Properties

Label 1045.2.a.j.1.3
Level $1045$
Weight $2$
Character 1045.1
Self dual yes
Analytic conductor $8.344$
Analytic rank $0$
Dimension $9$
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] = [1045,2,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, 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("1045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 11x^{7} + 11x^{6} + 34x^{5} - 20x^{4} - 36x^{3} + 13x^{2} + 10x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.65050\) of defining polynomial
Character \(\chi\) \(=\) 1045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.31671 q^{2} +2.24751 q^{3} -0.266275 q^{4} -1.00000 q^{5} -2.95932 q^{6} +2.74324 q^{7} +2.98403 q^{8} +2.05132 q^{9} +O(q^{10})\) \(q-1.31671 q^{2} +2.24751 q^{3} -0.266275 q^{4} -1.00000 q^{5} -2.95932 q^{6} +2.74324 q^{7} +2.98403 q^{8} +2.05132 q^{9} +1.31671 q^{10} -1.00000 q^{11} -0.598458 q^{12} -0.823772 q^{13} -3.61205 q^{14} -2.24751 q^{15} -3.39655 q^{16} +4.43351 q^{17} -2.70099 q^{18} +1.00000 q^{19} +0.266275 q^{20} +6.16547 q^{21} +1.31671 q^{22} +5.39996 q^{23} +6.70664 q^{24} +1.00000 q^{25} +1.08467 q^{26} -2.13218 q^{27} -0.730457 q^{28} +2.90610 q^{29} +2.95932 q^{30} -7.05785 q^{31} -1.49579 q^{32} -2.24751 q^{33} -5.83765 q^{34} -2.74324 q^{35} -0.546215 q^{36} +8.40534 q^{37} -1.31671 q^{38} -1.85144 q^{39} -2.98403 q^{40} -10.6533 q^{41} -8.11813 q^{42} +10.3406 q^{43} +0.266275 q^{44} -2.05132 q^{45} -7.11019 q^{46} +12.7551 q^{47} -7.63378 q^{48} +0.525367 q^{49} -1.31671 q^{50} +9.96438 q^{51} +0.219350 q^{52} +2.37381 q^{53} +2.80746 q^{54} +1.00000 q^{55} +8.18590 q^{56} +2.24751 q^{57} -3.82649 q^{58} +14.8451 q^{59} +0.598458 q^{60} -8.71368 q^{61} +9.29313 q^{62} +5.62726 q^{63} +8.76261 q^{64} +0.823772 q^{65} +2.95932 q^{66} -10.1782 q^{67} -1.18054 q^{68} +12.1365 q^{69} +3.61205 q^{70} +7.46427 q^{71} +6.12119 q^{72} -0.797840 q^{73} -11.0674 q^{74} +2.24751 q^{75} -0.266275 q^{76} -2.74324 q^{77} +2.43781 q^{78} -0.962752 q^{79} +3.39655 q^{80} -10.9460 q^{81} +14.0273 q^{82} +0.110739 q^{83} -1.64171 q^{84} -4.43351 q^{85} -13.6156 q^{86} +6.53150 q^{87} -2.98403 q^{88} +11.1632 q^{89} +2.70099 q^{90} -2.25980 q^{91} -1.43788 q^{92} -15.8626 q^{93} -16.7947 q^{94} -1.00000 q^{95} -3.36180 q^{96} +15.0623 q^{97} -0.691755 q^{98} -2.05132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 3 q^{2} + 3 q^{3} + 9 q^{4} - 9 q^{5} + 6 q^{6} - 9 q^{7} + 15 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 3 q^{2} + 3 q^{3} + 9 q^{4} - 9 q^{5} + 6 q^{6} - 9 q^{7} + 15 q^{8} + 16 q^{9} - 3 q^{10} - 9 q^{11} + 13 q^{12} - 3 q^{13} + 4 q^{14} - 3 q^{15} + 17 q^{16} + 5 q^{17} + 17 q^{18} + 9 q^{19} - 9 q^{20} - q^{21} - 3 q^{22} + 4 q^{23} + 21 q^{24} + 9 q^{25} + 20 q^{26} + 24 q^{27} - 24 q^{28} + 3 q^{29} - 6 q^{30} - q^{31} + 38 q^{32} - 3 q^{33} + 28 q^{34} + 9 q^{35} + 17 q^{36} + 5 q^{37} + 3 q^{38} - 15 q^{40} - 5 q^{41} - 43 q^{42} - 11 q^{43} - 9 q^{44} - 16 q^{45} + 2 q^{46} + 30 q^{47} + 54 q^{48} + 12 q^{49} + 3 q^{50} + 40 q^{51} - 3 q^{52} - q^{53} + 65 q^{54} + 9 q^{55} - 16 q^{56} + 3 q^{57} - 15 q^{58} + 59 q^{59} - 13 q^{60} - 21 q^{61} - 10 q^{62} - 12 q^{63} + 19 q^{64} + 3 q^{65} - 6 q^{66} - 2 q^{67} - 9 q^{68} - 22 q^{69} - 4 q^{70} + 34 q^{71} + 32 q^{72} - 34 q^{73} - 21 q^{74} + 3 q^{75} + 9 q^{76} + 9 q^{77} - 65 q^{78} - 13 q^{79} - 17 q^{80} + 57 q^{81} + 10 q^{82} + 51 q^{83} - 95 q^{84} - 5 q^{85} - 14 q^{86} + 8 q^{87} - 15 q^{88} + 8 q^{89} - 17 q^{90} + 62 q^{91} + 57 q^{92} - 18 q^{93} + 2 q^{94} - 9 q^{95} + 81 q^{96} - 8 q^{97} - 20 q^{98} - 16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.31671 −0.931054 −0.465527 0.885034i \(-0.654135\pi\)
−0.465527 + 0.885034i \(0.654135\pi\)
\(3\) 2.24751 1.29760 0.648801 0.760958i \(-0.275271\pi\)
0.648801 + 0.760958i \(0.275271\pi\)
\(4\) −0.266275 −0.133138
\(5\) −1.00000 −0.447214
\(6\) −2.95932 −1.20814
\(7\) 2.74324 1.03685 0.518424 0.855124i \(-0.326519\pi\)
0.518424 + 0.855124i \(0.326519\pi\)
\(8\) 2.98403 1.05501
\(9\) 2.05132 0.683773
\(10\) 1.31671 0.416380
\(11\) −1.00000 −0.301511
\(12\) −0.598458 −0.172760
\(13\) −0.823772 −0.228473 −0.114237 0.993454i \(-0.536442\pi\)
−0.114237 + 0.993454i \(0.536442\pi\)
\(14\) −3.61205 −0.965361
\(15\) −2.24751 −0.580306
\(16\) −3.39655 −0.849137
\(17\) 4.43351 1.07528 0.537642 0.843173i \(-0.319315\pi\)
0.537642 + 0.843173i \(0.319315\pi\)
\(18\) −2.70099 −0.636629
\(19\) 1.00000 0.229416
\(20\) 0.266275 0.0595410
\(21\) 6.16547 1.34542
\(22\) 1.31671 0.280723
\(23\) 5.39996 1.12597 0.562985 0.826467i \(-0.309653\pi\)
0.562985 + 0.826467i \(0.309653\pi\)
\(24\) 6.70664 1.36899
\(25\) 1.00000 0.200000
\(26\) 1.08467 0.212721
\(27\) −2.13218 −0.410338
\(28\) −0.730457 −0.138043
\(29\) 2.90610 0.539649 0.269825 0.962909i \(-0.413034\pi\)
0.269825 + 0.962909i \(0.413034\pi\)
\(30\) 2.95932 0.540296
\(31\) −7.05785 −1.26763 −0.633813 0.773486i \(-0.718511\pi\)
−0.633813 + 0.773486i \(0.718511\pi\)
\(32\) −1.49579 −0.264420
\(33\) −2.24751 −0.391242
\(34\) −5.83765 −1.00115
\(35\) −2.74324 −0.463692
\(36\) −0.546215 −0.0910359
\(37\) 8.40534 1.38183 0.690915 0.722936i \(-0.257208\pi\)
0.690915 + 0.722936i \(0.257208\pi\)
\(38\) −1.31671 −0.213599
\(39\) −1.85144 −0.296467
\(40\) −2.98403 −0.471816
