Properties

Label 4334.2.a.a.1.7
Level $4334$
Weight $2$
Character 4334.1
Self dual yes
Analytic conductor $34.607$
Analytic rank $1$
Dimension $15$
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] = [4334,2,Mod(1,4334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4334, 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("4334.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4334 = 2 \cdot 11 \cdot 197 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4334.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: \(34.6071642360\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 23 x^{13} + 19 x^{12} + 194 x^{11} - 124 x^{10} - 761 x^{9} + 353 x^{8} + 1417 x^{7} - 465 x^{6} - 1128 x^{5} + 288 x^{4} + 316 x^{3} - 79 x^{2} - 20 x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.415881\) of defining polynomial
Character \(\chi\) \(=\) 4334.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.415881 q^{3} +1.00000 q^{4} -2.24456 q^{5} +0.415881 q^{6} +1.94686 q^{7} -1.00000 q^{8} -2.82704 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.415881 q^{3} +1.00000 q^{4} -2.24456 q^{5} +0.415881 q^{6} +1.94686 q^{7} -1.00000 q^{8} -2.82704 q^{9} +2.24456 q^{10} +1.00000 q^{11} -0.415881 q^{12} +1.65329 q^{13} -1.94686 q^{14} +0.933471 q^{15} +1.00000 q^{16} -4.04145 q^{17} +2.82704 q^{18} -7.25266 q^{19} -2.24456 q^{20} -0.809661 q^{21} -1.00000 q^{22} +8.28397 q^{23} +0.415881 q^{24} +0.0380545 q^{25} -1.65329 q^{26} +2.42336 q^{27} +1.94686 q^{28} +10.1950 q^{29} -0.933471 q^{30} +2.78382 q^{31} -1.00000 q^{32} -0.415881 q^{33} +4.04145 q^{34} -4.36984 q^{35} -2.82704 q^{36} -0.827167 q^{37} +7.25266 q^{38} -0.687573 q^{39} +2.24456 q^{40} +5.69287 q^{41} +0.809661 q^{42} +7.31501 q^{43} +1.00000 q^{44} +6.34547 q^{45} -8.28397 q^{46} -7.73976 q^{47} -0.415881 q^{48} -3.20976 q^{49} -0.0380545 q^{50} +1.68076 q^{51} +1.65329 q^{52} -7.74921 q^{53} -2.42336 q^{54} -2.24456 q^{55} -1.94686 q^{56} +3.01625 q^{57} -10.1950 q^{58} +7.99838 q^{59} +0.933471 q^{60} -8.08683 q^{61} -2.78382 q^{62} -5.50384 q^{63} +1.00000 q^{64} -3.71092 q^{65} +0.415881 q^{66} -11.4984 q^{67} -4.04145 q^{68} -3.44515 q^{69} +4.36984 q^{70} -3.83205 q^{71} +2.82704 q^{72} +0.425688 q^{73} +0.827167 q^{74} -0.0158262 q^{75} -7.25266 q^{76} +1.94686 q^{77} +0.687573 q^{78} -13.8952 q^{79} -2.24456 q^{80} +7.47330 q^{81} -5.69287 q^{82} -2.39944 q^{83} -0.809661 q^{84} +9.07128 q^{85} -7.31501 q^{86} -4.23991 q^{87} -1.00000 q^{88} +10.9318 q^{89} -6.34547 q^{90} +3.21872 q^{91} +8.28397 q^{92} -1.15774 q^{93} +7.73976 q^{94} +16.2790 q^{95} +0.415881 q^{96} +13.3160 q^{97} +3.20976 q^{98} -2.82704 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9} + 7 q^{10} + 15 q^{11} - q^{12} - q^{13} - q^{14} - 6 q^{15} + 15 q^{16} - 6 q^{17} - 2 q^{18} - 14 q^{19} - 7 q^{20} - 3 q^{21} - 15 q^{22} + 2 q^{23} + q^{24} - 10 q^{25} + q^{26} - 7 q^{27} + q^{28} + 8 q^{29} + 6 q^{30} - 33 q^{31} - 15 q^{32} - q^{33} + 6 q^{34} - 8 q^{35} + 2 q^{36} - 9 q^{37} + 14 q^{38} - 9 q^{39} + 7 q^{40} - 10 q^{41} + 3 q^{42} - 6 q^{43} + 15 q^{44} - 20 q^{45} - 2 q^{46} - q^{47} - q^{48} - 30 q^{49} + 10 q^{50} + 12 q^{51} - q^{52} + 6 q^{53} + 7 q^{54} - 7 q^{55} - q^{56} - 24 q^{57} - 8 q^{58} - 15 q^{59} - 6 q^{60} - 25 q^{61} + 33 q^{62} + 12 q^{63} + 15 q^{64} + 31 q^{65} + q^{66} - 13 q^{67} - 6 q^{68} - 43 q^{69} + 8 q^{70} - 4 q^{71} - 2 q^{72} - 4 q^{73} + 9 q^{74} - 5 q^{75} - 14 q^{76} + q^{77} + 9 q^{78} - 20 q^{79} - 7 q^{80} + 11 q^{81} + 10 q^{82} + q^{83} - 3 q^{84} - q^{85} + 6 q^{86} + 22 q^{87} - 15 q^{88} - 41 q^{89} + 20 q^{90} - 31 q^{91} + 2 q^{92} + 14 q^{93} + q^{94} + 41 q^{95} + q^{96} - 57 q^{97} + 30 q^{98} + 2 q^{99}+O(q^{100}) \) 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
\(3\) −0.415881 −0.240109 −0.120055 0.992767i \(-0.538307\pi\)
−0.120055 + 0.992767i \(0.538307\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.24456 −1.00380 −0.501899 0.864926i \(-0.667365\pi\)
−0.501899 + 0.864926i \(0.667365\pi\)
\(6\) 0.415881 0.169783
\(7\) 1.94686 0.735842 0.367921 0.929857i \(-0.380070\pi\)
0.367921 + 0.929857i \(0.380070\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.82704 −0.942348
\(10\) 2.24456 0.709793
\(11\) 1.00000 0.301511
\(12\) −0.415881 −0.120055
\(13\) 1.65329 0.458541 0.229270 0.973363i \(-0.426366\pi\)
0.229270 + 0.973363i \(0.426366\pi\)
\(14\) −1.94686 −0.520319
\(15\) 0.933471 0.241021
\(16\) 1.00000 0.250000
\(17\) −4.04145 −0.980196 −0.490098 0.871667i \(-0.663039\pi\)
−0.490098 + 0.871667i \(0.663039\pi\)
\(18\) 2.82704 0.666340
\(19\) −7.25266 −1.66387 −0.831937 0.554870i \(-0.812768\pi\)
−0.831937 + 0.554870i \(0.812768\pi\)
\(20\) −2.24456 −0.501899
\(21\) −0.809661 −0.176682
\(22\) −1.00000 −0.213201
\(23\) 8.28397 1.72733 0.863663 0.504069i \(-0.168164\pi\)
0.863663 + 0.504069i \(0.168164\pi\)
\(24\) 0.415881 0.0848914
\(25\) 0.0380545 0.00761091
\(26\) −1.65329 −0.324237
\(27\) 2.42336 0.466375
\(28\) 1.94686 0.367921
\(29\) 10.1950 1.89316 0.946581 0.322466i \(-0.104512\pi\)
0.946581 + 0.322466i \(0.104512\pi\)
\(30\) −0.933471 −0.170428
\(31\) 2.78382 0.499989 0.249994 0.968247i \(-0.419571\pi\)
0.249994 + 0.968247i \(0.419571\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.415881 −0.0723956
\(34\) 4.04145 0.693103
\(35\) −4.36984 −0.738637
