571 + 0.991307i \(0.542002\pi\)
\(200\) 4.74505 0.335526
\(201\) 37.6467 2.65540
\(202\) 2.74730 0.193299
\(203\) 2.12016 0.148806
\(204\) −16.5762 −1.16057
\(205\) 4.32389 0.301993
\(206\) −0.112367 −0.00782895
\(207\) 3.47557 0.241569
\(208\) −5.05101 −0.350225
\(209\) 3.99354 0.276239
\(210\) −2.72417 −0.187986
\(211\) 4.72835 0.325513 0.162757 0.986666i \(-0.447961\pi\)
0.162757 + 0.986666i \(0.447961\pi\)
\(212\) 3.47576 0.238716
\(213\) 2.95697 0.202608
\(214\) 18.0811 1.23600
\(215\) −2.14174 −0.146065
\(216\) 1.21019 0.0823428
\(217\) 22.1759 1.50540
\(218\) 15.4803 1.04846
\(219\) 12.9683 0.876316
\(220\) 0.429431 0.0289522
\(221\) −32.9022 −2.21324
\(222\) 16.7592 1.12480
\(223\) −17.8577 −1.19584 −0.597920 0.801556i \(-0.704006\pi\)
−0.597920 + 0.801556i \(0.704006\pi\)
\(224\) −2.12016 −0.141659
\(225\) −16.4918 −1.09945
\(226\) 7.30730 0.486074
\(227\) −15.2508 −1.01223 −0.506117 0.862465i \(-0.668919\pi\)
−0.506117 + 0.862465i \(0.668919\pi\)
\(228\) 11.9490 0.791340
\(229\) −22.6368 −1.49588 −0.747940 0.663766i \(-0.768957\pi\)
−0.747940 + 0.663766i \(0.768957\pi\)
\(230\) 0.504925 0.0332937
\(231\) 4.58853 0.301903
\(232\) −1.00000 −0.0656532
\(233\) −7.62966 −0.499835 −0.249918 0.968267i \(-0.580404\pi\)
−0.249918 + 0.968267i \(0.580404\pi\)
\(234\) 17.5551 1.14762
\(235\) 0.456662 0.0297893
\(236\) 2.19195 0.142683
\(237\) −18.5786 −1.20681
\(238\) −13.8107 −0.895213
\(239\) −16.0383 −1.03743 −0.518716 0.854946i \(-0.673590\pi\)
−0.518716 + 0.854946i \(0.673590\pi\)
\(240\) 1.28489 0.0829392
\(241\) −27.6992 −1.78426 −0.892130 0.451778i \(-0.850790\pi\)
−0.892130 + 0.451778i \(0.850790\pi\)
\(242\) 10.2767 0.660610
\(243\) 22.3269 1.43227
\(244\) −12.9842 −0.831227
\(245\) 1.26480 0.0808050
\(246\) −21.7915 −1.38937
\(247\) 23.7176 1.50911
\(248\) −10.4596 −0.664183
\(249\) −15.5842 −0.987609
\(250\) −4.92052 −0.311201
\(251\) −14.2405 −0.898852 −0.449426 0.893318i \(-0.648371\pi\)
−0.449426 + 0.893318i \(0.648371\pi\)
\(252\) 7.36876 0.464188
\(253\) −0.850485 −0.0534695
\(254\) −1.40232 −0.0879896
\(255\) 8.36974 0.524134
\(256\) 1.00000 0.0625000
\(257\) 23.2522 1.45043 0.725216 0.688521i \(-0.241740\pi\)
0.725216 + 0.688521i \(0.241740\pi\)
\(258\) 10.7939 0.671999
\(259\) 13.9631 0.867626
\(260\) 2.55038 0.158168
\(261\) 3.47557 0.215132
\(262\) 17.9044 1.10614
\(263\) 11.5860 0.714426 0.357213 0.934023i \(-0.383727\pi\)
0.357213 + 0.934023i \(0.383727\pi\)
\(264\) −2.16424 −0.133200
\(265\) −1.75500 −0.107809
\(266\) 9.95543 0.610407
\(267\) −6.21600 −0.380413
\(268\) −14.7941 −0.903693
\(269\) −25.1049 −1.53067 −0.765336 0.643630i \(-0.777427\pi\)
−0.765336 + 0.643630i \(0.777427\pi\)
\(270\) −0.611053 −0.0371875
\(271\) 17.3946 1.05665 0.528323 0.849044i \(-0.322821\pi\)
0.528323 + 0.849044i \(0.322821\pi\)
\(272\) 6.51398 0.394968
\(273\) 27.2512 1.64932
\(274\) −21.6999 −1.31094
\(275\) 4.03559 0.243355
\(276\) −2.54471 −0.153174
\(277\) 14.8366 0.891448 0.445724 0.895171i \(-0.352946\pi\)
0.445724 + 0.895171i \(0.352946\pi\)
\(278\) 14.3121 0.858383
\(279\) 36.3529 2.17639
\(280\) 1.07052 0.0639758
\(281\) −17.4702 −1.04218 −0.521092 0.853500i \(-0.674475\pi\)
−0.521092 + 0.853500i \(0.674475\pi\)
\(282\) −2.30148 −0.137051
\(283\) 14.4926 0.861498 0.430749 0.902472i \(-0.358249\pi\)
0.430749 + 0.902472i \(0.358249\pi\)
\(284\) −1.16201 −0.0689523
\(285\) −6.03333 −0.357384
\(286\) −4.29581 −0.254017
\(287\) −18.1558 −1.07170
\(288\) −3.47557 −0.204800
\(289\) 25.4320 1.49600
\(290\) 0.504925 0.0296502
\(291\) 36.9625 2.16678
\(292\) −5.09617 −0.298231
\(293\) 5.92107 0.345913 0.172956 0.984929i \(-0.444668\pi\)
0.172956 + 0.984929i \(0.444668\pi\)
\(294\) −6.37432 −0.371758
\(295\) −1.10677 −0.0644385
\(296\) −6.58588 −0.382796
\(297\) 1.02925 0.0597229