\(41\) −10.6533 −1.66377 −0.831885 0.554947i \(-0.812738\pi\)
−0.831885 + 0.554947i \(0.812738\pi\)
\(42\) −8.11813 −1.25266
\(43\) 10.3406 1.57693 0.788463 0.615082i \(-0.210877\pi\)
0.788463 + 0.615082i \(0.210877\pi\)
\(44\) 0.266275 0.0401425
\(45\) −2.05132 −0.305792
\(46\) −7.11019 −1.04834
\(47\) 12.7551 1.86052 0.930259 0.366904i \(-0.119582\pi\)
0.930259 + 0.366904i \(0.119582\pi\)
\(48\) −7.63378 −1.10184
\(49\) 0.525367 0.0750524
\(50\) −1.31671 −0.186211
\(51\) 9.96438 1.39529
\(52\) 0.219350 0.0304184
\(53\) 2.37381 0.326068 0.163034 0.986620i \(-0.447872\pi\)
0.163034 + 0.986620i \(0.447872\pi\)
\(54\) 2.80746 0.382047
\(55\) 1.00000 0.134840
\(56\) 8.18590 1.09389
\(57\) 2.24751 0.297690
\(58\) −3.82649 −0.502443
\(59\) 14.8451 1.93266 0.966331 0.257303i \(-0.0828340\pi\)
0.966331 + 0.257303i \(0.0828340\pi\)
\(60\) 0.598458 0.0772605
\(61\) −8.71368 −1.11567 −0.557836 0.829951i \(-0.688368\pi\)
−0.557836 + 0.829951i \(0.688368\pi\)
\(62\) 9.29313 1.18023
\(63\) 5.62726 0.708968
\(64\) 8.76261 1.09533
\(65\) 0.823772 0.102176
\(66\) 2.95932 0.364267
\(67\) −10.1782 −1.24346 −0.621730 0.783231i \(-0.713570\pi\)
−0.621730 + 0.783231i \(0.713570\pi\)
\(68\) −1.18054 −0.143161
\(69\) 12.1365 1.46106
\(70\) 3.61205 0.431723
\(71\) 7.46427 0.885845 0.442923 0.896560i \(-0.353942\pi\)
0.442923 + 0.896560i \(0.353942\pi\)
\(72\) 6.12119 0.721389
\(73\) −0.797840 −0.0933801 −0.0466900 0.998909i \(-0.514867\pi\)
−0.0466900 + 0.998909i \(0.514867\pi\)
\(74\) −11.0674 −1.28656
\(75\) 2.24751 0.259521
\(76\) −0.266275 −0.0305439
\(77\) −2.74324 −0.312621
\(78\) 2.43781 0.276027
\(79\) −0.962752 −0.108318 −0.0541590 0.998532i \(-0.517248\pi\)
−0.0541590 + 0.998532i \(0.517248\pi\)
\(80\) 3.39655 0.379745
\(81\) −10.9460 −1.21623
\(82\) 14.0273 1.54906
\(83\) 0.110739 0.0121552 0.00607762 0.999982i \(-0.498065\pi\)
0.00607762 + 0.999982i \(0.498065\pi\)
\(84\) −1.64171 −0.179126
\(85\) −4.43351 −0.480882
\(86\) −13.6156 −1.46820
\(87\) 6.53150 0.700250
\(88\) −2.98403 −0.318098
\(89\) 11.1632 1.18330 0.591648 0.806197i \(-0.298478\pi\)
0.591648 + 0.806197i \(0.298478\pi\)
\(90\) 2.70099 0.284709
\(91\) −2.25980 −0.236892
\(92\) −1.43788 −0.149909
\(93\) −15.8626 −1.64488
\(94\) −16.7947 −1.73224
\(95\) −1.00000 −0.102598
\(96\) −3.36180 −0.343113
\(97\) 15.0623 1.52935 0.764675 0.644416i \(-0.222900\pi\)
0.764675 + 0.644416i \(0.222900\pi\)
\(98\) −0.691755 −0.0698778
\(99\) −2.05132 −0.206165
\(100\) −0.266275 −0.0266275
\(101\) 1.64259 0.163444 0.0817219 0.996655i \(-0.473958\pi\)
0.0817219 + 0.996655i \(0.473958\pi\)
\(102\) −13.1202 −1.29909
\(103\) −18.7841 −1.85085 −0.925426 0.378928i \(-0.876293\pi\)
−0.925426 + 0.378928i \(0.876293\pi\)
\(104\) −2.45816 −0.241042
\(105\) −6.16547 −0.601688
\(106\) −3.12562 −0.303587
\(107\) −19.0393 −1.84060 −0.920299 0.391215i \(-0.872055\pi\)
−0.920299 + 0.391215i \(0.872055\pi\)
\(108\) 0.567746 0.0546314
\(109\) 16.7693 1.60621 0.803104 0.595839i \(-0.203180\pi\)
0.803104 + 0.595839i \(0.203180\pi\)
\(110\) −1.31671 −0.125543
\(111\) 18.8911 1.79307
\(112\) −9.31754 −0.880425
\(113\) 10.8292 1.01873 0.509364 0.860551i \(-0.329881\pi\)
0.509364 + 0.860551i \(0.329881\pi\)
\(114\) −2.95932 −0.277166
\(115\) −5.39996 −0.503549
\(116\) −0.773823 −0.0718477
\(117\) −1.68982 −0.156224
\(118\) −19.5466 −1.79941
\(119\) 12.1622 1.11491
\(120\) −6.70664 −0.612230
\(121\) 1.00000 0.0909091
\(122\) 11.4734 1.03875
\(123\) −23.9435 −2.15891
\(124\) 1.87933 0.168769
\(125\) −1.00000 −0.0894427
\(126\) −7.40946 −0.660088
\(127\) −8.47608 −0.752130 −0.376065 0.926593i \(-0.622723\pi\)
−0.376065 + 0.926593i \(0.622723\pi\)
\(128\) −8.54624 −0.755388
\(129\) 23.2406 2.04622
\(130\) −1.08467 −0.0951317
\(131\) 10.7024 0.935071 0.467536 0.883974i \(-0.345142\pi\)
0.467536 + 0.883974i \(0.345142\pi\)
\(132\) 0.598458 0.0520890
\(133\) 2.74324 0.237869
\(134\) 13.4017 1.15773
\(135\) 2.13218 0.183509
\(136\) 13.2297 1.13444
\(137\) 19.2944 1.64843 0.824215 0.566277i \(-0.191617\pi\)
0.824215 + 0.566277i \(0.191617\pi\)
\(138\) −15.9802 −1.36033
\(139\) 5.32362 0.451544 0.225772 0.974180i \(-0.427510\pi\)
0.225772 + 0.974180i \(0.427510\pi\)
\(140\) 0.730457 0.0617349
\(141\) 28.6672 2.41421
\(142\) −9.82827 −0.824770
\(143\) 0.823772 0.0688873
\(144\) −6.96740 −0.580616
\(145\) −2.90610 −0.241338
\(146\) 1.05052 0.0869419
\(147\) 1.18077 0.0973882
\(148\) −2.23814 −0.183974
\(149\) −17.7842 −1.45694 −0.728470 0.685078i \(-0.759768\pi\)
−0.728470 + 0.685078i \(0.759768\pi\)
\(150\) −2.95932 −0.241628
\(151\) −9.70582 −0.789849 −0.394924 0.918714i \(-0.629229\pi\)
−0.394924 + 0.918714i \(0.629229\pi\)
\(152\) 2.98403 0.242037
\(153\) 9.09454 0.735250
\(154\) 3.61205 0.291067
\(155\) 7.05785 0.566900
\(156\) 0.492993 0.0394710
\(157\) 6.78280 0.541327 0.270663 0.962674i \(-0.412757\pi\)
0.270663 + 0.962674i \(0.412757\pi\)
\(158\) 1.26766 0.100850
\(159\) 5.33517 0.423106
\(160\) 1.49579 0.118252
\(161\) 14.8134 1.16746
\(162\) 14.4128 1.13237
\(163\) −16.7229 −1.30984 −0.654918 0.755700i \(-0.727297\pi\)
−0.654918 + 0.755700i \(0.727297\pi\)
\(164\) 2.83672 0.221511
\(165\) 2.24751 0.174969
\(166\) −0.145812 −0.0113172
\(167\) −9.05766 −0.700903 −0.350452 0.936581i \(-0.613972\pi\)
−0.350452 + 0.936581i \(0.613972\pi\)
