Properties

Label 2366.2.a.z.1.3
Level $2366$
Weight $2$
Character 2366.1
Self dual yes
Analytic conductor $18.893$
Analytic rank $1$
Dimension $3$
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] = [2366,2,Mod(1,2366)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2366, 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("2366.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.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: \(18.8926051182\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 2366.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.198062 q^{3} +1.00000 q^{4} +0.890084 q^{5} -0.198062 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.96077 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.198062 q^{3} +1.00000 q^{4} +0.890084 q^{5} -0.198062 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.96077 q^{9} +0.890084 q^{10} +0.664874 q^{11} -0.198062 q^{12} -1.00000 q^{14} -0.176292 q^{15} +1.00000 q^{16} -2.44504 q^{17} -2.96077 q^{18} -8.63102 q^{19} +0.890084 q^{20} +0.198062 q^{21} +0.664874 q^{22} -7.60388 q^{23} -0.198062 q^{24} -4.20775 q^{25} +1.18060 q^{27} -1.00000 q^{28} +3.60388 q^{29} -0.176292 q^{30} -1.50604 q^{31} +1.00000 q^{32} -0.131687 q^{33} -2.44504 q^{34} -0.890084 q^{35} -2.96077 q^{36} -5.60388 q^{37} -8.63102 q^{38} +0.890084 q^{40} +7.83877 q^{41} +0.198062 q^{42} +7.46681 q^{43} +0.664874 q^{44} -2.63533 q^{45} -7.60388 q^{46} -1.20775 q^{47} -0.198062 q^{48} +1.00000 q^{49} -4.20775 q^{50} +0.484271 q^{51} -4.89008 q^{53} +1.18060 q^{54} +0.591794 q^{55} -1.00000 q^{56} +1.70948 q^{57} +3.60388 q^{58} -12.8334 q^{59} -0.176292 q^{60} +7.70171 q^{61} -1.50604 q^{62} +2.96077 q^{63} +1.00000 q^{64} -0.131687 q^{66} +6.07606 q^{67} -2.44504 q^{68} +1.50604 q^{69} -0.890084 q^{70} -6.27413 q^{71} -2.96077 q^{72} +3.67994 q^{73} -5.60388 q^{74} +0.833397 q^{75} -8.63102 q^{76} -0.664874 q^{77} +4.37196 q^{79} +0.890084 q^{80} +8.64848 q^{81} +7.83877 q^{82} -2.92931 q^{83} +0.198062 q^{84} -2.17629 q^{85} +7.46681 q^{86} -0.713792 q^{87} +0.664874 q^{88} -12.5157 q^{89} -2.63533 q^{90} -7.60388 q^{92} +0.298290 q^{93} -1.20775 q^{94} -7.68233 q^{95} -0.198062 q^{96} -10.4450 q^{97} +1.00000 q^{98} -1.96854 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 5 q^{3} + 3 q^{4} + 2 q^{5} - 5 q^{6} - 3 q^{7} + 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 5 q^{3} + 3 q^{4} + 2 q^{5} - 5 q^{6} - 3 q^{7} + 3 q^{8} + 4 q^{9} + 2 q^{10} + 3 q^{11} - 5 q^{12} - 3 q^{14} - 8 q^{15} + 3 q^{16} - 7 q^{17} + 4 q^{18} - 11 q^{19} + 2 q^{20} + 5 q^{21} + 3 q^{22} - 14 q^{23} - 5 q^{24} + 5 q^{25} - 8 q^{27} - 3 q^{28} + 2 q^{29} - 8 q^{30} - 14 q^{31} + 3 q^{32} + 2 q^{33} - 7 q^{34} - 2 q^{35} + 4 q^{36} - 8 q^{37} - 11 q^{38} + 2 q^{40} - 9 q^{41} + 5 q^{42} + 19 q^{43} + 3 q^{44} + 26 q^{45} - 14 q^{46} + 14 q^{47} - 5 q^{48} + 3 q^{49} + 5 q^{50} + 14 q^{51} - 14 q^{53} - 8 q^{54} - 26 q^{55} - 3 q^{56} + 16 q^{57} + 2 q^{58} - 9 q^{59} - 8 q^{60} - 4 q^{61} - 14 q^{62} - 4 q^{63} + 3 q^{64} + 2 q^{66} + 3 q^{67} - 7 q^{68} + 14 q^{69} - 2 q^{70} - 8 q^{71} + 4 q^{72} - 13 q^{73} - 8 q^{74} - 27 q^{75} - 11 q^{76} - 3 q^{77} - 16 q^{79} + 2 q^{80} + 27 q^{81} - 9 q^{82} - 21 q^{83} + 5 q^{84} - 14 q^{85} + 19 q^{86} + 6 q^{87} + 3 q^{88} - 25 q^{89} + 26 q^{90} - 14 q^{92} + 28 q^{93} + 14 q^{94} - 40 q^{95} - 5 q^{96} - 31 q^{97} + 3 q^{98} - 31 q^{99}+O(q^{100}) \) 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
\(3\) −0.198062 −0.114351 −0.0571757 0.998364i \(-0.518210\pi\)
−0.0571757 + 0.998364i \(0.518210\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.890084 0.398058 0.199029 0.979994i \(-0.436221\pi\)
0.199029 + 0.979994i \(0.436221\pi\)
\(6\) −0.198062 −0.0808586
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.96077 −0.986924
\(10\) 0.890084 0.281469
\(11\) 0.664874 0.200467 0.100234 0.994964i \(-0.468041\pi\)
0.100234 + 0.994964i \(0.468041\pi\)
\(12\) −0.198062 −0.0571757
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) −0.176292 −0.0455184
\(16\) 1.00000 0.250000
\(17\) −2.44504 −0.593010 −0.296505 0.955031i \(-0.595821\pi\)
−0.296505 + 0.955031i \(0.595821\pi\)
\(18\) −2.96077 −0.697860
\(19\) −8.63102 −1.98009 −0.990046 0.140743i \(-0.955051\pi\)
−0.990046 + 0.140743i \(0.955051\pi\)
\(20\) 0.890084 0.199029
\(21\) 0.198062 0.0432207
\(22\) 0.664874 0.141752
\(23\) −7.60388 −1.58552 −0.792759 0.609535i \(-0.791356\pi\)
−0.792759 + 0.609535i \(0.791356\pi\)
\(24\) −0.198062 −0.0404293
\(25\) −4.20775 −0.841550
\(26\) 0 0
\(27\) 1.18060 0.227207
\(28\) −1.00000 −0.188982
\(29\) 3.60388 0.669223 0.334611 0.942356i \(-0.391395\pi\)
0.334611 + 0.942356i \(0.391395\pi\)
\(30\) −0.176292 −0.0321864
\(31\) −1.50604 −0.270493 −0.135246 0.990812i \(-0.543183\pi\)
−0.135246 + 0.990812i \(0.543183\pi\)
\(32\) 1.00000 0.176777
\(33\) −0.131687 −0.0229237
\(34\) −2.44504 −0.419321
\(35\) −0.890084 −0.150452
\(36\) −2.96077 −0.493462
\(37\) −5.60388 −0.921271 −0.460636 0.887589i \(-0.652379\pi\)
−0.460636 + 0.887589i \(0.652379\pi\)
\(38\) −8.63102 −1.40014
\(39\) 0 0
\(40\) 0.890084 0.140735
\(41\) 7.83877 1.22421 0.612105 0.790776i \(-0.290323\pi\)