\(36\) −2.82704 −0.471174
\(37\) −0.827167 −0.135985 −0.0679927 0.997686i \(-0.521659\pi\)
−0.0679927 + 0.997686i \(0.521659\pi\)
\(38\) 7.25266 1.17654
\(39\) −0.687573 −0.110100
\(40\) 2.24456 0.354896
\(41\) 5.69287 0.889077 0.444538 0.895760i \(-0.353368\pi\)
0.444538 + 0.895760i \(0.353368\pi\)
\(42\) 0.809661 0.124933
\(43\) 7.31501 1.11553 0.557764 0.830000i \(-0.311659\pi\)
0.557764 + 0.830000i \(0.311659\pi\)
\(44\) 1.00000 0.150756
\(45\) 6.34547 0.945927
\(46\) −8.28397 −1.22140
\(47\) −7.73976 −1.12896 −0.564480 0.825447i \(-0.690923\pi\)
−0.564480 + 0.825447i \(0.690923\pi\)
\(48\) −0.415881 −0.0600273
\(49\) −3.20976 −0.458536
\(50\) −0.0380545 −0.00538172
\(51\) 1.68076 0.235354
\(52\) 1.65329 0.229270
\(53\) −7.74921 −1.06444 −0.532218 0.846607i \(-0.678641\pi\)
−0.532218 + 0.846607i \(0.678641\pi\)
\(54\) −2.42336 −0.329777
\(55\) −2.24456 −0.302657
\(56\) −1.94686 −0.260159
\(57\) 3.01625 0.399512
\(58\) −10.1950 −1.33867
\(59\) 7.99838 1.04130 0.520650 0.853770i \(-0.325690\pi\)
0.520650 + 0.853770i \(0.325690\pi\)
\(60\) 0.933471 0.120511
\(61\) −8.08683 −1.03541 −0.517706 0.855559i \(-0.673214\pi\)
−0.517706 + 0.855559i \(0.673214\pi\)
\(62\) −2.78382 −0.353545
\(63\) −5.50384 −0.693419
\(64\) 1.00000 0.125000
\(65\) −3.71092 −0.460282
\(66\) 0.415881 0.0511915
\(67\) −11.4984 −1.40475 −0.702377 0.711805i \(-0.747878\pi\)
−0.702377 + 0.711805i \(0.747878\pi\)
\(68\) −4.04145 −0.490098
\(69\) −3.44515 −0.414747
\(70\) 4.36984 0.522295
\(71\) −3.83205 −0.454781 −0.227391 0.973804i \(-0.573019\pi\)
−0.227391 + 0.973804i \(0.573019\pi\)
\(72\) 2.82704 0.333170
\(73\) 0.425688 0.0498230 0.0249115 0.999690i \(-0.492070\pi\)
0.0249115 + 0.999690i \(0.492070\pi\)
\(74\) 0.827167 0.0961562
\(75\) −0.0158262 −0.00182745
\(76\) −7.25266 −0.831937
\(77\) 1.94686 0.221865
\(78\) 0.687573 0.0778524
\(79\) −13.8952 −1.56333 −0.781666 0.623698i \(-0.785630\pi\)
−0.781666 + 0.623698i \(0.785630\pi\)
\(80\) −2.24456 −0.250950
\(81\) 7.47330 0.830367
\(82\) −5.69287 −0.628672
\(83\) −2.39944 −0.263373 −0.131686 0.991291i \(-0.542039\pi\)
−0.131686 + 0.991291i \(0.542039\pi\)
\(84\) −0.809661 −0.0883412
\(85\) 9.07128 0.983919
\(86\) −7.31501 −0.788798
\(87\) −4.23991 −0.454566
\(88\) −1.00000 −0.106600
\(89\) 10.9318 1.15877 0.579387 0.815053i \(-0.303292\pi\)
0.579387 + 0.815053i \(0.303292\pi\)
\(90\) −6.34547 −0.668871
\(91\) 3.21872 0.337414
\(92\) 8.28397 0.863663
\(93\) −1.15774 −0.120052
\(94\) 7.73976 0.798295
\(95\) 16.2790 1.67019
\(96\) 0.415881 0.0424457
\(97\) 13.3160 1.35204 0.676020 0.736883i \(-0.263703\pi\)
0.676020 + 0.736883i \(0.263703\pi\)
\(98\) 3.20976 0.324234
\(99\) −2.82704 −0.284128
\(100\) 0.0380545 0.00380545
\(101\) 8.43947 0.839758 0.419879 0.907580i \(-0.362072\pi\)
0.419879 + 0.907580i \(0.362072\pi\)
\(102\) −1.68076 −0.166420
\(103\) −0.908357 −0.0895031 −0.0447516 0.998998i \(-0.514250\pi\)
−0.0447516 + 0.998998i \(0.514250\pi\)
\(104\) −1.65329 −0.162119
\(105\) 1.81733 0.177354
\(106\) 7.74921 0.752670
\(107\) 14.1911 1.37190 0.685951 0.727647i \(-0.259386\pi\)
0.685951 + 0.727647i \(0.259386\pi\)
\(108\) 2.42336 0.233188
\(109\) 4.23544 0.405681 0.202841 0.979212i \(-0.434983\pi\)
0.202841 + 0.979212i \(0.434983\pi\)
\(110\) 2.24456 0.214011
\(111\) 0.344003 0.0326513
\(112\) 1.94686 0.183961
\(113\) −7.13396 −0.671106 −0.335553 0.942021i \(-0.608923\pi\)
−0.335553 + 0.942021i \(0.608923\pi\)
\(114\) −3.01625 −0.282497
\(115\) −18.5939 −1.73389
\(116\) 10.1950 0.946581
\(117\) −4.67393 −0.432105
\(118\) −7.99838 −0.736310
\(119\) −7.86812 −0.721269
\(120\) −0.933471 −0.0852139
\(121\) 1.00000 0.0909091
\(122\) 8.08683 0.732147
\(123\) −2.36756 −0.213476
\(124\) 2.78382 0.249994
\(125\) 11.1374 0.996158
\(126\) 5.50384 0.490321
\(127\) −17.6578 −1.56687 −0.783436 0.621472i \(-0.786535\pi\)
−0.783436 + 0.621472i \(0.786535\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −3.04217 −0.267849
\(130\) 3.71092 0.325469
\(131\) −7.73493 −0.675804 −0.337902 0.941181i \(-0.609717\pi\)
−0.337902 + 0.941181i \(0.609717\pi\)
\(132\) −0.415881 −0.0361978
\(133\) −14.1199 −1.22435
\(134\) 11.4984 0.993312
\(135\) −5.43938 −0.468147
\(136\) 4.04145 0.346551
\(137\) 1.28592 0.109864 0.0549319 0.998490i \(-0.482506\pi\)
0.0549319 + 0.998490i \(0.482506\pi\)
\(138\) 3.44515 0.293270
\(139\) −15.5538 −1.31925 −0.659627 0.751593i \(-0.729286\pi\)
−0.659627 + 0.751593i \(0.729286\pi\)
\(140\) −4.36984 −0.369318
\(141\) 3.21882 0.271074
\(142\) 3.83205 0.321579
\(143\) 1.65329 0.138255
\(144\) −2.82704 −0.235587
\(145\) −22.8833 −1.90035
\(146\) −0.425688 −0.0352302
\(147\) 1.33488 0.110099
\(148\) −0.827167 −0.0679927
\(149\) −20.2290 −1.65722 −0.828611 0.559825i \(-0.810868\pi\)
−0.828611 + 0.559825i \(0.810868\pi\)
\(150\) 0.0158262 0.00129220
\(151\) 5.49776 0.447402 0.223701 0.974658i \(-0.428186\pi\)
0.223701 + 0.974658i \(0.428186\pi\)
\(152\) 7.25266 0.588269
\(153\) 11.4254 0.923685
\(154\) −1.94686 −0.156882
\(155\) −6.24845 −0.501888
\(156\) −0.687573 −0.0550499
\(157\) 2.13019 0.170008 0.0850039 0.996381i \(-0.472910\pi\)
0.0850039 + 0.996381i \(0.472910\pi\)
\(158\) 13.8952 1.10544
\(159\) 3.22275 0.255581
\(160\) 2.24456 0.177448
\(161\) 16.1277 1.27104
\(162\) −7.47330 −0.587158
\(163\) 20.4786 1.60401 0.802005 0.597317i \(-0.203767\pi\)