\(168\) 18.3979 1.41943
\(169\) −12.3214 −0.947800
\(170\) 5.83765 0.447727
\(171\) 2.05132 0.156868
\(172\) −2.75345 −0.209948
\(173\) 8.57947 0.652285 0.326142 0.945321i \(-0.394251\pi\)
0.326142 + 0.945321i \(0.394251\pi\)
\(174\) −8.60009 −0.651971
\(175\) 2.74324 0.207369
\(176\) 3.39655 0.256024
\(177\) 33.3645 2.50783
\(178\) −14.6987 −1.10171
\(179\) −9.13708 −0.682937 −0.341469 0.939893i \(-0.610924\pi\)
−0.341469 + 0.939893i \(0.610924\pi\)
\(180\) 0.546215 0.0407125
\(181\) −18.3843 −1.36650 −0.683248 0.730187i \(-0.739433\pi\)
−0.683248 + 0.730187i \(0.739433\pi\)
\(182\) 2.97551 0.220559
\(183\) −19.5841 −1.44770
\(184\) 16.1136 1.18791
\(185\) −8.40534 −0.617973
\(186\) 20.8864 1.53147
\(187\) −4.43351 −0.324211
\(188\) −3.39636 −0.247705
\(189\) −5.84907 −0.425457
\(190\) 1.31671 0.0955242
\(191\) −4.72197 −0.341670 −0.170835 0.985300i \(-0.554647\pi\)
−0.170835 + 0.985300i \(0.554647\pi\)
\(192\) 19.6941 1.42130
\(193\) 3.20004 0.230344 0.115172 0.993346i \(-0.463258\pi\)
0.115172 + 0.993346i \(0.463258\pi\)
\(194\) −19.8327 −1.42391
\(195\) 1.85144 0.132584
\(196\) −0.139892 −0.00999230
\(197\) −16.7733 −1.19505 −0.597525 0.801850i \(-0.703849\pi\)
−0.597525 + 0.801850i \(0.703849\pi\)
\(198\) 2.70099 0.191951
\(199\) −1.76433 −0.125070 −0.0625351 0.998043i \(-0.519919\pi\)
−0.0625351 + 0.998043i \(0.519919\pi\)
\(200\) 2.98403 0.211003
\(201\) −22.8756 −1.61352
\(202\) −2.16281 −0.152175
\(203\) 7.97213 0.559534
\(204\) −2.65327 −0.185766
\(205\) 10.6533 0.744061
\(206\) 24.7332 1.72324
\(207\) 11.0770 0.769908
\(208\) 2.79798 0.194005
\(209\) −1.00000 −0.0691714
\(210\) 8.11813 0.560204
\(211\) 23.8350 1.64087 0.820434 0.571741i \(-0.193732\pi\)
0.820434 + 0.571741i \(0.193732\pi\)
\(212\) −0.632087 −0.0434119
\(213\) 16.7760 1.14948
\(214\) 25.0692 1.71370
\(215\) −10.3406 −0.705223
\(216\) −6.36247 −0.432911
\(217\) −19.3614 −1.31434
\(218\) −22.0803 −1.49547
\(219\) −1.79316 −0.121170
\(220\) −0.266275 −0.0179523
\(221\) −3.65220 −0.245674
\(222\) −24.8741 −1.66944
\(223\) −13.4106 −0.898037 −0.449019 0.893522i \(-0.648226\pi\)
−0.449019 + 0.893522i \(0.648226\pi\)
\(224\) −4.10331 −0.274164
\(225\) 2.05132 0.136755
\(226\) −14.2590 −0.948492
\(227\) 4.43325 0.294245 0.147123 0.989118i \(-0.452999\pi\)
0.147123 + 0.989118i \(0.452999\pi\)
\(228\) −0.598458 −0.0396338
\(229\) 14.0125 0.925970 0.462985 0.886366i \(-0.346778\pi\)
0.462985 + 0.886366i \(0.346778\pi\)
\(230\) 7.11019 0.468832
\(231\) −6.16547 −0.405658
\(232\) 8.67188 0.569337
\(233\) −2.70680 −0.177329 −0.0886643 0.996062i \(-0.528260\pi\)
−0.0886643 + 0.996062i \(0.528260\pi\)
\(234\) 2.22500 0.145453
\(235\) −12.7551 −0.832049
\(236\) −3.95287 −0.257310
\(237\) −2.16380 −0.140554
\(238\) −16.0141 −1.03804
\(239\) −28.1731 −1.82237 −0.911183 0.412002i \(-0.864830\pi\)
−0.911183 + 0.412002i \(0.864830\pi\)
\(240\) 7.63378 0.492759
\(241\) −18.4455 −1.18818 −0.594091 0.804398i \(-0.702488\pi\)
−0.594091 + 0.804398i \(0.702488\pi\)
\(242\) −1.31671 −0.0846413
\(243\) −18.2049 −1.16784
\(244\) 2.32024 0.148538
\(245\) −0.525367 −0.0335644
\(246\) 31.5266 2.01007
\(247\) −0.823772 −0.0524154
\(248\) −21.0608 −1.33736
\(249\) 0.248888 0.0157727
\(250\) 1.31671 0.0832760
\(251\) −0.411540 −0.0259761 −0.0129881 0.999916i \(-0.504134\pi\)
−0.0129881 + 0.999916i \(0.504134\pi\)
\(252\) −1.49840 −0.0943903
\(253\) −5.39996 −0.339493
\(254\) 11.1605 0.700274
\(255\) −9.96438 −0.623994
\(256\) −6.27230 −0.392019
\(257\) −14.4753 −0.902944 −0.451472 0.892285i \(-0.649101\pi\)
−0.451472 + 0.892285i \(0.649101\pi\)
\(258\) −30.6012 −1.90515
\(259\) 23.0579 1.43275
\(260\) −0.219350 −0.0136035
\(261\) 5.96133 0.368997
\(262\) −14.0919 −0.870602
\(263\) 4.91687 0.303187 0.151594 0.988443i \(-0.451560\pi\)
0.151594 + 0.988443i \(0.451560\pi\)
\(264\) −6.70664 −0.412765
\(265\) −2.37381 −0.145822
\(266\) −3.61205 −0.221469
\(267\) 25.0894 1.53545
\(268\) 2.71020 0.165552
\(269\) 11.4936 0.700778 0.350389 0.936604i \(-0.386049\pi\)
0.350389 + 0.936604i \(0.386049\pi\)
\(270\) −2.80746 −0.170856
\(271\) −18.6741 −1.13437 −0.567185 0.823591i \(-0.691967\pi\)
−0.567185 + 0.823591i \(0.691967\pi\)
\(272\) −15.0586 −0.913064
\(273\) −5.07894 −0.307391
\(274\) −25.4051 −1.53478
\(275\) −1.00000 −0.0603023
\(276\) −3.23165 −0.194522
\(277\) −10.3277 −0.620529 −0.310265 0.950650i \(-0.600418\pi\)
−0.310265 + 0.950650i \(0.600418\pi\)
\(278\) −7.00967 −0.420412
\(279\) −14.4779 −0.866768
\(280\) −8.18590 −0.489201
\(281\) −19.5032 −1.16346 −0.581732 0.813380i \(-0.697625\pi\)
−0.581732 + 0.813380i \(0.697625\pi\)
\(282\) −37.7464 −2.24776
\(283\) 6.51668 0.387376 0.193688 0.981063i \(-0.437955\pi\)
0.193688 + 0.981063i \(0.437955\pi\)
\(284\) −1.98755 −0.117939
\(285\) −2.24751 −0.133131
\(286\) −1.08467 −0.0641378
\(287\) −29.2246 −1.72508
\(288\) −3.06834 −0.180803
\(289\) 2.65604 0.156237
\(290\) 3.82649 0.224699
\(291\) 33.8528 1.98449
\(292\) 0.212445 0.0124324
\(293\) −2.06584 −0.120687 −0.0603437 0.998178i \(-0.519220\pi\)
−0.0603437 + 0.998178i \(0.519220\pi\)
\(294\) −1.55473 −0.0906737
\(295\) −14.8451 −0.864312
\(296\) 25.0818 1.45785
\(297\) 2.13218 0.123721
\(298\) 23.4166 1.35649
\(299\) −4.44834 −0.257254
\(300\) −0.598458 −0.0345520
\(301\) 28.3668 1.63503
\(302\) 12.7798 0.735392
