Properties

Label 2001.2.a.h.1.2
Level $2001$
Weight $2$
Character 2001.1
Self dual yes
Analytic conductor $15.978$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.312617.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 11x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.762877\) 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)
 
\(f(q)\) \(=\) \(q-1.79920 q^{2} -1.00000 q^{3} +1.23712 q^{4} -2.60753 q^{5} +1.79920 q^{6} +1.37040 q^{7} +1.37257 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.79920 q^{2} -1.00000 q^{3} +1.23712 q^{4} -2.60753 q^{5} +1.79920 q^{6} +1.37040 q^{7} +1.37257 q^{8} +1.00000 q^{9} +4.69147 q^{10} -5.21722 q^{11} -1.23712 q^{12} -0.609692 q^{13} -2.46563 q^{14} +2.60753 q^{15} -4.94377 q^{16} +0.762877 q^{17} -1.79920 q^{18} +5.43444 q^{19} -3.22583 q^{20} -1.37040 q^{21} +9.38682 q^{22} +1.00000 q^{23} -1.37257 q^{24} +1.79920 q^{25} +1.09696 q^{26} -1.00000 q^{27} +1.69536 q^{28} +1.00000 q^{29} -4.69147 q^{30} +3.96881 q^{31} +6.14970 q^{32} +5.21722 q^{33} -1.37257 q^{34} -3.57337 q^{35} +1.23712 q^{36} -3.31418 q^{37} -9.77765 q^{38} +0.609692 q^{39} -3.57901 q^{40} -1.10557 q^{41} +2.46563 q^{42} +7.23148 q^{43} -6.45434 q^{44} -2.60753 q^{45} -1.79920 q^{46} -12.9585 q^{47} +4.94377 q^{48} -5.12199 q^{49} -3.23712 q^{50} -0.762877 q^{51} -0.754264 q^{52} +4.97097 q^{53} +1.79920 q^{54} +13.6040 q^{55} +1.88097 q^{56} -5.43444 q^{57} -1.79920 q^{58} +14.3220 q^{59} +3.22583 q^{60} +14.4747 q^{61} -7.14068 q^{62} +1.37040 q^{63} -1.17700 q^{64} +1.58979 q^{65} -9.38682 q^{66} +12.1060 q^{67} +0.943773 q^{68} -1.00000 q^{69} +6.42921 q^{70} +2.20809 q^{71} +1.37257 q^{72} +3.58894 q^{73} +5.96287 q^{74} -1.79920 q^{75} +6.72307 q^{76} -7.14970 q^{77} -1.09696 q^{78} -7.27909 q^{79} +12.8910 q^{80} +1.00000 q^{81} +1.98914 q^{82} -3.93854 q^{83} -1.69536 q^{84} -1.98922 q^{85} -13.0109 q^{86} -1.00000 q^{87} -7.16099 q^{88} -1.67033 q^{89} +4.69147 q^{90} -0.835524 q^{91} +1.23712 q^{92} -3.96881 q^{93} +23.3150 q^{94} -14.1704 q^{95} -6.14970 q^{96} -18.5166 q^{97} +9.21549 q^{98} -5.21722 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} - 5 q^{3} + 8 q^{4} - 3 q^{5} + 2 q^{6} - 5 q^{7} - 3 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} - 5 q^{3} + 8 q^{4} - 3 q^{5} + 2 q^{6} - 5 q^{7} - 3 q^{8} + 5 q^{9} + 9 q^{10} - 8 q^{11} - 8 q^{12} + 5 q^{13} - 2 q^{14} + 3 q^{15} - 10 q^{16} + 2 q^{17} - 2 q^{18} - 9 q^{19} - 12 q^{20} + 5 q^{21} + 10 q^{22} + 5 q^{23} + 3 q^{24} + 2 q^{25} - 3 q^{26} - 5 q^{27} - 14 q^{28} + 5 q^{29} - 9 q^{30} - 6 q^{31} - 8 q^{32} + 8 q^{33} + 3 q^{34} - 15 q^{35} + 8 q^{36} + 10 q^{37} - 10 q^{38} - 5 q^{39} + 26 q^{40} - 11 q^{41} + 2 q^{42} - 9 q^{43} - 16 q^{44} - 3 q^{45} - 2 q^{46} - 13 q^{47} + 10 q^{48} - 6 q^{49} - 18 q^{50} - 2 q^{51} + 12 q^{52} + q^{53} + 2 q^{54} + 13 q^{55} - 4 q^{56} + 9 q^{57} - 2 q^{58} + 6 q^{59} + 12 q^{60} + 23 q^{61} - 36 q^{62} - 5 q^{63} - q^{64} - 20 q^{65} - 10 q^{66} - 10 q^{67} - 10 q^{68} - 5 q^{69} - 16 q^{70} - 11 q^{71} - 3 q^{72} + 31 q^{73} - 18 q^{74} - 2 q^{75} - 8 q^{76} + 3 q^{77} + 3 q^{78} + 8 q^{79} - 8 q^{80} + 5 q^{81} + 16 q^{82} + 7 q^{83} + 14 q^{84} + 6 q^{85} - 36 q^{86} - 5 q^{87} - 3 q^{88} + 3 q^{89} + 9 q^{90} + 8 q^{91} + 8 q^{92} + 6 q^{93} - 39 q^{94} - 11 q^{95} + 8 q^{96} + 3 q^{97} + 38 q^{98} - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.79920 −1.27223 −0.636113 0.771596i \(-0.719459\pi\)
−0.636113 + 0.771596i \(0.719459\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.23712 0.618562
\(5\) −2.60753 −1.16612 −0.583061 0.812428i \(-0.698145\pi\)
−0.583061 + 0.812428i \(0.698145\pi\)
\(6\) 1.79920 0.734521
\(7\) 1.37040 0.517964 0.258982 0.965882i \(-0.416613\pi\)
0.258982 + 0.965882i \(0.416613\pi\)
\(8\) 1.37257 0.485276
\(9\) 1.00000 0.333333
\(10\) 4.69147 1.48357
\(11\) −5.21722 −1.57305 −0.786525 0.617558i \(-0.788122\pi\)
−0.786525 + 0.617558i \(0.788122\pi\)
\(12\) −1.23712 −0.357127
\(13\) −0.609692 −0.169098 −0.0845490 0.996419i \(-0.526945\pi\)
−0.0845490 + 0.996419i \(0.526945\pi\)
\(14\) −2.46563 −0.658968
\(15\) 2.60753 0.673261
\(16\) −4.94377 −1.23594
\(17\) 0.762877 0.185025 0.0925124 0.995712i \(-0.470510\pi\)
0.0925124 + 0.995712i \(0.470510\pi\)