\(298\) −2.45800 −0.142388
\(299\) −5.05101 −0.292108
\(300\) 12.0748 0.697139
\(301\) 8.99307 0.518352
\(302\) 19.9665 1.14895
\(303\) 6.99109 0.401628
\(304\) −4.69560 −0.269311
\(305\) 6.55604 0.375398
\(306\) −22.6398 −1.29423
\(307\) 17.5305 1.00052 0.500260 0.865875i \(-0.333238\pi\)
0.500260 + 0.865875i \(0.333238\pi\)
\(308\) −1.80316 −0.102745
\(309\) −0.285941 −0.0162666
\(310\) 5.28129 0.299957
\(311\) 23.7841 1.34867 0.674337 0.738424i \(-0.264429\pi\)
0.674337 + 0.738424i \(0.264429\pi\)
\(312\) −12.8534 −0.727680
\(313\) 17.6353 0.996808 0.498404 0.866945i \(-0.333920\pi\)
0.498404 + 0.866945i \(0.333920\pi\)
\(314\) 3.40867 0.192363
\(315\) −3.72067 −0.209636
\(316\) 7.30088 0.410707
\(317\) −16.7350 −0.939930 −0.469965 0.882685i \(-0.655734\pi\)
−0.469965 + 0.882685i \(0.655734\pi\)
\(318\) 8.84482 0.495993
\(319\) −0.850485 −0.0476180
\(320\) −0.504925 −0.0282261
\(321\) 46.0113 2.56810
\(322\) −2.12016 −0.118152
\(323\) −30.5871 −1.70191
\(324\) −7.34713 −0.408174
\(325\) 23.9673 1.32947
\(326\) −14.5175 −0.804049
\(327\) 39.3928 2.17843
\(328\) 8.56343 0.472836
\(329\) −1.91751 −0.105715
\(330\) 1.09278 0.0601555
\(331\) 34.5251 1.89767 0.948836 0.315770i \(-0.102263\pi\)
0.948836 + 0.315770i \(0.102263\pi\)
\(332\) 6.12415 0.336106
\(333\) 22.8897 1.25435
\(334\) −14.6764 −0.803056
\(335\) 7.46990 0.408124
\(336\) −5.39520 −0.294332
\(337\) −26.2706 −1.43105 −0.715526 0.698587i \(-0.753813\pi\)
−0.715526 + 0.698587i \(0.753813\pi\)
\(338\) −12.5127 −0.680604
\(339\) 18.5950 1.00994
\(340\) −3.28907 −0.178375
\(341\) −8.89570 −0.481729
\(342\) 16.3199 0.882479
\(343\) −20.1520 −1.08810
\(344\) −4.24169 −0.228697
\(345\) 1.28489 0.0691761
\(346\) 4.89401 0.263104
\(347\) −18.7646 −1.00734 −0.503669 0.863897i \(-0.668017\pi\)
−0.503669 + 0.863897i \(0.668017\pi\)
\(348\) −2.54471 −0.136411
\(349\) −21.6052 −1.15650 −0.578250 0.815860i \(-0.696264\pi\)
−0.578250 + 0.815860i \(0.696264\pi\)
\(350\) 10.0603 0.537744
\(351\) 6.11267 0.326270
\(352\) 0.850485 0.0453310
\(353\) −25.0509 −1.33332 −0.666662 0.745360i \(-0.732278\pi\)
−0.666662 + 0.745360i \(0.732278\pi\)
\(354\) 5.57787 0.296461
\(355\) 0.586725 0.0311401
\(356\) 2.44271 0.129463
\(357\) −35.1442 −1.86003
\(358\) −23.9294 −1.26471
\(359\) 14.9448 0.788754 0.394377 0.918949i \(-0.370960\pi\)
0.394377 + 0.918949i \(0.370960\pi\)
\(360\) 1.75490 0.0924914
\(361\) 3.04870 0.160458
\(362\) 20.2876 1.06629
\(363\) 26.1512 1.37258
\(364\) −10.7090 −0.561302
\(365\) 2.57318 0.134686
\(366\) −33.0410 −1.72708
\(367\) 19.3929 1.01230 0.506152 0.862444i \(-0.331068\pi\)
0.506152 + 0.862444i \(0.331068\pi\)
\(368\) 1.00000 0.0521286
\(369\) −29.7628 −1.54939
\(370\) 3.32537 0.172878
\(371\) 7.36917 0.382588
\(372\) −26.6166 −1.38001
\(373\) 2.82893 0.146477 0.0732383 0.997314i \(-0.476667\pi\)
0.0732383 + 0.997314i \(0.476667\pi\)
\(374\) 5.54004 0.286469
\(375\) −12.5213 −0.646598
\(376\) 0.904416 0.0466417
\(377\) −5.05101 −0.260140
\(378\) 2.56579 0.131970
\(379\) 23.6832 1.21652 0.608262 0.793736i \(-0.291867\pi\)
0.608262 + 0.793736i \(0.291867\pi\)
\(380\) 2.37093 0.121626
\(381\) −3.56851 −0.182820
\(382\) −3.15447 −0.161397
\(383\) −22.1269 −1.13063 −0.565317 0.824874i \(-0.691246\pi\)
−0.565317 + 0.824874i \(0.691246\pi\)
\(384\) 2.54471 0.129859
\(385\) 0.910462 0.0464014
\(386\) 12.8415 0.653616
\(387\) 14.7423 0.749393
\(388\) −14.5252 −0.737406
\(389\) 10.9909 0.557262 0.278631 0.960398i \(-0.410119\pi\)
0.278631 + 0.960398i \(0.410119\pi\)
\(390\) 6.48999 0.328633
\(391\) 6.51398 0.329426
\(392\) 2.50492 0.126518
\(393\) 45.5616 2.29828
\(394\) −19.3191 −0.973284
\(395\) −3.68639 −0.185483
\(396\) −2.95592 −0.148541
\(397\) 20.2250 1.01506 0.507531 0.861634i \(-0.330558\pi\)
0.507531 + 0.861634i \(0.330558\pi\)