\(303\) 3.69174 0.212085
\(304\) −3.39655 −0.194805
\(305\) 8.71368 0.498944
\(306\) −11.9749 −0.684558
\(307\) 6.49853 0.370890 0.185445 0.982655i \(-0.440627\pi\)
0.185445 + 0.982655i \(0.440627\pi\)
\(308\) 0.730457 0.0416217
\(309\) −42.2175 −2.40167
\(310\) −9.29313 −0.527815
\(311\) 17.2796 0.979834 0.489917 0.871769i \(-0.337027\pi\)
0.489917 + 0.871769i \(0.337027\pi\)
\(312\) −5.52474 −0.312777
\(313\) −22.9407 −1.29669 −0.648343 0.761348i \(-0.724538\pi\)
−0.648343 + 0.761348i \(0.724538\pi\)
\(314\) −8.93098 −0.504004
\(315\) −5.62726 −0.317060
\(316\) 0.256357 0.0144212
\(317\) −16.4474 −0.923778 −0.461889 0.886938i \(-0.652828\pi\)
−0.461889 + 0.886938i \(0.652828\pi\)
\(318\) −7.02487 −0.393935
\(319\) −2.90610 −0.162710
\(320\) −8.76261 −0.489845
\(321\) −42.7911 −2.38837
\(322\) −19.5049 −1.08697
\(323\) 4.43351 0.246687
\(324\) 2.91466 0.161926
\(325\) −0.823772 −0.0456946
\(326\) 22.0192 1.21953
\(327\) 37.6892 2.08422
\(328\) −31.7898 −1.75530
\(329\) 34.9902 1.92907
\(330\) −2.95932 −0.162905
\(331\) −30.8271 −1.69441 −0.847206 0.531265i \(-0.821717\pi\)
−0.847206 + 0.531265i \(0.821717\pi\)
\(332\) −0.0294872 −0.00161832
\(333\) 17.2420 0.944857
\(334\) 11.9263 0.652579
\(335\) 10.1782 0.556093
\(336\) −20.9413 −1.14244
\(337\) −18.9430 −1.03189 −0.515946 0.856621i \(-0.672559\pi\)
−0.515946 + 0.856621i \(0.672559\pi\)
\(338\) 16.2237 0.882453
\(339\) 24.3389 1.32191
\(340\) 1.18054 0.0640235
\(341\) 7.05785 0.382204
\(342\) −2.70099 −0.146053
\(343\) −17.7615 −0.959029
\(344\) 30.8566 1.66368
\(345\) −12.1365 −0.653407
\(346\) −11.2967 −0.607313
\(347\) −2.14728 −0.115272 −0.0576361 0.998338i \(-0.518356\pi\)
−0.0576361 + 0.998338i \(0.518356\pi\)
\(348\) −1.73918 −0.0932297
\(349\) 4.95553 0.265264 0.132632 0.991165i \(-0.457657\pi\)
0.132632 + 0.991165i \(0.457657\pi\)
\(350\) −3.61205 −0.193072
\(351\) 1.75643 0.0937512
\(352\) 1.49579 0.0797258
\(353\) 27.1851 1.44692 0.723459 0.690367i \(-0.242551\pi\)
0.723459 + 0.690367i \(0.242551\pi\)
\(354\) −43.9313 −2.33492
\(355\) −7.46427 −0.396162
\(356\) −2.97248 −0.157541
\(357\) 27.3347 1.44671
\(358\) 12.0309 0.635852
\(359\) −4.00177 −0.211205 −0.105603 0.994408i \(-0.533677\pi\)
−0.105603 + 0.994408i \(0.533677\pi\)
\(360\) −6.12119 −0.322615
\(361\) 1.00000 0.0526316
\(362\) 24.2068 1.27228
\(363\) 2.24751 0.117964
\(364\) 0.601730 0.0315392
\(365\) 0.797840 0.0417608
\(366\) 25.7866 1.34789
\(367\) −21.6919 −1.13231 −0.566153 0.824300i \(-0.691569\pi\)
−0.566153 + 0.824300i \(0.691569\pi\)
\(368\) −18.3412 −0.956103
\(369\) −21.8534 −1.13764
\(370\) 11.0674 0.575367
\(371\) 6.51193 0.338083
\(372\) 4.22382 0.218995
\(373\) −25.8940 −1.34074 −0.670371 0.742026i \(-0.733865\pi\)
−0.670371 + 0.742026i \(0.733865\pi\)
\(374\) 5.83765 0.301858
\(375\) −2.24751 −0.116061
\(376\) 38.0615 1.96287
\(377\) −2.39396 −0.123295
\(378\) 7.70153 0.396124
\(379\) 21.6754 1.11339 0.556694 0.830718i \(-0.312070\pi\)
0.556694 + 0.830718i \(0.312070\pi\)
\(380\) 0.266275 0.0136596
\(381\) −19.0501 −0.975966
\(382\) 6.21747 0.318113
\(383\) −7.45213 −0.380786 −0.190393 0.981708i \(-0.560976\pi\)
−0.190393 + 0.981708i \(0.560976\pi\)
\(384\) −19.2078 −0.980194
\(385\) 2.74324 0.139808
\(386\) −4.21352 −0.214463
\(387\) 21.2119 1.07826
\(388\) −4.01073 −0.203614
\(389\) −15.9821 −0.810323 −0.405162 0.914245i \(-0.632785\pi\)
−0.405162 + 0.914245i \(0.632785\pi\)
\(390\) −2.43781 −0.123443
\(391\) 23.9408 1.21074
\(392\) 1.56771 0.0791812
\(393\) 24.0537 1.21335
\(394\) 22.0856 1.11266
\(395\) 0.962752 0.0484413
\(396\) 0.546215 0.0274484
\(397\) −29.8765 −1.49946 −0.749729 0.661746i \(-0.769816\pi\)
−0.749729 + 0.661746i \(0.769816\pi\)
\(398\) 2.32311 0.116447
\(399\) 6.16547 0.308660
\(400\) −3.39655 −0.169827
\(401\) 7.73230 0.386133 0.193066 0.981186i \(-0.438157\pi\)
0.193066 + 0.981186i \(0.438157\pi\)
\(402\) 30.1205 1.50227
\(403\) 5.81406 0.289619
\(404\) −0.437381 −0.0217605
\(405\) 10.9460 0.543914
\(406\) −10.4970 −0.520956
\(407\) −8.40534 −0.416637
\(408\) 29.7340 1.47205
\(409\) −25.9708 −1.28417 −0.642087 0.766632i \(-0.721931\pi\)
−0.642087 + 0.766632i \(0.721931\pi\)
\(410\) −14.0273 −0.692761
\(411\) 43.3644 2.13901
\(412\) 5.00174 0.246418
\(413\) 40.7235 2.00387
\(414\) −14.5852 −0.716826
\(415\) −0.110739 −0.00543599
\(416\) 1.23219 0.0604130
\(417\) 11.9649 0.585925
\(418\) 1.31671 0.0644024
\(419\) −7.42032 −0.362507 −0.181253 0.983436i \(-0.558015\pi\)
−0.181253 + 0.983436i \(0.558015\pi\)
\(420\) 1.64171 0.0801074
\(421\) 16.8440 0.820928 0.410464 0.911877i \(-0.365367\pi\)
0.410464 + 0.911877i \(0.365367\pi\)
\(422\) −31.3838 −1.52774
\(423\) 26.1647 1.27217
\(424\) 7.08351 0.344006
\(425\) 4.43351 0.215057
\(426\) −22.0892 −1.07022
\(427\) −23.9037 −1.15678
\(428\) 5.06970 0.245053
\(429\) 1.85144 0.0893883
\(430\) 13.6156 0.656601
\(431\) 1.56149 0.0752144 0.0376072 0.999293i \(-0.488026\pi\)
0.0376072 + 0.999293i \(0.488026\pi\)
\(432\) 7.24204 0.348433
\(433\) 30.7097 1.47581 0.737907 0.674902i \(-0.235814\pi\)
0.737907 + 0.674902i \(0.235814\pi\)
\(434\) 25.4933 1.22372
\(435\) −6.53150 −0.313161
\(436\) −4.46525 −0.213847
\(437\) 5.39996 0.258315
\(438\) 2.36106 0.112816
\(439\) −38.5063 −1.83781 −0.918903 0.394484i \(-0.870923\pi\)
−0.918903 + 0.394484i \(0.870923\pi\)