0.612105 + 0.790776i \(0.290323\pi\)
\(42\) 0.198062 0.0305617
\(43\) 7.46681 1.13868 0.569339 0.822103i \(-0.307199\pi\)
0.569339 + 0.822103i \(0.307199\pi\)
\(44\) 0.664874 0.100234
\(45\) −2.63533 −0.392852
\(46\) −7.60388 −1.12113
\(47\) −1.20775 −0.176169 −0.0880843 0.996113i \(-0.528074\pi\)
−0.0880843 + 0.996113i \(0.528074\pi\)
\(48\) −0.198062 −0.0285878
\(49\) 1.00000 0.142857
\(50\) −4.20775 −0.595066
\(51\) 0.484271 0.0678114
\(52\) 0 0
\(53\) −4.89008 −0.671705 −0.335852 0.941915i \(-0.609024\pi\)
−0.335852 + 0.941915i \(0.609024\pi\)
\(54\) 1.18060 0.160660
\(55\) 0.591794 0.0797975
\(56\) −1.00000 −0.133631
\(57\) 1.70948 0.226426
\(58\) 3.60388 0.473212
\(59\) −12.8334 −1.67077 −0.835383 0.549668i \(-0.814754\pi\)
−0.835383 + 0.549668i \(0.814754\pi\)
\(60\) −0.176292 −0.0227592
\(61\) 7.70171 0.986103 0.493051 0.870000i \(-0.335881\pi\)
0.493051 + 0.870000i \(0.335881\pi\)
\(62\) −1.50604 −0.191267
\(63\) 2.96077 0.373022
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −0.131687 −0.0162095
\(67\) 6.07606 0.742309 0.371155 0.928571i \(-0.378962\pi\)
0.371155 + 0.928571i \(0.378962\pi\)
\(68\) −2.44504 −0.296505
\(69\) 1.50604 0.181306
\(70\) −0.890084 −0.106385
\(71\) −6.27413 −0.744602 −0.372301 0.928112i \(-0.621431\pi\)
−0.372301 + 0.928112i \(0.621431\pi\)
\(72\) −2.96077 −0.348930
\(73\) 3.67994 0.430704 0.215352 0.976536i \(-0.430910\pi\)
0.215352 + 0.976536i \(0.430910\pi\)
\(74\) −5.60388 −0.651437
\(75\) 0.833397 0.0962324
\(76\) −8.63102 −0.990046
\(77\) −0.664874 −0.0757695
\(78\) 0 0
\(79\) 4.37196 0.491884 0.245942 0.969285i \(-0.420903\pi\)
0.245942 + 0.969285i \(0.420903\pi\)
\(80\) 0.890084 0.0995144
\(81\) 8.64848 0.960942
\(82\) 7.83877 0.865648
\(83\) −2.92931 −0.321534 −0.160767 0.986992i \(-0.551397\pi\)
−0.160767 + 0.986992i \(0.551397\pi\)
\(84\) 0.198062 0.0216104
\(85\) −2.17629 −0.236052
\(86\) 7.46681 0.805167
\(87\) −0.713792 −0.0765265
\(88\) 0.664874 0.0708758
\(89\) −12.5157 −1.32666 −0.663332 0.748325i \(-0.730858\pi\)
−0.663332 + 0.748325i \(0.730858\pi\)
\(90\) −2.63533 −0.277789
\(91\) 0 0
\(92\) −7.60388 −0.792759
\(93\) 0.298290 0.0309312
\(94\) −1.20775 −0.124570
\(95\) −7.68233 −0.788191
\(96\) −0.198062 −0.0202146
\(97\) −10.4450 −1.06053 −0.530267 0.847831i \(-0.677908\pi\)
−0.530267 + 0.847831i \(0.677908\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.96854 −0.197846
\(100\) −4.20775 −0.420775
\(101\) −8.98792 −0.894331 −0.447166 0.894451i \(-0.647567\pi\)
−0.447166 + 0.894451i \(0.647567\pi\)
\(102\) 0.484271 0.0479499
\(103\) −8.27413 −0.815274 −0.407637 0.913144i \(-0.633647\pi\)
−0.407637 + 0.913144i \(0.633647\pi\)
\(104\) 0 0
\(105\) 0.176292 0.0172043
\(106\) −4.89008 −0.474967
\(107\) 14.5090 1.40264 0.701320 0.712846i \(-0.252594\pi\)
0.701320 + 0.712846i \(0.252594\pi\)
\(108\) 1.18060 0.113604
\(109\) 12.8659 1.23233 0.616166 0.787616i \(-0.288685\pi\)
0.616166 + 0.787616i \(0.288685\pi\)
\(110\) 0.591794 0.0564253
\(111\) 1.10992 0.105349
\(112\) −1.00000 −0.0944911
\(113\) −1.99761 −0.187919 −0.0939595 0.995576i \(-0.529952\pi\)
−0.0939595 + 0.995576i \(0.529952\pi\)
\(114\) 1.70948 0.160107
\(115\) −6.76809 −0.631127
\(116\) 3.60388 0.334611
\(117\) 0 0
\(118\) −12.8334 −1.18141
\(119\) 2.44504 0.224137
\(120\) −0.176292 −0.0160932
\(121\) −10.5579 −0.959813
\(122\) 7.70171 0.697280
\(123\) −1.55257 −0.139990
\(124\) −1.50604 −0.135246
\(125\) −8.19567 −0.733043
\(126\) 2.96077 0.263766
\(127\) 5.78017 0.512907 0.256453 0.966557i \(-0.417446\pi\)
0.256453 + 0.966557i \(0.417446\pi\)
\(128\) 1.00000 0.0883883
\(129\) −1.47889 −0.130209
\(130\) 0 0
\(131\) 2.05861 0.179861 0.0899306 0.995948i \(-0.471335\pi\)
0.0899306 + 0.995948i \(0.471335\pi\)
\(132\) −0.131687 −0.0114618
\(133\) 8.63102 0.748405
\(134\) 6.07606 0.524892
\(135\) 1.05084 0.0904416
\(136\) −2.44504 −0.209661
\(137\) 6.24160 0.533256 0.266628 0.963800i \(-0.414090\pi\)
0.266628 + 0.963800i \(0.414090\pi\)
\(138\) 1.50604 0.128203
\(139\) −7.55496 −0.640803 −0.320402 0.947282i \(-0.603818\pi\)
−0.320402 + 0.947282i \(0.603818\pi\)
\(140\) −0.890084 −0.0752258
\(141\) 0.239210 0.0201451
\(142\) −6.27413 −0.526513
\(143\) 0 0
\(144\) −2.96077 −0.246731
\(145\) 3.20775 0.266389
\(146\) 3.67994 0.304554
\(147\) −0.198062 −0.0163359
\(148\) −5.60388 −0.460636
\(149\) −16.8901 −1.38369 −0.691845 0.722046i \(-0.743202\pi\)
−0.691845 + 0.722046i \(0.743202\pi\)
\(150\) 0.833397 0.0680466
\(151\) −11.5254 −0.937925 −0.468963 0.883218i \(-0.655372\pi\)
−0.468963 + 0.883218i \(0.655372\pi\)
\(152\) −8.63102 −0.700068
\(153\) 7.23921 0.585255
\(154\) −0.664874 −0.0535771
\(155\) −1.34050 −0.107672
\(156\) 0 0
\(157\) 11.7017 0.933898 0.466949 0.884284i \(-0.345353\pi\)
0.466949 + 0.884284i \(0.345353\pi\)
\(158\) 4.37196 0.347815
\(159\) 0.968541 0.0768103
\(160\) 0.890084 0.0703673
\(161\) 7.60388 0.599269
\(162\) 8.64848 0.679489
\(163\) −20.5187 −1.60715 −0.803575 0.595204i \(-0.797071\pi\)
−0.803575 + 0.595204i \(0.797071\pi\)
\(164\) 7.83877 0.612105
\(165\) −0.117212 −0.00912494
\(166\) −2.92931 −0.227359
\(167\) 6.46980 0.500648 0.250324 0.968162i \(-0.419463\pi\)
0.250324 + 0.968162i \(0.419463\pi\)
\(168\) 0.198062 0.0152808