0.802005 + 0.597317i \(0.203767\pi\)
\(164\) 5.69287 0.444538
\(165\) 0.933471 0.0726706
\(166\) 2.39944 0.186233
\(167\) −2.86254 −0.221510 −0.110755 0.993848i \(-0.535327\pi\)
−0.110755 + 0.993848i \(0.535327\pi\)
\(168\) 0.809661 0.0624667
\(169\) −10.2666 −0.789740
\(170\) −9.07128 −0.695735
\(171\) 20.5036 1.56795
\(172\) 7.31501 0.557764
\(173\) −16.6986 −1.26957 −0.634784 0.772690i \(-0.718911\pi\)
−0.634784 + 0.772690i \(0.718911\pi\)
\(174\) 4.23991 0.321426
\(175\) 0.0740866 0.00560042
\(176\) 1.00000 0.0753778
\(177\) −3.32637 −0.250026
\(178\) −10.9318 −0.819377
\(179\) −23.1531 −1.73055 −0.865273 0.501301i \(-0.832855\pi\)
−0.865273 + 0.501301i \(0.832855\pi\)
\(180\) 6.34547 0.472963
\(181\) 8.37604 0.622586 0.311293 0.950314i \(-0.399238\pi\)
0.311293 + 0.950314i \(0.399238\pi\)
\(182\) −3.21872 −0.238587
\(183\) 3.36316 0.248612
\(184\) −8.28397 −0.610702
\(185\) 1.85663 0.136502
\(186\) 1.15774 0.0848895
\(187\) −4.04145 −0.295540
\(188\) −7.73976 −0.564480
\(189\) 4.71793 0.343179
\(190\) −16.2790 −1.18101
\(191\) 8.39612 0.607522 0.303761 0.952748i \(-0.401758\pi\)
0.303761 + 0.952748i \(0.401758\pi\)
\(192\) −0.415881 −0.0300136
\(193\) −21.4162 −1.54157 −0.770784 0.637096i \(-0.780135\pi\)
−0.770784 + 0.637096i \(0.780135\pi\)
\(194\) −13.3160 −0.956036
\(195\) 1.54330 0.110518
\(196\) −3.20976 −0.229268
\(197\) 1.00000 0.0712470
\(198\) 2.82704 0.200909
\(199\) 2.37949 0.168678 0.0843389 0.996437i \(-0.473122\pi\)
0.0843389 + 0.996437i \(0.473122\pi\)
\(200\) −0.0380545 −0.00269086
\(201\) 4.78197 0.337294
\(202\) −8.43947 −0.593799
\(203\) 19.8482 1.39307
\(204\) 1.68076 0.117677
\(205\) −12.7780 −0.892454
\(206\) 0.908357 0.0632883
\(207\) −23.4191 −1.62774
\(208\) 1.65329 0.114635
\(209\) −7.25266 −0.501677
\(210\) −1.81733 −0.125408
\(211\) 12.5858 0.866441 0.433220 0.901288i \(-0.357377\pi\)
0.433220 + 0.901288i \(0.357377\pi\)
\(212\) −7.74921 −0.532218
\(213\) 1.59368 0.109197
\(214\) −14.1911 −0.970082
\(215\) −16.4190 −1.11977
\(216\) −2.42336 −0.164889
\(217\) 5.41969 0.367913
\(218\) −4.23544 −0.286860
\(219\) −0.177036 −0.0119630
\(220\) −2.24456 −0.151328
\(221\) −6.68170 −0.449460
\(222\) −0.344003 −0.0230880
\(223\) −17.6630 −1.18280 −0.591402 0.806377i \(-0.701425\pi\)
−0.591402 + 0.806377i \(0.701425\pi\)
\(224\) −1.94686 −0.130080
\(225\) −0.107582 −0.00717212
\(226\) 7.13396 0.474544
\(227\) −18.5462 −1.23096 −0.615478 0.788154i \(-0.711037\pi\)
−0.615478 + 0.788154i \(0.711037\pi\)
\(228\) 3.01625 0.199756
\(229\) −21.5774 −1.42587 −0.712936 0.701229i \(-0.752635\pi\)
−0.712936 + 0.701229i \(0.752635\pi\)
\(230\) 18.5939 1.22604
\(231\) −0.809661 −0.0532718
\(232\) −10.1950 −0.669334
\(233\) 22.8480 1.49682 0.748412 0.663234i \(-0.230817\pi\)
0.748412 + 0.663234i \(0.230817\pi\)
\(234\) 4.67393 0.305544
\(235\) 17.3724 1.13325
\(236\) 7.99838 0.520650
\(237\) 5.77875 0.375370
\(238\) 7.86812 0.510014
\(239\) 15.0535 0.973733 0.486866 0.873476i \(-0.338140\pi\)
0.486866 + 0.873476i \(0.338140\pi\)
\(240\) 0.933471 0.0602553
\(241\) −5.12265 −0.329979 −0.164989 0.986295i \(-0.552759\pi\)
−0.164989 + 0.986295i \(0.552759\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −10.3781 −0.665754
\(244\) −8.08683 −0.517706
\(245\) 7.20449 0.460278
\(246\) 2.36756 0.150950
\(247\) −11.9908 −0.762954
\(248\) −2.78382 −0.176773
\(249\) 0.997882 0.0632382
\(250\) −11.1374 −0.704390
\(251\) −7.34662 −0.463715 −0.231857 0.972750i \(-0.574480\pi\)
−0.231857 + 0.972750i \(0.574480\pi\)
\(252\) −5.50384 −0.346709
\(253\) 8.28397 0.520808
\(254\) 17.6578 1.10795
\(255\) −3.77258 −0.236248
\(256\) 1.00000 0.0625000
\(257\) −23.0169 −1.43576 −0.717878 0.696169i \(-0.754886\pi\)
−0.717878 + 0.696169i \(0.754886\pi\)
\(258\) 3.04217 0.189398
\(259\) −1.61037 −0.100064
\(260\) −3.71092 −0.230141
\(261\) −28.8217 −1.78402
\(262\) 7.73493 0.477865
\(263\) −12.3679 −0.762636 −0.381318 0.924444i \(-0.624530\pi\)
−0.381318 + 0.924444i \(0.624530\pi\)
\(264\) 0.415881 0.0255957
\(265\) 17.3936 1.06848
\(266\) 14.1199 0.865745
\(267\) −4.54635 −0.278232
\(268\) −11.4984 −0.702377
\(269\) 19.0593 1.16206 0.581032 0.813881i \(-0.302649\pi\)
0.581032 + 0.813881i \(0.302649\pi\)
\(270\) 5.43938 0.331030
\(271\) −27.2184 −1.65340 −0.826700 0.562643i \(-0.809784\pi\)
−0.826700 + 0.562643i \(0.809784\pi\)
\(272\) −4.04145 −0.245049
\(273\) −1.33861 −0.0810161
\(274\) −1.28592 −0.0776854
\(275\) 0.0380545 0.00229477
\(276\) −3.44515 −0.207373
\(277\) −9.80168 −0.588926 −0.294463 0.955663i \(-0.595141\pi\)
−0.294463 + 0.955663i \(0.595141\pi\)
\(278\) 15.5538 0.932854
\(279\) −7.86998 −0.471163
\(280\) 4.36984 0.261148
\(281\) 10.1543 0.605753 0.302876 0.953030i \(-0.402053\pi\)
0.302876 + 0.953030i \(0.402053\pi\)
\(282\) −3.21882 −0.191678
\(283\) −9.36458 −0.556667 −0.278333 0.960485i \(-0.589782\pi\)
−0.278333 + 0.960485i \(0.589782\pi\)
\(284\) −3.83205 −0.227391
\(285\) −6.77015 −0.401029
\(286\) −1.65329 −0.0977612
\(287\) 11.0832 0.654220
\(288\) 2.82704 0.166585
\(289\) −0.666685 −0.0392167
\(290\) 22.8833 1.34375
\(291\) −5.53789 −0.324637
\(292\) 0.425688 0.0249115
\(293\) −6.59187 −0.385101 −0.192550 0.981287i \(-0.561676\pi\)
−0.192550 + 0.981287i \(0.561676\pi\)
\(294\) −1.33488 −0.0778516
\(295\) −17.9528 −1.04525
\(296\) 0.827167 0.0480781
\(297\) 2.42336 0.140618
\(298\) 20.2290 1.17183