\(440\) 2.98403 0.142258
\(441\) 1.07769 0.0513188
\(442\) 4.80889 0.228736
\(443\) 21.8516 1.03820 0.519101 0.854713i \(-0.326267\pi\)
0.519101 + 0.854713i \(0.326267\pi\)
\(444\) −5.03024 −0.238725
\(445\) −11.1632 −0.529186
\(446\) 17.6578 0.836122
\(447\) −39.9703 −1.89053
\(448\) 24.0379 1.13569
\(449\) −6.60543 −0.311730 −0.155865 0.987778i \(-0.549816\pi\)
−0.155865 + 0.987778i \(0.549816\pi\)
\(450\) −2.70099 −0.127326
\(451\) 10.6533 0.501646
\(452\) −2.88356 −0.135631
\(453\) −21.8140 −1.02491
\(454\) −5.83730 −0.273958
\(455\) 2.25980 0.105941
\(456\) 6.70664 0.314067
\(457\) 6.48912 0.303548 0.151774 0.988415i \(-0.451501\pi\)
0.151774 + 0.988415i \(0.451501\pi\)
\(458\) −18.4504 −0.862129
\(459\) −9.45303 −0.441230
\(460\) 1.43788 0.0670414
\(461\) 33.7065 1.56987 0.784933 0.619580i \(-0.212697\pi\)
0.784933 + 0.619580i \(0.212697\pi\)
\(462\) 8.11813 0.377690
\(463\) 12.0591 0.560436 0.280218 0.959936i \(-0.409593\pi\)
0.280218 + 0.959936i \(0.409593\pi\)
\(464\) −9.87070 −0.458236
\(465\) 15.8626 0.735611
\(466\) 3.56408 0.165103
\(467\) 7.68943 0.355824 0.177912 0.984046i \(-0.443066\pi\)
0.177912 + 0.984046i \(0.443066\pi\)
\(468\) 0.449957 0.0207993
\(469\) −27.9212 −1.28928
\(470\) 16.7947 0.774683
\(471\) 15.2444 0.702427
\(472\) 44.2980 2.03898
\(473\) −10.3406 −0.475461
\(474\) 2.84909 0.130863
\(475\) 1.00000 0.0458831
\(476\) −3.23849 −0.148436
\(477\) 4.86944 0.222956
\(478\) 37.0958 1.69672
\(479\) 11.8771 0.542680 0.271340 0.962484i \(-0.412533\pi\)
0.271340 + 0.962484i \(0.412533\pi\)
\(480\) 3.36180 0.153445
\(481\) −6.92409 −0.315711
\(482\) 24.2874 1.10626
\(483\) 33.2933 1.51490
\(484\) −0.266275 −0.0121034
\(485\) −15.0623 −0.683946
\(486\) 23.9705 1.08732
\(487\) 20.8445 0.944556 0.472278 0.881450i \(-0.343432\pi\)
0.472278 + 0.881450i \(0.343432\pi\)
\(488\) −26.0019 −1.17705
\(489\) −37.5849 −1.69965
\(490\) 0.691755 0.0312503
\(491\) −16.4510 −0.742425 −0.371212 0.928548i \(-0.621058\pi\)
−0.371212 + 0.928548i \(0.621058\pi\)
\(492\) 6.37557 0.287433
\(493\) 12.8842 0.580277
\(494\) 1.08467 0.0488015
\(495\) 2.05132 0.0921999
\(496\) 23.9723 1.07639
\(497\) 20.4763 0.918486
\(498\) −0.327714 −0.0146852
\(499\) −5.40939 −0.242158 −0.121079 0.992643i \(-0.538635\pi\)
−0.121079 + 0.992643i \(0.538635\pi\)
\(500\) 0.266275 0.0119082
\(501\) −20.3572 −0.909494
\(502\) 0.541878 0.0241852
\(503\) −2.39558 −0.106814 −0.0534068 0.998573i \(-0.517008\pi\)
−0.0534068 + 0.998573i \(0.517008\pi\)
\(504\) 16.7919 0.747970
\(505\) −1.64259 −0.0730943
\(506\) 7.11019 0.316086
\(507\) −27.6925 −1.22987
\(508\) 2.25697 0.100137
\(509\) −10.7037 −0.474435 −0.237217 0.971457i \(-0.576235\pi\)
−0.237217 + 0.971457i \(0.576235\pi\)
\(510\) 13.1202 0.580972
\(511\) −2.18867 −0.0968209
\(512\) 25.3513 1.12038
\(513\) −2.13218 −0.0941379
\(514\) 19.0598 0.840690
\(515\) 18.7841 0.827726
\(516\) −6.18841 −0.272430
\(517\) −12.7551 −0.560967
\(518\) −30.3605 −1.33397
\(519\) 19.2825 0.846407
\(520\) 2.45816 0.107797
\(521\) 41.3484 1.81151 0.905754 0.423805i \(-0.139306\pi\)
0.905754 + 0.423805i \(0.139306\pi\)
\(522\) −7.84935 −0.343557
\(523\) 26.7932 1.17158 0.585792 0.810461i \(-0.300783\pi\)
0.585792 + 0.810461i \(0.300783\pi\)
\(524\) −2.84978 −0.124493
\(525\) 6.16547 0.269083
\(526\) −6.47409 −0.282284
\(527\) −31.2910 −1.36306
\(528\) 7.63378 0.332218
\(529\) 6.15961 0.267809
\(530\) 3.12562 0.135768
\(531\) 30.4519 1.32150
\(532\) −0.730457 −0.0316693
\(533\) 8.77591 0.380127
\(534\) −33.0355 −1.42959
\(535\) 19.0393 0.823141
\(536\) −30.3719 −1.31187
\(537\) −20.5357 −0.886181
\(538\) −15.1337 −0.652462
\(539\) −0.525367 −0.0226291
\(540\) −0.567746 −0.0244319
\(541\) 4.24816 0.182643 0.0913214 0.995821i \(-0.470891\pi\)
0.0913214 + 0.995821i \(0.470891\pi\)
\(542\) 24.5883 1.05616
\(543\) −41.3190 −1.77317
\(544\) −6.63159 −0.284327
\(545\) −16.7693 −0.718318
\(546\) 6.68749 0.286198
\(547\) −34.9167 −1.49293 −0.746466 0.665424i \(-0.768251\pi\)
−0.746466 + 0.665424i \(0.768251\pi\)
\(548\) −5.13762 −0.219468
\(549\) −17.8745 −0.762866
\(550\) 1.31671 0.0561447
\(551\) 2.90610 0.123804
\(552\) 36.2156 1.54144
\(553\) −2.64106 −0.112309
\(554\) 13.5985 0.577746
\(555\) −18.8911 −0.801884
\(556\) −1.41755 −0.0601175
\(557\) 15.7363 0.666768 0.333384 0.942791i \(-0.391809\pi\)
0.333384 + 0.942791i \(0.391809\pi\)
\(558\) 19.0632 0.807008
\(559\) −8.51830 −0.360286
\(560\) 9.31754 0.393738
\(561\) −9.96438 −0.420696
\(562\) 25.6801 1.08325
\(563\) 25.6865 1.08256 0.541278 0.840844i \(-0.317941\pi\)
0.541278 + 0.840844i \(0.317941\pi\)
\(564\) −7.63337 −0.321423
\(565\) −10.8292 −0.455589
\(566\) −8.58058 −0.360669
\(567\) −30.0276 −1.26104
\(568\) 22.2736 0.934578
\(569\) −7.78096 −0.326195 −0.163097 0.986610i \(-0.552149\pi\)
−0.163097 + 0.986610i \(0.552149\pi\)
\(570\) 2.95932 0.123952
\(571\) −3.39607 −0.142121 −0.0710605 0.997472i \(-0.522638\pi\)
−0.0710605 + 0.997472i \(0.522638\pi\)
\(572\) −0.219350 −0.00917149
\(573\) −10.6127 −0.443352
\(574\) 38.4804 1.60614
\(575\) 5.39996 0.225194
\(576\) 17.9749 0.748954
\(577\) 26.6095 1.10777 0.553885 0.832593i \(-0.313145\pi\)
0.553885 + 0.832593i \(0.313145\pi\)
\(578\) −3.49723 −0.145466
\(579\) 7.19213 0.298895
\(580\) 0.773823 0.0321312
\(581\) 0.303785 0.0126031
\(582\) −44.5744 −1.84767