\(169\) 0 0
\(170\) −2.17629 −0.166914
\(171\) 25.5545 1.95420
\(172\) 7.46681 0.569339
\(173\) 24.5133 1.86371 0.931857 0.362825i \(-0.118188\pi\)
0.931857 + 0.362825i \(0.118188\pi\)
\(174\) −0.713792 −0.0541124
\(175\) 4.20775 0.318076
\(176\) 0.664874 0.0501168
\(177\) 2.54181 0.191054
\(178\) −12.5157 −0.938094
\(179\) 3.67696 0.274829 0.137414 0.990514i \(-0.456121\pi\)
0.137414 + 0.990514i \(0.456121\pi\)
\(180\) −2.63533 −0.196426
\(181\) 13.7995 1.02571 0.512856 0.858475i \(-0.328587\pi\)
0.512856 + 0.858475i \(0.328587\pi\)
\(182\) 0 0
\(183\) −1.52542 −0.112762
\(184\) −7.60388 −0.560565
\(185\) −4.98792 −0.366719
\(186\) 0.298290 0.0218717
\(187\) −1.62565 −0.118879
\(188\) −1.20775 −0.0880843
\(189\) −1.18060 −0.0858763
\(190\) −7.68233 −0.557335
\(191\) 20.0737 1.45248 0.726240 0.687441i \(-0.241266\pi\)
0.726240 + 0.687441i \(0.241266\pi\)
\(192\) −0.198062 −0.0142939
\(193\) 3.74094 0.269279 0.134639 0.990895i \(-0.457012\pi\)
0.134639 + 0.990895i \(0.457012\pi\)
\(194\) −10.4450 −0.749910
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 0.0978347 0.00697043 0.00348522 0.999994i \(-0.498891\pi\)
0.00348522 + 0.999994i \(0.498891\pi\)
\(198\) −1.96854 −0.139898
\(199\) −20.4892 −1.45244 −0.726219 0.687463i \(-0.758724\pi\)
−0.726219 + 0.687463i \(0.758724\pi\)
\(200\) −4.20775 −0.297533
\(201\) −1.20344 −0.0848840
\(202\) −8.98792 −0.632388
\(203\) −3.60388 −0.252942
\(204\) 0.484271 0.0339057
\(205\) 6.97716 0.487306
\(206\) −8.27413 −0.576486
\(207\) 22.5133 1.56479
\(208\) 0 0
\(209\) −5.73855 −0.396944
\(210\) 0.176292 0.0121653
\(211\) 22.1903 1.52764 0.763821 0.645428i \(-0.223321\pi\)
0.763821 + 0.645428i \(0.223321\pi\)
\(212\) −4.89008 −0.335852
\(213\) 1.24267 0.0851462
\(214\) 14.5090 0.991817
\(215\) 6.64609 0.453259
\(216\) 1.18060 0.0803299
\(217\) 1.50604 0.102237
\(218\) 12.8659 0.871390
\(219\) −0.728857 −0.0492516
\(220\) 0.591794 0.0398987
\(221\) 0 0
\(222\) 1.10992 0.0744927
\(223\) −22.3478 −1.49652 −0.748260 0.663406i \(-0.769110\pi\)
−0.748260 + 0.663406i \(0.769110\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 12.4582 0.830546
\(226\) −1.99761 −0.132879
\(227\) −2.61356 −0.173468 −0.0867342 0.996231i \(-0.527643\pi\)
−0.0867342 + 0.996231i \(0.527643\pi\)
\(228\) 1.70948 0.113213
\(229\) −16.0978 −1.06377 −0.531887 0.846815i \(-0.678517\pi\)
−0.531887 + 0.846815i \(0.678517\pi\)
\(230\) −6.76809 −0.446274
\(231\) 0.131687 0.00866434
\(232\) 3.60388 0.236606
\(233\) −1.80194 −0.118049 −0.0590244 0.998257i \(-0.518799\pi\)
−0.0590244 + 0.998257i \(0.518799\pi\)
\(234\) 0 0
\(235\) −1.07500 −0.0701252
\(236\) −12.8334 −0.835383
\(237\) −0.865921 −0.0562476
\(238\) 2.44504 0.158489
\(239\) −3.94571 −0.255226 −0.127613 0.991824i \(-0.540732\pi\)
−0.127613 + 0.991824i \(0.540732\pi\)
\(240\) −0.176292 −0.0113796
\(241\) −2.79225 −0.179865 −0.0899323 0.995948i \(-0.528665\pi\)
−0.0899323 + 0.995948i \(0.528665\pi\)
\(242\) −10.5579 −0.678690
\(243\) −5.25475 −0.337092
\(244\) 7.70171 0.493051
\(245\) 0.890084 0.0568654
\(246\) −1.55257 −0.0989879
\(247\) 0 0
\(248\) −1.50604 −0.0956337
\(249\) 0.580186 0.0367678
\(250\) −8.19567 −0.518340
\(251\) −10.6649 −0.673161 −0.336580 0.941655i \(-0.609270\pi\)
−0.336580 + 0.941655i \(0.609270\pi\)
\(252\) 2.96077 0.186511
\(253\) −5.05562 −0.317844
\(254\) 5.78017 0.362680
\(255\) 0.431041 0.0269929
\(256\) 1.00000 0.0625000
\(257\) −25.4620 −1.58828 −0.794139 0.607736i \(-0.792078\pi\)
−0.794139 + 0.607736i \(0.792078\pi\)
\(258\) −1.47889 −0.0920719
\(259\) 5.60388 0.348208
\(260\) 0 0
\(261\) −10.6703 −0.660472
\(262\) 2.05861 0.127181
\(263\) 4.81163 0.296698 0.148349 0.988935i \(-0.452604\pi\)
0.148349 + 0.988935i \(0.452604\pi\)
\(264\) −0.131687 −0.00810475
\(265\) −4.35258 −0.267377
\(266\) 8.63102 0.529202
\(267\) 2.47889 0.151706
\(268\) 6.07606 0.371155
\(269\) 24.5133 1.49460 0.747302 0.664484i \(-0.231349\pi\)
0.747302 + 0.664484i \(0.231349\pi\)
\(270\) 1.05084 0.0639519
\(271\) 22.5241 1.36824 0.684121 0.729369i \(-0.260186\pi\)
0.684121 + 0.729369i \(0.260186\pi\)
\(272\) −2.44504 −0.148252
\(273\) 0 0
\(274\) 6.24160 0.377069
\(275\) −2.79763 −0.168703
\(276\) 1.50604 0.0906530
\(277\) 6.45042 0.387568 0.193784 0.981044i \(-0.437924\pi\)
0.193784 + 0.981044i \(0.437924\pi\)
\(278\) −7.55496 −0.453116
\(279\) 4.45904 0.266956
\(280\) −0.890084 −0.0531927
\(281\) −30.6165 −1.82643 −0.913215 0.407478i \(-0.866408\pi\)
−0.913215 + 0.407478i \(0.866408\pi\)
\(282\) 0.239210 0.0142447
\(283\) −16.9922 −1.01008 −0.505042 0.863095i \(-0.668523\pi\)
−0.505042 + 0.863095i \(0.668523\pi\)
\(284\) −6.27413 −0.372301
\(285\) 1.52158 0.0901306
\(286\) 0 0
\(287\) −7.83877 −0.462708
\(288\) −2.96077 −0.174465
\(289\) −11.0218 −0.648339
\(290\) 3.20775 0.188366
\(291\) 2.06877 0.121273
\(292\) 3.67994 0.215352
\(293\) 24.8116 1.44951 0.724755 0.689006i \(-0.241953\pi\)
0.724755 + 0.689006i \(0.241953\pi\)
\(294\) −0.198062 −0.0115512
\(295\) −11.4228 −0.665061
\(296\) −5.60388 −0.325719
\(297\) 0.784953 0.0455476
\(298\) −16.8901 −0.978416
\(299\) 0 0
\(300\) 0.833397 0.0481162
\(301\) −7.46681 −0.430380
\(302\) −11.5254 −0.663213
\(303\) 1.78017 0.102268
\(304\) −8.63102 −0.495023