\(18\) −1.79920 −0.424076
\(19\) 5.43444 1.24675 0.623373 0.781925i \(-0.285762\pi\)
0.623373 + 0.781925i \(0.285762\pi\)
\(20\) −3.22583 −0.721318
\(21\) −1.37040 −0.299047
\(22\) 9.38682 2.00128
\(23\) 1.00000 0.208514
\(24\) −1.37257 −0.280174
\(25\) 1.79920 0.359840
\(26\) 1.09696 0.215131
\(27\) −1.00000 −0.192450
\(28\) 1.69536 0.320393
\(29\) 1.00000 0.185695
\(30\) −4.69147 −0.856541
\(31\) 3.96881 0.712819 0.356409 0.934330i \(-0.384001\pi\)
0.356409 + 0.934330i \(0.384001\pi\)
\(32\) 6.14970 1.08712
\(33\) 5.21722 0.908201
\(34\) −1.37257 −0.235394
\(35\) −3.57337 −0.604009
\(36\) 1.23712 0.206187
\(37\) −3.31418 −0.544847 −0.272424 0.962177i \(-0.587825\pi\)
−0.272424 + 0.962177i \(0.587825\pi\)
\(38\) −9.77765 −1.58614
\(39\) 0.609692 0.0976288
\(40\) −3.57901 −0.565891
\(41\) −1.10557 −0.172661 −0.0863306 0.996267i \(-0.527514\pi\)
−0.0863306 + 0.996267i \(0.527514\pi\)
\(42\) 2.46563 0.380455
\(43\) 7.23148 1.10279 0.551395 0.834244i \(-0.314096\pi\)
0.551395 + 0.834244i \(0.314096\pi\)
\(44\) −6.45434 −0.973029
\(45\) −2.60753 −0.388707
\(46\) −1.79920 −0.265278
\(47\) −12.9585 −1.89020 −0.945099 0.326785i \(-0.894035\pi\)
−0.945099 + 0.326785i \(0.894035\pi\)
\(48\) 4.94377 0.713572
\(49\) −5.12199 −0.731713
\(50\) −3.23712 −0.457798
\(51\) −0.762877 −0.106824
\(52\) −0.754264 −0.104598
\(53\) 4.97097 0.682815 0.341408 0.939915i \(-0.389096\pi\)
0.341408 + 0.939915i \(0.389096\pi\)
\(54\) 1.79920 0.244840
\(55\) 13.6040 1.83437
\(56\) 1.88097 0.251356
\(57\) −5.43444 −0.719809
\(58\) −1.79920 −0.236247
\(59\) 14.3220 1.86456 0.932282 0.361733i \(-0.117815\pi\)
0.932282 + 0.361733i \(0.117815\pi\)
\(60\) 3.22583 0.416453
\(61\) 14.4747 1.85329 0.926645 0.375938i \(-0.122679\pi\)
0.926645 + 0.375938i \(0.122679\pi\)
\(62\) −7.14068 −0.906867
\(63\) 1.37040 0.172655
\(64\) −1.17700 −0.147125
\(65\) 1.58979 0.197189
\(66\) −9.38682 −1.15544
\(67\) 12.1060 1.47898 0.739492 0.673166i \(-0.235066\pi\)
0.739492 + 0.673166i \(0.235066\pi\)
\(68\) 0.943773 0.114449
\(69\) −1.00000 −0.120386
\(70\) 6.42921 0.768437
\(71\) 2.20809 0.262052 0.131026 0.991379i \(-0.458173\pi\)
0.131026 + 0.991379i \(0.458173\pi\)
\(72\) 1.37257 0.161759
\(73\) 3.58894 0.420054 0.210027 0.977696i \(-0.432645\pi\)
0.210027 + 0.977696i \(0.432645\pi\)
\(74\) 5.96287 0.693170
\(75\) −1.79920 −0.207754
\(76\) 6.72307 0.771189
\(77\) −7.14970 −0.814784
\(78\) −1.09696 −0.124206
\(79\) −7.27909 −0.818962 −0.409481 0.912319i \(-0.634290\pi\)
−0.409481 + 0.912319i \(0.634290\pi\)
\(80\) 12.8910 1.44126
\(81\) 1.00000 0.111111
\(82\) 1.98914 0.219664
\(83\) −3.93854 −0.432311 −0.216155 0.976359i \(-0.569352\pi\)
−0.216155 + 0.976359i \(0.569352\pi\)
\(84\) −1.69536 −0.184979
\(85\) −1.98922 −0.215761
\(86\) −13.0109 −1.40300
\(87\) −1.00000 −0.107211
\(88\) −7.16099 −0.763364
\(89\) −1.67033 −0.177054 −0.0885271 0.996074i \(-0.528216\pi\)
−0.0885271 + 0.996074i \(0.528216\pi\)
\(90\) 4.69147 0.494524
\(91\) −0.835524 −0.0875868
\(92\) 1.23712 0.128979
\(93\) −3.96881 −0.411546
\(94\) 23.3150 2.40476
\(95\) −14.1704 −1.45386
\(96\) −6.14970 −0.627651
\(97\) −18.5166 −1.88008 −0.940039 0.341067i \(-0.889212\pi\)
−0.940039 + 0.341067i \(0.889212\pi\)
\(98\) 9.21549 0.930905
\(99\) −5.21722 −0.524350
\(100\) 2.22583 0.222583
\(101\) −1.06925 −0.106394 −0.0531970 0.998584i \(-0.516941\pi\)
−0.0531970 + 0.998584i \(0.516941\pi\)
\(102\) 1.37257 0.135905
\(103\) −15.5989 −1.53701 −0.768503 0.639846i \(-0.778998\pi\)
−0.768503 + 0.639846i \(0.778998\pi\)
\(104\) −0.836844 −0.0820593
\(105\) 3.57337 0.348725
\(106\) −8.94377 −0.868696
\(107\) −7.81735 −0.755732 −0.377866 0.925860i \(-0.623342\pi\)
−0.377866 + 0.925860i \(0.623342\pi\)
\(108\) −1.23712 −0.119042
\(109\) −1.85985 −0.178142 −0.0890709 0.996025i \(-0.528390\pi\)
−0.0890709 + 0.996025i \(0.528390\pi\)
\(110\) −24.4764 −2.33373
\(111\) 3.31418 0.314568
\(112\) −6.77497 −0.640174
\(113\) 12.2467 1.15207 0.576035 0.817425i \(-0.304599\pi\)
0.576035 + 0.817425i \(0.304599\pi\)
\(114\) 9.77765 0.915760
\(115\) −2.60753 −0.243153
\(116\) 1.23712 0.114864
\(117\) −0.609692 −0.0563660
\(118\) −25.7681 −2.37215
\(119\) 1.04545 0.0958363
\(120\) 3.57901 0.326717
\(121\) 16.2194 1.47449
