Properties

Label 2001.2.a.n.1.3
Level $2001$
Weight $2$
Character 2001.1
Self dual yes
Analytic conductor $15.978$
Analytic rank $0$
Dimension $16$
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] = [2001,2,Mod(1,2001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, 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("2001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.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: \(15.9780654445\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + 5778 x^{8} - 5124 x^{7} - 9405 x^{6} + 8288 x^{5} + 6405 x^{4} - 6032 x^{3} + \cdots - 192 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.18082\) of defining polynomial
Character \(\chi\) \(=\) 2001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.18082 q^{2} +1.00000 q^{3} +2.75597 q^{4} -3.64804 q^{5} -2.18082 q^{6} +2.94825 q^{7} -1.64864 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.18082 q^{2} +1.00000 q^{3} +2.75597 q^{4} -3.64804 q^{5} -2.18082 q^{6} +2.94825 q^{7} -1.64864 q^{8} +1.00000 q^{9} +7.95571 q^{10} -2.90138 q^{11} +2.75597 q^{12} +1.83459 q^{13} -6.42960 q^{14} -3.64804 q^{15} -1.91657 q^{16} +6.16079 q^{17} -2.18082 q^{18} -7.26232 q^{19} -10.0539 q^{20} +2.94825 q^{21} +6.32738 q^{22} -1.00000 q^{23} -1.64864 q^{24} +8.30818 q^{25} -4.00091 q^{26} +1.00000 q^{27} +8.12529 q^{28} -1.00000 q^{29} +7.95571 q^{30} +9.78276 q^{31} +7.47695 q^{32} -2.90138 q^{33} -13.4356 q^{34} -10.7553 q^{35} +2.75597 q^{36} -1.23929 q^{37} +15.8378 q^{38} +1.83459 q^{39} +6.01428 q^{40} -6.05736 q^{41} -6.42960 q^{42} +6.93454 q^{43} -7.99611 q^{44} -3.64804 q^{45} +2.18082 q^{46} -6.40585 q^{47} -1.91657 q^{48} +1.69218 q^{49} -18.1186 q^{50} +6.16079 q^{51} +5.05607 q^{52} +4.62451 q^{53} -2.18082 q^{54} +10.5843 q^{55} -4.86059 q^{56} -7.26232 q^{57} +2.18082 q^{58} -3.54179 q^{59} -10.0539 q^{60} -9.97508 q^{61} -21.3344 q^{62} +2.94825 q^{63} -12.4728 q^{64} -6.69265 q^{65} +6.32738 q^{66} +5.17176 q^{67} +16.9789 q^{68} -1.00000 q^{69} +23.4554 q^{70} -4.85140 q^{71} -1.64864 q^{72} +7.56931 q^{73} +2.70268 q^{74} +8.30818 q^{75} -20.0148 q^{76} -8.55398 q^{77} -4.00091 q^{78} +7.61876 q^{79} +6.99170 q^{80} +1.00000 q^{81} +13.2100 q^{82} +11.2522 q^{83} +8.12529 q^{84} -22.4748 q^{85} -15.1230 q^{86} -1.00000 q^{87} +4.78331 q^{88} +3.11532 q^{89} +7.95571 q^{90} +5.40883 q^{91} -2.75597 q^{92} +9.78276 q^{93} +13.9700 q^{94} +26.4932 q^{95} +7.47695 q^{96} -1.36579 q^{97} -3.69033 q^{98} -2.90138 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9} + 11 q^{10} + 8 q^{11} + 25 q^{12} + 19 q^{13} + 16 q^{14} + 3 q^{15} + 31 q^{16} - 4 q^{17} - q^{18} + 19 q^{19} + 16 q^{20} + 13 q^{21} + 6 q^{22} - 16 q^{23} + 23 q^{25} - 15 q^{26} + 16 q^{27} + 18 q^{28} - 16 q^{29} + 11 q^{30} + 24 q^{31} - 21 q^{32} + 8 q^{33} - 9 q^{34} - 13 q^{35} + 25 q^{36} + 26 q^{37} + 19 q^{39} - 22 q^{40} - 15 q^{41} + 16 q^{42} + 33 q^{43} + 6 q^{44} + 3 q^{45} + q^{46} + 13 q^{47} + 31 q^{48} + 41 q^{49} + 13 q^{50} - 4 q^{51} - 26 q^{52} + 5 q^{53} - q^{54} + 9 q^{55} + 40 q^{56} + 19 q^{57} + q^{58} + 2 q^{59} + 16 q^{60} + 29 q^{61} - 32 q^{62} + 13 q^{63} + 28 q^{64} + 18 q^{65} + 6 q^{66} + 32 q^{67} - 26 q^{68} - 16 q^{69} + 18 q^{70} + 29 q^{71} + 19 q^{73} - 16 q^{74} + 23 q^{75} + 64 q^{76} - 21 q^{77} - 15 q^{78} + 56 q^{79} + 16 q^{81} + 14 q^{82} + 5 q^{83} + 18 q^{84} + 16 q^{85} - 20 q^{86} - 16 q^{87} + q^{88} + 7 q^{89} + 11 q^{90} - 6 q^{91} - 25 q^{92} + 24 q^{93} - 11 q^{94} + 39 q^{95} - 21 q^{96} + 35 q^{97} - 109 q^{98} + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.18082 −1.54207 −0.771036 0.636792i \(-0.780261\pi\)
−0.771036 + 0.636792i \(0.780261\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.75597 1.37799
\(5\) −3.64804 −1.63145 −0.815726 0.578439i \(-0.803662\pi\)
−0.815726 + 0.578439i \(0.803662\pi\)
\(6\) −2.18082 −0.890316
\(7\) 2.94825 1.11433 0.557167 0.830401i \(-0.311888\pi\)
0.557167 + 0.830401i \(0.311888\pi\)
\(8\) −1.64864 −0.582881
\(9\) 1.00000 0.333333
\(10\) 7.95571 2.51582
\(11\) −2.90138 −0.874798 −0.437399 0.899268i \(-0.644100\pi\)
−0.437399 + 0.899268i \(0.644100\pi\)
\(12\) 2.75597 0.795580
\(13\) 1.83459 0.508823 0.254412 0.967096i \(-0.418118\pi\)
0.254412 + 0.967096i \(0.418118\pi\)
\(14\) −6.42960 −1.71838
\(15\) −3.64804 −0.941919
\(16\) −1.91657 −0.479142
\(17\) 6.16079 1.49421 0.747105 0.664706i \(-0.231443\pi\)
0.747105 + 0.664706i \(0.231443\pi\)
\(18\) −2.18082 −0.514024
\(19\) −7.26232 −1.66609 −0.833046 0.553204i \(-0.813405\pi\)
−0.833046 + 0.553204i \(0.813405\pi\)
\(20\) −10.0539 −2.24812
\(21\) 2.94825 0.643361
\(22\) 6.32738 1.34900
\(23\) −1.00000 −0.208514
\(24\) −1.64864 −0.336526
\(25\) 8.30818 1.66164
\(26\) −4.00091 −0.784642
\(27\) 1.00000 0.192450
\(28\) 8.12529 1.53554
\(29\) −1.00000 −0.185695
\(30\) 7.95571 1.45251
\(31\) 9.78276 1.75704 0.878518 0.477710i \(-0.158533\pi\)
0.878518 + 0.477710i \(0.158533\pi\)
\(32\) 7.47695 1.32175
\(33\) −2.90138 −0.505065
\(34\) −13.4356 −2.30418
\(35\) −10.7553 −1.81798
\(36\) 2.75597 0.459328
\(37\) −1.23929 −0.203739 −0.101869 0.994798i \(-0.532482\pi\)
−0.101869 + 0.994798i \(0.532482\pi\)
\(38\) 15.8378 2.56923
\(39\) 1.83459 0.293769
\(40\) 6.01428 0.950942