\(299\) 13.6958 0.792049
\(300\) −0.0158262 −0.000913724 0
\(301\) 14.2413 0.820853
\(302\) −5.49776 −0.316361
\(303\) −3.50982 −0.201634
\(304\) −7.25266 −0.415969
\(305\) 18.1514 1.03934
\(306\) −11.4254 −0.653144
\(307\) −6.82122 −0.389308 −0.194654 0.980872i \(-0.562358\pi\)
−0.194654 + 0.980872i \(0.562358\pi\)
\(308\) 1.94686 0.110932
\(309\) 0.377769 0.0214905
\(310\) 6.24845 0.354888
\(311\) −12.7217 −0.721384 −0.360692 0.932685i \(-0.617459\pi\)
−0.360692 + 0.932685i \(0.617459\pi\)
\(312\) 0.687573 0.0389262
\(313\) −2.87908 −0.162735 −0.0813676 0.996684i \(-0.525929\pi\)
−0.0813676 + 0.996684i \(0.525929\pi\)
\(314\) −2.13019 −0.120214
\(315\) 12.3537 0.696053
\(316\) −13.8952 −0.781666
\(317\) 1.25368 0.0704135 0.0352067 0.999380i \(-0.488791\pi\)
0.0352067 + 0.999380i \(0.488791\pi\)
\(318\) −3.22275 −0.180723
\(319\) 10.1950 0.570810
\(320\) −2.24456 −0.125475
\(321\) −5.90180 −0.329406
\(322\) −16.1277 −0.898761
\(323\) 29.3113 1.63092
\(324\) 7.47330 0.415183
\(325\) 0.0629153 0.00348991
\(326\) −20.4786 −1.13421
\(327\) −1.76144 −0.0974078
\(328\) −5.69287 −0.314336
\(329\) −15.0682 −0.830736
\(330\) −0.933471 −0.0513859
\(331\) −30.0110 −1.64955 −0.824777 0.565458i \(-0.808700\pi\)
−0.824777 + 0.565458i \(0.808700\pi\)
\(332\) −2.39944 −0.131686
\(333\) 2.33844 0.128145
\(334\) 2.86254 0.156631
\(335\) 25.8089 1.41009
\(336\) −0.809661 −0.0441706
\(337\) −17.6720 −0.962654 −0.481327 0.876541i \(-0.659845\pi\)
−0.481327 + 0.876541i \(0.659845\pi\)
\(338\) 10.2666 0.558431
\(339\) 2.96688 0.161139
\(340\) 9.07128 0.491959
\(341\) 2.78382 0.150752
\(342\) −20.5036 −1.10871
\(343\) −19.8769 −1.07325
\(344\) −7.31501 −0.394399
\(345\) 7.73284 0.416322
\(346\) 16.6986 0.897720
\(347\) −0.243786 −0.0130871 −0.00654355 0.999979i \(-0.502083\pi\)
−0.00654355 + 0.999979i \(0.502083\pi\)
\(348\) −4.23991 −0.227283
\(349\) −34.9736 −1.87210 −0.936048 0.351872i \(-0.885545\pi\)
−0.936048 + 0.351872i \(0.885545\pi\)
\(350\) −0.0740866 −0.00396010
\(351\) 4.00652 0.213852
\(352\) −1.00000 −0.0533002
\(353\) 24.5306 1.30563 0.652816 0.757517i \(-0.273588\pi\)
0.652816 + 0.757517i \(0.273588\pi\)
\(354\) 3.32637 0.176795
\(355\) 8.60128 0.456508
\(356\) 10.9318 0.579387
\(357\) 3.27220 0.173183
\(358\) 23.1531 1.22368
\(359\) 34.5941 1.82581 0.912903 0.408177i \(-0.133835\pi\)
0.912903 + 0.408177i \(0.133835\pi\)
\(360\) −6.34547 −0.334436
\(361\) 33.6011 1.76848
\(362\) −8.37604 −0.440235
\(363\) −0.415881 −0.0218281
\(364\) 3.21872 0.168707
\(365\) −0.955483 −0.0500123
\(366\) −3.36316 −0.175795
\(367\) −16.5904 −0.866013 −0.433006 0.901391i \(-0.642547\pi\)
−0.433006 + 0.901391i \(0.642547\pi\)
\(368\) 8.28397 0.431832
\(369\) −16.0940 −0.837819
\(370\) −1.85663 −0.0965214
\(371\) −15.0866 −0.783257
\(372\) −1.15774 −0.0600259
\(373\) 31.2666 1.61892 0.809462 0.587172i \(-0.199759\pi\)
0.809462 + 0.587172i \(0.199759\pi\)
\(374\) 4.04145 0.208978
\(375\) −4.63183 −0.239187
\(376\) 7.73976 0.399148
\(377\) 16.8553 0.868092
\(378\) −4.71793 −0.242664
\(379\) −7.45385 −0.382878 −0.191439 0.981504i \(-0.561315\pi\)
−0.191439 + 0.981504i \(0.561315\pi\)
\(380\) 16.2790 0.835097
\(381\) 7.34353 0.376221
\(382\) −8.39612 −0.429583
\(383\) −19.2295 −0.982582 −0.491291 0.870996i \(-0.663475\pi\)
−0.491291 + 0.870996i \(0.663475\pi\)
\(384\) 0.415881 0.0212229
\(385\) −4.36984 −0.222707
\(386\) 21.4162 1.09005
\(387\) −20.6798 −1.05122
\(388\) 13.3160 0.676020
\(389\) −18.3410 −0.929926 −0.464963 0.885330i \(-0.653932\pi\)
−0.464963 + 0.885330i \(0.653932\pi\)
\(390\) −1.54330 −0.0781481
\(391\) −33.4792 −1.69312
\(392\) 3.20976 0.162117
\(393\) 3.21681 0.162267
\(394\) −1.00000 −0.0503793
\(395\) 31.1886 1.56927
\(396\) −2.82704 −0.142064
\(397\) −5.37974 −0.270001 −0.135001 0.990846i \(-0.543104\pi\)
−0.135001 + 0.990846i \(0.543104\pi\)
\(398\) −2.37949 −0.119273
\(399\) 5.87219 0.293977
\(400\) 0.0380545 0.00190273
\(401\) −38.0579 −1.90052 −0.950261 0.311456i \(-0.899183\pi\)
−0.950261 + 0.311456i \(0.899183\pi\)
\(402\) −4.78197 −0.238503
\(403\) 4.60247 0.229265
\(404\) 8.43947 0.419879
\(405\) −16.7743 −0.833520
\(406\) −19.8482 −0.985048
\(407\) −0.827167 −0.0410011
\(408\) −1.68076 −0.0832102
\(409\) 25.1927 1.24570 0.622850 0.782341i \(-0.285975\pi\)
0.622850 + 0.782341i \(0.285975\pi\)
\(410\) 12.7780 0.631060
\(411\) −0.534791 −0.0263793
\(412\) −0.908357 −0.0447516
\(413\) 15.5717 0.766232
\(414\) 23.4191 1.15099
\(415\) 5.38569 0.264373
\(416\) −1.65329 −0.0810593
\(417\) 6.46853 0.316765
\(418\) 7.25266 0.354739
\(419\) 3.41451 0.166809 0.0834047 0.996516i \(-0.473421\pi\)
0.0834047 + 0.996516i \(0.473421\pi\)
\(420\) 1.81733 0.0886768
\(421\) 20.5210 1.00013 0.500067 0.865987i \(-0.333309\pi\)
0.500067 + 0.865987i \(0.333309\pi\)
\(422\) −12.5858 −0.612666
\(423\) 21.8806 1.06387
\(424\) 7.74921 0.376335
\(425\) −0.153795 −0.00746018
\(426\) −1.59368 −0.0772140
\(427\) −15.7439 −0.761900
\(428\) 14.1911 0.685951
\(429\) −0.687573 −0.0331964
\(430\) 16.4190 0.791794
\(431\) 7.43175 0.357975 0.178987 0.983851i \(-0.442718\pi\)
0.178987 + 0.983851i \(0.442718\pi\)
\(432\) 2.42336 0.116594
\(433\) −32.3343 −1.55389 −0.776943 0.629571i \(-0.783231\pi\)
−0.776943 + 0.629571i \(0.783231\pi\)
\(434\) −5.41969 −0.260154
\(435\) 9.51673 0.456292
\(436\) 4.23544 0.202841