\(122\) −26.0428 −2.35781
\(123\) 1.10557 0.0996860
\(124\) 4.90990 0.440922
\(125\) 8.34617 0.746504
\(126\) −2.46563 −0.219656
\(127\) 7.53521 0.668642 0.334321 0.942459i \(-0.391493\pi\)
0.334321 + 0.942459i \(0.391493\pi\)
\(128\) −10.1817 −0.899947
\(129\) −7.23148 −0.636696
\(130\) −2.86035 −0.250869
\(131\) −3.77922 −0.330192 −0.165096 0.986278i \(-0.552793\pi\)
−0.165096 + 0.986278i \(0.552793\pi\)
\(132\) 6.45434 0.561778
\(133\) 7.44738 0.645770
\(134\) −21.7811 −1.88160
\(135\) 2.60753 0.224420
\(136\) 1.04710 0.0897882
\(137\) −7.73550 −0.660888 −0.330444 0.943826i \(-0.607199\pi\)
−0.330444 + 0.943826i \(0.607199\pi\)
\(138\) 1.79920 0.153158
\(139\) −12.3033 −1.04355 −0.521777 0.853082i \(-0.674731\pi\)
−0.521777 + 0.853082i \(0.674731\pi\)
\(140\) −4.42070 −0.373617
\(141\) 12.9585 1.09131
\(142\) −3.97280 −0.333390
\(143\) 3.18090 0.266000
\(144\) −4.94377 −0.411981
\(145\) −2.60753 −0.216543
\(146\) −6.45723 −0.534404
\(147\) 5.12199 0.422455
\(148\) −4.10005 −0.337022
\(149\) −13.5656 −1.11133 −0.555667 0.831405i \(-0.687537\pi\)
−0.555667 + 0.831405i \(0.687537\pi\)
\(150\) 3.23712 0.264310
\(151\) −7.82160 −0.636513 −0.318256 0.948005i \(-0.603097\pi\)
−0.318256 + 0.948005i \(0.603097\pi\)
\(152\) 7.45914 0.605016
\(153\) 0.762877 0.0616749
\(154\) 12.8637 1.03659
\(155\) −10.3488 −0.831233
\(156\) 0.754264 0.0603894
\(157\) −3.37164 −0.269086 −0.134543 0.990908i \(-0.542957\pi\)
−0.134543 + 0.990908i \(0.542957\pi\)
\(158\) 13.0965 1.04191
\(159\) −4.97097 −0.394224
\(160\) −16.0355 −1.26772
\(161\) 1.37040 0.108003
\(162\) −1.79920 −0.141359
\(163\) 1.25282 0.0981285 0.0490642 0.998796i \(-0.484376\pi\)
0.0490642 + 0.998796i \(0.484376\pi\)
\(164\) −1.36773 −0.106802
\(165\) −13.6040 −1.05907
\(166\) 7.08622 0.549998
\(167\) −16.6491 −1.28834 −0.644171 0.764881i \(-0.722798\pi\)
−0.644171 + 0.764881i \(0.722798\pi\)
\(168\) −1.88097 −0.145120
\(169\) −12.6283 −0.971406
\(170\) 3.57901 0.274498
\(171\) 5.43444 0.415582
\(172\) 8.94623 0.682143
\(173\) −8.21773 −0.624783 −0.312391 0.949954i \(-0.601130\pi\)
−0.312391 + 0.949954i \(0.601130\pi\)
\(174\) 1.79920 0.136397
\(175\) 2.46563 0.186384
\(176\) 25.7927 1.94420
\(177\) −14.3220 −1.07651
\(178\) 3.00525 0.225253
\(179\) −18.3073 −1.36835 −0.684176 0.729317i \(-0.739838\pi\)
−0.684176 + 0.729317i \(0.739838\pi\)
\(180\) −3.22583 −0.240439
\(181\) −17.8131 −1.32403 −0.662017 0.749489i \(-0.730299\pi\)
−0.662017 + 0.749489i \(0.730299\pi\)
\(182\) 1.50328 0.111430
\(183\) −14.4747 −1.07000
\(184\) 1.37257 0.101187
\(185\) 8.64181 0.635358
\(186\) 7.14068 0.523580
\(187\) −3.98010 −0.291053
\(188\) −16.0313 −1.16920
\(189\) −1.37040 −0.0996823
\(190\) 25.4955 1.84964
\(191\) −26.5437 −1.92064 −0.960319 0.278904i \(-0.910029\pi\)
−0.960319 + 0.278904i \(0.910029\pi\)
\(192\) 1.17700 0.0849428
\(193\) 22.4928 1.61907 0.809535 0.587072i \(-0.199720\pi\)
0.809535 + 0.587072i \(0.199720\pi\)
\(194\) 33.3151 2.39189
\(195\) −1.58979 −0.113847
\(196\) −6.33653 −0.452610
\(197\) 8.10815 0.577681 0.288841 0.957377i \(-0.406730\pi\)
0.288841 + 0.957377i \(0.406730\pi\)
\(198\) 9.38682 0.667093
\(199\) −3.13145 −0.221983 −0.110991 0.993821i \(-0.535403\pi\)
−0.110991 + 0.993821i \(0.535403\pi\)
\(200\) 2.46953 0.174622
\(201\) −12.1060 −0.853891
\(202\) 1.92379 0.135357
\(203\) 1.37040 0.0961835
\(204\) −0.943773 −0.0660773
\(205\) 2.88281 0.201344
\(206\) 28.0656 1.95542
\(207\) 1.00000 0.0695048
\(208\) 3.01418 0.208996
\(209\) −28.3527 −1.96119
\(210\) −6.42921 −0.443657
\(211\) −1.93803 −0.133419 −0.0667096 0.997772i \(-0.521250\pi\)
−0.0667096 + 0.997772i \(0.521250\pi\)
\(212\) 6.14970 0.422363
\(213\) −2.20809 −0.151296
\(214\) 14.0650 0.961462
\(215\) −18.8563 −1.28599
\(216\) −1.37257 −0.0933915
\(217\) 5.43887 0.369215
\(218\) 3.34625 0.226637
\(219\) −3.58894 −0.242518
\(220\) 16.8299 1.13467
\(221\) −0.465120 −0.0312873
\(222\) −5.96287 −0.400202
\(223\) 23.5288 1.57560 0.787801 0.615930i \(-0.211219\pi\)
0.787801 + 0.615930i \(0.211219\pi\)
\(224\) 8.42758 0.563091
\(225\) 1.79920 0.119947
\(226\) −22.0342 −1.46570
\(227\) 16.2471 1.07836 0.539179 0.842191i \(-0.318735\pi\)
0.539179 + 0.842191i \(0.318735\pi\)
\(228\) −6.72307 −0.445246