\(41\) −6.05736 −0.946001 −0.473001 0.881062i \(-0.656829\pi\)
−0.473001 + 0.881062i \(0.656829\pi\)
\(42\) −6.42960 −0.992109
\(43\) 6.93454 1.05751 0.528754 0.848775i \(-0.322660\pi\)
0.528754 + 0.848775i \(0.322660\pi\)
\(44\) −7.99611 −1.20546
\(45\) −3.64804 −0.543817
\(46\) 2.18082 0.321544
\(47\) −6.40585 −0.934389 −0.467194 0.884155i \(-0.654735\pi\)
−0.467194 + 0.884155i \(0.654735\pi\)
\(48\) −1.91657 −0.276632
\(49\) 1.69218 0.241739
\(50\) −18.1186 −2.56236
\(51\) 6.16079 0.862683
\(52\) 5.05607 0.701151
\(53\) 4.62451 0.635226 0.317613 0.948220i \(-0.397119\pi\)
0.317613 + 0.948220i \(0.397119\pi\)
\(54\) −2.18082 −0.296772
\(55\) 10.5843 1.42719
\(56\) −4.86059 −0.649524
\(57\) −7.26232 −0.961918
\(58\) 2.18082 0.286356
\(59\) −3.54179 −0.461101 −0.230551 0.973060i \(-0.574053\pi\)
−0.230551 + 0.973060i \(0.574053\pi\)
\(60\) −10.0539 −1.29795
\(61\) −9.97508 −1.27718 −0.638589 0.769548i \(-0.720482\pi\)
−0.638589 + 0.769548i \(0.720482\pi\)
\(62\) −21.3344 −2.70947
\(63\) 2.94825 0.371445
\(64\) −12.4728 −1.55909
\(65\) −6.69265 −0.830121
\(66\) 6.32738 0.778846
\(67\) 5.17176 0.631830 0.315915 0.948787i \(-0.397688\pi\)
0.315915 + 0.948787i \(0.397688\pi\)
\(68\) 16.9789 2.05900
\(69\) −1.00000 −0.120386
\(70\) 23.4554 2.80346
\(71\) −4.85140 −0.575755 −0.287877 0.957667i \(-0.592950\pi\)
−0.287877 + 0.957667i \(0.592950\pi\)
\(72\) −1.64864 −0.194294
\(73\) 7.56931 0.885921 0.442961 0.896541i \(-0.353928\pi\)
0.442961 + 0.896541i \(0.353928\pi\)
\(74\) 2.70268 0.314180
\(75\) 8.30818 0.959346
\(76\) −20.0148 −2.29585
\(77\) −8.55398 −0.974817
\(78\) −4.00091 −0.453013
\(79\) 7.61876 0.857178 0.428589 0.903500i \(-0.359011\pi\)
0.428589 + 0.903500i \(0.359011\pi\)
\(80\) 6.99170 0.781696
\(81\) 1.00000 0.111111
\(82\) 13.2100 1.45880
\(83\) 11.2522 1.23508 0.617542 0.786538i \(-0.288129\pi\)
0.617542 + 0.786538i \(0.288129\pi\)
\(84\) 8.12529 0.886542
\(85\) −22.4748 −2.43773
\(86\) −15.1230 −1.63075
\(87\) −1.00000 −0.107211
\(88\) 4.78331 0.509903
\(89\) 3.11532 0.330223 0.165111 0.986275i \(-0.447202\pi\)
0.165111 + 0.986275i \(0.447202\pi\)
\(90\) 7.95571 0.838605
\(91\) 5.40883 0.566999
\(92\) −2.75597 −0.287330
\(93\) 9.78276 1.01442
\(94\) 13.9700 1.44089
\(95\) 26.4932 2.71815
\(96\) 7.47695 0.763114
\(97\) −1.36579 −0.138674 −0.0693372 0.997593i \(-0.522088\pi\)
−0.0693372 + 0.997593i \(0.522088\pi\)
\(98\) −3.69033 −0.372780
\(99\) −2.90138 −0.291599
\(100\) 22.8971 2.28971
\(101\) −3.09336 −0.307801 −0.153900 0.988086i \(-0.549184\pi\)
−0.153900 + 0.988086i \(0.549184\pi\)
\(102\) −13.4356 −1.33032
\(103\) −5.80530 −0.572013 −0.286007 0.958228i \(-0.592328\pi\)
−0.286007 + 0.958228i \(0.592328\pi\)
\(104\) −3.02457 −0.296583
\(105\) −10.7553 −1.04961
\(106\) −10.0852 −0.979564
\(107\) 13.4507 1.30033 0.650163 0.759795i \(-0.274701\pi\)
0.650163 + 0.759795i \(0.274701\pi\)
\(108\) 2.75597 0.265193
\(109\) 1.09175 0.104571 0.0522853 0.998632i \(-0.483349\pi\)
0.0522853 + 0.998632i \(0.483349\pi\)
\(110\) −23.0825 −2.20083
\(111\) −1.23929 −0.117629
\(112\) −5.65052 −0.533923
\(113\) 17.7325 1.66814 0.834068 0.551661i \(-0.186006\pi\)
0.834068 + 0.551661i \(0.186006\pi\)
\(114\) 15.8378 1.48335
\(115\) 3.64804 0.340181
\(116\) −2.75597 −0.255885
\(117\) 1.83459 0.169608
\(118\) 7.72399 0.711051
\(119\) 18.1635 1.66505
\(120\) 6.01428 0.549027
\(121\) −2.58201 −0.234728
\(122\) 21.7538 1.96950
\(123\) −6.05736 −0.546174
\(124\) 26.9610 2.42117
\(125\) −12.0684 −1.07943
\(126\) −6.42960 −0.572794
\(127\) 15.4962 1.37507 0.687533 0.726154i \(-0.258694\pi\)
0.687533 + 0.726154i \(0.258694\pi\)
\(128\) 12.2469 1.08248
\(129\) 6.93454 0.610552
\(130\) 14.5955 1.28011
\(131\) 10.5488 0.921657 0.460829 0.887489i \(-0.347552\pi\)
0.460829 + 0.887489i \(0.347552\pi\)
\(132\) −7.99611 −0.695972
\(133\) −21.4111 −1.85658
\(134\) −11.2787 −0.974328
\(135\) −3.64804 −0.313973
\(136\) −10.1569 −0.870946
\(137\) −0.813436 −0.0694965 −0.0347483 0.999396i \(-0.511063\pi\)
−0.0347483 + 0.999396i \(0.511063\pi\)
\(138\) 2.18082 0.185644
\(139\) −17.3423 −1.47096 −0.735478 0.677549i \(-0.763042\pi\)
−0.735478 + 0.677549i \(0.763042\pi\)
\(140\) −29.6414 −2.50515
\(141\) −6.40585 −0.539470
\(142\) 10.5800 0.887855
\(143\) −5.32283 −0.445118
\(144\) −1.91657 −0.159714
\(145\) 3.64804 0.302953
\(146\) −16.5073 −1.36615
\(147\) 1.69218 0.139568
\(148\) −3.41546 −0.280749
\(149\) 21.0736 1.72641 0.863207 0.504850i \(-0.168452\pi\)
0.863207 + 0.504850i \(0.168452\pi\)
\(150\) −18.1186 −1.47938
\(151\) −17.4071 −1.41657 −0.708286 0.705925i \(-0.750531\pi\)
−0.708286 + 0.705925i \(0.750531\pi\)
\(152\) 11.9729 0.971132
\(153\) 6.16079 0.498070
\(154\) 18.6547 1.50324
\(155\) −35.6879 −2.86652
\(156\) 5.05607 0.404810
\(157\) 20.9775 1.67419 0.837095 0.547057i \(-0.184252\pi\)
0.837095 + 0.547057i \(0.184252\pi\)
\(158\) −16.6151 −1.32183
\(159\) 4.62451 0.366748
\(160\) −27.2762 −2.15637
\(161\) −2.94825 −0.232355
\(162\) −2.18082 −0.171341
\(163\) 3.29012 0.257702 0.128851 0.991664i \(-0.458871\pi\)
0.128851 + 0.991664i \(0.458871\pi\)
\(164\) −16.6939 −1.30358
\(165\) 10.5843 0.823989
\(166\) −24.5389 −1.90459
\(167\) 19.1189 1.47947 0.739733 0.672901i \(-0.234952\pi\)
0.739733 + 0.672901i \(0.234952\pi\)