\(398\) 3.71208 0.186070
\(399\) 25.3337 1.26827
\(400\) −4.74505 −0.237253
\(401\) 14.6135 0.729761 0.364881 0.931054i \(-0.381110\pi\)
0.364881 + 0.931054i \(0.381110\pi\)
\(402\) −37.6467 −1.87765
\(403\) −52.8314 −2.63172
\(404\) −2.74730 −0.136683
\(405\) 3.70975 0.184339
\(406\) −2.12016 −0.105222
\(407\) −5.60119 −0.277641
\(408\) 16.5762 0.820645
\(409\) −27.4662 −1.35812 −0.679058 0.734085i \(-0.737611\pi\)
−0.679058 + 0.734085i \(0.737611\pi\)
\(410\) −4.32389 −0.213542
\(411\) −55.2200 −2.72380
\(412\) 0.112367 0.00553590
\(413\) 4.64727 0.228677
\(414\) −3.47557 −0.170815
\(415\) −3.09223 −0.151792
\(416\) 5.05101 0.247646
\(417\) 36.4202 1.78351
\(418\) −3.99354 −0.195330
\(419\) 6.96837 0.340427 0.170213 0.985407i \(-0.445554\pi\)
0.170213 + 0.985407i \(0.445554\pi\)
\(420\) 2.72417 0.132926
\(421\) −4.95289 −0.241389 −0.120695 0.992690i \(-0.538512\pi\)
−0.120695 + 0.992690i \(0.538512\pi\)
\(422\) −4.72835 −0.230173
\(423\) −3.14336 −0.152835
\(424\) −3.47576 −0.168798
\(425\) −30.9092 −1.49932
\(426\) −2.95697 −0.143266
\(427\) −27.5285 −1.33220
\(428\) −18.0811 −0.873984
\(429\) −10.9316 −0.527783
\(430\) 2.14174 0.103284
\(431\) −26.2304 −1.26347 −0.631736 0.775183i \(-0.717657\pi\)
−0.631736 + 0.775183i \(0.717657\pi\)
\(432\) −1.21019 −0.0582251
\(433\) −13.7429 −0.660440 −0.330220 0.943904i \(-0.607123\pi\)
−0.330220 + 0.943904i \(0.607123\pi\)
\(434\) −22.1759 −1.06448
\(435\) 1.28489 0.0616057
\(436\) −15.4803 −0.741370
\(437\) −4.69560 −0.224621
\(438\) −12.9683 −0.619649
\(439\) −12.9060 −0.615968 −0.307984 0.951392i \(-0.599654\pi\)
−0.307984 + 0.951392i \(0.599654\pi\)
\(440\) −0.429431 −0.0204723
\(441\) −8.70604 −0.414573
\(442\) 32.9022 1.56500
\(443\) −0.0726725 −0.00345277 −0.00172639 0.999999i \(-0.500550\pi\)
−0.00172639 + 0.999999i \(0.500550\pi\)
\(444\) −16.7592 −0.795355
\(445\) −1.23339 −0.0584681
\(446\) 17.8577 0.845587
\(447\) −6.25489 −0.295846
\(448\) 2.12016 0.100168
\(449\) 26.4195 1.24681 0.623406 0.781898i \(-0.285748\pi\)
0.623406 + 0.781898i \(0.285748\pi\)
\(450\) 16.4918 0.777429
\(451\) 7.28307 0.342946
\(452\) −7.30730 −0.343706
\(453\) 50.8091 2.38722
\(454\) 15.2508 0.715757
\(455\) 5.40722 0.253494
\(456\) −11.9490 −0.559562
\(457\) 12.5126 0.585313 0.292657 0.956218i \(-0.405461\pi\)
0.292657 + 0.956218i \(0.405461\pi\)
\(458\) 22.6368 1.05775
\(459\) −7.88313 −0.367953
\(460\) −0.504925 −0.0235422
\(461\) −15.3188 −0.713468 −0.356734 0.934206i \(-0.616110\pi\)
−0.356734 + 0.934206i \(0.616110\pi\)
\(462\) −4.58853 −0.213478
\(463\) 20.5657 0.955767 0.477884 0.878423i \(-0.341404\pi\)
0.477884 + 0.878423i \(0.341404\pi\)
\(464\) 1.00000 0.0464238
\(465\) 13.4394 0.623236
\(466\) 7.62966 0.353437
\(467\) −23.5091 −1.08787 −0.543935 0.839127i \(-0.683066\pi\)
−0.543935 + 0.839127i \(0.683066\pi\)
\(468\) −17.5551 −0.811487
\(469\) −31.3658 −1.44834
\(470\) −0.456662 −0.0210642
\(471\) 8.67410 0.399681
\(472\) −2.19195 −0.100892
\(473\) −3.60750 −0.165873
\(474\) 18.5786 0.853346
\(475\) 22.2809 1.02232
\(476\) 13.8107 0.633011
\(477\) 12.0802 0.553116
\(478\) 16.0383 0.733576
\(479\) 23.0640 1.05382 0.526910 0.849921i \(-0.323350\pi\)
0.526910 + 0.849921i \(0.323350\pi\)
\(480\) −1.28489 −0.0586469
\(481\) −33.2654 −1.51677
\(482\) 27.6992 1.26166
\(483\) −5.39520 −0.245490
\(484\) −10.2767 −0.467122
\(485\) 7.33414 0.333026
\(486\) −22.3269 −1.01277
\(487\) −25.9056 −1.17389 −0.586947 0.809625i \(-0.699670\pi\)
−0.586947 + 0.809625i \(0.699670\pi\)
\(488\) 12.9842 0.587766
\(489\) −36.9428 −1.67061
\(490\) −1.26480 −0.0571378
\(491\) −11.9550 −0.539520 −0.269760 0.962928i \(-0.586944\pi\)
−0.269760 + 0.962928i \(0.586944\pi\)
\(492\) 21.7915 0.982436
\(493\) 6.51398 0.293375
\(494\) −23.7176 −1.06710
\(495\) 1.49252 0.0670836