\(229\) −13.5040 −0.892371 −0.446185 0.894941i \(-0.647218\pi\)
−0.446185 + 0.894941i \(0.647218\pi\)
\(230\) 4.69147 0.309346
\(231\) 7.14970 0.470416
\(232\) 1.37257 0.0901135
\(233\) −12.4171 −0.813471 −0.406735 0.913546i \(-0.633333\pi\)
−0.406735 + 0.913546i \(0.633333\pi\)
\(234\) 1.09696 0.0717104
\(235\) 33.7898 2.20420
\(236\) 17.7181 1.15335
\(237\) 7.27909 0.472828
\(238\) −1.88097 −0.121925
\(239\) 6.13547 0.396870 0.198435 0.980114i \(-0.436414\pi\)
0.198435 + 0.980114i \(0.436414\pi\)
\(240\) −12.8910 −0.832112
\(241\) −21.1558 −1.36276 −0.681382 0.731928i \(-0.738621\pi\)
−0.681382 + 0.731928i \(0.738621\pi\)
\(242\) −29.1819 −1.87588
\(243\) −1.00000 −0.0641500
\(244\) 17.9069 1.14637
\(245\) 13.3557 0.853267
\(246\) −1.98914 −0.126823
\(247\) −3.31333 −0.210822
\(248\) 5.44746 0.345914
\(249\) 3.93854 0.249595
\(250\) −15.0164 −0.949723
\(251\) −25.5336 −1.61166 −0.805832 0.592144i \(-0.798282\pi\)
−0.805832 + 0.592144i \(0.798282\pi\)
\(252\) 1.69536 0.106798
\(253\) −5.21722 −0.328004
\(254\) −13.5574 −0.850664
\(255\) 1.98922 0.124570
\(256\) 20.6730 1.29206
\(257\) 24.1304 1.50521 0.752607 0.658470i \(-0.228796\pi\)
0.752607 + 0.658470i \(0.228796\pi\)
\(258\) 13.0109 0.810022
\(259\) −4.54176 −0.282211
\(260\) 1.96676 0.121973
\(261\) 1.00000 0.0618984
\(262\) 6.79957 0.420079
\(263\) 16.0047 0.986892 0.493446 0.869776i \(-0.335737\pi\)
0.493446 + 0.869776i \(0.335737\pi\)
\(264\) 7.16099 0.440729
\(265\) −12.9619 −0.796246
\(266\) −13.3993 −0.821566
\(267\) 1.67033 0.102222
\(268\) 14.9766 0.914842
\(269\) 8.58719 0.523570 0.261785 0.965126i \(-0.415689\pi\)
0.261785 + 0.965126i \(0.415689\pi\)
\(270\) −4.69147 −0.285514
\(271\) −5.11668 −0.310816 −0.155408 0.987850i \(-0.549669\pi\)
−0.155408 + 0.987850i \(0.549669\pi\)
\(272\) −3.77149 −0.228680
\(273\) 0.835524 0.0505682
\(274\) 13.9177 0.840800
\(275\) −9.38682 −0.566047
\(276\) −1.23712 −0.0744661
\(277\) −27.1069 −1.62870 −0.814348 0.580377i \(-0.802905\pi\)
−0.814348 + 0.580377i \(0.802905\pi\)
\(278\) 22.1361 1.32764
\(279\) 3.96881 0.237606
\(280\) −4.90469 −0.293111
\(281\) −27.2727 −1.62695 −0.813477 0.581597i \(-0.802428\pi\)
−0.813477 + 0.581597i \(0.802428\pi\)
\(282\) −23.3150 −1.38839
\(283\) −26.2431 −1.55999 −0.779995 0.625786i \(-0.784778\pi\)
−0.779995 + 0.625786i \(0.784778\pi\)
\(284\) 2.73168 0.162096
\(285\) 14.1704 0.839385
\(286\) −5.72307 −0.338412
\(287\) −1.51508 −0.0894323
\(288\) 6.14970 0.362375
\(289\) −16.4180 −0.965766
\(290\) 4.69147 0.275492
\(291\) 18.5166 1.08546
\(292\) 4.43996 0.259829
\(293\) 24.4461 1.42815 0.714077 0.700067i \(-0.246847\pi\)
0.714077 + 0.700067i \(0.246847\pi\)
\(294\) −9.21549 −0.537458
\(295\) −37.3450 −2.17431
\(296\) −4.54894 −0.264402
\(297\) 5.21722 0.302734
\(298\) 24.4072 1.41387
\(299\) −0.609692 −0.0352594
\(300\) −2.22583 −0.128509
\(301\) 9.91005 0.571206
\(302\) 14.0726 0.809789
\(303\) 1.06925 0.0614267
\(304\) −26.8666 −1.54091
\(305\) −37.7431 −2.16116
\(306\) −1.37257 −0.0784645
\(307\) 21.4171 1.22234 0.611170 0.791500i \(-0.290699\pi\)
0.611170 + 0.791500i \(0.290699\pi\)
\(308\) −8.84506 −0.503994
\(309\) 15.5989 0.887391
\(310\) 18.6195 1.05752
\(311\) −3.00084 −0.170162 −0.0850811 0.996374i \(-0.527115\pi\)
−0.0850811 + 0.996374i \(0.527115\pi\)
\(312\) 0.836844 0.0473770
\(313\) 28.8668 1.63165 0.815825 0.578299i \(-0.196283\pi\)
0.815825 + 0.578299i \(0.196283\pi\)
\(314\) 6.06626 0.342339
\(315\) −3.57337 −0.201336
\(316\) −9.00513 −0.506578
\(317\) 7.14197 0.401133 0.200567 0.979680i \(-0.435722\pi\)
0.200567 + 0.979680i \(0.435722\pi\)
\(318\) 8.94377 0.501542
\(319\) −5.21722 −0.292108
\(320\) 3.06907 0.171566
\(321\) 7.81735 0.436322
\(322\) −2.46563 −0.137404
\(323\) 4.14581 0.230679
\(324\) 1.23712 0.0687291
\(325\) −1.09696 −0.0608483
\(326\) −2.25407 −0.124842
\(327\) 1.85985 0.102850
\(328\) −1.51747 −0.0837884
\(329\) −17.7584 −0.979055
\(330\) 24.4764 1.34738
\(331\) 2.92127 0.160568 0.0802838 0.996772i \(-0.474417\pi\)
0.0802838 + 0.996772i \(0.474417\pi\)
\(332\) −4.87246 −0.267411
\(333\) −3.31418 −0.181616
\(334\) 29.9550 1.63906
\(335\) −31.5667 −1.72467
\(336\) 6.77497 0.369605
\(337\) −18.1590 −0.989184 −0.494592 0.869125i \(-0.664683\pi\)
−0.494592 + 0.869125i \(0.664683\pi\)