\(168\) −4.86059 −0.375003
\(169\) −9.63428 −0.741099
\(170\) 49.0134 3.75916
\(171\) −7.26232 −0.555364
\(172\) 19.1114 1.45723
\(173\) −4.38468 −0.333361 −0.166681 0.986011i \(-0.553305\pi\)
−0.166681 + 0.986011i \(0.553305\pi\)
\(174\) 2.18082 0.165327
\(175\) 24.4946 1.85162
\(176\) 5.56068 0.419152
\(177\) −3.54179 −0.266217
\(178\) −6.79394 −0.509227
\(179\) −21.2216 −1.58618 −0.793090 0.609104i \(-0.791529\pi\)
−0.793090 + 0.609104i \(0.791529\pi\)
\(180\) −10.0539 −0.749372
\(181\) 19.4211 1.44356 0.721781 0.692121i \(-0.243324\pi\)
0.721781 + 0.692121i \(0.243324\pi\)
\(182\) −11.7957 −0.874353
\(183\) −9.97508 −0.737379
\(184\) 1.64864 0.121539
\(185\) 4.52099 0.332390
\(186\) −21.3344 −1.56432
\(187\) −17.8748 −1.30713
\(188\) −17.6543 −1.28757
\(189\) 2.94825 0.214454
\(190\) −57.7769 −4.19158
\(191\) 2.72682 0.197306 0.0986530 0.995122i \(-0.468547\pi\)
0.0986530 + 0.995122i \(0.468547\pi\)
\(192\) −12.4728 −0.900143
\(193\) −2.27863 −0.164019 −0.0820097 0.996632i \(-0.526134\pi\)
−0.0820097 + 0.996632i \(0.526134\pi\)
\(194\) 2.97853 0.213846
\(195\) −6.69265 −0.479270
\(196\) 4.66359 0.333113
\(197\) 11.3746 0.810408 0.405204 0.914226i \(-0.367201\pi\)
0.405204 + 0.914226i \(0.367201\pi\)
\(198\) 6.32738 0.449667
\(199\) 15.0746 1.06861 0.534306 0.845291i \(-0.320573\pi\)
0.534306 + 0.845291i \(0.320573\pi\)
\(200\) −13.6972 −0.968535
\(201\) 5.17176 0.364787
\(202\) 6.74606 0.474651
\(203\) −2.94825 −0.206927
\(204\) 16.9789 1.18876
\(205\) 22.0975 1.54336
\(206\) 12.6603 0.882086
\(207\) −1.00000 −0.0695048
\(208\) −3.51611 −0.243798
\(209\) 21.0707 1.45749
\(210\) 23.4554 1.61858
\(211\) 20.6210 1.41961 0.709804 0.704399i \(-0.248783\pi\)
0.709804 + 0.704399i \(0.248783\pi\)
\(212\) 12.7450 0.875332
\(213\) −4.85140 −0.332412
\(214\) −29.3335 −2.00520
\(215\) −25.2975 −1.72527
\(216\) −1.64864 −0.112175
\(217\) 28.8420 1.95792
\(218\) −2.38091 −0.161255
\(219\) 7.56931 0.511487
\(220\) 29.1701 1.96665
\(221\) 11.3025 0.760289
\(222\) 2.70268 0.181392
\(223\) 1.95442 0.130877 0.0654387 0.997857i \(-0.479155\pi\)
0.0654387 + 0.997857i \(0.479155\pi\)
\(224\) 22.0439 1.47287
\(225\) 8.30818 0.553878
\(226\) −38.6714 −2.57239
\(227\) 21.4195 1.42166 0.710831 0.703362i \(-0.248319\pi\)
0.710831 + 0.703362i \(0.248319\pi\)
\(228\) −20.0148 −1.32551
\(229\) 4.10165 0.271045 0.135522 0.990774i \(-0.456729\pi\)
0.135522 + 0.990774i \(0.456729\pi\)
\(230\) −7.95571 −0.524584
\(231\) −8.55398 −0.562811
\(232\) 1.64864 0.108238
\(233\) 16.6505 1.09081 0.545407 0.838171i \(-0.316375\pi\)
0.545407 + 0.838171i \(0.316375\pi\)
\(234\) −4.00091 −0.261547
\(235\) 23.3688 1.52441
\(236\) −9.76106 −0.635391
\(237\) 7.61876 0.494892
\(238\) −39.6114 −2.56762
\(239\) 15.1886 0.982467 0.491234 0.871028i \(-0.336546\pi\)
0.491234 + 0.871028i \(0.336546\pi\)
\(240\) 6.99170 0.451313
\(241\) −6.49318 −0.418262 −0.209131 0.977888i \(-0.567064\pi\)
−0.209131 + 0.977888i \(0.567064\pi\)
\(242\) 5.63090 0.361968
\(243\) 1.00000 0.0641500
\(244\) −27.4910 −1.75993
\(245\) −6.17312 −0.394386
\(246\) 13.2100 0.842240
\(247\) −13.3234 −0.847746
\(248\) −16.1282 −1.02414
\(249\) 11.2522 0.713076
\(250\) 26.3189 1.66455
\(251\) 26.2268 1.65542 0.827710 0.561157i \(-0.189644\pi\)
0.827710 + 0.561157i \(0.189644\pi\)
\(252\) 8.12529 0.511845
\(253\) 2.90138 0.182408
\(254\) −33.7944 −2.12045
\(255\) −22.4748 −1.40743
\(256\) −1.76277 −0.110173
\(257\) 0.394588 0.0246137 0.0123069 0.999924i \(-0.496083\pi\)
0.0123069 + 0.999924i \(0.496083\pi\)
\(258\) −15.1230 −0.941515
\(259\) −3.65375 −0.227033
\(260\) −18.4447 −1.14389
\(261\) −1.00000 −0.0618984
\(262\) −23.0051 −1.42126
\(263\) 27.4775 1.69434 0.847168 0.531325i \(-0.178306\pi\)
0.847168 + 0.531325i \(0.178306\pi\)
\(264\) 4.78331 0.294393
\(265\) −16.8704 −1.03634
\(266\) 46.6938 2.86298
\(267\) 3.11532 0.190654
\(268\) 14.2532 0.870653
\(269\) −12.5639 −0.766037 −0.383019 0.923741i \(-0.625115\pi\)
−0.383019 + 0.923741i \(0.625115\pi\)
\(270\) 7.95571 0.484169
\(271\) 3.16176 0.192063 0.0960316 0.995378i \(-0.469385\pi\)
0.0960316 + 0.995378i \(0.469385\pi\)
\(272\) −11.8076 −0.715938
\(273\) 5.40883 0.327357
\(274\) 1.77396 0.107169
\(275\) −24.1052 −1.45360
\(276\) −2.75597 −0.165890
\(277\) 9.84864 0.591747 0.295874 0.955227i \(-0.404389\pi\)
0.295874 + 0.955227i \(0.404389\pi\)
\(278\) 37.8204 2.26832
\(279\) 9.78276 0.585678
\(280\) 17.7316 1.05967
\(281\) 18.5477 1.10646 0.553232 0.833027i \(-0.313394\pi\)
0.553232 + 0.833027i \(0.313394\pi\)
\(282\) 13.9700 0.831901
\(283\) 2.64756 0.157381 0.0786905 0.996899i \(-0.474926\pi\)
0.0786905 + 0.996899i \(0.474926\pi\)
\(284\) −13.3703 −0.793382
\(285\) 26.4932 1.56932
\(286\) 11.6081 0.686403
\(287\) −17.8586 −1.05416
\(288\) 7.47695 0.440584
\(289\) 20.9553 1.23266
\(290\) −7.95571 −0.467175
\(291\) −1.36579 −0.0800637
\(292\) 20.8608 1.22079
\(293\) −27.9274 −1.63154 −0.815768 0.578379i \(-0.803685\pi\)
−0.815768 + 0.578379i \(0.803685\pi\)
\(294\) −3.69033 −0.215224
\(295\) 12.9206 0.752264
\(296\) 2.04315 0.118755
\(297\) −2.90138 −0.168355
\(298\) −45.9576 −2.66225
\(299\) −1.83459 −0.106097
\(300\) 22.8971 1.32196
\(301\) 20.4447 1.17842
\(302\) 37.9618 2.18446
\(303\) −3.09336 −0.177709
\(304\) 13.9187 0.798293