\(583\) −2.37381 −0.0983132
\(584\) −2.38077 −0.0985172
\(585\) 1.68982 0.0698654
\(586\) 2.72011 0.112367
\(587\) 40.0612 1.65350 0.826751 0.562568i \(-0.190186\pi\)
0.826751 + 0.562568i \(0.190186\pi\)
\(588\) −0.314410 −0.0129660
\(589\) −7.05785 −0.290813
\(590\) 19.5466 0.804722
\(591\) −37.6983 −1.55070
\(592\) −28.5491 −1.17336
\(593\) 13.6756 0.561590 0.280795 0.959768i \(-0.409402\pi\)
0.280795 + 0.959768i \(0.409402\pi\)
\(594\) −2.80746 −0.115191
\(595\) −12.1622 −0.498601
\(596\) 4.73550 0.193974
\(597\) −3.96536 −0.162291
\(598\) 5.85717 0.239518
\(599\) −19.1403 −0.782051 −0.391026 0.920380i \(-0.627880\pi\)
−0.391026 + 0.920380i \(0.627880\pi\)
\(600\) 6.70664 0.273797
\(601\) −28.2509 −1.15238 −0.576189 0.817316i \(-0.695461\pi\)
−0.576189 + 0.817316i \(0.695461\pi\)
\(602\) −37.3508 −1.52230
\(603\) −20.8787 −0.850244
\(604\) 2.58442 0.105159
\(605\) −1.00000 −0.0406558
\(606\) −4.86095 −0.197463
\(607\) 11.6697 0.473660 0.236830 0.971551i \(-0.423892\pi\)
0.236830 + 0.971551i \(0.423892\pi\)
\(608\) −1.49579 −0.0606622
\(609\) 17.9175 0.726053
\(610\) −11.4734 −0.464544
\(611\) −10.5073 −0.425078
\(612\) −2.42165 −0.0978895
\(613\) 28.0921 1.13463 0.567314 0.823502i \(-0.307983\pi\)
0.567314 + 0.823502i \(0.307983\pi\)
\(614\) −8.55667 −0.345319
\(615\) 23.9435 0.965495
\(616\) −8.18590 −0.329819
\(617\) 20.7862 0.836822 0.418411 0.908258i \(-0.362587\pi\)
0.418411 + 0.908258i \(0.362587\pi\)
\(618\) 55.5882 2.23609
\(619\) 44.7320 1.79793 0.898965 0.438021i \(-0.144321\pi\)
0.898965 + 0.438021i \(0.144321\pi\)
\(620\) −1.87933 −0.0754757
\(621\) −11.5137 −0.462028
\(622\) −22.7522 −0.912279
\(623\) 30.6233 1.22690
\(624\) 6.28850 0.251741
\(625\) 1.00000 0.0400000
\(626\) 30.2063 1.20729
\(627\) −2.24751 −0.0897570
\(628\) −1.80609 −0.0720710
\(629\) 37.2652 1.48586
\(630\) 7.40946 0.295200
\(631\) −5.71483 −0.227504 −0.113752 0.993509i \(-0.536287\pi\)
−0.113752 + 0.993509i \(0.536287\pi\)
\(632\) −2.87288 −0.114277
\(633\) 53.5695 2.12920
\(634\) 21.6565 0.860088
\(635\) 8.47608 0.336363
\(636\) −1.42062 −0.0563314
\(637\) −0.432782 −0.0171475
\(638\) 3.82649 0.151492
\(639\) 15.3116 0.605717
\(640\) 8.54624 0.337820
\(641\) 9.13627 0.360861 0.180431 0.983588i \(-0.442251\pi\)
0.180431 + 0.983588i \(0.442251\pi\)
\(642\) 56.3434 2.22370
\(643\) 38.5090 1.51865 0.759324 0.650712i \(-0.225530\pi\)
0.759324 + 0.650712i \(0.225530\pi\)
\(644\) −3.94444 −0.155433
\(645\) −23.2406 −0.915099
\(646\) −5.83765 −0.229679
\(647\) 5.06693 0.199202 0.0996009 0.995027i \(-0.468243\pi\)
0.0996009 + 0.995027i \(0.468243\pi\)
\(648\) −32.6633 −1.28314
\(649\) −14.8451 −0.582719
\(650\) 1.08467 0.0425442
\(651\) −43.5149 −1.70548
\(652\) 4.45289 0.174389
\(653\) −43.3175 −1.69515 −0.847573 0.530678i \(-0.821937\pi\)
−0.847573 + 0.530678i \(0.821937\pi\)
\(654\) −49.6258 −1.94052
\(655\) −10.7024 −0.418177
\(656\) 36.1845 1.41277
\(657\) −1.63662 −0.0638507
\(658\) −46.0720 −1.79607
\(659\) 48.8179 1.90168 0.950838 0.309689i \(-0.100225\pi\)
0.950838 + 0.309689i \(0.100225\pi\)
\(660\) −0.598458 −0.0232949
\(661\) 3.07287 0.119521 0.0597605 0.998213i \(-0.480966\pi\)
0.0597605 + 0.998213i \(0.480966\pi\)
\(662\) 40.5904 1.57759
\(663\) −8.20838 −0.318787
\(664\) 0.330449 0.0128239
\(665\) −2.74324 −0.106378
\(666\) −22.7027 −0.879714
\(667\) 15.6928 0.607629
\(668\) 2.41183 0.0933166
\(669\) −30.1404 −1.16530
\(670\) −13.4017 −0.517752
\(671\) 8.71368 0.336388
\(672\) −9.22223 −0.355755
\(673\) −21.3366 −0.822465 −0.411233 0.911530i \(-0.634902\pi\)
−0.411233 + 0.911530i \(0.634902\pi\)
\(674\) 24.9424 0.960747
\(675\) −2.13218 −0.0820675
\(676\) 3.28089 0.126188
\(677\) −16.2317 −0.623836 −0.311918 0.950109i \(-0.600971\pi\)
−0.311918 + 0.950109i \(0.600971\pi\)
\(678\) −32.0472 −1.23077
\(679\) 41.3196 1.58570
\(680\) −13.2297 −0.507337
\(681\) 9.96379 0.381813
\(682\) −9.29313 −0.355853
\(683\) 13.0211 0.498238 0.249119 0.968473i \(-0.419859\pi\)
0.249119 + 0.968473i \(0.419859\pi\)
\(684\) −0.546215 −0.0208851
\(685\) −19.2944 −0.737201
\(686\) 23.3867 0.892909
\(687\) 31.4932 1.20154
\(688\) −35.1223 −1.33903
\(689\) −1.95548 −0.0744978
\(690\) 15.9802 0.608357
\(691\) −12.8934 −0.490489 −0.245245 0.969461i \(-0.578868\pi\)
−0.245245 + 0.969461i \(0.578868\pi\)
\(692\) −2.28450 −0.0868437
\(693\) −5.62726 −0.213762
\(694\) 2.82735 0.107325
\(695\) −5.32362 −0.201937
\(696\) 19.4902 0.738773
\(697\) −47.2317 −1.78903
\(698\) −6.52500 −0.246975
\(699\) −6.08358 −0.230102
\(700\) −0.730457 −0.0276087
\(701\) −18.4505 −0.696865 −0.348433 0.937334i \(-0.613286\pi\)
−0.348433 + 0.937334i \(0.613286\pi\)
\(702\) −2.31271 −0.0872874
\(703\) 8.40534 0.317014
\(704\) −8.76261 −0.330253
\(705\) −28.6672 −1.07967
\(706\) −35.7949 −1.34716
\(707\) 4.50602 0.169466
\(708\) −8.88413 −0.333886
\(709\) 40.8332 1.53352 0.766762 0.641932i \(-0.221867\pi\)
0.766762 + 0.641932i \(0.221867\pi\)
\(710\) 9.82827 0.368848
\(711\) −1.97491 −0.0740649
\(712\) 33.3113 1.24839
\(713\) −38.1121 −1.42731
\(714\) −35.9919 −1.34696
\(715\) −0.823772 −0.0308073
\(716\) 2.43298 0.0909247
\(717\) −63.3194 −2.36471
\(718\) 5.26916 0.196643
\(719\) −45.9049 −1.71196 −0.855982 0.517005i \(-0.827047\pi\)
−0.855982 + 0.517005i \(0.827047\pi\)
\(720\) 6.96740 0.259660
\(721\) −51.5293 −1.91905