\(338\) 22.7208 1.23585
\(339\) −12.2467 −0.665148
\(340\) −2.46091 −0.133462
\(341\) −20.7061 −1.12130
\(342\) −9.77765 −0.528715
\(343\) −16.6120 −0.896965
\(344\) 9.92570 0.535158
\(345\) 2.60753 0.140385
\(346\) 14.7854 0.794865
\(347\) 19.3473 1.03862 0.519310 0.854586i \(-0.326189\pi\)
0.519310 + 0.854586i \(0.326189\pi\)
\(348\) −1.23712 −0.0663168
\(349\) 15.1619 0.811598 0.405799 0.913962i \(-0.366993\pi\)
0.405799 + 0.913962i \(0.366993\pi\)
\(350\) −4.43617 −0.237123
\(351\) 0.609692 0.0325429
\(352\) −32.0843 −1.71010
\(353\) −32.7282 −1.74195 −0.870973 0.491331i \(-0.836511\pi\)
−0.870973 + 0.491331i \(0.836511\pi\)
\(354\) 25.7681 1.36956
\(355\) −5.75766 −0.305585
\(356\) −2.06640 −0.109519
\(357\) −1.04545 −0.0553311
\(358\) 32.9385 1.74085
\(359\) −13.2280 −0.698148 −0.349074 0.937095i \(-0.613504\pi\)
−0.349074 + 0.937095i \(0.613504\pi\)
\(360\) −3.57901 −0.188630
\(361\) 10.5331 0.554375
\(362\) 32.0493 1.68447
\(363\) −16.2194 −0.851297
\(364\) −1.03365 −0.0541778
\(365\) −9.35827 −0.489834
\(366\) 26.0428 1.36128
\(367\) 2.98259 0.155690 0.0778450 0.996965i \(-0.475196\pi\)
0.0778450 + 0.996965i \(0.475196\pi\)
\(368\) −4.94377 −0.257712
\(369\) −1.10557 −0.0575537
\(370\) −15.5483 −0.808320
\(371\) 6.81224 0.353674
\(372\) −4.90990 −0.254567
\(373\) −5.90327 −0.305660 −0.152830 0.988253i \(-0.548839\pi\)
−0.152830 + 0.988253i \(0.548839\pi\)
\(374\) 7.16099 0.370286
\(375\) −8.34617 −0.430995
\(376\) −17.7865 −0.917268
\(377\) −0.609692 −0.0314007
\(378\) 2.46563 0.126818
\(379\) 3.75455 0.192858 0.0964292 0.995340i \(-0.469258\pi\)
0.0964292 + 0.995340i \(0.469258\pi\)
\(380\) −17.5306 −0.899300
\(381\) −7.53521 −0.386041
\(382\) 47.7575 2.44349
\(383\) 7.76452 0.396749 0.198374 0.980126i \(-0.436434\pi\)
0.198374 + 0.980126i \(0.436434\pi\)
\(384\) 10.1817 0.519585
\(385\) 18.6430 0.950138
\(386\) −40.4691 −2.05982
\(387\) 7.23148 0.367597
\(388\) −22.9073 −1.16294
\(389\) 18.8378 0.955116 0.477558 0.878600i \(-0.341522\pi\)
0.477558 + 0.878600i \(0.341522\pi\)
\(390\) 2.86035 0.144839
\(391\) 0.762877 0.0385803
\(392\) −7.03028 −0.355083
\(393\) 3.77922 0.190636
\(394\) −14.5882 −0.734942
\(395\) 18.9804 0.955009
\(396\) −6.45434 −0.324343
\(397\) 8.72995 0.438144 0.219072 0.975709i \(-0.429697\pi\)
0.219072 + 0.975709i \(0.429697\pi\)
\(398\) 5.63411 0.282412
\(399\) −7.44738 −0.372835
\(400\) −8.89484 −0.444742
\(401\) 9.57017 0.477912 0.238956 0.971030i \(-0.423195\pi\)
0.238956 + 0.971030i \(0.423195\pi\)
\(402\) 21.7811 1.08634
\(403\) −2.41975 −0.120536
\(404\) −1.32279 −0.0658113
\(405\) −2.60753 −0.129569
\(406\) −2.46563 −0.122367
\(407\) 17.2908 0.857073
\(408\) −1.04710 −0.0518392
\(409\) −8.16414 −0.403691 −0.201845 0.979417i \(-0.564694\pi\)
−0.201845 + 0.979417i \(0.564694\pi\)
\(410\) −5.18675 −0.256155
\(411\) 7.73550 0.381564
\(412\) −19.2978 −0.950733
\(413\) 19.6269 0.965777
\(414\) −1.79920 −0.0884259
\(415\) 10.2699 0.504127
\(416\) −3.74942 −0.183831
\(417\) 12.3033 0.602496
\(418\) 51.0121 2.49508
\(419\) 20.4469 0.998895 0.499447 0.866344i \(-0.333536\pi\)
0.499447 + 0.866344i \(0.333536\pi\)
\(420\) 4.42070 0.215708
\(421\) 17.2632 0.841356 0.420678 0.907210i \(-0.361792\pi\)
0.420678 + 0.907210i \(0.361792\pi\)
\(422\) 3.48690 0.169740
\(423\) −12.9585 −0.630066
\(424\) 6.82300 0.331354
\(425\) 1.37257 0.0665794
\(426\) 3.97280 0.192483
\(427\) 19.8361 0.959938
\(428\) −9.67102 −0.467467
\(429\) −3.18090 −0.153575
\(430\) 33.9262 1.63607
\(431\) −2.96320 −0.142732 −0.0713661 0.997450i \(-0.522736\pi\)
−0.0713661 + 0.997450i \(0.522736\pi\)
\(432\) 4.94377 0.237857
\(433\) 21.8920 1.05206 0.526032 0.850465i \(-0.323679\pi\)
0.526032 + 0.850465i \(0.323679\pi\)
\(434\) −9.78562 −0.469725
\(435\) 2.60753 0.125021
\(436\) −2.30087 −0.110192
\(437\) 5.43444 0.259964
\(438\) 6.45723 0.308538
\(439\) −29.4328 −1.40475 −0.702375 0.711807i \(-0.747877\pi\)
−0.702375 + 0.711807i \(0.747877\pi\)
\(440\) 18.6725 0.890176
\(441\) −5.12199 −0.243904
\(442\) 0.836844 0.0398046
\(443\) −13.9192 −0.661320 −0.330660 0.943750i \(-0.607271\pi\)
−0.330660 + 0.943750i \(0.607271\pi\)
\(444\) 4.10005 0.194580
\(445\) 4.35542 0.206467
\(446\) −42.3330 −2.00452
\(447\) 13.5656 0.641629
\(448\) −1.61297 −0.0762056