\(305\) 36.3895 2.08365
\(306\) −13.4356 −0.768060
\(307\) −22.5768 −1.28853 −0.644263 0.764804i \(-0.722836\pi\)
−0.644263 + 0.764804i \(0.722836\pi\)
\(308\) −23.5745 −1.34328
\(309\) −5.80530 −0.330252
\(310\) 77.8288 4.42038
\(311\) 5.39270 0.305792 0.152896 0.988242i \(-0.451140\pi\)
0.152896 + 0.988242i \(0.451140\pi\)
\(312\) −3.02457 −0.171232
\(313\) −3.71684 −0.210088 −0.105044 0.994468i \(-0.533498\pi\)
−0.105044 + 0.994468i \(0.533498\pi\)
\(314\) −45.7482 −2.58172
\(315\) −10.7553 −0.605994
\(316\) 20.9971 1.18118
\(317\) −27.1940 −1.52736 −0.763682 0.645592i \(-0.776611\pi\)
−0.763682 + 0.645592i \(0.776611\pi\)
\(318\) −10.0852 −0.565552
\(319\) 2.90138 0.162446
\(320\) 45.5011 2.54359
\(321\) 13.4507 0.750744
\(322\) 6.42960 0.358308
\(323\) −44.7416 −2.48949
\(324\) 2.75597 0.153109
\(325\) 15.2421 0.845479
\(326\) −7.17515 −0.397395
\(327\) 1.09175 0.0603738
\(328\) 9.98639 0.551406
\(329\) −18.8860 −1.04122
\(330\) −23.0825 −1.27065
\(331\) −13.7728 −0.757020 −0.378510 0.925597i \(-0.623563\pi\)
−0.378510 + 0.925597i \(0.623563\pi\)
\(332\) 31.0106 1.70193
\(333\) −1.23929 −0.0679129
\(334\) −41.6949 −2.28144
\(335\) −18.8668 −1.03080
\(336\) −5.65052 −0.308261
\(337\) −5.87686 −0.320133 −0.160066 0.987106i \(-0.551171\pi\)
−0.160066 + 0.987106i \(0.551171\pi\)
\(338\) 21.0106 1.14283
\(339\) 17.7325 0.963099
\(340\) −61.9398 −3.35916
\(341\) −28.3835 −1.53705
\(342\) 15.8378 0.856411
\(343\) −15.6488 −0.844955
\(344\) −11.4325 −0.616401
\(345\) 3.64804 0.196404
\(346\) 9.56220 0.514067
\(347\) −15.4892 −0.831502 −0.415751 0.909478i \(-0.636481\pi\)
−0.415751 + 0.909478i \(0.636481\pi\)
\(348\) −2.75597 −0.147736
\(349\) 16.3066 0.872874 0.436437 0.899735i \(-0.356240\pi\)
0.436437 + 0.899735i \(0.356240\pi\)
\(350\) −53.4182 −2.85532
\(351\) 1.83459 0.0979231
\(352\) −21.6935 −1.15627
\(353\) −3.27199 −0.174151 −0.0870753 0.996202i \(-0.527752\pi\)
−0.0870753 + 0.996202i \(0.527752\pi\)
\(354\) 7.72399 0.410526
\(355\) 17.6981 0.939316
\(356\) 8.58572 0.455042
\(357\) 18.1635 0.961316
\(358\) 46.2806 2.44600
\(359\) 29.6747 1.56617 0.783085 0.621914i \(-0.213645\pi\)
0.783085 + 0.621914i \(0.213645\pi\)
\(360\) 6.01428 0.316981
\(361\) 33.7413 1.77586
\(362\) −42.3540 −2.22608
\(363\) −2.58201 −0.135520
\(364\) 14.9066 0.781316
\(365\) −27.6131 −1.44534
\(366\) 21.7538 1.13709
\(367\) 14.2811 0.745467 0.372733 0.927938i \(-0.378421\pi\)
0.372733 + 0.927938i \(0.378421\pi\)
\(368\) 1.91657 0.0999079
\(369\) −6.05736 −0.315334
\(370\) −9.85947 −0.512569
\(371\) 13.6342 0.707854
\(372\) 26.9610 1.39786
\(373\) 22.9700 1.18934 0.594672 0.803968i \(-0.297282\pi\)
0.594672 + 0.803968i \(0.297282\pi\)
\(374\) 38.9816 2.01569
\(375\) −12.0684 −0.623207
\(376\) 10.5609 0.544637
\(377\) −1.83459 −0.0944861
\(378\) −6.42960 −0.330703
\(379\) 0.488397 0.0250873 0.0125436 0.999921i \(-0.496007\pi\)
0.0125436 + 0.999921i \(0.496007\pi\)
\(380\) 73.0146 3.74557
\(381\) 15.4962 0.793894
\(382\) −5.94671 −0.304260
\(383\) −18.6541 −0.953182 −0.476591 0.879125i \(-0.658128\pi\)
−0.476591 + 0.879125i \(0.658128\pi\)
\(384\) 12.2469 0.624972
\(385\) 31.2053 1.59037
\(386\) 4.96928 0.252930
\(387\) 6.93454 0.352502
\(388\) −3.76406 −0.191091
\(389\) −10.6579 −0.540375 −0.270188 0.962808i \(-0.587086\pi\)
−0.270188 + 0.962808i \(0.587086\pi\)
\(390\) 14.5955 0.739069
\(391\) −6.16079 −0.311564
\(392\) −2.78978 −0.140905
\(393\) 10.5488 0.532119
\(394\) −24.8060 −1.24971
\(395\) −27.7935 −1.39844
\(396\) −7.99611 −0.401820
\(397\) −18.4482 −0.925890 −0.462945 0.886387i \(-0.653207\pi\)
−0.462945 + 0.886387i \(0.653207\pi\)
\(398\) −32.8750 −1.64788
\(399\) −21.4111 −1.07190
\(400\) −15.9232 −0.796158
\(401\) 14.9013 0.744134 0.372067 0.928206i \(-0.378649\pi\)
0.372067 + 0.928206i \(0.378649\pi\)
\(402\) −11.2787 −0.562529
\(403\) 17.9473 0.894020
\(404\) −8.52521 −0.424145
\(405\) −3.64804 −0.181272
\(406\) 6.42960 0.319096
\(407\) 3.59566 0.178230
\(408\) −10.1569 −0.502841
\(409\) 17.7732 0.878828 0.439414 0.898285i \(-0.355186\pi\)
0.439414 + 0.898285i \(0.355186\pi\)
\(410\) −48.1906 −2.37996
\(411\) −0.813436 −0.0401238
\(412\) −15.9992 −0.788226
\(413\) −10.4421 −0.513821
\(414\) 2.18082 0.107181
\(415\) −41.0483 −2.01498
\(416\) 13.7171 0.672538
\(417\) −17.3423 −0.849256
\(418\) −45.9515 −2.24756
\(419\) −27.0503 −1.32149 −0.660746 0.750609i \(-0.729760\pi\)
−0.660746 + 0.750609i \(0.729760\pi\)
\(420\) −29.6414 −1.44635
\(421\) −16.2889 −0.793872 −0.396936 0.917846i \(-0.629927\pi\)
−0.396936 + 0.917846i \(0.629927\pi\)
\(422\) −44.9707 −2.18914
\(423\) −6.40585 −0.311463
\(424\) −7.62414 −0.370261
\(425\) 51.1849 2.48283
\(426\) 10.5800 0.512603
\(427\) −29.4090 −1.42320
\(428\) 37.0697 1.79183
\(429\) −5.32283 −0.256989
\(430\) 55.1692 2.66049
\(431\) −9.61562 −0.463168 −0.231584 0.972815i \(-0.574391\pi\)
−0.231584 + 0.972815i \(0.574391\pi\)
\(432\) −1.91657 −0.0922108
\(433\) 24.8003 1.19183 0.595913 0.803049i \(-0.296790\pi\)
0.595913 + 0.803049i \(0.296790\pi\)
\(434\) −62.8992 −3.01926
\(435\) 3.64804 0.174910
\(436\) 3.00883 0.144097
\(437\) 7.26232 0.347404
\(438\) −16.5073 −0.788750
\(439\) −36.6776 −1.75053 −0.875263 0.483646i \(-0.839312\pi\)
−0.875263 + 0.483646i \(0.839312\pi\)