\(305\) 6.85517 0.392526
\(306\) 7.23921 0.413838
\(307\) −14.4209 −0.823043 −0.411522 0.911400i \(-0.635002\pi\)
−0.411522 + 0.911400i \(0.635002\pi\)
\(308\) −0.664874 −0.0378847
\(309\) 1.63879 0.0932276
\(310\) −1.34050 −0.0761354
\(311\) −22.1521 −1.25613 −0.628066 0.778160i \(-0.716153\pi\)
−0.628066 + 0.778160i \(0.716153\pi\)
\(312\) 0 0
\(313\) 25.5550 1.44445 0.722226 0.691657i \(-0.243119\pi\)
0.722226 + 0.691657i \(0.243119\pi\)
\(314\) 11.7017 0.660366
\(315\) 2.63533 0.148484
\(316\) 4.37196 0.245942
\(317\) 13.3297 0.748673 0.374337 0.927293i \(-0.377870\pi\)
0.374337 + 0.927293i \(0.377870\pi\)
\(318\) 0.968541 0.0543131
\(319\) 2.39612 0.134157
\(320\) 0.890084 0.0497572
\(321\) −2.87369 −0.160394
\(322\) 7.60388 0.423747
\(323\) 21.1032 1.17421
\(324\) 8.64848 0.480471
\(325\) 0 0
\(326\) −20.5187 −1.13643
\(327\) −2.54825 −0.140919
\(328\) 7.83877 0.432824
\(329\) 1.20775 0.0665855
\(330\) −0.117212 −0.00645231
\(331\) 18.9638 1.04234 0.521171 0.853452i \(-0.325495\pi\)
0.521171 + 0.853452i \(0.325495\pi\)
\(332\) −2.92931 −0.160767
\(333\) 16.5918 0.909225
\(334\) 6.46980 0.354011
\(335\) 5.40821 0.295482
\(336\) 0.198062 0.0108052
\(337\) 29.8823 1.62779 0.813897 0.581010i \(-0.197342\pi\)
0.813897 + 0.581010i \(0.197342\pi\)
\(338\) 0 0
\(339\) 0.395651 0.0214888
\(340\) −2.17629 −0.118026
\(341\) −1.00133 −0.0542249
\(342\) 25.5545 1.38183
\(343\) −1.00000 −0.0539949
\(344\) 7.46681 0.402584
\(345\) 1.34050 0.0721702
\(346\) 24.5133 1.31785
\(347\) −5.78017 −0.310296 −0.155148 0.987891i \(-0.549585\pi\)
−0.155148 + 0.987891i \(0.549585\pi\)
\(348\) −0.713792 −0.0382633
\(349\) 7.12929 0.381622 0.190811 0.981627i \(-0.438888\pi\)
0.190811 + 0.981627i \(0.438888\pi\)
\(350\) 4.20775 0.224914
\(351\) 0 0
\(352\) 0.664874 0.0354379
\(353\) −7.09113 −0.377423 −0.188711 0.982033i \(-0.560431\pi\)
−0.188711 + 0.982033i \(0.560431\pi\)
\(354\) 2.54181 0.135096
\(355\) −5.58450 −0.296394
\(356\) −12.5157 −0.663332
\(357\) −0.484271 −0.0256303
\(358\) 3.67696 0.194333
\(359\) 25.9081 1.36738 0.683689 0.729773i \(-0.260374\pi\)
0.683689 + 0.729773i \(0.260374\pi\)
\(360\) −2.63533 −0.138894
\(361\) 55.4946 2.92077
\(362\) 13.7995 0.725288
\(363\) 2.09113 0.109756
\(364\) 0 0
\(365\) 3.27545 0.171445
\(366\) −1.52542 −0.0797349
\(367\) −30.4698 −1.59051 −0.795255 0.606275i \(-0.792663\pi\)
−0.795255 + 0.606275i \(0.792663\pi\)
\(368\) −7.60388 −0.396379
\(369\) −23.2088 −1.20820
\(370\) −4.98792 −0.259310
\(371\) 4.89008 0.253880
\(372\) 0.298290 0.0154656
\(373\) −7.42758 −0.384586 −0.192293 0.981338i \(-0.561592\pi\)
−0.192293 + 0.981338i \(0.561592\pi\)
\(374\) −1.62565 −0.0840601
\(375\) 1.62325 0.0838244
\(376\) −1.20775 −0.0622850
\(377\) 0 0
\(378\) −1.18060 −0.0607237
\(379\) −27.5472 −1.41500 −0.707502 0.706711i \(-0.750178\pi\)
−0.707502 + 0.706711i \(0.750178\pi\)
\(380\) −7.68233 −0.394095
\(381\) −1.14483 −0.0586516
\(382\) 20.0737 1.02706
\(383\) −30.8853 −1.57817 −0.789083 0.614287i \(-0.789444\pi\)
−0.789083 + 0.614287i \(0.789444\pi\)
\(384\) −0.198062 −0.0101073
\(385\) −0.591794 −0.0301606
\(386\) 3.74094 0.190409
\(387\) −22.1075 −1.12379
\(388\) −10.4450 −0.530267
\(389\) 32.0301 1.62399 0.811996 0.583663i \(-0.198381\pi\)
0.811996 + 0.583663i \(0.198381\pi\)
\(390\) 0 0
\(391\) 18.5918 0.940227
\(392\) 1.00000 0.0505076
\(393\) −0.407732 −0.0205674
\(394\) 0.0978347 0.00492884
\(395\) 3.89141 0.195798
\(396\) −1.96854 −0.0989229
\(397\) −7.06638 −0.354651 −0.177326 0.984152i \(-0.556745\pi\)
−0.177326 + 0.984152i \(0.556745\pi\)
\(398\) −20.4892 −1.02703
\(399\) −1.70948 −0.0855810
\(400\) −4.20775 −0.210388
\(401\) −22.6058 −1.12888 −0.564440 0.825474i \(-0.690908\pi\)
−0.564440 + 0.825474i \(0.690908\pi\)
\(402\) −1.20344 −0.0600221
\(403\) 0 0
\(404\) −8.98792 −0.447166
\(405\) 7.69787 0.382510
\(406\) −3.60388 −0.178857
\(407\) −3.72587 −0.184685
\(408\) 0.484271 0.0239750
\(409\) −14.8006 −0.731843 −0.365922 0.930646i \(-0.619246\pi\)
−0.365922 + 0.930646i \(0.619246\pi\)
\(410\) 6.97716 0.344578
\(411\) −1.23623 −0.0609785
\(412\) −8.27413 −0.407637
\(413\) 12.8334 0.631490
\(414\) 22.5133 1.10647
\(415\) −2.60733 −0.127989
\(416\) 0 0
\(417\) 1.49635 0.0732767
\(418\) −5.73855 −0.280681
\(419\) −4.79225 −0.234117 −0.117058 0.993125i \(-0.537346\pi\)
−0.117058 + 0.993125i \(0.537346\pi\)
\(420\) 0.176292 0.00860217
\(421\) 7.78017 0.379182 0.189591 0.981863i \(-0.439284\pi\)
0.189591 + 0.981863i \(0.439284\pi\)
\(422\) 22.1903 1.08021
\(423\) 3.57587 0.173865
\(424\) −4.89008 −0.237483
\(425\) 10.2881 0.499047
\(426\) 1.24267 0.0602074
\(427\) −7.70171 −0.372712
\(428\) 14.5090 0.701320
\(429\) 0 0
\(430\) 6.64609 0.320503
\(431\) −9.46250 −0.455793 −0.227896 0.973685i \(-0.573185\pi\)
−0.227896 + 0.973685i \(0.573185\pi\)
\(432\) 1.18060 0.0568018
\(433\) −10.5351 −0.506285 −0.253142 0.967429i \(-0.581464\pi\)
−0.253142 + 0.967429i \(0.581464\pi\)
\(434\) 1.50604 0.0722923
\(435\) −0.635334 −0.0304620
\(436\) 12.8659 0.616166
\(437\) 65.6292 3.13947
\(438\) −0.728857 −0.0348261
\(439\) −2.08708 −0.0996109 −0.0498055 0.998759i \(-0.515860\pi\)
−0.0498055 + 0.998759i \(0.515860\pi\)
\(440\) 0.591794 0.0282127