\(449\) 17.3235 0.817544 0.408772 0.912636i \(-0.365957\pi\)
0.408772 + 0.912636i \(0.365957\pi\)
\(450\) −3.23712 −0.152599
\(451\) 5.76801 0.271605
\(452\) 15.1506 0.712627
\(453\) 7.82160 0.367491
\(454\) −29.2318 −1.37192
\(455\) 2.17865 0.102137
\(456\) −7.45914 −0.349306
\(457\) −13.5547 −0.634062 −0.317031 0.948415i \(-0.602686\pi\)
−0.317031 + 0.948415i \(0.602686\pi\)
\(458\) 24.2964 1.13530
\(459\) −0.762877 −0.0356080
\(460\) −3.22583 −0.150405
\(461\) −13.5638 −0.631731 −0.315866 0.948804i \(-0.602295\pi\)
−0.315866 + 0.948804i \(0.602295\pi\)
\(462\) −12.8637 −0.598476
\(463\) −27.0376 −1.25654 −0.628272 0.777994i \(-0.716237\pi\)
−0.628272 + 0.777994i \(0.716237\pi\)
\(464\) −4.94377 −0.229509
\(465\) 10.3488 0.479913
\(466\) 22.3408 1.03492
\(467\) 28.2895 1.30908 0.654540 0.756027i \(-0.272862\pi\)
0.654540 + 0.756027i \(0.272862\pi\)
\(468\) −0.754264 −0.0348659
\(469\) 16.5901 0.766061
\(470\) −60.7946 −2.80424
\(471\) 3.37164 0.155357
\(472\) 19.6579 0.904829
\(473\) −37.7282 −1.73474
\(474\) −13.0965 −0.601544
\(475\) 9.77765 0.448629
\(476\) 1.29335 0.0592806
\(477\) 4.97097 0.227605
\(478\) −11.0389 −0.504909
\(479\) 1.22365 0.0559099 0.0279550 0.999609i \(-0.491100\pi\)
0.0279550 + 0.999609i \(0.491100\pi\)
\(480\) 16.0355 0.731918
\(481\) 2.02063 0.0921326
\(482\) 38.0635 1.73374
\(483\) −1.37040 −0.0623556
\(484\) 20.0654 0.912062
\(485\) 48.2826 2.19240
\(486\) 1.79920 0.0816134
\(487\) −28.8149 −1.30573 −0.652864 0.757475i \(-0.726433\pi\)
−0.652864 + 0.757475i \(0.726433\pi\)
\(488\) 19.8675 0.899358
\(489\) −1.25282 −0.0566545
\(490\) −24.0296 −1.08555
\(491\) −28.8091 −1.30014 −0.650068 0.759876i \(-0.725259\pi\)
−0.650068 + 0.759876i \(0.725259\pi\)
\(492\) 1.36773 0.0616619
\(493\) 0.762877 0.0343582
\(494\) 5.96135 0.268214
\(495\) 13.6040 0.611456
\(496\) −19.6209 −0.881003
\(497\) 3.02598 0.135734
\(498\) −7.08622 −0.317541
\(499\) −8.61789 −0.385790 −0.192895 0.981219i \(-0.561788\pi\)
−0.192895 + 0.981219i \(0.561788\pi\)
\(500\) 10.3252 0.461759
\(501\) 16.6491 0.743825
\(502\) 45.9400 2.05040
\(503\) −42.6447 −1.90143 −0.950717 0.310059i \(-0.899651\pi\)
−0.950717 + 0.310059i \(0.899651\pi\)
\(504\) 1.88097 0.0837853
\(505\) 2.78809 0.124068
\(506\) 9.38682 0.417295
\(507\) 12.6283 0.560841
\(508\) 9.32198 0.413596
\(509\) 15.6150 0.692123 0.346061 0.938212i \(-0.387519\pi\)
0.346061 + 0.938212i \(0.387519\pi\)
\(510\) −3.57901 −0.158481
\(511\) 4.91830 0.217573
\(512\) −16.8314 −0.743849
\(513\) −5.43444 −0.239936
\(514\) −43.4154 −1.91497
\(515\) 40.6746 1.79234
\(516\) −8.94623 −0.393836
\(517\) 67.6076 2.97338
\(518\) 8.17154 0.359037
\(519\) 8.21773 0.360718
\(520\) 2.18209 0.0956911
\(521\) 26.8010 1.17417 0.587086 0.809525i \(-0.300275\pi\)
0.587086 + 0.809525i \(0.300275\pi\)
\(522\) −1.79920 −0.0787489
\(523\) −39.2125 −1.71464 −0.857321 0.514783i \(-0.827873\pi\)
−0.857321 + 0.514783i \(0.827873\pi\)
\(524\) −4.67536 −0.204244
\(525\) −2.46563 −0.107609
\(526\) −28.7957 −1.25555
\(527\) 3.02771 0.131889
\(528\) −25.7927 −1.12249
\(529\) 1.00000 0.0434783
\(530\) 23.3211 1.01301
\(531\) 14.3220 0.621521
\(532\) 9.21333 0.399448
\(533\) 0.674058 0.0291967
\(534\) −3.00525 −0.130050
\(535\) 20.3840 0.881275
\(536\) 16.6163 0.717716
\(537\) 18.3073 0.790018
\(538\) −15.4501 −0.666100
\(539\) 26.7226 1.15102
\(540\) 3.22583 0.138818
\(541\) −1.11216 −0.0478156 −0.0239078 0.999714i \(-0.507611\pi\)
−0.0239078 + 0.999714i \(0.507611\pi\)
\(542\) 9.20593 0.395429
\(543\) 17.8131 0.764432
\(544\) 4.69147 0.201145
\(545\) 4.84962 0.207735
\(546\) −1.50328 −0.0643343
\(547\) 22.3385 0.955125 0.477563 0.878598i \(-0.341520\pi\)
0.477563 + 0.878598i \(0.341520\pi\)
\(548\) −9.56976 −0.408800
\(549\) 14.4747 0.617763
\(550\) 16.8888 0.720140
\(551\) 5.43444 0.231515
\(552\) −1.37257 −0.0584204
\(553\) −9.97530 −0.424193
\(554\) 48.7708 2.07207
\(555\) −8.64181 −0.366824
\(556\) −15.2207 −0.645502
\(557\) 4.70470 0.199344 0.0996722 0.995020i \(-0.468221\pi\)
0.0996722 + 0.995020i \(0.468221\pi\)
\(558\) −7.14068 −0.302289
\(559\) −4.40897 −0.186480
\(560\) 17.6659 0.746521
\(561\) 3.98010 0.168040
\(562\) 49.0691 2.06985
\(563\) 17.7092 0.746355 0.373177 0.927760i \(-0.378268\pi\)