\(440\) −17.4497 −0.831882
\(441\) 1.69218 0.0805798
\(442\) −24.6487 −1.17242
\(443\) −8.45059 −0.401499 −0.200750 0.979643i \(-0.564338\pi\)
−0.200750 + 0.979643i \(0.564338\pi\)
\(444\) −3.41546 −0.162091
\(445\) −11.3648 −0.538743
\(446\) −4.26223 −0.201822
\(447\) 21.0736 0.996746
\(448\) −36.7728 −1.73735
\(449\) 10.7809 0.508782 0.254391 0.967102i \(-0.418125\pi\)
0.254391 + 0.967102i \(0.418125\pi\)
\(450\) −18.1186 −0.854120
\(451\) 17.5747 0.827560
\(452\) 48.8704 2.29867
\(453\) −17.4071 −0.817858
\(454\) −46.7121 −2.19231
\(455\) −19.7316 −0.925032
\(456\) 11.9729 0.560684
\(457\) −17.4004 −0.813955 −0.406978 0.913438i \(-0.633417\pi\)
−0.406978 + 0.913438i \(0.633417\pi\)
\(458\) −8.94495 −0.417970
\(459\) 6.16079 0.287561
\(460\) 10.0539 0.468765
\(461\) −12.1558 −0.566151 −0.283076 0.959098i \(-0.591355\pi\)
−0.283076 + 0.959098i \(0.591355\pi\)
\(462\) 18.6547 0.867895
\(463\) −32.2609 −1.49929 −0.749645 0.661840i \(-0.769776\pi\)
−0.749645 + 0.661840i \(0.769776\pi\)
\(464\) 1.91657 0.0889743
\(465\) −35.6879 −1.65499
\(466\) −36.3118 −1.68211
\(467\) 3.87350 0.179244 0.0896222 0.995976i \(-0.471434\pi\)
0.0896222 + 0.995976i \(0.471434\pi\)
\(468\) 5.05607 0.233717
\(469\) 15.2476 0.704070
\(470\) −50.9631 −2.35075
\(471\) 20.9775 0.966594
\(472\) 5.83911 0.268767
\(473\) −20.1197 −0.925105
\(474\) −16.6151 −0.763159
\(475\) −60.3367 −2.76844
\(476\) 50.0582 2.29441
\(477\) 4.62451 0.211742
\(478\) −33.1235 −1.51504
\(479\) −22.3385 −1.02067 −0.510337 0.859975i \(-0.670479\pi\)
−0.510337 + 0.859975i \(0.670479\pi\)
\(480\) −27.2762 −1.24498
\(481\) −2.27360 −0.103667
\(482\) 14.1604 0.644990
\(483\) −2.94825 −0.134150
\(484\) −7.11595 −0.323452
\(485\) 4.98244 0.226241
\(486\) −2.18082 −0.0989240
\(487\) 11.6918 0.529804 0.264902 0.964275i \(-0.414660\pi\)
0.264902 + 0.964275i \(0.414660\pi\)
\(488\) 16.4453 0.744442
\(489\) 3.29012 0.148784
\(490\) 13.4625 0.608172
\(491\) 32.6810 1.47488 0.737438 0.675415i \(-0.236036\pi\)
0.737438 + 0.675415i \(0.236036\pi\)
\(492\) −16.6939 −0.752620
\(493\) −6.16079 −0.277468
\(494\) 29.0559 1.30729
\(495\) 10.5843 0.475730
\(496\) −18.7493 −0.841868
\(497\) −14.3031 −0.641583
\(498\) −24.5389 −1.09961
\(499\) −26.3934 −1.18153 −0.590765 0.806844i \(-0.701174\pi\)
−0.590765 + 0.806844i \(0.701174\pi\)
\(500\) −33.2600 −1.48743
\(501\) 19.1189 0.854170
\(502\) −57.1958 −2.55278
\(503\) 30.8951 1.37754 0.688772 0.724978i \(-0.258150\pi\)
0.688772 + 0.724978i \(0.258150\pi\)
\(504\) −4.86059 −0.216508
\(505\) 11.2847 0.502162
\(506\) −6.32738 −0.281286
\(507\) −9.63428 −0.427874
\(508\) 42.7071 1.89482
\(509\) 33.1618 1.46987 0.734936 0.678136i \(-0.237212\pi\)
0.734936 + 0.678136i \(0.237212\pi\)
\(510\) 49.0134 2.17035
\(511\) 22.3162 0.987212
\(512\) −20.6495 −0.912588
\(513\) −7.26232 −0.320639
\(514\) −0.860526 −0.0379562
\(515\) 21.1780 0.933212
\(516\) 19.1114 0.841332
\(517\) 18.5858 0.817402
\(518\) 7.96817 0.350101
\(519\) −4.38468 −0.192466
\(520\) 11.0337 0.483861
\(521\) −13.2878 −0.582151 −0.291075 0.956700i \(-0.594013\pi\)
−0.291075 + 0.956700i \(0.594013\pi\)
\(522\) 2.18082 0.0954518
\(523\) −12.3035 −0.537995 −0.268997 0.963141i \(-0.586692\pi\)
−0.268997 + 0.963141i \(0.586692\pi\)
\(524\) 29.0723 1.27003
\(525\) 24.4946 1.06903
\(526\) −59.9235 −2.61279
\(527\) 60.2695 2.62538
\(528\) 5.56068 0.241998
\(529\) 1.00000 0.0434783
\(530\) 36.7913 1.59811
\(531\) −3.54179 −0.153700
\(532\) −59.0085 −2.55834
\(533\) −11.1128 −0.481347
\(534\) −6.79394 −0.294003
\(535\) −49.0686 −2.12142
\(536\) −8.52634 −0.368282
\(537\) −21.2216 −0.915782
\(538\) 27.3997 1.18128
\(539\) −4.90964 −0.211473
\(540\) −10.0539 −0.432650
\(541\) 5.59393 0.240502 0.120251 0.992744i \(-0.461630\pi\)
0.120251 + 0.992744i \(0.461630\pi\)
\(542\) −6.89522 −0.296175
\(543\) 19.4211 0.833441
\(544\) 46.0639 1.97497
\(545\) −3.98274 −0.170602
\(546\) −11.7957 −0.504808
\(547\) −22.5204 −0.962904 −0.481452 0.876472i \(-0.659890\pi\)
−0.481452 + 0.876472i \(0.659890\pi\)
\(548\) −2.24181 −0.0957652
\(549\) −9.97508 −0.425726
\(550\) 52.5690 2.24155
\(551\) 7.26232 0.309385
\(552\) 1.64864 0.0701706
\(553\) 22.4620 0.955182
\(554\) −21.4781 −0.912517
\(555\) 4.52099 0.191905
\(556\) −47.7949 −2.02696
\(557\) −21.5581 −0.913448 −0.456724 0.889608i \(-0.650977\pi\)
−0.456724 + 0.889608i \(0.650977\pi\)
\(558\) −21.3344 −0.903158
\(559\) 12.7220 0.538084
\(560\) 20.6133 0.871071
\(561\) −17.8748 −0.754673
\(562\) −40.4492 −1.70625
\(563\) −21.7684 −0.917430 −0.458715 0.888583i \(-0.651690\pi\)
−0.458715 + 0.888583i \(0.651690\pi\)
\(564\) −17.6543 −0.743381
\(565\) −64.6890 −2.72148
\(566\) −5.77385 −0.242693
\(567\) 2.94825 0.123815
\(568\) 7.99818 0.335596
\(569\) −15.5221 −0.650719 −0.325360 0.945590i \(-0.605485\pi\)
−0.325360 + 0.945590i \(0.605485\pi\)
\(570\) −57.7769 −2.42001
\(571\) −13.0573 −0.546430 −0.273215 0.961953i \(-0.588087\pi\)
−0.273215 + 0.961953i \(0.588087\pi\)
\(572\) −14.6696 −0.613366
\(573\) 2.72682 0.113915
\(574\) 38.9464 1.62559
\(575\) −8.30818 −0.346475
\(576\) −12.4728 −0.519698
\(577\) −4.59902 −0.191460 −0.0957299 0.995407i \(-0.530519\pi\)
−0.0957299 + 0.995407i \(0.530519\pi\)
\(578\) −45.6997 −1.90086
\(579\) −2.27863 −0.0946966