\(441\) −2.96077 −0.140989
\(442\) 0 0
\(443\) 17.6635 0.839220 0.419610 0.907704i \(-0.362167\pi\)
0.419610 + 0.907704i \(0.362167\pi\)
\(444\) 1.10992 0.0526743
\(445\) −11.1400 −0.528089
\(446\) −22.3478 −1.05820
\(447\) 3.34529 0.158227
\(448\) −1.00000 −0.0472456
\(449\) 29.9124 1.41166 0.705828 0.708383i \(-0.250575\pi\)
0.705828 + 0.708383i \(0.250575\pi\)
\(450\) 12.4582 0.587285
\(451\) 5.21180 0.245414
\(452\) −1.99761 −0.0939595
\(453\) 2.28275 0.107253
\(454\) −2.61356 −0.122661
\(455\) 0 0
\(456\) 1.70948 0.0800537
\(457\) 37.5120 1.75474 0.877369 0.479816i \(-0.159297\pi\)
0.877369 + 0.479816i \(0.159297\pi\)
\(458\) −16.0978 −0.752202
\(459\) −2.88663 −0.134736
\(460\) −6.76809 −0.315564
\(461\) 36.9288 1.71995 0.859974 0.510338i \(-0.170480\pi\)
0.859974 + 0.510338i \(0.170480\pi\)
\(462\) 0.131687 0.00612661
\(463\) −23.8237 −1.10718 −0.553591 0.832789i \(-0.686743\pi\)
−0.553591 + 0.832789i \(0.686743\pi\)
\(464\) 3.60388 0.167306
\(465\) 0.265503 0.0123124
\(466\) −1.80194 −0.0834732
\(467\) −19.4010 −0.897772 −0.448886 0.893589i \(-0.648179\pi\)
−0.448886 + 0.893589i \(0.648179\pi\)
\(468\) 0 0
\(469\) −6.07606 −0.280567
\(470\) −1.07500 −0.0495860
\(471\) −2.31767 −0.106792
\(472\) −12.8334 −0.590705
\(473\) 4.96449 0.228268
\(474\) −0.865921 −0.0397730
\(475\) 36.3172 1.66635
\(476\) 2.44504 0.112068
\(477\) 14.4784 0.662921
\(478\) −3.94571 −0.180472
\(479\) 20.7224 0.946831 0.473416 0.880839i \(-0.343021\pi\)
0.473416 + 0.880839i \(0.343021\pi\)
\(480\) −0.176292 −0.00804659
\(481\) 0 0
\(482\) −2.79225 −0.127183
\(483\) −1.50604 −0.0685272
\(484\) −10.5579 −0.479906
\(485\) −9.29696 −0.422153
\(486\) −5.25475 −0.238360
\(487\) 31.9758 1.44896 0.724482 0.689294i \(-0.242079\pi\)
0.724482 + 0.689294i \(0.242079\pi\)
\(488\) 7.70171 0.348640
\(489\) 4.06398 0.183780
\(490\) 0.890084 0.0402099
\(491\) −9.24027 −0.417008 −0.208504 0.978022i \(-0.566859\pi\)
−0.208504 + 0.978022i \(0.566859\pi\)
\(492\) −1.55257 −0.0699950
\(493\) −8.81163 −0.396856
\(494\) 0 0
\(495\) −1.75217 −0.0787540
\(496\) −1.50604 −0.0676232
\(497\) 6.27413 0.281433
\(498\) 0.580186 0.0259988
\(499\) 9.94630 0.445257 0.222629 0.974903i \(-0.428536\pi\)
0.222629 + 0.974903i \(0.428536\pi\)
\(500\) −8.19567 −0.366521
\(501\) −1.28142 −0.0572497
\(502\) −10.6649 −0.475997
\(503\) 21.2379 0.946950 0.473475 0.880807i \(-0.342999\pi\)
0.473475 + 0.880807i \(0.342999\pi\)
\(504\) 2.96077 0.131883
\(505\) −8.00000 −0.355995
\(506\) −5.05562 −0.224750
\(507\) 0 0
\(508\) 5.78017 0.256453
\(509\) −8.29350 −0.367603 −0.183802 0.982963i \(-0.558840\pi\)
−0.183802 + 0.982963i \(0.558840\pi\)
\(510\) 0.431041 0.0190868
\(511\) −3.67994 −0.162791
\(512\) 1.00000 0.0441942
\(513\) −10.1898 −0.449891
\(514\) −25.4620 −1.12308
\(515\) −7.36467 −0.324526
\(516\) −1.47889 −0.0651047
\(517\) −0.803003 −0.0353160
\(518\) 5.60388 0.246220
\(519\) −4.85517 −0.213118
\(520\) 0 0
\(521\) 7.33214 0.321227 0.160613 0.987017i \(-0.448653\pi\)
0.160613 + 0.987017i \(0.448653\pi\)
\(522\) −10.6703 −0.467024
\(523\) −36.5816 −1.59960 −0.799802 0.600265i \(-0.795062\pi\)
−0.799802 + 0.600265i \(0.795062\pi\)
\(524\) 2.05861 0.0899306
\(525\) −0.833397 −0.0363724
\(526\) 4.81163 0.209797
\(527\) 3.68233 0.160405
\(528\) −0.131687 −0.00573092
\(529\) 34.8189 1.51387
\(530\) −4.35258 −0.189064
\(531\) 37.9968 1.64892
\(532\) 8.63102 0.374202
\(533\) 0 0
\(534\) 2.47889 0.107272
\(535\) 12.9142 0.558332
\(536\) 6.07606 0.262446
\(537\) −0.728266 −0.0314270
\(538\) 24.5133 1.05684
\(539\) 0.664874 0.0286382
\(540\) 1.05084 0.0452208
\(541\) 19.7125 0.847505 0.423753 0.905778i \(-0.360713\pi\)
0.423753 + 0.905778i \(0.360713\pi\)
\(542\) 22.5241 0.967493
\(543\) −2.73317 −0.117292
\(544\) −2.44504 −0.104830
\(545\) 11.4517 0.490539
\(546\) 0 0
\(547\) 2.66786 0.114069 0.0570347 0.998372i \(-0.481835\pi\)
0.0570347 + 0.998372i \(0.481835\pi\)
\(548\) 6.24160 0.266628
\(549\) −22.8030 −0.973208
\(550\) −2.79763 −0.119291
\(551\) −31.1051 −1.32512
\(552\) 1.50604 0.0641014
\(553\) −4.37196 −0.185915
\(554\) 6.45042 0.274052
\(555\) 0.987918 0.0419348
\(556\) −7.55496 −0.320402
\(557\) 16.1655 0.684956 0.342478 0.939526i \(-0.388734\pi\)
0.342478 + 0.939526i \(0.388734\pi\)
\(558\) 4.45904 0.188766
\(559\) 0 0
\(560\) −0.890084 −0.0376129
\(561\) 0.321979 0.0135940
\(562\) −30.6165 −1.29148
\(563\) −34.5241 −1.45502 −0.727508 0.686099i \(-0.759322\pi\)
−0.727508 + 0.686099i \(0.759322\pi\)
\(564\) 0.239210 0.0100726
\(565\) −1.77804 −0.0748026
\(566\) −16.9922 −0.714237
\(567\) −8.64848 −0.363202
\(568\) −6.27413 −0.263257
\(569\) 3.66919 0.153820 0.0769101 0.997038i \(-0.475495\pi\)
0.0769101 + 0.997038i \(0.475495\pi\)
\(570\) 1.52158 0.0637320
\(571\) 8.29590 0.347172 0.173586 0.984819i \(-0.444464\pi\)
0.173586 + 0.984819i \(0.444464\pi\)
\(572\) 0 0
\(573\) −3.97584 −0.166093
\(574\) −7.83877 −0.327184
\(575\) 31.9952 1.33429
\(576\) −2.96077 −0.123365
\(577\) −15.0949 −0.628407 −0.314203 0.949356i \(-0.601737\pi\)
−0.314203 + 0.949356i \(0.601737\pi\)
\(578\) −11.0218 −0.458445
\(579\) −0.740939 −0.0307924
\(580\) 3.20775 0.133195
\(581\) 2.92931 0.121528
\(582\) 2.06877 0.0857532
\(583\) −3.25129 −0.134655