\(496\) 10.4596 0.469648
\(497\) −2.46364 −0.110509
\(498\) 15.5842 0.698345
\(499\) 11.1509 0.499181 0.249591 0.968351i \(-0.419704\pi\)
0.249591 + 0.968351i \(0.419704\pi\)
\(500\) 4.92052 0.220052
\(501\) −37.3472 −1.66855
\(502\) 14.2405 0.635584
\(503\) 17.9630 0.800932 0.400466 0.916312i \(-0.368848\pi\)
0.400466 + 0.916312i \(0.368848\pi\)
\(504\) −7.36876 −0.328231
\(505\) 1.38718 0.0617287
\(506\) 0.850485 0.0378087
\(507\) −31.8414 −1.41412
\(508\) 1.40232 0.0622181
\(509\) 41.1077 1.82207 0.911033 0.412334i \(-0.135286\pi\)
0.911033 + 0.412334i \(0.135286\pi\)
\(510\) −8.36974 −0.370618
\(511\) −10.8047 −0.477971
\(512\) −1.00000 −0.0441942
\(513\) 5.68256 0.250891
\(514\) −23.2522 −1.02561
\(515\) −0.0567367 −0.00250012
\(516\) −10.7939 −0.475175
\(517\) 0.769192 0.0338290
\(518\) −13.9631 −0.613504
\(519\) 12.4539 0.546663
\(520\) −2.55038 −0.111842
\(521\) −17.8595 −0.782437 −0.391219 0.920298i \(-0.627946\pi\)
−0.391219 + 0.920298i \(0.627946\pi\)
\(522\) −3.47557 −0.152122
\(523\) 27.7568 1.21372 0.606861 0.794808i \(-0.292428\pi\)
0.606861 + 0.794808i \(0.292428\pi\)
\(524\) −17.9044 −0.782157
\(525\) 25.6005 1.11730
\(526\) −11.5860 −0.505176
\(527\) 68.1334 2.96794
\(528\) 2.16424 0.0941865
\(529\) 1.00000 0.0434783
\(530\) 1.75500 0.0762322
\(531\) 7.61826 0.330604
\(532\) −9.95543 −0.431623
\(533\) 43.2540 1.87354
\(534\) 6.21600 0.268993
\(535\) 9.12961 0.394707
\(536\) 14.7941 0.639007
\(537\) −60.8934 −2.62775
\(538\) 25.1049 1.08235
\(539\) 2.13040 0.0917628
\(540\) 0.611053 0.0262955
\(541\) 0.962692 0.0413894 0.0206947 0.999786i \(-0.493412\pi\)
0.0206947 + 0.999786i \(0.493412\pi\)
\(542\) −17.3946 −0.747161
\(543\) 51.6261 2.21549
\(544\) −6.51398 −0.279285
\(545\) 7.81636 0.334816
\(546\) −27.2512 −1.16624
\(547\) −6.40185 −0.273723 −0.136862 0.990590i \(-0.543702\pi\)
−0.136862 + 0.990590i \(0.543702\pi\)
\(548\) 21.6999 0.926973
\(549\) −45.1274 −1.92599
\(550\) −4.03559 −0.172078
\(551\) −4.69560 −0.200039
\(552\) 2.54471 0.108310
\(553\) 15.4790 0.658235
\(554\) −14.8366 −0.630349
\(555\) 8.46212 0.359197
\(556\) −14.3121 −0.606968
\(557\) −0.225527 −0.00955588 −0.00477794 0.999989i \(-0.501521\pi\)
−0.00477794 + 0.999989i \(0.501521\pi\)
\(558\) −36.3529 −1.53894
\(559\) −21.4249 −0.906175
\(560\) −1.07052 −0.0452378
\(561\) 14.0978 0.595211
\(562\) 17.4702 0.736935
\(563\) 4.86110 0.204871 0.102435 0.994740i \(-0.467336\pi\)
0.102435 + 0.994740i \(0.467336\pi\)
\(564\) 2.30148 0.0969097
\(565\) 3.68963 0.155224
\(566\) −14.4926 −0.609171
\(567\) −15.5771 −0.654176
\(568\) 1.16201 0.0487567
\(569\) 23.8190 0.998543 0.499272 0.866446i \(-0.333601\pi\)
0.499272 + 0.866446i \(0.333601\pi\)
\(570\) 6.03333 0.252708
\(571\) 15.4499 0.646559 0.323280 0.946304i \(-0.395215\pi\)
0.323280 + 0.946304i \(0.395215\pi\)
\(572\) 4.29581 0.179617
\(573\) −8.02723 −0.335342
\(574\) 18.1558 0.757810
\(575\) −4.74505 −0.197882
\(576\) 3.47557 0.144815
\(577\) −40.4140 −1.68246 −0.841228 0.540680i \(-0.818167\pi\)
−0.841228 + 0.540680i \(0.818167\pi\)
\(578\) −25.4320 −1.05783
\(579\) 32.6780 1.35805
\(580\) −0.504925 −0.0209659
\(581\) 12.9842 0.538674
\(582\) −36.9625 −1.53214
\(583\) −2.95608 −0.122428
\(584\) 5.09617 0.210881
\(585\) 8.86403 0.366482
\(586\) −5.92107 −0.244597
\(587\) 16.5023 0.681122 0.340561 0.940222i \(-0.389383\pi\)
0.340561 + 0.940222i \(0.389383\pi\)
\(588\) 6.37432 0.262872
\(589\) −49.1140 −2.02371
\(590\) 1.10677 0.0455649
\(591\) −49.1617 −2.02224
\(592\) 6.58588 0.270678
\(593\) −24.9126 −1.02304 −0.511519 0.859272i \(-0.670917\pi\)
−0.511519 + 0.859272i \(0.670917\pi\)
\(594\) −1.02925 −0.0422304
\(595\) −6.97335 −0.285880
\(596\) 2.45800 0.100683
\(597\) 9.44619 0.386607
\(598\) 5.05101 0.206551
\(599\) −15.0912 −0.616609 −0.308305 0.951288i \(-0.599762\pi\)