\(580\) 10.0539 0.417465
\(581\) 33.1742 1.37630
\(582\) 2.97853 0.123464
\(583\) −13.4175 −0.555694
\(584\) −12.4790 −0.516386
\(585\) −6.69265 −0.276707
\(586\) 60.9046 2.51595
\(587\) −9.84048 −0.406160 −0.203080 0.979162i \(-0.565095\pi\)
−0.203080 + 0.979162i \(0.565095\pi\)
\(588\) 4.66359 0.192323
\(589\) −71.0455 −2.92738
\(590\) −28.1774 −1.16005
\(591\) 11.3746 0.467889
\(592\) 2.37519 0.0976197
\(593\) 43.2061 1.77426 0.887130 0.461519i \(-0.152695\pi\)
0.887130 + 0.461519i \(0.152695\pi\)
\(594\) 6.32738 0.259615
\(595\) −66.2612 −2.71645
\(596\) 58.0781 2.37897
\(597\) 15.0746 0.616963
\(598\) 4.00091 0.163609
\(599\) 46.1400 1.88523 0.942615 0.333881i \(-0.108359\pi\)
0.942615 + 0.333881i \(0.108359\pi\)
\(600\) −13.6972 −0.559184
\(601\) 31.5835 1.28832 0.644159 0.764891i \(-0.277207\pi\)
0.644159 + 0.764891i \(0.277207\pi\)
\(602\) −44.5863 −1.81720
\(603\) 5.17176 0.210610
\(604\) −47.9735 −1.95202
\(605\) 9.41927 0.382948
\(606\) 6.74606 0.274040
\(607\) 34.6230 1.40530 0.702652 0.711534i \(-0.251999\pi\)
0.702652 + 0.711534i \(0.251999\pi\)
\(608\) −54.3001 −2.20216
\(609\) −2.94825 −0.119469
\(610\) −79.3588 −3.21314
\(611\) −11.7521 −0.475439
\(612\) 16.9789 0.686333
\(613\) −17.1548 −0.692877 −0.346438 0.938073i \(-0.612609\pi\)
−0.346438 + 0.938073i \(0.612609\pi\)
\(614\) 49.2359 1.98700
\(615\) 22.0975 0.891057
\(616\) 14.1024 0.568202
\(617\) −37.5396 −1.51129 −0.755643 0.654984i \(-0.772676\pi\)
−0.755643 + 0.654984i \(0.772676\pi\)
\(618\) 12.6603 0.509272
\(619\) −23.3422 −0.938204 −0.469102 0.883144i \(-0.655422\pi\)
−0.469102 + 0.883144i \(0.655422\pi\)
\(620\) −98.3547 −3.95002
\(621\) −1.00000 −0.0401286
\(622\) −11.7605 −0.471553
\(623\) 9.18473 0.367978
\(624\) −3.51611 −0.140757
\(625\) 2.48491 0.0993965
\(626\) 8.10576 0.323971
\(627\) 21.0707 0.841484
\(628\) 57.8135 2.30701
\(629\) −7.63503 −0.304429
\(630\) 23.4554 0.934486
\(631\) 41.4409 1.64973 0.824867 0.565326i \(-0.191250\pi\)
0.824867 + 0.565326i \(0.191250\pi\)
\(632\) −12.5606 −0.499632
\(633\) 20.6210 0.819611
\(634\) 59.3051 2.35531
\(635\) −56.5307 −2.24335
\(636\) 12.7450 0.505373
\(637\) 3.10445 0.123003
\(638\) −6.32738 −0.250503
\(639\) −4.85140 −0.191918
\(640\) −44.6772 −1.76602
\(641\) −33.3001 −1.31527 −0.657637 0.753335i \(-0.728444\pi\)
−0.657637 + 0.753335i \(0.728444\pi\)
\(642\) −29.3335 −1.15770
\(643\) −17.4466 −0.688025 −0.344012 0.938965i \(-0.611786\pi\)
−0.344012 + 0.938965i \(0.611786\pi\)
\(644\) −8.12529 −0.320181
\(645\) −25.2975 −0.996086
\(646\) 97.5734 3.83897
\(647\) −16.1523 −0.635012 −0.317506 0.948256i \(-0.602845\pi\)
−0.317506 + 0.948256i \(0.602845\pi\)
\(648\) −1.64864 −0.0647645
\(649\) 10.2761 0.403370
\(650\) −33.2402 −1.30379
\(651\) 28.8420 1.13041
\(652\) 9.06747 0.355110
\(653\) −36.5084 −1.42868 −0.714342 0.699797i \(-0.753274\pi\)
−0.714342 + 0.699797i \(0.753274\pi\)
\(654\) −2.38091 −0.0931008
\(655\) −38.4826 −1.50364
\(656\) 11.6093 0.453268
\(657\) 7.56931 0.295307
\(658\) 41.1870 1.60564
\(659\) −34.0056 −1.32467 −0.662336 0.749207i \(-0.730435\pi\)
−0.662336 + 0.749207i \(0.730435\pi\)
\(660\) 29.1701 1.13545
\(661\) 4.58617 0.178381 0.0891906 0.996015i \(-0.471572\pi\)
0.0891906 + 0.996015i \(0.471572\pi\)
\(662\) 30.0359 1.16738
\(663\) 11.3025 0.438953
\(664\) −18.5507 −0.719907
\(665\) 78.1086 3.02892
\(666\) 2.70268 0.104727
\(667\) 1.00000 0.0387202
\(668\) 52.6911 2.03868
\(669\) 1.95442 0.0755621
\(670\) 41.1450 1.58957
\(671\) 28.9415 1.11727
\(672\) 22.0439 0.850363
\(673\) −10.9084 −0.420488 −0.210244 0.977649i \(-0.567426\pi\)
−0.210244 + 0.977649i \(0.567426\pi\)
\(674\) 12.8164 0.493668
\(675\) 8.30818 0.319782
\(676\) −26.5518 −1.02122
\(677\) 3.91537 0.150480 0.0752400 0.997165i \(-0.476028\pi\)
0.0752400 + 0.997165i \(0.476028\pi\)
\(678\) −38.6714 −1.48517
\(679\) −4.02668 −0.154530
\(680\) 37.0527 1.42091
\(681\) 21.4195 0.820798
\(682\) 61.8992 2.37024
\(683\) −22.2311 −0.850650 −0.425325 0.905041i \(-0.639840\pi\)
−0.425325 + 0.905041i \(0.639840\pi\)
\(684\) −20.0148 −0.765283
\(685\) 2.96745 0.113380
\(686\) 34.1272 1.30298
\(687\) 4.10165 0.156488
\(688\) −13.2905 −0.506696
\(689\) 8.48408 0.323218
\(690\) −7.95571 −0.302869
\(691\) 32.7261 1.24496 0.622479 0.782637i \(-0.286126\pi\)
0.622479 + 0.782637i \(0.286126\pi\)
\(692\) −12.0841 −0.459367
\(693\) −8.55398 −0.324939
\(694\) 33.7791 1.28224
\(695\) 63.2654 2.39979
\(696\) 1.64864 0.0624914
\(697\) −37.3181 −1.41352
\(698\) −35.5618 −1.34603
\(699\) 16.6505 0.629782
\(700\) 67.5063 2.55150
\(701\) 19.8786 0.750805 0.375402 0.926862i \(-0.377505\pi\)
0.375402 + 0.926862i \(0.377505\pi\)
\(702\) −4.00091 −0.151004
\(703\) 9.00016 0.339447
\(704\) 36.1882 1.36389
\(705\) 23.3688 0.880119
\(706\) 7.13562 0.268553
\(707\) −9.12000 −0.342993
\(708\) −9.76106 −0.366843
\(709\) −34.8409 −1.30848 −0.654239 0.756288i \(-0.727011\pi\)
−0.654239 + 0.756288i \(0.727011\pi\)
\(710\) −38.5963 −1.44849
\(711\) 7.61876 0.285726
\(712\) −5.13602 −0.192481
\(713\) −9.78276 −0.366367
\(714\) −39.6114 −1.48242
\(715\) 19.4179 0.726188
\(716\) −58.4862 −2.18573
\(717\) 15.1886 0.567228
\(718\) −64.7151 −2.41515
\(719\) −42.3706 −1.58016 −0.790079 0.613005i \(-0.789960\pi\)