\(584\) 3.67994 0.152277
\(585\) 0 0
\(586\) 24.8116 1.02496
\(587\) −19.0465 −0.786134 −0.393067 0.919510i \(-0.628586\pi\)
−0.393067 + 0.919510i \(0.628586\pi\)
\(588\) −0.198062 −0.00816795
\(589\) 12.9987 0.535601
\(590\) −11.4228 −0.470269
\(591\) −0.0193774 −0.000797078 0
\(592\) −5.60388 −0.230318
\(593\) 23.6770 0.972296 0.486148 0.873876i \(-0.338402\pi\)
0.486148 + 0.873876i \(0.338402\pi\)
\(594\) 0.784953 0.0322070
\(595\) 2.17629 0.0892193
\(596\) −16.8901 −0.691845
\(597\) 4.05813 0.166088
\(598\) 0 0
\(599\) −5.41896 −0.221413 −0.110706 0.993853i \(-0.535311\pi\)
−0.110706 + 0.993853i \(0.535311\pi\)
\(600\) 0.833397 0.0340233
\(601\) −43.9861 −1.79423 −0.897116 0.441796i \(-0.854342\pi\)
−0.897116 + 0.441796i \(0.854342\pi\)
\(602\) −7.46681 −0.304325
\(603\) −17.9898 −0.732603
\(604\) −11.5254 −0.468963
\(605\) −9.39745 −0.382061
\(606\) 1.78017 0.0723144
\(607\) −35.6426 −1.44669 −0.723345 0.690487i \(-0.757396\pi\)
−0.723345 + 0.690487i \(0.757396\pi\)
\(608\) −8.63102 −0.350034
\(609\) 0.713792 0.0289243
\(610\) 6.85517 0.277558
\(611\) 0 0
\(612\) 7.23921 0.292628
\(613\) −18.7573 −0.757602 −0.378801 0.925478i \(-0.623663\pi\)
−0.378801 + 0.925478i \(0.623663\pi\)
\(614\) −14.4209 −0.581979
\(615\) −1.38191 −0.0557241
\(616\) −0.664874 −0.0267886
\(617\) 35.2247 1.41809 0.709047 0.705161i \(-0.249125\pi\)
0.709047 + 0.705161i \(0.249125\pi\)
\(618\) 1.63879 0.0659219
\(619\) 44.5870 1.79210 0.896052 0.443950i \(-0.146423\pi\)
0.896052 + 0.443950i \(0.146423\pi\)
\(620\) −1.34050 −0.0538359
\(621\) −8.97716 −0.360241
\(622\) −22.1521 −0.888219
\(623\) 12.5157 0.501432
\(624\) 0 0
\(625\) 13.7439 0.549757
\(626\) 25.5550 1.02138
\(627\) 1.13659 0.0453910
\(628\) 11.7017 0.466949
\(629\) 13.7017 0.546323
\(630\) 2.63533 0.104994
\(631\) −22.9745 −0.914601 −0.457300 0.889312i \(-0.651184\pi\)
−0.457300 + 0.889312i \(0.651184\pi\)
\(632\) 4.37196 0.173907
\(633\) −4.39506 −0.174688
\(634\) 13.3297 0.529392
\(635\) 5.14483 0.204166
\(636\) 0.968541 0.0384052
\(637\) 0 0
\(638\) 2.39612 0.0948635
\(639\) 18.5763 0.734865
\(640\) 0.890084 0.0351836
\(641\) 9.32544 0.368333 0.184166 0.982895i \(-0.441041\pi\)
0.184166 + 0.982895i \(0.441041\pi\)
\(642\) −2.87369 −0.113416
\(643\) 19.5948 0.772743 0.386371 0.922343i \(-0.373728\pi\)
0.386371 + 0.922343i \(0.373728\pi\)
\(644\) 7.60388 0.299635
\(645\) −1.31634 −0.0518308
\(646\) 21.1032 0.830295
\(647\) −46.5569 −1.83034 −0.915170 0.403068i \(-0.867944\pi\)
−0.915170 + 0.403068i \(0.867944\pi\)
\(648\) 8.64848 0.339744
\(649\) −8.53260 −0.334934
\(650\) 0 0
\(651\) −0.298290 −0.0116909
\(652\) −20.5187 −0.803575
\(653\) −13.3357 −0.521867 −0.260933 0.965357i \(-0.584030\pi\)
−0.260933 + 0.965357i \(0.584030\pi\)
\(654\) −2.54825 −0.0996446
\(655\) 1.83233 0.0715951
\(656\) 7.83877 0.306053
\(657\) −10.8955 −0.425072
\(658\) 1.20775 0.0470830
\(659\) −16.6872 −0.650042 −0.325021 0.945707i \(-0.605371\pi\)
−0.325021 + 0.945707i \(0.605371\pi\)
\(660\) −0.117212 −0.00456247
\(661\) −47.5943 −1.85120 −0.925602 0.378498i \(-0.876441\pi\)
−0.925602 + 0.378498i \(0.876441\pi\)
\(662\) 18.9638 0.737047
\(663\) 0 0
\(664\) −2.92931 −0.113679
\(665\) 7.68233 0.297908
\(666\) 16.5918 0.642919
\(667\) −27.4034 −1.06106
\(668\) 6.46980 0.250324
\(669\) 4.42626 0.171129
\(670\) 5.40821 0.208937
\(671\) 5.12067 0.197681
\(672\) 0.198062 0.00764042
\(673\) 43.2683 1.66787 0.833935 0.551863i \(-0.186083\pi\)
0.833935 + 0.551863i \(0.186083\pi\)
\(674\) 29.8823 1.15102
\(675\) −4.96769 −0.191206
\(676\) 0 0
\(677\) −18.9772 −0.729352 −0.364676 0.931135i \(-0.618820\pi\)
−0.364676 + 0.931135i \(0.618820\pi\)
\(678\) 0.395651 0.0151949
\(679\) 10.4450 0.400844
\(680\) −2.17629 −0.0834570
\(681\) 0.517648 0.0198363
\(682\) −1.00133 −0.0383428
\(683\) 26.2435 1.00418 0.502090 0.864815i \(-0.332565\pi\)
0.502090 + 0.864815i \(0.332565\pi\)
\(684\) 25.5545 0.977100
\(685\) 5.55555 0.212267
\(686\) −1.00000 −0.0381802
\(687\) 3.18837 0.121644
\(688\) 7.46681 0.284670
\(689\) 0 0
\(690\) 1.34050 0.0510321
\(691\) −16.1129 −0.612964 −0.306482 0.951877i \(-0.599152\pi\)
−0.306482 + 0.951877i \(0.599152\pi\)
\(692\) 24.5133 0.931857
\(693\) 1.96854 0.0747787
\(694\) −5.78017 −0.219412
\(695\) −6.72455 −0.255077
\(696\) −0.713792 −0.0270562
\(697\) −19.1661 −0.725969
\(698\) 7.12929 0.269848
\(699\) 0.356896 0.0134990
\(700\) 4.20775 0.159038
\(701\) −27.9409 −1.05531 −0.527657 0.849458i \(-0.676929\pi\)
−0.527657 + 0.849458i \(0.676929\pi\)
\(702\) 0 0
\(703\) 48.3672 1.82420
\(704\) 0.664874 0.0250584
\(705\) 0.212917 0.00801891
\(706\) −7.09113 −0.266878
\(707\) 8.98792 0.338025
\(708\) 2.54181 0.0955271
\(709\) −37.6823 −1.41519 −0.707595 0.706618i \(-0.750220\pi\)
−0.707595 + 0.706618i \(0.750220\pi\)
\(710\) −5.58450 −0.209582
\(711\) −12.9444 −0.485452
\(712\) −12.5157 −0.469047
\(713\) 11.4517 0.428871
\(714\) −0.484271 −0.0181234
\(715\) 0 0
\(716\) 3.67696 0.137414
\(717\) 0.781495 0.0291855
\(718\) 25.9081 0.966883
\(719\) 9.53020 0.355417 0.177708 0.984083i \(-0.443132\pi\)
0.177708 + 0.984083i \(0.443132\pi\)
\(720\) −2.63533 −0.0982131
\(721\) 8.27413 0.308145
\(722\) 55.4946 2.06529
\(723\) 0.553039 0.0205677