\(437\) −60.0808 −2.87405
\(438\) 0.177036 0.00845909
\(439\) 6.87171 0.327969 0.163984 0.986463i \(-0.447565\pi\)
0.163984 + 0.986463i \(0.447565\pi\)
\(440\) 2.24456 0.107005
\(441\) 9.07412 0.432101
\(442\) 6.68170 0.317816
\(443\) −2.12632 −0.101025 −0.0505124 0.998723i \(-0.516085\pi\)
−0.0505124 + 0.998723i \(0.516085\pi\)
\(444\) 0.344003 0.0163257
\(445\) −24.5372 −1.16317
\(446\) 17.6630 0.836369
\(447\) 8.41285 0.397914
\(448\) 1.94686 0.0919803
\(449\) −29.2391 −1.37988 −0.689939 0.723868i \(-0.742363\pi\)
−0.689939 + 0.723868i \(0.742363\pi\)
\(450\) 0.107582 0.00507145
\(451\) 5.69287 0.268067
\(452\) −7.13396 −0.335553
\(453\) −2.28642 −0.107425
\(454\) 18.5462 0.870417
\(455\) −7.22461 −0.338695
\(456\) −3.01625 −0.141249
\(457\) −26.0219 −1.21725 −0.608626 0.793458i \(-0.708279\pi\)
−0.608626 + 0.793458i \(0.708279\pi\)
\(458\) 21.5774 1.00824
\(459\) −9.79388 −0.457139
\(460\) −18.5939 −0.866944
\(461\) −22.1268 −1.03055 −0.515275 0.857025i \(-0.672310\pi\)
−0.515275 + 0.857025i \(0.672310\pi\)
\(462\) 0.809661 0.0376688
\(463\) −18.9937 −0.882711 −0.441356 0.897332i \(-0.645502\pi\)
−0.441356 + 0.897332i \(0.645502\pi\)
\(464\) 10.1950 0.473291
\(465\) 2.59861 0.120508
\(466\) −22.8480 −1.05841
\(467\) 20.0636 0.928433 0.464217 0.885722i \(-0.346336\pi\)
0.464217 + 0.885722i \(0.346336\pi\)
\(468\) −4.67393 −0.216052
\(469\) −22.3857 −1.03368
\(470\) −17.3724 −0.801327
\(471\) −0.885907 −0.0408204
\(472\) −7.99838 −0.368155
\(473\) 7.31501 0.336344
\(474\) −5.77875 −0.265427
\(475\) −0.275997 −0.0126636
\(476\) −7.86812 −0.360635
\(477\) 21.9074 1.00307
\(478\) −15.0535 −0.688533
\(479\) −30.9779 −1.41542 −0.707708 0.706505i \(-0.750271\pi\)
−0.707708 + 0.706505i \(0.750271\pi\)
\(480\) −0.933471 −0.0426069
\(481\) −1.36755 −0.0623548
\(482\) 5.12265 0.233330
\(483\) −6.70720 −0.305188
\(484\) 1.00000 0.0454545
\(485\) −29.8887 −1.35717
\(486\) 10.3781 0.470759
\(487\) −40.2405 −1.82347 −0.911735 0.410778i \(-0.865257\pi\)
−0.911735 + 0.410778i \(0.865257\pi\)
\(488\) 8.08683 0.366073
\(489\) −8.51668 −0.385137
\(490\) −7.20449 −0.325466
\(491\) 23.9956 1.08291 0.541453 0.840731i \(-0.317874\pi\)
0.541453 + 0.840731i \(0.317874\pi\)
\(492\) −2.36756 −0.106738
\(493\) −41.2025 −1.85567
\(494\) 11.9908 0.539490
\(495\) 6.34547 0.285208
\(496\) 2.78382 0.124997
\(497\) −7.46045 −0.334647
\(498\) −0.997882 −0.0447162
\(499\) 10.6271 0.475733 0.237867 0.971298i \(-0.423552\pi\)
0.237867 + 0.971298i \(0.423552\pi\)
\(500\) 11.1374 0.498079
\(501\) 1.19048 0.0531866
\(502\) 7.34662 0.327896
\(503\) −12.7231 −0.567297 −0.283648 0.958928i \(-0.591545\pi\)
−0.283648 + 0.958928i \(0.591545\pi\)
\(504\) 5.50384 0.245161
\(505\) −18.9429 −0.842948
\(506\) −8.28397 −0.368267
\(507\) 4.26970 0.189624
\(508\) −17.6578 −0.783436
\(509\) 1.21159 0.0537026 0.0268513 0.999639i \(-0.491452\pi\)
0.0268513 + 0.999639i \(0.491452\pi\)
\(510\) 3.77258 0.167052
\(511\) 0.828753 0.0366619
\(512\) −1.00000 −0.0441942
\(513\) −17.5758 −0.775990
\(514\) 23.0169 1.01523
\(515\) 2.03886 0.0898431
\(516\) −3.04217 −0.133924
\(517\) −7.73976 −0.340394
\(518\) 1.61037 0.0707557
\(519\) 6.94462 0.304835
\(520\) 3.71092 0.162734
\(521\) 10.2745 0.450133 0.225067 0.974343i \(-0.427740\pi\)
0.225067 + 0.974343i \(0.427740\pi\)
\(522\) 28.8217 1.26149
\(523\) 38.8472 1.69867 0.849334 0.527855i \(-0.177004\pi\)
0.849334 + 0.527855i \(0.177004\pi\)
\(524\) −7.73493 −0.337902
\(525\) −0.0308113 −0.00134471
\(526\) 12.3679 0.539265
\(527\) −11.2507 −0.490087
\(528\) −0.415881 −0.0180989
\(529\) 45.6241 1.98366
\(530\) −17.3936 −0.755529
\(531\) −22.6117 −0.981266
\(532\) −14.1199 −0.612174
\(533\) 9.41198 0.407678
\(534\) 4.54635 0.196740
\(535\) −31.8527 −1.37711
\(536\) 11.4984 0.496656
\(537\) 9.62895 0.415520
\(538\) −19.0593 −0.821703
\(539\) −3.20976 −0.138254
\(540\) −5.43938 −0.234073
\(541\) 14.0073 0.602221 0.301111 0.953589i \(-0.402643\pi\)
0.301111 + 0.953589i \(0.402643\pi\)
\(542\) 27.2184 1.16913
\(543\) −3.48344 −0.149489
\(544\) 4.04145 0.173276
\(545\) −9.50670 −0.407222
\(546\) 1.33861 0.0572870
\(547\) 41.4133 1.77070 0.885352 0.464922i \(-0.153918\pi\)
0.885352 + 0.464922i \(0.153918\pi\)
\(548\) 1.28592 0.0549319
\(549\) 22.8618 0.975718
\(550\) −0.0380545 −0.00162265
\(551\) −73.9408 −3.14998
\(552\) 3.44515 0.146635
\(553\) −27.0519 −1.15036
\(554\) 9.80168 0.416433
\(555\) −0.772136 −0.0327753
\(556\) −15.5538 −0.659627
\(557\) 12.2680 0.519814 0.259907 0.965634i \(-0.416308\pi\)
0.259907 + 0.965634i \(0.416308\pi\)
\(558\) 7.86998 0.333163
\(559\) 12.0938 0.511515
\(560\) −4.36984 −0.184659
\(561\) 1.68076 0.0709619
\(562\) −10.1543 −0.428332
\(563\) 14.1117 0.594738 0.297369 0.954763i \(-0.403891\pi\)
0.297369 + 0.954763i \(0.403891\pi\)
\(564\) 3.21882 0.135537
\(565\) 16.0126 0.673655
\(566\) 9.36458 0.393623
\(567\) 14.5494 0.611019
\(568\) 3.83205 0.160789
\(569\) 1.27004 0.0532428 0.0266214 0.999646i \(-0.491525\pi\)
0.0266214 + 0.999646i \(0.491525\pi\)
\(570\) 6.77015 0.283570
\(571\) −29.1426 −1.21958 −0.609789 0.792564i \(-0.708746\pi\)
−0.609789 + 0.792564i \(0.708746\pi\)
\(572\) 1.65329 0.0691276
\(573\) −3.49179 −0.145872
\(574\) −11.0832 −0.462603
\(575\) 0.315242 0.0131465
\(576\) −2.82704 −0.117793
\(577\) 17.4190 0.725164 0.362582 0.931952i \(-0.381895\pi\)
0.362582 + 0.931952i \(0.381895\pi\)