−0.790079 + 0.613005i \(0.789960\pi\)
\(720\) 6.99170 0.260565
\(721\) −17.1155 −0.637414
\(722\) −73.5837 −2.73850
\(723\) −6.49318 −0.241484
\(724\) 53.5241 1.98921
\(725\) −8.30818 −0.308558
\(726\) 5.63090 0.208982
\(727\) −7.10224 −0.263408 −0.131704 0.991289i \(-0.542045\pi\)
−0.131704 + 0.991289i \(0.542045\pi\)
\(728\) −8.91718 −0.330493
\(729\) 1.00000 0.0370370
\(730\) 60.2193 2.22881
\(731\) 42.7222 1.58014
\(732\) −27.4910 −1.01610
\(733\) −2.19649 −0.0811293 −0.0405646 0.999177i \(-0.512916\pi\)
−0.0405646 + 0.999177i \(0.512916\pi\)
\(734\) −31.1445 −1.14956
\(735\) −6.17312 −0.227699
\(736\) −7.47695 −0.275604
\(737\) −15.0052 −0.552724
\(738\) 13.2100 0.486267
\(739\) 7.65800 0.281704 0.140852 0.990031i \(-0.455016\pi\)
0.140852 + 0.990031i \(0.455016\pi\)
\(740\) 12.4597 0.458029
\(741\) −13.3234 −0.489446
\(742\) −29.7338 −1.09156
\(743\) −0.670199 −0.0245872 −0.0122936 0.999924i \(-0.503913\pi\)
−0.0122936 + 0.999924i \(0.503913\pi\)
\(744\) −16.1282 −0.591289
\(745\) −76.8772 −2.81656
\(746\) −50.0935 −1.83405
\(747\) 11.2522 0.411695
\(748\) −49.2623 −1.80121
\(749\) 39.6560 1.44900
\(750\) 26.3189 0.961030
\(751\) −17.0871 −0.623516 −0.311758 0.950162i \(-0.600918\pi\)
−0.311758 + 0.950162i \(0.600918\pi\)
\(752\) 12.2772 0.447705
\(753\) 26.2268 0.955757
\(754\) 4.00091 0.145704
\(755\) 63.5019 2.31107
\(756\) 8.12529 0.295514
\(757\) 34.3726 1.24929 0.624646 0.780908i \(-0.285243\pi\)
0.624646 + 0.780908i \(0.285243\pi\)
\(758\) −1.06511 −0.0386864
\(759\) 2.90138 0.105313
\(760\) −43.6777 −1.58436
\(761\) −19.7508 −0.715964 −0.357982 0.933728i \(-0.616535\pi\)
−0.357982 + 0.933728i \(0.616535\pi\)
\(762\) −33.7944 −1.22424
\(763\) 3.21875 0.116526
\(764\) 7.51505 0.271885
\(765\) −22.4748 −0.812577
\(766\) 40.6813 1.46988
\(767\) −6.49772 −0.234619
\(768\) −1.76277 −0.0636086
\(769\) −24.9364 −0.899231 −0.449616 0.893222i \(-0.648439\pi\)
−0.449616 + 0.893222i \(0.648439\pi\)
\(770\) −68.0530 −2.45246
\(771\) 0.394588 0.0142108
\(772\) −6.27984 −0.226016
\(773\) 42.9566 1.54504 0.772520 0.634990i \(-0.218996\pi\)
0.772520 + 0.634990i \(0.218996\pi\)
\(774\) −15.1230 −0.543584
\(775\) 81.2769 2.91955
\(776\) 2.25168 0.0808307
\(777\) −3.65375 −0.131078
\(778\) 23.2429 0.833297
\(779\) 43.9905 1.57612
\(780\) −18.4447 −0.660428
\(781\) 14.0757 0.503669
\(782\) 13.4356 0.480455
\(783\) −1.00000 −0.0357371
\(784\) −3.24317 −0.115827
\(785\) −76.5269 −2.73136
\(786\) −23.0051 −0.820566
\(787\) −0.0246370 −0.000878213 0 −0.000439106 1.00000i \(-0.500140\pi\)
−0.000439106 1.00000i \(0.500140\pi\)
\(788\) 31.3481 1.11673
\(789\) 27.4775 0.978225
\(790\) 60.6127 2.15650
\(791\) 52.2799 1.85886
\(792\) 4.78331 0.169968
\(793\) −18.3002 −0.649858
\(794\) 40.2323 1.42779
\(795\) −16.8704 −0.598332
\(796\) 41.5452 1.47253
\(797\) 44.0032 1.55867 0.779337 0.626605i \(-0.215556\pi\)
0.779337 + 0.626605i \(0.215556\pi\)
\(798\) 46.6938 1.65294
\(799\) −39.4651 −1.39617
\(800\) 62.1199 2.19627
\(801\) 3.11532 0.110074
\(802\) −32.4970 −1.14751
\(803\) −21.9614 −0.775002
\(804\) 14.2532 0.502672
\(805\) 10.7553 0.379075
\(806\) −39.1399 −1.37864
\(807\) −12.5639 −0.442272
\(808\) 5.09982 0.179411
\(809\) −23.4278 −0.823676 −0.411838 0.911257i \(-0.635113\pi\)
−0.411838 + 0.911257i \(0.635113\pi\)
\(810\) 7.95571 0.279535
\(811\) 43.9171 1.54214 0.771069 0.636751i \(-0.219722\pi\)
0.771069 + 0.636751i \(0.219722\pi\)
\(812\) −8.12529 −0.285142
\(813\) 3.16176 0.110888
\(814\) −7.84149 −0.274844
\(815\) −12.0025 −0.420428
\(816\) −11.8076 −0.413347
\(817\) −50.3609 −1.76190
\(818\) −38.7601 −1.35522
\(819\) 5.40883 0.189000
\(820\) 60.9000 2.12672
\(821\) −9.21652 −0.321659 −0.160829 0.986982i \(-0.551417\pi\)
−0.160829 + 0.986982i \(0.551417\pi\)
\(822\) 1.77396 0.0618739
\(823\) 22.0721 0.769385 0.384692 0.923045i \(-0.374308\pi\)
0.384692 + 0.923045i \(0.374308\pi\)
\(824\) 9.57083 0.333416
\(825\) −24.1052 −0.839234
\(826\) 22.7723 0.792348
\(827\) 6.11448 0.212621 0.106311 0.994333i \(-0.466096\pi\)
0.106311 + 0.994333i \(0.466096\pi\)
\(828\) −2.75597 −0.0957766
\(829\) 0.991392 0.0344325 0.0172162 0.999852i \(-0.494520\pi\)
0.0172162 + 0.999852i \(0.494520\pi\)
\(830\) 89.5189 3.10724
\(831\) 9.84864 0.341646
\(832\) −22.8824 −0.793303
\(833\) 10.4251 0.361209
\(834\) 37.8204 1.30961
\(835\) −69.7465 −2.41368
\(836\) 58.0703 2.00840
\(837\) 9.78276 0.338142
\(838\) 58.9918 2.03784
\(839\) 48.3667 1.66980 0.834902 0.550399i \(-0.185524\pi\)
0.834902 + 0.550399i \(0.185524\pi\)
\(840\) 17.7316 0.611799
\(841\) 1.00000 0.0344828
\(842\) 35.5231 1.22421
\(843\) 18.5477 0.638818
\(844\) 56.8309 1.95620
\(845\) 35.1462 1.20907
\(846\) 13.9700 0.480298
\(847\) −7.61241 −0.261566
\(848\) −8.86319 −0.304363
\(849\) 2.64756 0.0908640
\(850\) −111.625 −3.82871
\(851\) 1.23929 0.0424825
\(852\) −13.3703 −0.458059
\(853\) −25.5102 −0.873452 −0.436726 0.899595i \(-0.643862\pi\)
−0.436726 + 0.899595i \(0.643862\pi\)
\(854\) 64.1358 2.19468
\(855\) 26.4932 0.906049
\(856\) −22.1753 −0.757935
\(857\) −50.0291 −1.70896 −0.854481 0.519483i \(-0.826124\pi\)
−0.854481 + 0.519483i \(0.826124\pi\)
\(858\) 11.6081 0.396295
\(859\) 2.36514 0.0806974 0.0403487 0.999186i \(-0.487153\pi\)
0.0403487 + 0.999186i \(0.487153\pi\)