\(722\) −1.31671 −0.0490029
\(723\) −41.4566 −1.54179
\(724\) 4.89529 0.181932
\(725\) 2.90610 0.107930
\(726\) −2.95932 −0.109831
\(727\) 46.7486 1.73381 0.866905 0.498473i \(-0.166106\pi\)
0.866905 + 0.498473i \(0.166106\pi\)
\(728\) −6.74332 −0.249924
\(729\) −8.07753 −0.299168
\(730\) −1.05052 −0.0388816
\(731\) 45.8452 1.69565
\(732\) 5.21477 0.192743
\(733\) 7.20742 0.266212 0.133106 0.991102i \(-0.457505\pi\)
0.133106 + 0.991102i \(0.457505\pi\)
\(734\) 28.5619 1.05424
\(735\) −1.18077 −0.0435533
\(736\) −8.07720 −0.297730
\(737\) 10.1782 0.374918
\(738\) 28.7745 1.05921
\(739\) −15.8711 −0.583828 −0.291914 0.956445i \(-0.594292\pi\)
−0.291914 + 0.956445i \(0.594292\pi\)
\(740\) 2.23814 0.0822755
\(741\) −1.85144 −0.0680143
\(742\) −8.57432 −0.314773
\(743\) −18.8696 −0.692258 −0.346129 0.938187i \(-0.612504\pi\)
−0.346129 + 0.938187i \(0.612504\pi\)
\(744\) −47.3344 −1.73536
\(745\) 17.7842 0.651563
\(746\) 34.0949 1.24830
\(747\) 0.227162 0.00831141
\(748\) 1.18054 0.0431646
\(749\) −52.2294 −1.90842
\(750\) 2.95932 0.108059
\(751\) 20.6044 0.751866 0.375933 0.926647i \(-0.377322\pi\)
0.375933 + 0.926647i \(0.377322\pi\)
\(752\) −43.3232 −1.57983
\(753\) −0.924941 −0.0337067
\(754\) 3.15216 0.114795
\(755\) 9.70582 0.353231
\(756\) 1.55746 0.0566444
\(757\) 2.85131 0.103633 0.0518164 0.998657i \(-0.483499\pi\)
0.0518164 + 0.998657i \(0.483499\pi\)
\(758\) −28.5402 −1.03663
\(759\) −12.1365 −0.440527
\(760\) −2.98403 −0.108242
\(761\) −8.39479 −0.304311 −0.152155 0.988357i \(-0.548621\pi\)
−0.152155 + 0.988357i \(0.548621\pi\)
\(762\) 25.0835 0.908678
\(763\) 46.0022 1.66539
\(764\) 1.25735 0.0454892
\(765\) −9.09454 −0.328814
\(766\) 9.81230 0.354533
\(767\) −12.2289 −0.441561
\(768\) −14.0971 −0.508685
\(769\) 2.19228 0.0790557 0.0395278 0.999218i \(-0.487415\pi\)
0.0395278 + 0.999218i \(0.487415\pi\)
\(770\) −3.61205 −0.130169
\(771\) −32.5334 −1.17166
\(772\) −0.852091 −0.0306674
\(773\) −43.6273 −1.56916 −0.784582 0.620025i \(-0.787123\pi\)
−0.784582 + 0.620025i \(0.787123\pi\)
\(774\) −27.9299 −1.00392
\(775\) −7.05785 −0.253525
\(776\) 44.9465 1.61348
\(777\) 51.8229 1.85914
\(778\) 21.0437 0.754455
\(779\) −10.6533 −0.381695
\(780\) −0.492993 −0.0176520
\(781\) −7.46427 −0.267092
\(782\) −31.5231 −1.12726
\(783\) −6.19632 −0.221438
\(784\) −1.78443 −0.0637297
\(785\) −6.78280 −0.242089
\(786\) −31.6718 −1.12970
\(787\) 12.4440 0.443582 0.221791 0.975094i \(-0.428810\pi\)
0.221791 + 0.975094i \(0.428810\pi\)
\(788\) 4.46632 0.159106
\(789\) 11.0507 0.393416
\(790\) −1.26766 −0.0451015
\(791\) 29.7072 1.05627
\(792\) −6.12119 −0.217507
\(793\) 7.17808 0.254901
\(794\) 39.3386 1.39608
\(795\) −5.33517 −0.189219
\(796\) 0.469798 0.0166516
\(797\) −32.4735 −1.15027 −0.575134 0.818059i \(-0.695050\pi\)
−0.575134 + 0.818059i \(0.695050\pi\)
\(798\) −8.11813 −0.287379
\(799\) 56.5497 2.00059
\(800\) −1.49579 −0.0528841
\(801\) 22.8992 0.809105
\(802\) −10.1812 −0.359510
\(803\) 0.797840 0.0281551
\(804\) 6.09120 0.214820
\(805\) −14.8134 −0.522104
\(806\) −7.65542 −0.269651
\(807\) 25.8320 0.909331
\(808\) 4.90153 0.172435
\(809\) −6.49725 −0.228431 −0.114215 0.993456i \(-0.536435\pi\)
−0.114215 + 0.993456i \(0.536435\pi\)
\(810\) −14.4128 −0.506413
\(811\) 19.0752 0.669820 0.334910 0.942250i \(-0.391294\pi\)
0.334910 + 0.942250i \(0.391294\pi\)
\(812\) −2.12278 −0.0744951
\(813\) −41.9702 −1.47196
\(814\) 11.0674 0.387912
\(815\) 16.7229 0.585777
\(816\) −33.8445 −1.18479
\(817\) 10.3406 0.361772
\(818\) 34.1960 1.19563
\(819\) −4.63558 −0.161980
\(820\) −2.83672 −0.0990626
\(821\) −3.43435 −0.119860 −0.0599299 0.998203i \(-0.519088\pi\)
−0.0599299 + 0.998203i \(0.519088\pi\)
\(822\) −57.0983 −1.99153
\(823\) 9.46472 0.329919 0.164960 0.986300i \(-0.447251\pi\)
0.164960 + 0.986300i \(0.447251\pi\)
\(824\) −56.0523 −1.95267
\(825\) −2.24751 −0.0782484
\(826\) −53.6211 −1.86572
\(827\) −7.05327 −0.245266 −0.122633 0.992452i \(-0.539134\pi\)
−0.122633 + 0.992452i \(0.539134\pi\)
\(828\) −2.94954 −0.102504
\(829\) −7.70745 −0.267691 −0.133845 0.991002i \(-0.542733\pi\)
−0.133845 + 0.991002i \(0.542733\pi\)
\(830\) 0.145812 0.00506120
\(831\) −23.2116 −0.805200
\(832\) −7.21839 −0.250253
\(833\) 2.32922 0.0807027
\(834\) −15.7543 −0.545528
\(835\) 9.05766 0.313453
\(836\) 0.266275 0.00920933
\(837\) 15.0486 0.520155
\(838\) 9.77041 0.337513
\(839\) −3.64475 −0.125831 −0.0629154 0.998019i \(-0.520040\pi\)
−0.0629154 + 0.998019i \(0.520040\pi\)
\(840\) −18.3979 −0.634789
\(841\) −20.5546 −0.708779
\(842\) −22.1787 −0.764329
\(843\) −43.8338 −1.50971
\(844\) −6.34667 −0.218462
\(845\) 12.3214 0.423869
\(846\) −34.4513 −1.18446
\(847\) 2.74324 0.0942588
\(848\) −8.06276 −0.276876
\(849\) 14.6463 0.502661
\(850\) −5.83765 −0.200230
\(851\) 45.3886 1.55590
\(852\) −4.46705 −0.153038
\(853\) 37.9565 1.29960 0.649802 0.760103i \(-0.274852\pi\)
0.649802 + 0.760103i \(0.274852\pi\)
\(854\) 31.4743 1.07703
\(855\) −2.05132 −0.0701536
\(856\) −56.8138 −1.94186
\(857\) −45.6479 −1.55930 −0.779652 0.626214i \(-0.784604\pi\)
−0.779652 + 0.626214i \(0.784604\pi\)
\(858\) −2.43781 −0.0832254
\(859\) 6.68610 0.228127 0.114063 0.993473i \(-0.463613\pi\)
0.114063 + 0.993473i \(0.463613\pi\)
\(860\) 2.75345 0.0938918
\(861\) −65.6828 −2.23846
\(862\) −2.05603 −0.0700287