\(578\) 0.666685 0.0277304
\(579\) 8.90658 0.370145
\(580\) −22.8833 −0.950177
\(581\) −4.67136 −0.193801
\(582\) 5.53789 0.229553
\(583\) −7.74921 −0.320940
\(584\) −0.425688 −0.0176151
\(585\) 10.4909 0.433746
\(586\) 6.59187 0.272307
\(587\) 23.4923 0.969629 0.484815 0.874617i \(-0.338887\pi\)
0.484815 + 0.874617i \(0.338887\pi\)
\(588\) 1.33488 0.0550494
\(589\) −20.1901 −0.831919
\(590\) 17.9528 0.739107
\(591\) −0.415881 −0.0171071
\(592\) −0.827167 −0.0339963
\(593\) −31.0519 −1.27515 −0.637575 0.770389i \(-0.720062\pi\)
−0.637575 + 0.770389i \(0.720062\pi\)
\(594\) −2.42336 −0.0994316
\(595\) 17.6605 0.724009
\(596\) −20.2290 −0.828611
\(597\) −0.989587 −0.0405011
\(598\) −13.6958 −0.560064
\(599\) 11.6353 0.475407 0.237703 0.971338i \(-0.423605\pi\)
0.237703 + 0.971338i \(0.423605\pi\)
\(600\) 0.0158262 0.000646101 0
\(601\) −21.8978 −0.893230 −0.446615 0.894726i \(-0.647371\pi\)
−0.446615 + 0.894726i \(0.647371\pi\)
\(602\) −14.2413 −0.580430
\(603\) 32.5065 1.32377
\(604\) 5.49776 0.223701
\(605\) −2.24456 −0.0912544
\(606\) 3.50982 0.142577
\(607\) −3.60391 −0.146278 −0.0731391 0.997322i \(-0.523302\pi\)
−0.0731391 + 0.997322i \(0.523302\pi\)
\(608\) 7.25266 0.294134
\(609\) −8.25448 −0.334489
\(610\) −18.1514 −0.734928
\(611\) −12.7961 −0.517674
\(612\) 11.4254 0.461842
\(613\) −11.8355 −0.478031 −0.239016 0.971016i \(-0.576825\pi\)
−0.239016 + 0.971016i \(0.576825\pi\)
\(614\) 6.82122 0.275282
\(615\) 5.31413 0.214286
\(616\) −1.94686 −0.0784410
\(617\) −11.4542 −0.461129 −0.230564 0.973057i \(-0.574057\pi\)
−0.230564 + 0.973057i \(0.574057\pi\)
\(618\) −0.377769 −0.0151961
\(619\) −26.2561 −1.05532 −0.527660 0.849455i \(-0.676931\pi\)
−0.527660 + 0.849455i \(0.676931\pi\)
\(620\) −6.24845 −0.250944
\(621\) 20.0750 0.805583
\(622\) 12.7217 0.510095
\(623\) 21.2827 0.852674
\(624\) −0.687573 −0.0275250
\(625\) −25.1888 −1.00755
\(626\) 2.87908 0.115071
\(627\) 3.01625 0.120457
\(628\) 2.13019 0.0850039
\(629\) 3.34295 0.133292
\(630\) −12.3537 −0.492184
\(631\) 11.1616 0.444336 0.222168 0.975008i \(-0.428687\pi\)
0.222168 + 0.975008i \(0.428687\pi\)
\(632\) 13.8952 0.552721
\(633\) −5.23419 −0.208040
\(634\) −1.25368 −0.0497899
\(635\) 39.6339 1.57282
\(636\) 3.22275 0.127790
\(637\) −5.30666 −0.210258
\(638\) −10.1950 −0.403624
\(639\) 10.8334 0.428562
\(640\) 2.24456 0.0887241
\(641\) 43.2448 1.70807 0.854033 0.520219i \(-0.174150\pi\)
0.854033 + 0.520219i \(0.174150\pi\)
\(642\) 5.90180 0.232926
\(643\) 49.6160 1.95666 0.978332 0.207040i \(-0.0663830\pi\)
0.978332 + 0.207040i \(0.0663830\pi\)
\(644\) 16.1277 0.635520
\(645\) 6.82835 0.268866
\(646\) −29.3113 −1.15324
\(647\) −28.2651 −1.11121 −0.555607 0.831445i \(-0.687514\pi\)
−0.555607 + 0.831445i \(0.687514\pi\)
\(648\) −7.47330 −0.293579
\(649\) 7.99838 0.313964
\(650\) −0.0629153 −0.00246774
\(651\) −2.25395 −0.0883392
\(652\) 20.4786 0.802005
\(653\) −15.1579 −0.593175 −0.296588 0.955006i \(-0.595849\pi\)
−0.296588 + 0.955006i \(0.595849\pi\)
\(654\) 1.76144 0.0688777
\(655\) 17.3615 0.678371
\(656\) 5.69287 0.222269
\(657\) −1.20344 −0.0469506
\(658\) 15.0682 0.587419
\(659\) −9.71715 −0.378526 −0.189263 0.981926i \(-0.560610\pi\)
−0.189263 + 0.981926i \(0.560610\pi\)
\(660\) 0.933471 0.0363353
\(661\) 31.9873 1.24416 0.622080 0.782953i \(-0.286288\pi\)
0.622080 + 0.782953i \(0.286288\pi\)
\(662\) 30.0110 1.16641
\(663\) 2.77879 0.107919
\(664\) 2.39944 0.0931164
\(665\) 31.6929 1.22900
\(666\) −2.33844 −0.0906125
\(667\) 84.4550 3.27011
\(668\) −2.86254 −0.110755
\(669\) 7.34572 0.284002
\(670\) −25.8089 −0.997084
\(671\) −8.08683 −0.312188
\(672\) 0.809661 0.0312333
\(673\) −1.26061 −0.0485930 −0.0242965 0.999705i \(-0.507735\pi\)
−0.0242965 + 0.999705i \(0.507735\pi\)
\(674\) 17.6720 0.680699
\(675\) 0.0922197 0.00354954
\(676\) −10.2666 −0.394870
\(677\) −4.77936 −0.183686 −0.0918428 0.995774i \(-0.529276\pi\)
−0.0918428 + 0.995774i \(0.529276\pi\)
\(678\) −2.96688 −0.113942
\(679\) 25.9244 0.994888
\(680\) −9.07128 −0.347868
\(681\) 7.71302 0.295564
\(682\) −2.78382 −0.106598
\(683\) 29.3451 1.12286 0.561430 0.827524i \(-0.310251\pi\)
0.561430 + 0.827524i \(0.310251\pi\)
\(684\) 20.5036 0.783974
\(685\) −2.88633 −0.110281
\(686\) 19.8769 0.758904
\(687\) 8.97362 0.342365
\(688\) 7.31501 0.278882
\(689\) −12.8117 −0.488087
\(690\) −7.73284 −0.294384
\(691\) −34.2141 −1.30157 −0.650783 0.759263i \(-0.725559\pi\)
−0.650783 + 0.759263i \(0.725559\pi\)
\(692\) −16.6986 −0.634784
\(693\) −5.50384 −0.209074
\(694\) 0.243786 0.00925398
\(695\) 34.9114 1.32427
\(696\) 4.23991 0.160713
\(697\) −23.0074 −0.871469
\(698\) 34.9736 1.32377
\(699\) −9.50207 −0.359401
\(700\) 0.0740866 0.00280021
\(701\) −11.5370 −0.435745 −0.217873 0.975977i \(-0.569912\pi\)
−0.217873 + 0.975977i \(0.569912\pi\)
\(702\) −4.00652 −0.151216
\(703\) 5.99916 0.226263
\(704\) 1.00000 0.0376889
\(705\) −7.22484 −0.272103
\(706\) −24.5306 −0.923221
\(707\) 16.4304 0.617929
\(708\) −3.32637 −0.125013
\(709\) −21.8738 −0.821489 −0.410745 0.911750i \(-0.634731\pi\)
−0.410745 + 0.911750i \(0.634731\pi\)
\(710\) −8.60128 −0.322800
\(711\) 39.2823 1.47320
\(712\) −10.9318 −0.409688
\(713\) 23.0611 0.863644
\(714\) −3.27220 −0.122459
\(715\) −3.71092 −0.138780
\(716\) −23.1531 −0.865273
\(717\) −6.26049 −0.233802
\(718\) −34.5941 −1.29104