0.373177 + 0.927760i \(0.378268\pi\)
\(564\) 16.0313 0.675040
\(565\) −31.9336 −1.34345
\(566\) 47.2166 1.98466
\(567\) 1.37040 0.0575516
\(568\) 3.03076 0.127168
\(569\) 24.6412 1.03301 0.516506 0.856284i \(-0.327232\pi\)
0.516506 + 0.856284i \(0.327232\pi\)
\(570\) −25.4955 −1.06789
\(571\) −1.47211 −0.0616057 −0.0308029 0.999525i \(-0.509806\pi\)
−0.0308029 + 0.999525i \(0.509806\pi\)
\(572\) 3.93516 0.164537
\(573\) 26.5437 1.10888
\(574\) 2.72593 0.113778
\(575\) 1.79920 0.0750319
\(576\) −1.17700 −0.0490418
\(577\) −32.6363 −1.35867 −0.679333 0.733830i \(-0.737731\pi\)
−0.679333 + 0.733830i \(0.737731\pi\)
\(578\) 29.5393 1.22867
\(579\) −22.4928 −0.934770
\(580\) −3.22583 −0.133945
\(581\) −5.39739 −0.223922
\(582\) −33.3151 −1.38096
\(583\) −25.9346 −1.07410
\(584\) 4.92607 0.203842
\(585\) 1.58979 0.0657297
\(586\) −43.9834 −1.81694
\(587\) 4.74693 0.195927 0.0979633 0.995190i \(-0.468767\pi\)
0.0979633 + 0.995190i \(0.468767\pi\)
\(588\) 6.33653 0.261314
\(589\) 21.5682 0.888704
\(590\) 67.1911 2.76621
\(591\) −8.10815 −0.333525
\(592\) 16.3845 0.673400
\(593\) −29.1757 −1.19810 −0.599051 0.800711i \(-0.704455\pi\)
−0.599051 + 0.800711i \(0.704455\pi\)
\(594\) −9.38682 −0.385146
\(595\) −2.72604 −0.111757
\(596\) −16.7823 −0.687428
\(597\) 3.13145 0.128162
\(598\) 1.09696 0.0448579
\(599\) −2.07025 −0.0845883 −0.0422942 0.999105i \(-0.513467\pi\)
−0.0422942 + 0.999105i \(0.513467\pi\)
\(600\) −2.46953 −0.100818
\(601\) −16.5876 −0.676622 −0.338311 0.941034i \(-0.609856\pi\)
−0.338311 + 0.941034i \(0.609856\pi\)
\(602\) −17.8302 −0.726703
\(603\) 12.1060 0.492994
\(604\) −9.67628 −0.393722
\(605\) −42.2925 −1.71943
\(606\) −1.92379 −0.0781486
\(607\) 1.22240 0.0496156 0.0248078 0.999692i \(-0.492103\pi\)
0.0248078 + 0.999692i \(0.492103\pi\)
\(608\) 33.4202 1.35537
\(609\) −1.37040 −0.0555316
\(610\) 67.9074 2.74949
\(611\) 7.90072 0.319629
\(612\) 0.943773 0.0381497
\(613\) 14.7569 0.596025 0.298013 0.954562i \(-0.403676\pi\)
0.298013 + 0.954562i \(0.403676\pi\)
\(614\) −38.5337 −1.55509
\(615\) −2.88281 −0.116246
\(616\) −9.81346 −0.395395
\(617\) 40.4655 1.62908 0.814540 0.580107i \(-0.196989\pi\)
0.814540 + 0.580107i \(0.196989\pi\)
\(618\) −28.0656 −1.12896
\(619\) −11.2915 −0.453842 −0.226921 0.973913i \(-0.572866\pi\)
−0.226921 + 0.973913i \(0.572866\pi\)
\(620\) −12.8027 −0.514169
\(621\) −1.00000 −0.0401286
\(622\) 5.39912 0.216485
\(623\) −2.28902 −0.0917077
\(624\) −3.01418 −0.120664
\(625\) −30.7589 −1.23036
\(626\) −51.9372 −2.07583
\(627\) 28.3527 1.13230
\(628\) −4.17113 −0.166446
\(629\) −2.52831 −0.100810
\(630\) 6.42921 0.256146
\(631\) 12.7883 0.509095 0.254547 0.967060i \(-0.418074\pi\)
0.254547 + 0.967060i \(0.418074\pi\)
\(632\) −9.99105 −0.397423
\(633\) 1.93803 0.0770296
\(634\) −12.8498 −0.510332
\(635\) −19.6483 −0.779718
\(636\) −6.14970 −0.243852
\(637\) 3.12284 0.123731
\(638\) 9.38682 0.371628
\(639\) 2.20809 0.0873508
\(640\) 26.5492 1.04945
\(641\) 20.0048 0.790144 0.395072 0.918650i \(-0.370720\pi\)
0.395072 + 0.918650i \(0.370720\pi\)
\(642\) −14.0650 −0.555101
\(643\) −19.5252 −0.769998 −0.384999 0.922917i \(-0.625798\pi\)
−0.384999 + 0.922917i \(0.625798\pi\)
\(644\) 1.69536 0.0668065
\(645\) 18.8563 0.742465
\(646\) −7.45914 −0.293476
\(647\) 5.45125 0.214311 0.107155 0.994242i \(-0.465826\pi\)
0.107155 + 0.994242i \(0.465826\pi\)
\(648\) 1.37257 0.0539196
\(649\) −74.7209 −2.93305
\(650\) 1.97365 0.0774128
\(651\) −5.43887 −0.213166
\(652\) 1.54989 0.0606985
\(653\) −29.8135 −1.16669 −0.583346 0.812224i \(-0.698257\pi\)
−0.583346 + 0.812224i \(0.698257\pi\)
\(654\) −3.34625 −0.130849
\(655\) 9.85442 0.385044
\(656\) 5.46569 0.213399
\(657\) 3.58894 0.140018
\(658\) 31.9510 1.24558
\(659\) −5.57380 −0.217124 −0.108562 0.994090i \(-0.534625\pi\)
−0.108562 + 0.994090i \(0.534625\pi\)
\(660\) −16.8299 −0.655102
\(661\) −22.8573 −0.889046 −0.444523 0.895768i \(-0.646627\pi\)
−0.444523 + 0.895768i \(0.646627\pi\)
\(662\) −5.25595 −0.204278
\(663\) 0.465120 0.0180638
\(664\) −5.40592 −0.209790
\(665\) −19.4193 −0.753046
\(666\) 5.96287 0.231057
\(667\) 1.00000 0.0387202
\(668\) −20.5969 −0.796919
\(669\) −23.5288 −0.909674
\(670\) 56.7949 2.19418
\(671\) −75.5175 −2.91532
\(672\) −8.42758 −0.325101