\(724\) 13.7995 0.512856
\(725\) −15.1642 −0.563185
\(726\) 2.09113 0.0776091
\(727\) 1.32975 0.0493177 0.0246588 0.999696i \(-0.492150\pi\)
0.0246588 + 0.999696i \(0.492150\pi\)
\(728\) 0 0
\(729\) −24.9047 −0.922395
\(730\) 3.27545 0.121230
\(731\) −18.2567 −0.675247
\(732\) −1.52542 −0.0563811
\(733\) 32.9590 1.21737 0.608684 0.793413i \(-0.291698\pi\)
0.608684 + 0.793413i \(0.291698\pi\)
\(734\) −30.4698 −1.12466
\(735\) −0.176292 −0.00650263
\(736\) −7.60388 −0.280283
\(737\) 4.03982 0.148809
\(738\) −23.2088 −0.854328
\(739\) 10.0277 0.368876 0.184438 0.982844i \(-0.440953\pi\)
0.184438 + 0.982844i \(0.440953\pi\)
\(740\) −4.98792 −0.183360
\(741\) 0 0
\(742\) 4.89008 0.179521
\(743\) 8.52888 0.312894 0.156447 0.987686i \(-0.449996\pi\)
0.156447 + 0.987686i \(0.449996\pi\)
\(744\) 0.298290 0.0109358
\(745\) −15.0336 −0.550788
\(746\) −7.42758 −0.271943
\(747\) 8.67302 0.317329
\(748\) −1.62565 −0.0594395
\(749\) −14.5090 −0.530148
\(750\) 1.62325 0.0592728
\(751\) 3.05084 0.111327 0.0556633 0.998450i \(-0.482273\pi\)
0.0556633 + 0.998450i \(0.482273\pi\)
\(752\) −1.20775 −0.0440421
\(753\) 2.11231 0.0769768
\(754\) 0 0
\(755\) −10.2586 −0.373348
\(756\) −1.18060 −0.0429381
\(757\) −45.5206 −1.65448 −0.827238 0.561852i \(-0.810089\pi\)
−0.827238 + 0.561852i \(0.810089\pi\)
\(758\) −27.5472 −1.00056
\(759\) 1.00133 0.0363459
\(760\) −7.68233 −0.278667
\(761\) 36.9101 1.33799 0.668994 0.743268i \(-0.266725\pi\)
0.668994 + 0.743268i \(0.266725\pi\)
\(762\) −1.14483 −0.0414729
\(763\) −12.8659 −0.465778
\(764\) 20.0737 0.726240
\(765\) 6.44350 0.232965
\(766\) −30.8853 −1.11593
\(767\) 0 0
\(768\) −0.198062 −0.00714696
\(769\) −45.6528 −1.64628 −0.823141 0.567837i \(-0.807780\pi\)
−0.823141 + 0.567837i \(0.807780\pi\)
\(770\) −0.591794 −0.0213268
\(771\) 5.04307 0.181622
\(772\) 3.74094 0.134639
\(773\) −22.7380 −0.817827 −0.408914 0.912573i \(-0.634092\pi\)
−0.408914 + 0.912573i \(0.634092\pi\)
\(774\) −22.1075 −0.794639
\(775\) 6.33704 0.227633
\(776\) −10.4450 −0.374955
\(777\) −1.10992 −0.0398180
\(778\) 32.0301 1.14834
\(779\) −67.6566 −2.42405
\(780\) 0 0
\(781\) −4.17151 −0.149268
\(782\) 18.5918 0.664841
\(783\) 4.25475 0.152052
\(784\) 1.00000 0.0357143
\(785\) 10.4155 0.371745
\(786\) −0.407732 −0.0145433
\(787\) 8.72779 0.311112 0.155556 0.987827i \(-0.450283\pi\)
0.155556 + 0.987827i \(0.450283\pi\)
\(788\) 0.0978347 0.00348522
\(789\) −0.953002 −0.0339278
\(790\) 3.89141 0.138450
\(791\) 1.99761 0.0710267
\(792\) −1.96854 −0.0699491
\(793\) 0 0
\(794\) −7.06638 −0.250776
\(795\) 0.862083 0.0305749
\(796\) −20.4892 −0.726219
\(797\) 52.6956 1.86657 0.933287 0.359132i \(-0.116927\pi\)
0.933287 + 0.359132i \(0.116927\pi\)
\(798\) −1.70948 −0.0605149
\(799\) 2.95300 0.104470
\(800\) −4.20775 −0.148766
\(801\) 37.0562 1.30932
\(802\) −22.6058 −0.798238
\(803\) 2.44670 0.0863421
\(804\) −1.20344 −0.0424420
\(805\) 6.76809 0.238544
\(806\) 0 0
\(807\) −4.85517 −0.170910
\(808\) −8.98792 −0.316194
\(809\) −1.09006 −0.0383246 −0.0191623 0.999816i \(-0.506100\pi\)
−0.0191623 + 0.999816i \(0.506100\pi\)
\(810\) 7.69787 0.270476
\(811\) −34.2747 −1.20355 −0.601774 0.798666i \(-0.705539\pi\)
−0.601774 + 0.798666i \(0.705539\pi\)
\(812\) −3.60388 −0.126471
\(813\) −4.46117 −0.156460
\(814\) −3.72587 −0.130592
\(815\) −18.2634 −0.639738
\(816\) 0.484271 0.0169529
\(817\) −64.4462 −2.25469
\(818\) −14.8006 −0.517491
\(819\) 0 0
\(820\) 6.97716 0.243653
\(821\) −31.4034 −1.09599 −0.547993 0.836483i \(-0.684608\pi\)
−0.547993 + 0.836483i \(0.684608\pi\)
\(822\) −1.23623 −0.0431183
\(823\) −30.1715 −1.05171 −0.525856 0.850573i \(-0.676255\pi\)
−0.525856 + 0.850573i \(0.676255\pi\)
\(824\) −8.27413 −0.288243
\(825\) 0.554104 0.0192914
\(826\) 12.8334 0.446531
\(827\) −28.2040 −0.980750 −0.490375 0.871511i \(-0.663140\pi\)
−0.490375 + 0.871511i \(0.663140\pi\)
\(828\) 22.5133 0.782393
\(829\) −34.2452 −1.18938 −0.594692 0.803954i \(-0.702726\pi\)
−0.594692 + 0.803954i \(0.702726\pi\)
\(830\) −2.60733 −0.0905019
\(831\) −1.27758 −0.0443189
\(832\) 0 0
\(833\) −2.44504 −0.0847157
\(834\) 1.49635 0.0518144
\(835\) 5.75866 0.199287
\(836\) −5.73855 −0.198472
\(837\) −1.77804 −0.0614580
\(838\) −4.79225 −0.165545
\(839\) −22.4590 −0.775372 −0.387686 0.921791i \(-0.626726\pi\)
−0.387686 + 0.921791i \(0.626726\pi\)
\(840\) 0.176292 0.00608265
\(841\) −16.0121 −0.552141
\(842\) 7.78017 0.268122
\(843\) 6.06398 0.208855
\(844\) 22.1903 0.763821
\(845\) 0 0
\(846\) 3.57587 0.122941
\(847\) 10.5579 0.362775
\(848\) −4.89008 −0.167926
\(849\) 3.36552 0.115504
\(850\) 10.2881 0.352880
\(851\) 42.6112 1.46069
\(852\) 1.24267 0.0425731
\(853\) −22.9288 −0.785068 −0.392534 0.919737i \(-0.628402\pi\)
−0.392534 + 0.919737i \(0.628402\pi\)
\(854\) −7.70171 −0.263547
\(855\) 22.7456 0.777884
\(856\) 14.5090 0.495908
\(857\) −3.95971 −0.135261 −0.0676305 0.997710i \(-0.521544\pi\)
−0.0676305 + 0.997710i \(0.521544\pi\)
\(858\) 0 0
\(859\) 3.00969 0.102689 0.0513446 0.998681i \(-0.483649\pi\)
0.0513446 + 0.998681i \(0.483649\pi\)
\(860\) 6.64609 0.226630
\(861\) 1.55257 0.0529113
\(862\) −9.46250 −0.322294
\(863\) −20.0978 −0.684138 −0.342069 0.939675i \(-0.611128\pi\)