\(719\) −13.8885 −0.517953 −0.258976 0.965884i \(-0.583385\pi\)
−0.258976 + 0.965884i \(0.583385\pi\)
\(720\) 6.34547 0.236482
\(721\) −1.76844 −0.0658601
\(722\) −33.6011 −1.25050
\(723\) 2.13041 0.0792309
\(724\) 8.37604 0.311293
\(725\) 0.387966 0.0144087
\(726\) 0.415881 0.0154348
\(727\) 13.0964 0.485720 0.242860 0.970061i \(-0.421914\pi\)
0.242860 + 0.970061i \(0.421914\pi\)
\(728\) −3.21872 −0.119294
\(729\) −18.1038 −0.670513
\(730\) 0.955483 0.0353640
\(731\) −29.5632 −1.09344
\(732\) 3.36316 0.124306
\(733\) 13.6829 0.505388 0.252694 0.967546i \(-0.418683\pi\)
0.252694 + 0.967546i \(0.418683\pi\)
\(734\) 16.5904 0.612364
\(735\) −2.99621 −0.110517
\(736\) −8.28397 −0.305351
\(737\) −11.4984 −0.423549
\(738\) 16.0940 0.592428
\(739\) −24.5625 −0.903545 −0.451773 0.892133i \(-0.649208\pi\)
−0.451773 + 0.892133i \(0.649208\pi\)
\(740\) 1.85663 0.0682509
\(741\) 4.98674 0.183192
\(742\) 15.0866 0.553846
\(743\) 31.4640 1.15430 0.577151 0.816637i \(-0.304164\pi\)
0.577151 + 0.816637i \(0.304164\pi\)
\(744\) 1.15774 0.0424448
\(745\) 45.4051 1.66352
\(746\) −31.2666 −1.14475
\(747\) 6.78332 0.248189
\(748\) −4.04145 −0.147770
\(749\) 27.6280 1.00950
\(750\) 4.63183 0.169131
\(751\) 1.88471 0.0687740 0.0343870 0.999409i \(-0.489052\pi\)
0.0343870 + 0.999409i \(0.489052\pi\)
\(752\) −7.73976 −0.282240
\(753\) 3.05532 0.111342
\(754\) −16.8553 −0.613834
\(755\) −12.3401 −0.449101
\(756\) 4.71793 0.171589
\(757\) 9.73585 0.353856 0.176928 0.984224i \(-0.443384\pi\)
0.176928 + 0.984224i \(0.443384\pi\)
\(758\) 7.45385 0.270736
\(759\) −3.44515 −0.125051
\(760\) −16.2790 −0.590503
\(761\) −3.78408 −0.137173 −0.0685863 0.997645i \(-0.521849\pi\)
−0.0685863 + 0.997645i \(0.521849\pi\)
\(762\) −7.34353 −0.266028
\(763\) 8.24578 0.298517
\(764\) 8.39612 0.303761
\(765\) −25.6449 −0.927193
\(766\) 19.2295 0.694790
\(767\) 13.2237 0.477478
\(768\) −0.415881 −0.0150068
\(769\) 22.9503 0.827608 0.413804 0.910366i \(-0.364200\pi\)
0.413804 + 0.910366i \(0.364200\pi\)
\(770\) 4.36984 0.157478
\(771\) 9.57231 0.344738
\(772\) −21.4162 −0.770784
\(773\) −24.2586 −0.872523 −0.436261 0.899820i \(-0.643698\pi\)
−0.436261 + 0.899820i \(0.643698\pi\)
\(774\) 20.6798 0.743321
\(775\) 0.105937 0.00380537
\(776\) −13.3160 −0.478018
\(777\) 0.669724 0.0240262
\(778\) 18.3410 0.657557
\(779\) −41.2885 −1.47931
\(780\) 1.54330 0.0552590
\(781\) −3.83205 −0.137122
\(782\) 33.4792 1.19721
\(783\) 24.7061 0.882925
\(784\) −3.20976 −0.114634
\(785\) −4.78135 −0.170654
\(786\) −3.21681 −0.114740
\(787\) −46.0669 −1.64211 −0.821054 0.570850i \(-0.806614\pi\)
−0.821054 + 0.570850i \(0.806614\pi\)
\(788\) 1.00000 0.0356235
\(789\) 5.14357 0.183116
\(790\) −31.1886 −1.10964
\(791\) −13.8888 −0.493828
\(792\) 2.82704 0.100455
\(793\) −13.3699 −0.474779
\(794\) 5.37974 0.190920
\(795\) −7.23366 −0.256552
\(796\) 2.37949 0.0843389
\(797\) 43.5713 1.54337 0.771687 0.636003i \(-0.219413\pi\)
0.771687 + 0.636003i \(0.219413\pi\)
\(798\) −5.87219 −0.207873
\(799\) 31.2798 1.10660
\(800\) −0.0380545 −0.00134543
\(801\) −30.9048 −1.09197
\(802\) 38.0579 1.34387
\(803\) 0.425688 0.0150222
\(804\) 4.78197 0.168647
\(805\) −36.1996 −1.27587
\(806\) −4.60247 −0.162115
\(807\) −7.92639 −0.279022
\(808\) −8.43947 −0.296899
\(809\) −15.7310 −0.553071 −0.276536 0.961004i \(-0.589186\pi\)
−0.276536 + 0.961004i \(0.589186\pi\)
\(810\) 16.7743 0.589388
\(811\) 20.9510 0.735689 0.367845 0.929887i \(-0.380096\pi\)
0.367845 + 0.929887i \(0.380096\pi\)
\(812\) 19.8482 0.696534
\(813\) 11.3196 0.396996
\(814\) 0.827167 0.0289922
\(815\) −45.9655 −1.61010
\(816\) 1.68076 0.0588385
\(817\) −53.0533 −1.85610
\(818\) −25.1927 −0.880843
\(819\) −9.09946 −0.317961
\(820\) −12.7780 −0.446227
\(821\) 21.6596 0.755924 0.377962 0.925821i \(-0.376625\pi\)
0.377962 + 0.925821i \(0.376625\pi\)
\(822\) 0.534791 0.0186530
\(823\) 51.1991 1.78469 0.892344 0.451355i \(-0.149059\pi\)
0.892344 + 0.451355i \(0.149059\pi\)
\(824\) 0.908357 0.0316441
\(825\) −0.0158262 −0.000550996 0
\(826\) −15.5717 −0.541808
\(827\) 39.9638 1.38968 0.694839 0.719166i \(-0.255476\pi\)
0.694839 + 0.719166i \(0.255476\pi\)
\(828\) −23.4191 −0.813871
\(829\) 8.72748 0.303118 0.151559 0.988448i \(-0.451571\pi\)
0.151559 + 0.988448i \(0.451571\pi\)
\(830\) −5.38569 −0.186940
\(831\) 4.07634 0.141406
\(832\) 1.65329 0.0573176
\(833\) 12.9721 0.449455
\(834\) −6.46853 −0.223987
\(835\) 6.42514 0.222351
\(836\) −7.25266 −0.250839
\(837\) 6.74619 0.233183
\(838\) −3.41451 −0.117952
\(839\) −28.0173 −0.967265 −0.483632 0.875271i \(-0.660683\pi\)
−0.483632 + 0.875271i \(0.660683\pi\)
\(840\) −1.81733 −0.0627039
\(841\) 74.9378 2.58406
\(842\) −20.5210 −0.707202
\(843\) −4.22297 −0.145447
\(844\) 12.5858 0.433220
\(845\) 23.0441 0.792740
\(846\) −21.8806 −0.752271
\(847\) 1.94686 0.0668947
\(848\) −7.74921 −0.266109
\(849\) 3.89456 0.133661
\(850\) 0.153795 0.00527514
\(851\) −6.85222 −0.234891
\(852\) 1.59368 0.0545985
\(853\) 35.2276 1.20617 0.603085 0.797677i \(-0.293938\pi\)
0.603085 + 0.797677i \(0.293938\pi\)
\(854\) 15.7439 0.538744
\(855\) −46.0215 −1.57390
\(856\) −14.1911 −0.485041
\(857\) −17.6481 −0.602848 −0.301424 0.953490i \(-0.597462\pi\)
−0.301424 + 0.953490i \(0.597462\pi\)
\(858\) 0.687573 0.0234734
\(859\) −0.698490 −0.0238322 −0.0119161 0.999929i \(-0.503793\pi\)