\(860\) −69.7190 −2.37740
\(861\) −17.8586 −0.608620
\(862\) 20.9699 0.714239
\(863\) −33.5614 −1.14244 −0.571222 0.820796i \(-0.693530\pi\)
−0.571222 + 0.820796i \(0.693530\pi\)
\(864\) 7.47695 0.254371
\(865\) 15.9955 0.543863
\(866\) −54.0849 −1.83788
\(867\) 20.9553 0.711679
\(868\) 79.4877 2.69799
\(869\) −22.1049 −0.749858
\(870\) −7.95571 −0.269724
\(871\) 9.48804 0.321490
\(872\) −1.79990 −0.0609521
\(873\) −1.36579 −0.0462248
\(874\) −15.8378 −0.535722
\(875\) −35.5805 −1.20284
\(876\) 20.8608 0.704822
\(877\) −7.13680 −0.240993 −0.120496 0.992714i \(-0.538449\pi\)
−0.120496 + 0.992714i \(0.538449\pi\)
\(878\) 79.9872 2.69944
\(879\) −27.9274 −0.941968
\(880\) −20.2856 −0.683826
\(881\) −44.4613 −1.49794 −0.748970 0.662604i \(-0.769451\pi\)
−0.748970 + 0.662604i \(0.769451\pi\)
\(882\) −3.69033 −0.124260
\(883\) 40.1969 1.35274 0.676368 0.736564i \(-0.263553\pi\)
0.676368 + 0.736564i \(0.263553\pi\)
\(884\) 31.1494 1.04767
\(885\) 12.9206 0.434320
\(886\) 18.4292 0.619141
\(887\) 17.9446 0.602520 0.301260 0.953542i \(-0.402593\pi\)
0.301260 + 0.953542i \(0.402593\pi\)
\(888\) 2.04315 0.0685635
\(889\) 45.6867 1.53228
\(890\) 24.7845 0.830780
\(891\) −2.90138 −0.0971998
\(892\) 5.38631 0.180347
\(893\) 46.5213 1.55678
\(894\) −45.9576 −1.53705
\(895\) 77.4174 2.58778
\(896\) 36.1069 1.20625
\(897\) −1.83459 −0.0612551
\(898\) −23.5112 −0.784578
\(899\) −9.78276 −0.326273
\(900\) 22.8971 0.763236
\(901\) 28.4906 0.949161
\(902\) −38.3272 −1.27616
\(903\) 20.4447 0.680359
\(904\) −29.2345 −0.972325
\(905\) −70.8491 −2.35510
\(906\) 37.9618 1.26120
\(907\) 32.2495 1.07083 0.535414 0.844590i \(-0.320156\pi\)
0.535414 + 0.844590i \(0.320156\pi\)
\(908\) 59.0316 1.95903
\(909\) −3.09336 −0.102600
\(910\) 43.0310 1.42647
\(911\) 53.0013 1.75601 0.878005 0.478652i \(-0.158874\pi\)
0.878005 + 0.478652i \(0.158874\pi\)
\(912\) 13.9187 0.460895
\(913\) −32.6467 −1.08045
\(914\) 37.9471 1.25518
\(915\) 36.3895 1.20300
\(916\) 11.3040 0.373495
\(917\) 31.1006 1.02703
\(918\) −13.4356 −0.443439
\(919\) −44.2461 −1.45954 −0.729772 0.683691i \(-0.760374\pi\)
−0.729772 + 0.683691i \(0.760374\pi\)
\(920\) −6.01428 −0.198285
\(921\) −22.5768 −0.743931
\(922\) 26.5096 0.873046
\(923\) −8.90032 −0.292957
\(924\) −23.5745 −0.775545
\(925\) −10.2963 −0.338540
\(926\) 70.3552 2.31201
\(927\) −5.80530 −0.190671
\(928\) −7.47695 −0.245443
\(929\) 53.1570 1.74403 0.872013 0.489483i \(-0.162814\pi\)
0.872013 + 0.489483i \(0.162814\pi\)
\(930\) 77.8288 2.55211
\(931\) −12.2891 −0.402760
\(932\) 45.8884 1.50313
\(933\) 5.39270 0.176549
\(934\) −8.44741 −0.276408
\(935\) 65.2078 2.13252
\(936\) −3.02457 −0.0988611
\(937\) 44.8870 1.46640 0.733198 0.680016i \(-0.238027\pi\)
0.733198 + 0.680016i \(0.238027\pi\)
\(938\) −33.2523 −1.08573
\(939\) −3.71684 −0.121295
\(940\) 64.4037 2.10062
\(941\) 7.11766 0.232029 0.116015 0.993248i \(-0.462988\pi\)
0.116015 + 0.993248i \(0.462988\pi\)
\(942\) −45.7482 −1.49056
\(943\) 6.05736 0.197255
\(944\) 6.78807 0.220933
\(945\) −10.7553 −0.349871
\(946\) 43.8774 1.42658
\(947\) −25.8079 −0.838644 −0.419322 0.907838i \(-0.637732\pi\)
−0.419322 + 0.907838i \(0.637732\pi\)
\(948\) 20.9971 0.681954
\(949\) 13.8866 0.450777
\(950\) 131.583 4.26913
\(951\) −27.1940 −0.881824
\(952\) −29.9450 −0.970525
\(953\) −2.86492 −0.0928040 −0.0464020 0.998923i \(-0.514776\pi\)
−0.0464020 + 0.998923i \(0.514776\pi\)
\(954\) −10.0852 −0.326521
\(955\) −9.94755 −0.321895
\(956\) 41.8593 1.35383
\(957\) 2.90138 0.0937882
\(958\) 48.7163 1.57395
\(959\) −2.39821 −0.0774423
\(960\) 45.5011 1.46854
\(961\) 64.7023 2.08717
\(962\) 4.95830 0.159862
\(963\) 13.4507 0.433442
\(964\) −17.8950 −0.576359
\(965\) 8.31253 0.267590
\(966\) 6.42960 0.206869
\(967\) 58.3742 1.87719 0.938594 0.345023i \(-0.112129\pi\)
0.938594 + 0.345023i \(0.112129\pi\)
\(968\) 4.25680 0.136819
\(969\) −44.7416 −1.43731
\(970\) −10.8658 −0.348879
\(971\) −39.9542 −1.28219 −0.641095 0.767461i \(-0.721520\pi\)
−0.641095 + 0.767461i \(0.721520\pi\)
\(972\) 2.75597 0.0883978
\(973\) −51.1295 −1.63913
\(974\) −25.4976 −0.816996
\(975\) 15.2421 0.488137
\(976\) 19.1179 0.611949
\(977\) −4.45659 −0.142579 −0.0712894 0.997456i \(-0.522711\pi\)
−0.0712894 + 0.997456i \(0.522711\pi\)
\(978\) −7.17515 −0.229436
\(979\) −9.03871 −0.288878
\(980\) −17.0129 −0.543459
\(981\) 1.09175 0.0348568
\(982\) −71.2714 −2.27436
\(983\) 58.3721 1.86178 0.930891 0.365298i \(-0.119033\pi\)
0.930891 + 0.365298i \(0.119033\pi\)
\(984\) 9.98639 0.318354
\(985\) −41.4950 −1.32214
\(986\) 13.4356 0.427875
\(987\) −18.8860 −0.601149
\(988\) −36.7188 −1.16818
\(989\) −6.93454 −0.220505
\(990\) −23.0825 −0.733610
\(991\) 12.6190 0.400856 0.200428 0.979708i \(-0.435767\pi\)
0.200428 + 0.979708i \(0.435767\pi\)
\(992\) 73.1452 2.32236
\(993\) −13.7728 −0.437066
\(994\) 31.1925 0.989367
\(995\) −54.9928 −1.74339
\(996\) 31.0106 0.982609
\(997\) 20.7247 0.656357 0.328178 0.944616i \(-0.393565\pi\)
0.328178 + 0.944616i \(0.393565\pi\)
\(998\) 57.5591 1.82200
\(999\) −1.23929 −0.0392095
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 2001.2.a.n.1.3 16
3.2 odd 2 6003.2.a.r.1.14 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.n.1.3 16 1.1 even 1 trivial
6003.2.a.r.1.14 16 3.2 odd 2