−0.342069 + 0.939675i \(0.611128\pi\)
\(864\) 1.18060 0.0401650
\(865\) 21.8189 0.741866
\(866\) −10.5351 −0.357998
\(867\) 2.18300 0.0741385
\(868\) 1.50604 0.0511184
\(869\) 2.90681 0.0986066
\(870\) −0.635334 −0.0215399
\(871\) 0 0
\(872\) 12.8659 0.435695
\(873\) 30.9254 1.04667
\(874\) 65.6292 2.21994
\(875\) 8.19567 0.277064
\(876\) −0.728857 −0.0246258
\(877\) −34.7633 −1.17387 −0.586937 0.809633i \(-0.699666\pi\)
−0.586937 + 0.809633i \(0.699666\pi\)
\(878\) −2.08708 −0.0704356
\(879\) −4.91425 −0.165753
\(880\) 0.591794 0.0199494
\(881\) 39.5415 1.33219 0.666094 0.745868i \(-0.267965\pi\)
0.666094 + 0.745868i \(0.267965\pi\)
\(882\) −2.96077 −0.0996944
\(883\) 32.4704 1.09272 0.546358 0.837552i \(-0.316014\pi\)
0.546358 + 0.837552i \(0.316014\pi\)
\(884\) 0 0
\(885\) 2.26243 0.0760506
\(886\) 17.6635 0.593418
\(887\) 19.0422 0.639375 0.319687 0.947523i \(-0.396422\pi\)
0.319687 + 0.947523i \(0.396422\pi\)
\(888\) 1.10992 0.0372464
\(889\) −5.78017 −0.193861
\(890\) −11.1400 −0.373415
\(891\) 5.75015 0.192637
\(892\) −22.3478 −0.748260
\(893\) 10.4241 0.348830
\(894\) 3.34529 0.111883
\(895\) 3.27280 0.109398
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 29.9124 0.998191
\(899\) −5.42758 −0.181020
\(900\) 12.4582 0.415273
\(901\) 11.9565 0.398327
\(902\) 5.21180 0.173534
\(903\) 1.47889 0.0492145
\(904\) −1.99761 −0.0664394
\(905\) 12.2828 0.408292
\(906\) 2.28275 0.0758393
\(907\) −13.8369 −0.459445 −0.229723 0.973256i \(-0.573782\pi\)
−0.229723 + 0.973256i \(0.573782\pi\)
\(908\) −2.61356 −0.0867342
\(909\) 26.6112 0.882637
\(910\) 0 0
\(911\) −50.3564 −1.66838 −0.834191 0.551475i \(-0.814065\pi\)
−0.834191 + 0.551475i \(0.814065\pi\)
\(912\) 1.70948 0.0566065
\(913\) −1.94762 −0.0644570
\(914\) 37.5120 1.24079
\(915\) −1.35775 −0.0448858
\(916\) −16.0978 −0.531887
\(917\) −2.05861 −0.0679812
\(918\) −2.88663 −0.0952729
\(919\) −45.4965 −1.50079 −0.750395 0.660990i \(-0.770137\pi\)
−0.750395 + 0.660990i \(0.770137\pi\)
\(920\) −6.76809 −0.223137
\(921\) 2.85623 0.0941160
\(922\) 36.9288 1.21619
\(923\) 0 0
\(924\) 0.131687 0.00433217
\(925\) 23.5797 0.775296
\(926\) −23.8237 −0.782896
\(927\) 24.4978 0.804613
\(928\) 3.60388 0.118303
\(929\) −29.8931 −0.980760 −0.490380 0.871509i \(-0.663142\pi\)
−0.490380 + 0.871509i \(0.663142\pi\)
\(930\) 0.265503 0.00870618
\(931\) −8.63102 −0.282870
\(932\) −1.80194 −0.0590244
\(933\) 4.38750 0.143640
\(934\) −19.4010 −0.634821
\(935\) −1.44696 −0.0473207
\(936\) 0 0
\(937\) 23.4077 0.764697 0.382349 0.924018i \(-0.375115\pi\)
0.382349 + 0.924018i \(0.375115\pi\)
\(938\) −6.07606 −0.198391
\(939\) −5.06147 −0.165175
\(940\) −1.07500 −0.0350626
\(941\) −28.0650 −0.914894 −0.457447 0.889237i \(-0.651236\pi\)
−0.457447 + 0.889237i \(0.651236\pi\)
\(942\) −2.31767 −0.0755137
\(943\) −59.6051 −1.94101
\(944\) −12.8334 −0.417691
\(945\) −1.05084 −0.0341837
\(946\) 4.96449 0.161410
\(947\) 0.972853 0.0316135 0.0158067 0.999875i \(-0.494968\pi\)
0.0158067 + 0.999875i \(0.494968\pi\)
\(948\) −0.865921 −0.0281238
\(949\) 0 0
\(950\) 36.3172 1.17829
\(951\) −2.64012 −0.0856118
\(952\) 2.44504 0.0792443
\(953\) 52.5719 1.70297 0.851486 0.524377i \(-0.175702\pi\)
0.851486 + 0.524377i \(0.175702\pi\)
\(954\) 14.4784 0.468756
\(955\) 17.8672 0.578171
\(956\) −3.94571 −0.127613
\(957\) −0.474582 −0.0153411
\(958\) 20.7224 0.669511
\(959\) −6.24160 −0.201552
\(960\) −0.176292 −0.00568980
\(961\) −28.7318 −0.926834
\(962\) 0 0
\(963\) −42.9579 −1.38430
\(964\) −2.79225 −0.0899323
\(965\) 3.32975 0.107188
\(966\) −1.50604 −0.0484561
\(967\) 6.74392 0.216870 0.108435 0.994104i \(-0.465416\pi\)
0.108435 + 0.994104i \(0.465416\pi\)
\(968\) −10.5579 −0.339345
\(969\) −4.17975 −0.134273
\(970\) −9.29696 −0.298507
\(971\) 42.0586 1.34972 0.674862 0.737944i \(-0.264203\pi\)
0.674862 + 0.737944i \(0.264203\pi\)
\(972\) −5.25475 −0.168546
\(973\) 7.55496 0.242201
\(974\) 31.9758 1.02457
\(975\) 0 0
\(976\) 7.70171 0.246526
\(977\) −24.8461 −0.794896 −0.397448 0.917625i \(-0.630104\pi\)
−0.397448 + 0.917625i \(0.630104\pi\)
\(978\) 4.06398 0.129952
\(979\) −8.32139 −0.265953
\(980\) 0.890084 0.0284327
\(981\) −38.0930 −1.21622
\(982\) −9.24027 −0.294869
\(983\) 19.8888 0.634353 0.317176 0.948367i \(-0.397265\pi\)
0.317176 + 0.948367i \(0.397265\pi\)
\(984\) −1.55257 −0.0494940
\(985\) 0.0870811 0.00277463
\(986\) −8.81163 −0.280619
\(987\) −0.239210 −0.00761413
\(988\) 0 0
\(989\) −56.7767 −1.80539
\(990\) −1.75217 −0.0556875
\(991\) 36.3564 1.15490 0.577450 0.816426i \(-0.304048\pi\)
0.577450 + 0.816426i \(0.304048\pi\)
\(992\) −1.50604 −0.0478168
\(993\) −3.75600 −0.119193
\(994\) 6.27413 0.199003
\(995\) −18.2371 −0.578154
\(996\) 0.580186 0.0183839
\(997\) −30.0000 −0.950110 −0.475055 0.879956i \(-0.657572\pi\)
−0.475055 + 0.879956i \(0.657572\pi\)
\(998\) 9.94630 0.314845
\(999\) −6.61596 −0.209320
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 2366.2.a.z.1.3 yes 3
13.5 odd 4 2366.2.d.m.337.3 6
13.8 odd 4 2366.2.d.m.337.6 6
13.12 even 2 2366.2.a.u.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2366.2.a.u.1.3 3 13.12 even 2
2366.2.a.z.1.3 yes 3 1.1 even 1 trivial
2366.2.d.m.337.3 6 13.5 odd 4
2366.2.d.m.337.6 6 13.8 odd 4