−0.308305 + 0.951288i \(0.599762\pi\)
\(600\) −12.0748 −0.492952
\(601\) −2.31951 −0.0946146 −0.0473073 0.998880i \(-0.515064\pi\)
−0.0473073 + 0.998880i \(0.515064\pi\)
\(602\) −8.99307 −0.366530
\(603\) −51.4179 −2.09390
\(604\) −19.9665 −0.812427
\(605\) 5.18895 0.210961
\(606\) −6.99109 −0.283994
\(607\) 26.3854 1.07095 0.535475 0.844551i \(-0.320132\pi\)
0.535475 + 0.844551i \(0.320132\pi\)
\(608\) 4.69560 0.190432
\(609\) −5.39520 −0.218624
\(610\) −6.55604 −0.265446
\(611\) 4.56822 0.184810
\(612\) 22.6398 0.915159
\(613\) 4.14453 0.167396 0.0836979 0.996491i \(-0.473327\pi\)
0.0836979 + 0.996491i \(0.473327\pi\)
\(614\) −17.5305 −0.707475
\(615\) −11.0031 −0.443686
\(616\) 1.80316 0.0726515
\(617\) −2.44707 −0.0985154 −0.0492577 0.998786i \(-0.515686\pi\)
−0.0492577 + 0.998786i \(0.515686\pi\)
\(618\) 0.285941 0.0115022
\(619\) −18.7390 −0.753186 −0.376593 0.926379i \(-0.622904\pi\)
−0.376593 + 0.926379i \(0.622904\pi\)
\(620\) −5.28129 −0.212102
\(621\) −1.21019 −0.0485631
\(622\) −23.7841 −0.953657
\(623\) 5.17894 0.207490
\(624\) 12.8534 0.514547
\(625\) 21.2408 0.849631
\(626\) −17.6353 −0.704849
\(627\) −10.1624 −0.405848
\(628\) −3.40867 −0.136021
\(629\) 42.9003 1.71055
\(630\) 3.72067 0.148235
\(631\) −39.8482 −1.58633 −0.793166 0.609005i \(-0.791569\pi\)
−0.793166 + 0.609005i \(0.791569\pi\)
\(632\) −7.30088 −0.290413
\(633\) −12.0323 −0.478241
\(634\) 16.7350 0.664631
\(635\) −0.708068 −0.0280988
\(636\) −8.84482 −0.350720
\(637\) 12.6524 0.501307
\(638\) 0.850485 0.0336710
\(639\) −4.03863 −0.159766
\(640\) 0.504925 0.0199589
\(641\) −39.7599 −1.57042 −0.785210 0.619229i \(-0.787445\pi\)
−0.785210 + 0.619229i \(0.787445\pi\)
\(642\) −46.0113 −1.81592
\(643\) −27.1807 −1.07190 −0.535951 0.844249i \(-0.680047\pi\)
−0.535951 + 0.844249i \(0.680047\pi\)
\(644\) 2.12016 0.0835460
\(645\) 5.45011 0.214598
\(646\) 30.5871 1.20343
\(647\) −32.2716 −1.26873 −0.634364 0.773035i \(-0.718738\pi\)
−0.634364 + 0.773035i \(0.718738\pi\)
\(648\) 7.34713 0.288622
\(649\) −1.86422 −0.0731769
\(650\) −23.9673 −0.940076
\(651\) −56.4314 −2.21172
\(652\) 14.5175 0.568549
\(653\) −0.535483 −0.0209551 −0.0104775 0.999945i \(-0.503335\pi\)
−0.0104775 + 0.999945i \(0.503335\pi\)
\(654\) −39.3928 −1.54038
\(655\) 9.04037 0.353237
\(656\) −8.56343 −0.334346
\(657\) −17.7121 −0.691014
\(658\) 1.91751 0.0747521
\(659\) −39.0365 −1.52065 −0.760324 0.649544i \(-0.774960\pi\)
−0.760324 + 0.649544i \(0.774960\pi\)
\(660\) −1.09278 −0.0425363
\(661\) 44.2521 1.72121 0.860603 0.509276i \(-0.170087\pi\)
0.860603 + 0.509276i \(0.170087\pi\)
\(662\) −34.5251 −1.34186
\(663\) 83.7267 3.25168
\(664\) −6.12415 −0.237663
\(665\) 5.02674 0.194929
\(666\) −22.8897 −0.886957
\(667\) 1.00000 0.0387202
\(668\) 14.6764 0.567846
\(669\) 45.4427 1.75692
\(670\) −7.46990 −0.288587
\(671\) 11.0429 0.426305
\(672\) 5.39520 0.208124
\(673\) 41.7056 1.60763 0.803816 0.594879i \(-0.202800\pi\)
0.803816 + 0.594879i \(0.202800\pi\)
\(674\) 26.2706 1.01191
\(675\) 5.74240 0.221025
\(676\) 12.5127 0.481259
\(677\) −33.1375 −1.27358 −0.636789 0.771038i \(-0.719738\pi\)
−0.636789 + 0.771038i \(0.719738\pi\)
\(678\) −18.5950 −0.714136
\(679\) −30.7958 −1.18183
\(680\) 3.28907 0.126130
\(681\) 38.8090 1.48716
\(682\) 8.89570 0.340634
\(683\) −22.2003 −0.849472 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(684\) −16.3199 −0.624007
\(685\) −10.9568 −0.418638
\(686\) 20.1520 0.769405
\(687\) 57.6041 2.19773
\(688\) 4.24169 0.161713
\(689\) −17.5561 −0.668835
\(690\) −1.28489 −0.0489149
\(691\) −2.71861 −0.103421 −0.0517104 0.998662i \(-0.516467\pi\)
−0.0517104 + 0.998662i \(0.516467\pi\)
\(692\) −4.89401 −0.186042
\(693\) −6.26702 −0.238064
\(694\) 18.7646 0.712295
\(695\) 7.22653 0.274118