\(863\) −57.8059 −1.96774 −0.983868 0.178898i \(-0.942747\pi\)
−0.983868 + 0.178898i \(0.942747\pi\)
\(864\) 3.18928 0.108502
\(865\) −8.57947 −0.291711
\(866\) −40.4358 −1.37406
\(867\) 5.96948 0.202734
\(868\) 5.15546 0.174988
\(869\) 0.962752 0.0326591
\(870\) 8.60009 0.291570
\(871\) 8.38449 0.284098
\(872\) 50.0400 1.69457
\(873\) 30.8977 1.04573
\(874\) −7.11019 −0.240506
\(875\) −2.74324 −0.0927384
\(876\) 0.477473 0.0161323
\(877\) −1.92045 −0.0648490 −0.0324245 0.999474i \(-0.510323\pi\)
−0.0324245 + 0.999474i \(0.510323\pi\)
\(878\) 50.7016 1.71110
\(879\) −4.64300 −0.156604
\(880\) −3.39655 −0.114498
\(881\) −46.0047 −1.54994 −0.774969 0.631999i \(-0.782235\pi\)
−0.774969 + 0.631999i \(0.782235\pi\)
\(882\) −1.41901 −0.0477806
\(883\) 2.94257 0.0990252 0.0495126 0.998773i \(-0.484233\pi\)
0.0495126 + 0.998773i \(0.484233\pi\)
\(884\) 0.972492 0.0327084
\(885\) −33.3645 −1.12153
\(886\) −28.7723 −0.966623
\(887\) 2.48521 0.0834453 0.0417227 0.999129i \(-0.486715\pi\)
0.0417227 + 0.999129i \(0.486715\pi\)
\(888\) 56.3716 1.89171
\(889\) −23.2519 −0.779844
\(890\) 14.6987 0.492701
\(891\) 10.9460 0.366706
\(892\) 3.57090 0.119563
\(893\) 12.7551 0.426832
\(894\) 52.6292 1.76019
\(895\) 9.13708 0.305419
\(896\) −23.4444 −0.783222
\(897\) −9.99770 −0.333814
\(898\) 8.69743 0.290237
\(899\) −20.5108 −0.684074
\(900\) −0.546215 −0.0182072
\(901\) 10.5243 0.350616
\(902\) −14.0273 −0.467059
\(903\) 63.7547 2.12162
\(904\) 32.3147 1.07477
\(905\) 18.3843 0.611115
\(906\) 28.7227 0.954247
\(907\) −46.8633 −1.55607 −0.778036 0.628220i \(-0.783784\pi\)
−0.778036 + 0.628220i \(0.783784\pi\)
\(908\) −1.18046 −0.0391751
\(909\) 3.36947 0.111758
\(910\) −2.97551 −0.0986371
\(911\) −32.7237 −1.08418 −0.542092 0.840319i \(-0.682367\pi\)
−0.542092 + 0.840319i \(0.682367\pi\)
\(912\) −7.63378 −0.252780
\(913\) −0.110739 −0.00366494
\(914\) −8.54429 −0.282620
\(915\) 19.5841 0.647431
\(916\) −3.73118 −0.123282
\(917\) 29.3592 0.969526
\(918\) 12.4469 0.410809
\(919\) 59.2895 1.95578 0.977890 0.209121i \(-0.0670603\pi\)
0.977890 + 0.209121i \(0.0670603\pi\)
\(920\) −16.1136 −0.531251
\(921\) 14.6055 0.481268
\(922\) −44.3816 −1.46163
\(923\) −6.14885 −0.202392
\(924\) 1.64171 0.0540084
\(925\) 8.40534 0.276366
\(926\) −15.8784 −0.521797
\(927\) −38.5322 −1.26556
\(928\) −4.34691 −0.142694
\(929\) 24.5262 0.804679 0.402339 0.915491i \(-0.368197\pi\)
0.402339 + 0.915491i \(0.368197\pi\)
\(930\) −20.8864 −0.684894
\(931\) 0.525367 0.0172182
\(932\) 0.720756 0.0236091
\(933\) 38.8361 1.27144
\(934\) −10.1248 −0.331292
\(935\) 4.43351 0.144991
\(936\) −5.04246 −0.164818
\(937\) 10.1189 0.330571 0.165285 0.986246i \(-0.447145\pi\)
0.165285 + 0.986246i \(0.447145\pi\)
\(938\) 36.7641 1.20039
\(939\) −51.5596 −1.68258
\(940\) 3.39636 0.110777
\(941\) −50.0082 −1.63022 −0.815110 0.579307i \(-0.803323\pi\)
−0.815110 + 0.579307i \(0.803323\pi\)
\(942\) −20.0725 −0.653997
\(943\) −57.5276 −1.87336
\(944\) −50.4219 −1.64109
\(945\) 5.84907 0.190270
\(946\) 13.6156 0.442680
\(947\) −8.93511 −0.290352 −0.145176 0.989406i \(-0.546375\pi\)
−0.145176 + 0.989406i \(0.546375\pi\)
\(948\) 0.576166 0.0187130
\(949\) 0.657238 0.0213348
\(950\) −1.31671 −0.0427197
\(951\) −36.9658 −1.19870
\(952\) 36.2923 1.17624
\(953\) −23.7421 −0.769082 −0.384541 0.923108i \(-0.625640\pi\)
−0.384541 + 0.923108i \(0.625640\pi\)
\(954\) −6.41164 −0.207584
\(955\) 4.72197 0.152799
\(956\) 7.50180 0.242626
\(957\) −6.53150 −0.211133
\(958\) −15.6387 −0.505264
\(959\) 52.9291 1.70917
\(960\) −19.6941 −0.635624
\(961\) 18.8132 0.606877
\(962\) 9.11701 0.293944
\(963\) −39.0557 −1.25855
\(964\) 4.91159 0.158192
\(965\) −3.20004 −0.103013
\(966\) −43.8376 −1.41045
\(967\) 3.55415 0.114294 0.0571469 0.998366i \(-0.481800\pi\)
0.0571469 + 0.998366i \(0.481800\pi\)
\(968\) 2.98403 0.0959103
\(969\) 9.96438 0.320102
\(970\) 19.8327 0.636791
\(971\) −5.12760 −0.164553 −0.0822763 0.996610i \(-0.526219\pi\)
−0.0822763 + 0.996610i \(0.526219\pi\)
\(972\) 4.84751 0.155484
\(973\) 14.6040 0.468182
\(974\) −27.4462 −0.879433
\(975\) −1.85144 −0.0592935
\(976\) 29.5964 0.947358
\(977\) 25.3934 0.812407 0.406203 0.913783i \(-0.366852\pi\)
0.406203 + 0.913783i \(0.366852\pi\)
\(978\) 49.4884 1.58246
\(979\) −11.1632 −0.356777
\(980\) 0.139892 0.00446869
\(981\) 34.3992 1.09828
\(982\) 21.6612 0.691238
\(983\) 17.1989 0.548560 0.274280 0.961650i \(-0.411561\pi\)
0.274280 + 0.961650i \(0.411561\pi\)
\(984\) −71.4481 −2.27768
\(985\) 16.7733 0.534443
\(986\) −16.9648 −0.540269
\(987\) 78.6410 2.50317
\(988\) 0.219350 0.00697846
\(989\) 55.8389 1.77557
\(990\) −2.70099 −0.0858431
\(991\) 6.07473 0.192970 0.0964851 0.995334i \(-0.469240\pi\)
0.0964851 + 0.995334i \(0.469240\pi\)
\(992\) 10.5570 0.335186
\(993\) −69.2844 −2.19867
\(994\) −26.9613 −0.855161
\(995\) 1.76433 0.0559331
\(996\) −0.0662729 −0.00209994
\(997\) 20.9511 0.663527 0.331763 0.943363i \(-0.392356\pi\)
0.331763 + 0.943363i \(0.392356\pi\)
\(998\) 7.12260 0.225462
\(999\) −17.9217 −0.567017
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 1045.2.a.j.1.3 9
3.2 odd 2 9405.2.a.bi.1.7 9
5.4 even 2 5225.2.a.q.1.7 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.j.1.3 9 1.1 even 1 trivial
5225.2.a.q.1.7 9 5.4 even 2
9405.2.a.bi.1.7 9 3.2 odd 2