−0.0119161 + 0.999929i \(0.503793\pi\)
\(860\) −16.4190 −0.559883
\(861\) −4.60929 −0.157084
\(862\) −7.43175 −0.253126
\(863\) 5.07840 0.172871 0.0864354 0.996257i \(-0.472452\pi\)
0.0864354 + 0.996257i \(0.472452\pi\)
\(864\) −2.42336 −0.0824443
\(865\) 37.4809 1.27439
\(866\) 32.3343 1.09876
\(867\) 0.277262 0.00941630
\(868\) 5.41969 0.183956
\(869\) −13.8952 −0.471362
\(870\) −9.51673 −0.322647
\(871\) −19.0102 −0.644137
\(872\) −4.23544 −0.143430
\(873\) −37.6450 −1.27409
\(874\) 60.0808 2.03226
\(875\) 21.6829 0.733015
\(876\) −0.177036 −0.00598148
\(877\) 3.11728 0.105263 0.0526315 0.998614i \(-0.483239\pi\)
0.0526315 + 0.998614i \(0.483239\pi\)
\(878\) −6.87171 −0.231909
\(879\) 2.74143 0.0924663
\(880\) −2.24456 −0.0756641
\(881\) −54.2647 −1.82822 −0.914112 0.405462i \(-0.867111\pi\)
−0.914112 + 0.405462i \(0.867111\pi\)
\(882\) −9.07412 −0.305541
\(883\) 43.3020 1.45723 0.728614 0.684924i \(-0.240165\pi\)
0.728614 + 0.684924i \(0.240165\pi\)
\(884\) −6.68170 −0.224730
\(885\) 7.46625 0.250975
\(886\) 2.12632 0.0714353
\(887\) 3.68238 0.123642 0.0618210 0.998087i \(-0.480309\pi\)
0.0618210 + 0.998087i \(0.480309\pi\)
\(888\) −0.344003 −0.0115440
\(889\) −34.3771 −1.15297
\(890\) 24.5372 0.822489
\(891\) 7.47330 0.250365
\(892\) −17.6630 −0.591402
\(893\) 56.1338 1.87845
\(894\) −8.41285 −0.281368
\(895\) 51.9686 1.73712
\(896\) −1.94686 −0.0650399
\(897\) −5.69583 −0.190178
\(898\) 29.2391 0.975721
\(899\) 28.3810 0.946560
\(900\) −0.107582 −0.00358606
\(901\) 31.3180 1.04336
\(902\) −5.69287 −0.189552
\(903\) −5.92267 −0.197094
\(904\) 7.13396 0.237272
\(905\) −18.8005 −0.624951
\(906\) 2.28642 0.0759611
\(907\) −9.34926 −0.310437 −0.155218 0.987880i \(-0.549608\pi\)
−0.155218 + 0.987880i \(0.549608\pi\)
\(908\) −18.5462 −0.615478
\(909\) −23.8587 −0.791344
\(910\) 7.22461 0.239494
\(911\) −2.06437 −0.0683955 −0.0341977 0.999415i \(-0.510888\pi\)
−0.0341977 + 0.999415i \(0.510888\pi\)
\(912\) 3.01625 0.0998779
\(913\) −2.39944 −0.0794099
\(914\) 26.0219 0.860727
\(915\) −7.54882 −0.249556
\(916\) −21.5774 −0.712936
\(917\) −15.0588 −0.497285
\(918\) 9.79388 0.323246
\(919\) 38.7581 1.27851 0.639256 0.768994i \(-0.279242\pi\)
0.639256 + 0.768994i \(0.279242\pi\)
\(920\) 18.5939 0.613022
\(921\) 2.83682 0.0934764
\(922\) 22.1268 0.728708
\(923\) −6.33550 −0.208536
\(924\) −0.809661 −0.0266359
\(925\) −0.0314774 −0.00103497
\(926\) 18.9937 0.624171
\(927\) 2.56796 0.0843430
\(928\) −10.1950 −0.334667
\(929\) −22.1175 −0.725653 −0.362826 0.931857i \(-0.618188\pi\)
−0.362826 + 0.931857i \(0.618188\pi\)
\(930\) −2.59861 −0.0852119
\(931\) 23.2793 0.762947
\(932\) 22.8480 0.748412
\(933\) 5.29073 0.173211
\(934\) −20.0636 −0.656502
\(935\) 9.07128 0.296663
\(936\) 4.67393 0.152772
\(937\) 36.0675 1.17828 0.589138 0.808033i \(-0.299468\pi\)
0.589138 + 0.808033i \(0.299468\pi\)
\(938\) 22.3857 0.730920
\(939\) 1.19736 0.0390742
\(940\) 17.3724 0.566624
\(941\) −45.9104 −1.49664 −0.748319 0.663339i \(-0.769139\pi\)
−0.748319 + 0.663339i \(0.769139\pi\)
\(942\) 0.885907 0.0288644
\(943\) 47.1595 1.53573
\(944\) 7.99838 0.260325
\(945\) −10.5897 −0.344482
\(946\) −7.31501 −0.237831
\(947\) −29.7380 −0.966356 −0.483178 0.875522i \(-0.660518\pi\)
−0.483178 + 0.875522i \(0.660518\pi\)
\(948\) 5.77875 0.187685
\(949\) 0.703787 0.0228459
\(950\) 0.275997 0.00895451
\(951\) −0.521381 −0.0169069
\(952\) 7.86812 0.255007
\(953\) −17.7427 −0.574741 −0.287370 0.957820i \(-0.592781\pi\)
−0.287370 + 0.957820i \(0.592781\pi\)
\(954\) −21.9074 −0.709277
\(955\) −18.8456 −0.609829
\(956\) 15.0535 0.486866
\(957\) −4.23991 −0.137057
\(958\) 30.9779 1.00085
\(959\) 2.50351 0.0808424
\(960\) 0.933471 0.0301276
\(961\) −23.2503 −0.750011
\(962\) 1.36755 0.0440915
\(963\) −40.1188 −1.29281
\(964\) −5.12265 −0.164989
\(965\) 48.0699 1.54742
\(966\) 6.70720 0.215801
\(967\) 56.9254 1.83060 0.915299 0.402774i \(-0.131954\pi\)
0.915299 + 0.402774i \(0.131954\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −12.1900 −0.391599
\(970\) 29.8887 0.959668
\(971\) −14.3546 −0.460662 −0.230331 0.973112i \(-0.573981\pi\)
−0.230331 + 0.973112i \(0.573981\pi\)
\(972\) −10.3781 −0.332877
\(973\) −30.2810 −0.970763
\(974\) 40.2405 1.28939
\(975\) −0.0261653 −0.000837960 0
\(976\) −8.08683 −0.258853
\(977\) −9.26264 −0.296338 −0.148169 0.988962i \(-0.547338\pi\)
−0.148169 + 0.988962i \(0.547338\pi\)
\(978\) 8.51668 0.272333
\(979\) 10.9318 0.349383
\(980\) 7.20449 0.230139
\(981\) −11.9738 −0.382293
\(982\) −23.9956 −0.765731
\(983\) −25.0803 −0.799938 −0.399969 0.916529i \(-0.630979\pi\)
−0.399969 + 0.916529i \(0.630979\pi\)
\(984\) 2.36756 0.0754750
\(985\) −2.24456 −0.0715177
\(986\) 41.2025 1.31216
\(987\) 6.26658 0.199467
\(988\) −11.9908 −0.381477
\(989\) 60.5973 1.92688
\(990\) −6.34547 −0.201672
\(991\) −22.6092 −0.718206 −0.359103 0.933298i \(-0.616917\pi\)
−0.359103 + 0.933298i \(0.616917\pi\)
\(992\) −2.78382 −0.0883864
\(993\) 12.4810 0.396073
\(994\) 7.46045 0.236631
\(995\) −5.34092 −0.169318
\(996\) 0.997882 0.0316191
\(997\) −15.6951 −0.497070 −0.248535 0.968623i \(-0.579949\pi\)
−0.248535 + 0.968623i \(0.579949\pi\)
\(998\) −10.6271 −0.336394
\(999\) −2.00452 −0.0634202
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 4334.2.a.a.1.7 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4334.2.a.a.1.7 15 1.1 even 1 trivial