\(696\) 2.54471 0.0964571
\(697\) −55.7820 −2.11289
\(698\) 21.6052 0.817769
\(699\) 19.4153 0.734354
\(700\) −10.0603 −0.380242
\(701\) 41.9653 1.58501 0.792504 0.609867i \(-0.208777\pi\)
0.792504 + 0.609867i \(0.208777\pi\)
\(702\) −6.11267 −0.230708
\(703\) −30.9247 −1.16635
\(704\) −0.850485 −0.0320539
\(705\) −1.16207 −0.0437662
\(706\) 25.0509 0.942803
\(707\) −5.82471 −0.219061
\(708\) −5.57787 −0.209629
\(709\) 8.11517 0.304772 0.152386 0.988321i \(-0.451304\pi\)
0.152386 + 0.988321i \(0.451304\pi\)
\(710\) −0.586725 −0.0220194
\(711\) 25.3747 0.951626
\(712\) −2.44271 −0.0915445
\(713\) 10.4596 0.391714
\(714\) 35.1442 1.31524
\(715\) −2.16906 −0.0811183
\(716\) 23.9294 0.894283
\(717\) 40.8129 1.52419
\(718\) −14.9448 −0.557734
\(719\) 50.3076 1.87616 0.938079 0.346421i \(-0.112603\pi\)
0.938079 + 0.346421i \(0.112603\pi\)
\(720\) −1.75490 −0.0654013
\(721\) 0.238235 0.00887234
\(722\) −3.04870 −0.113461
\(723\) 70.4865 2.62142
\(724\) −20.2876 −0.753983
\(725\) −4.74505 −0.176227
\(726\) −26.1512 −0.970562
\(727\) −17.3323 −0.642818 −0.321409 0.946941i \(-0.604156\pi\)
−0.321409 + 0.946941i \(0.604156\pi\)
\(728\) 10.7090 0.396900
\(729\) −34.7742 −1.28793
\(730\) −2.57318 −0.0952377
\(731\) 27.6303 1.02194
\(732\) 33.0410 1.22123
\(733\) −30.2353 −1.11677 −0.558383 0.829583i \(-0.688578\pi\)
−0.558383 + 0.829583i \(0.688578\pi\)
\(734\) −19.3929 −0.715807
\(735\) −3.21855 −0.118718
\(736\) −1.00000 −0.0368605
\(737\) 12.5821 0.463469
\(738\) 29.7628 1.09558
\(739\) 18.1462 0.667517 0.333759 0.942659i \(-0.391683\pi\)
0.333759 + 0.942659i \(0.391683\pi\)
\(740\) −3.32537 −0.122243
\(741\) −60.3544 −2.21717
\(742\) −7.36917 −0.270531
\(743\) −10.1890 −0.373797 −0.186898 0.982379i \(-0.559843\pi\)
−0.186898 + 0.982379i \(0.559843\pi\)
\(744\) 26.6166 0.975812
\(745\) −1.24110 −0.0454705
\(746\) −2.82893 −0.103575
\(747\) 21.2849 0.778774
\(748\) −5.54004 −0.202564
\(749\) −38.3349 −1.40073
\(750\) 12.5213 0.457213
\(751\) −20.5941 −0.751488 −0.375744 0.926723i \(-0.622613\pi\)
−0.375744 + 0.926723i \(0.622613\pi\)
\(752\) −0.904416 −0.0329806
\(753\) 36.2380 1.32059
\(754\) 5.05101 0.183947
\(755\) 10.0816 0.366907
\(756\) −2.56579 −0.0933168
\(757\) 32.2007 1.17036 0.585178 0.810905i \(-0.301025\pi\)
0.585178 + 0.810905i \(0.301025\pi\)
\(758\) −23.6832 −0.860213
\(759\) 2.16424 0.0785570
\(760\) −2.37093 −0.0860025
\(761\) −3.44428 −0.124855 −0.0624275 0.998050i \(-0.519884\pi\)
−0.0624275 + 0.998050i \(0.519884\pi\)
\(762\) 3.56851 0.129274
\(763\) −32.8206 −1.18819
\(764\) 3.15447 0.114125
\(765\) −11.4314 −0.413303
\(766\) 22.1269 0.799479
\(767\) −11.0715 −0.399770
\(768\) −2.54471 −0.0918245
\(769\) 6.46099 0.232989 0.116495 0.993191i \(-0.462834\pi\)
0.116495 + 0.993191i \(0.462834\pi\)
\(770\) −0.910462 −0.0328108
\(771\) −59.1702 −2.13096
\(772\) −12.8415 −0.462176
\(773\) 1.61735 0.0581720 0.0290860 0.999577i \(-0.490740\pi\)
0.0290860 + 0.999577i \(0.490740\pi\)
\(774\) −14.7423 −0.529901
\(775\) −49.6312 −1.78280
\(776\) 14.5252 0.521425
\(777\) −35.5321 −1.27471
\(778\) −10.9909 −0.394044
\(779\) 40.2105 1.44069
\(780\) −6.48999 −0.232379
\(781\) 0.988268 0.0353630
\(782\) −6.51398 −0.232940
\(783\) −1.21019 −0.0432485
\(784\) −2.50492 −0.0894616
\(785\) 1.72112 0.0614295
\(786\) −45.5616 −1.62513
\(787\) −4.54522 −0.162020 −0.0810098 0.996713i \(-0.525814\pi\)
−0.0810098 + 0.996713i \(0.525814\pi\)
\(788\) 19.3191 0.688216
\(789\) −29.4832 −1.04963
\(790\) 3.68639 0.131156
\(791\) −15.4926 −0.550855
\(792\) 2.95592 0.105034
\(793\) 65.5833 2.32893
\(794\) −20.2250 −0.717757
\(795\) 4.46597 0.158392
\(796\) −3.71208 −0.131571
\(797\) 5.36370 0.189992 0.0949960 0.995478i \(-0.469716\pi\)
0.0949960 + 0.995478i \(0.469716\pi\)
\(798\) −25.3337 −0.896804
\(799\) −5.89135 −0.208421
\(800\) 4.74505 0.167763
\(801\) 8.48981 0.299973
\(802\) −14.6135 −0.516019
\(803\) 4.33421 0.152951
\(804\) 37.6467 1.32770
\(805\) −1.07052 −0.0377309
\(806\) 52.8314 1.86091
\(807\) 63.8848 2.24885
\(808\) 2.74730 0.0966497
\(809\) 46.9420 1.65039 0.825197 0.564846i \(-0.191064\pi\)
0.825197 + 0.564846i \(0.191064\pi\)
\(810\) −3.70975 −0.130347
\(811\) −43.0804 −1.51276 −0.756379 0.654134i \(-0.773033\pi\)
−0.756379 + 0.654134i \(0.773033\pi\)
\(812\) 2.12016 0.0744030
\(813\) −44.2642 −1.55241
\(814\) 5.60119 0.196322
\(815\) −7.33024 −0.256767
\(816\) −16.5762 −0.580284
\(817\) −19.9173 −0.696819
\(818\) 27.4662 0.960333
\(819\) −37.2197 −1.30056
\(820\) 4.32389 0.150997
\(821\) 43.2641 1.50993 0.754963 0.655767i \(-0.227655\pi\)
0.754963 + 0.655767i \(0.227655\pi\)
\(822\) 55.2200 1.92602
\(823\) −2.41919 −0.0843276 −0.0421638 0.999111i \(-0.513425\pi\)
−0.0421638 + 0.999111i \(0.513425\pi\)
\(824\) −0.112367 −0.00391448
\(825\) −10.2694 −0.357536
\(826\) −4.64727 −0.161699
\(827\) −49.7698 −1.73066 −0.865332 0.501199i \(-0.832892\pi\)
−0.865332 + 0.501199i \(0.832892\pi\)
\(828\) 3.47557 0.120784
\(829\) 14.1282 0.490694 0.245347 0.969435i \(-0.421098\pi\)
0.245347 + 0.969435i \(0.421098\pi\)
\(830\) 3.09223 0.107333
\(831\) −37.7550 −1.30971
\(832\) −5.05101 −0.175112
\(833\) −16.3170 −0.565352
\(834\) −36.4202 −1.26113
\(835\) −7.41047 −0.256450
\(836\) 3.99354 0.138119
\(837\) −12.6580 −0.437525
\(838\) −6.96837 −0.240718
\(839\) −44.5225 −1.53709 −0.768544 0.639797i \(-0.779018\pi\)
−0.768544 + 0.639797i \(0.779018\pi\)
\(840\) −2.72417 −0.0939928
\(841\) 1.00000 0.0344828
\(842\) 4.95289 0.170688
\(843\) 44.4566 1.53117
\(844\) 4.72835 0.162757
\(845\) −6.31799 −0.217346
\(846\) 3.14336 0.108071
\(847\) −21.7882 −0.748651
\(848\) 3.47576 0.119358
\(849\) −36.8796 −1.26571
\(850\) 30.9092 1.06018
\(851\) 6.58588 0.225761
\(852\) 2.95697 0.101304
\(853\) 37.3437 1.27862 0.639312 0.768947i \(-0.279219\pi\)
0.639312 + 0.768947i \(0.279219\pi\)
\(854\) 27.5285 0.942007
\(855\) 8.24032 0.281813
\(856\) 18.0811 0.618000
\(857\) −17.8963 −0.611325 −0.305663 0.952140i \(-0.598878\pi\)
−0.305663 + 0.952140i \(0.598878\pi\)
\(858\) 10.9316 0.373199
\(859\) 3.37267 0.115074 0.0575370 0.998343i \(-0.481675\pi\)
0.0575370 + 0.998343i \(0.481675\pi\)
\(860\) −2.14174 −0.0730326
\(861\) 46.2014 1.57454
\(862\) 26.2304 0.893410
\(863\) 25.3401 0.862587 0.431294 0.902212i \(-0.358057\pi\)
0.431294 + 0.902212i \(0.358057\pi\)
\(864\) 1.21019 0.0411714
\(865\) 2.47111 0.0840201
\(866\) 13.7429 0.467002
\(867\) −64.7171 −2.19791
\(868\) 22.1759 0.752700
\(869\) −6.20929 −0.210636
\(870\) −1.28489 −0.0435618
\(871\) 74.7252 2.53197
\(872\) 15.4803 0.524228
\(873\) −50.4834 −1.70860
\(874\) 4.69560 0.158831
\(875\) 10.4323 0.352675
\(876\) 12.9683 0.438158
\(877\) 15.7169 0.530723 0.265362 0.964149i \(-0.414509\pi\)
0.265362 + 0.964149i \(0.414509\pi\)
\(878\) 12.9060 0.435555
\(879\) −15.0674 −0.508212
\(880\) 0.429431 0.0144761
\(881\) 12.5620 0.423225 0.211613 0.977354i \(-0.432129\pi\)
0.211613 + 0.977354i \(0.432129\pi\)
\(882\) 8.70604 0.293148
\(883\) 22.3539 0.752267 0.376134 0.926565i \(-0.377253\pi\)
0.376134 + 0.926565i \(0.377253\pi\)
\(884\) −32.9022 −1.10662
\(885\) 2.81641 0.0946724
\(886\) 0.0726725 0.00244148
\(887\) −24.9506 −0.837760 −0.418880 0.908042i \(-0.637577\pi\)
−0.418880 + 0.908042i \(0.637577\pi\)
\(888\) 16.7592 0.562401
\(889\) 2.97315 0.0997162
\(890\) 1.23339 0.0413432
\(891\) 6.24862 0.209337
\(892\) −17.8577 −0.597920
\(893\) 4.24678 0.142113
\(894\) 6.25489 0.209195
\(895\) −12.0825 −0.403875
\(896\) −2.12016 −0.0708295
\(897\) 12.8534 0.429162
\(898\) −26.4195 −0.881630
\(899\) 10.4596 0.348846
\(900\) −16.4918 −0.549725
\(901\) 22.6410 0.754283
\(902\) −7.28307 −0.242500
\(903\) −22.8848 −0.761558
\(904\) 7.30730 0.243037
\(905\) 10.2437 0.340512
\(906\) −50.8091 −1.68802
\(907\) 21.9243 0.727983 0.363991 0.931402i \(-0.381414\pi\)
0.363991 + 0.931402i \(0.381414\pi\)
\(908\) −15.2508 −0.506117
\(909\) −9.54843 −0.316701
\(910\) −5.40722 −0.179247
\(911\) −3.44740 −0.114218 −0.0571088 0.998368i \(-0.518188\pi\)
−0.0571088 + 0.998368i \(0.518188\pi\)
\(912\) 11.9490 0.395670
\(913\) −5.20850 −0.172376
\(914\) −12.5126 −0.413879
\(915\) −16.6832 −0.551531
\(916\) −22.6368 −0.747940
\(917\) −37.9602 −1.25356
\(918\) 7.88313 0.260182
\(919\) 21.5035 0.709335 0.354668 0.934992i \(-0.384594\pi\)
0.354668 + 0.934992i \(0.384594\pi\)
\(920\) 0.504925 0.0166469
\(921\) −44.6102 −1.46996
\(922\) 15.3188 0.504498
\(923\) 5.86930 0.193191
\(924\) 4.58853 0.150952
\(925\) −31.2503 −1.02750
\(926\) −20.5657 −0.675830
\(927\) 0.390538 0.0128269
\(928\) −1.00000 −0.0328266
\(929\) −16.5557 −0.543176 −0.271588 0.962414i \(-0.587549\pi\)
−0.271588 + 0.962414i \(0.587549\pi\)
\(930\) −13.4394 −0.440694
\(931\) 11.7621 0.385488
\(932\) −7.62966 −0.249918
\(933\) −60.5238 −1.98146
\(934\) 23.5091 0.769240
\(935\) 2.79730 0.0914816
\(936\) 17.5551 0.573808
\(937\) 15.9139 0.519883 0.259942 0.965624i \(-0.416297\pi\)
0.259942 + 0.965624i \(0.416297\pi\)
\(938\) 31.3658 1.02413
\(939\) −44.8769 −1.46450
\(940\) 0.456662 0.0148947
\(941\) −26.4156 −0.861124 −0.430562 0.902561i \(-0.641685\pi\)
−0.430562 + 0.902561i \(0.641685\pi\)
\(942\) −8.67410 −0.282617
\(943\) −8.56343 −0.278864
\(944\) 2.19195 0.0713417
\(945\) 1.29553 0.0421436
\(946\) 3.60750 0.117290
\(947\) 49.7387 1.61629 0.808146 0.588982i \(-0.200471\pi\)
0.808146 + 0.588982i \(0.200471\pi\)
\(948\) −18.5786 −0.603406
\(949\) 25.7408 0.835582
\(950\) −22.2809 −0.722887
\(951\) 42.5858 1.38094
\(952\) −13.8107 −0.447607
\(953\) −6.91727 −0.224072 −0.112036 0.993704i \(-0.535737\pi\)
−0.112036 + 0.993704i \(0.535737\pi\)
\(954\) −12.0802 −0.391112
\(955\) −1.59277 −0.0515409
\(956\) −16.0383 −0.518716
\(957\) 2.16424 0.0699600
\(958\) −23.0640 −0.745163
\(959\) 46.0072 1.48565
\(960\) 1.28489 0.0414696
\(961\) 78.4025 2.52911
\(962\) 33.2654 1.07252
\(963\) −62.8422 −2.02506
\(964\) −27.6992 −0.892130
\(965\) 6.48400 0.208727
\(966\) 5.39520 0.173588
\(967\) −11.7585 −0.378126 −0.189063 0.981965i \(-0.560545\pi\)
−0.189063 + 0.981965i \(0.560545\pi\)
\(968\) 10.2767 0.330305
\(969\) 77.8354 2.50043
\(970\) −7.33414 −0.235485
\(971\) 21.3961 0.686634 0.343317 0.939220i \(-0.388449\pi\)
0.343317 + 0.939220i \(0.388449\pi\)
\(972\) 22.3269 0.716136
\(973\) −30.3439 −0.972782
\(974\) 25.9056 0.830068
\(975\) −60.9900 −1.95324
\(976\) −12.9842 −0.415614
\(977\) 1.60423 0.0513238 0.0256619 0.999671i \(-0.491831\pi\)
0.0256619 + 0.999671i \(0.491831\pi\)
\(978\) 36.9428 1.18130
\(979\) −2.07749 −0.0663969
\(980\) 1.26480 0.0404025
\(981\) −53.8027 −1.71779
\(982\) 11.9550 0.381498
\(983\) −2.69905 −0.0860863 −0.0430431 0.999073i \(-0.513705\pi\)
−0.0430431 + 0.999073i \(0.513705\pi\)
\(984\) −21.7915 −0.694687
\(985\) −9.75471 −0.310811
\(986\) −6.51398 −0.207447
\(987\) 4.87950 0.155316
\(988\) 23.7176 0.754556
\(989\) 4.24169 0.134878
\(990\) −1.49252 −0.0474353
\(991\) −31.3642 −0.996317 −0.498158 0.867086i \(-0.665990\pi\)
−0.498158 + 0.867086i \(0.665990\pi\)
\(992\) −10.4596 −0.332091
\(993\) −87.8565 −2.78804
\(994\) 2.46364 0.0781418
\(995\) 1.87432 0.0594200
\(996\) −15.5842 −0.493804
\(997\) −4.19811 −0.132956 −0.0664778 0.997788i \(-0.521176\pi\)
−0.0664778 + 0.997788i \(0.521176\pi\)
\(998\) −11.1509 −0.352974
\(999\) −7.97014 −0.252164
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 1334.2.a.f.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1334.2.a.f.1.1 5 1.1 even 1 trivial