\(673\) 27.5285 1.06115 0.530573 0.847639i \(-0.321977\pi\)
0.530573 + 0.847639i \(0.321977\pi\)
\(674\) 32.6717 1.25847
\(675\) −1.79920 −0.0692513
\(676\) −15.6227 −0.600874
\(677\) −20.7318 −0.796789 −0.398394 0.917214i \(-0.630432\pi\)
−0.398394 + 0.917214i \(0.630432\pi\)
\(678\) 22.0342 0.846220
\(679\) −25.3753 −0.973813
\(680\) −2.73034 −0.104704
\(681\) −16.2471 −0.622590
\(682\) 37.2545 1.42655
\(683\) 8.54864 0.327105 0.163552 0.986535i \(-0.447705\pi\)
0.163552 + 0.986535i \(0.447705\pi\)
\(684\) 6.72307 0.257063
\(685\) 20.1705 0.770676
\(686\) 29.8884 1.14114
\(687\) 13.5040 0.515210
\(688\) −35.7508 −1.36299
\(689\) −3.03076 −0.115463
\(690\) −4.69147 −0.178601
\(691\) −46.0282 −1.75100 −0.875498 0.483223i \(-0.839466\pi\)
−0.875498 + 0.483223i \(0.839466\pi\)
\(692\) −10.1663 −0.386467
\(693\) −7.14970 −0.271595
\(694\) −34.8098 −1.32136
\(695\) 32.0813 1.21691
\(696\) −1.37257 −0.0520271
\(697\) −0.843415 −0.0319466
\(698\) −27.2793 −1.03254
\(699\) 12.4171 0.469658
\(700\) 3.05029 0.115290
\(701\) 20.3401 0.768234 0.384117 0.923284i \(-0.374506\pi\)
0.384117 + 0.923284i \(0.374506\pi\)
\(702\) −1.09696 −0.0414020
\(703\) −18.0107 −0.679286
\(704\) 6.14068 0.231436
\(705\) −33.7898 −1.27260
\(706\) 58.8846 2.21615
\(707\) −1.46530 −0.0551083
\(708\) −17.7181 −0.665885
\(709\) −31.7725 −1.19324 −0.596620 0.802524i \(-0.703490\pi\)
−0.596620 + 0.802524i \(0.703490\pi\)
\(710\) 10.3592 0.388774
\(711\) −7.27909 −0.272987
\(712\) −2.29264 −0.0859202
\(713\) 3.96881 0.148633
\(714\) 1.88097 0.0703937
\(715\) −8.29427 −0.310188
\(716\) −22.6484 −0.846410
\(717\) −6.13547 −0.229133
\(718\) 23.7998 0.888202
\(719\) 2.96699 0.110650 0.0553251 0.998468i \(-0.482380\pi\)
0.0553251 + 0.998468i \(0.482380\pi\)
\(720\) 12.8910 0.480420
\(721\) −21.3768 −0.796115
\(722\) −18.9512 −0.705291
\(723\) 21.1558 0.786792
\(724\) −22.0370 −0.818997
\(725\) 1.79920 0.0668206
\(726\) 29.1819 1.08304
\(727\) 52.8858 1.96143 0.980713 0.195452i \(-0.0626172\pi\)
0.980713 + 0.195452i \(0.0626172\pi\)
\(728\) −1.14681 −0.0425038
\(729\) 1.00000 0.0370370
\(730\) 16.8374 0.623180
\(731\) 5.51673 0.204043
\(732\) −17.9069 −0.661859
\(733\) 49.9915 1.84648 0.923239 0.384227i \(-0.125532\pi\)
0.923239 + 0.384227i \(0.125532\pi\)
\(734\) −5.36628 −0.198073
\(735\) −13.3557 −0.492634
\(736\) 6.14970 0.226681
\(737\) −63.1597 −2.32652
\(738\) 1.98914 0.0732214
\(739\) −4.91398 −0.180764 −0.0903819 0.995907i \(-0.528809\pi\)
−0.0903819 + 0.995907i \(0.528809\pi\)
\(740\) 10.6910 0.393008
\(741\) 3.31333 0.121718
\(742\) −12.2566 −0.449953
\(743\) 0.172797 0.00633930 0.00316965 0.999995i \(-0.498991\pi\)
0.00316965 + 0.999995i \(0.498991\pi\)
\(744\) −5.44746 −0.199714
\(745\) 35.3726 1.29595
\(746\) 10.6212 0.388869
\(747\) −3.93854 −0.144104
\(748\) −4.92387 −0.180034
\(749\) −10.7129 −0.391442
\(750\) 15.0164 0.548323
\(751\) −20.1407 −0.734944 −0.367472 0.930035i \(-0.619777\pi\)
−0.367472 + 0.930035i \(0.619777\pi\)
\(752\) 64.0641 2.33618
\(753\) 25.5336 0.930495
\(754\) 1.09696 0.0399488
\(755\) 20.3950 0.742251
\(756\) −1.69536 −0.0616596
\(757\) 51.7908 1.88237 0.941184 0.337895i \(-0.109715\pi\)
0.941184 + 0.337895i \(0.109715\pi\)
\(758\) −6.75519 −0.245360
\(759\) 5.21722 0.189373
\(760\) −19.4499 −0.705523
\(761\) 0.526341 0.0190798 0.00953992 0.999954i \(-0.496963\pi\)
0.00953992 + 0.999954i \(0.496963\pi\)
\(762\) 13.5574 0.491131
\(763\) −2.54875 −0.0922711
\(764\) −32.8379 −1.18803
\(765\) −1.98922 −0.0719205
\(766\) −13.9699 −0.504754
\(767\) −8.73200 −0.315294
\(768\) −20.6730 −0.745973
\(769\) −31.9202 −1.15107 −0.575536 0.817777i \(-0.695207\pi\)
−0.575536 + 0.817777i \(0.695207\pi\)
\(770\) −33.5426 −1.20879
\(771\) −24.1304 −0.869035
\(772\) 27.8264 1.00149
\(773\) 19.6213 0.705731 0.352865 0.935674i \(-0.385207\pi\)
0.352865 + 0.935674i \(0.385207\pi\)
\(774\) −13.0109 −0.467666
\(775\) 7.14068 0.256501
\(776\) −25.4153 −0.912357
\(777\) 4.54176 0.162935
\(778\) −33.8930 −1.21512
\(779\) −6.00816 −0.215265
\(780\) −1.96676 −0.0704214
\(781\) −11.5201 −0.412222
\(782\) −1.37257 −0.0490830
\(783\) −1.00000 −0.0357371
\(784\) 25.3220 0.904356
\(785\) 8.79165 0.313787
\(786\) −6.79957 −0.242533
\(787\) −9.96730 −0.355296 −0.177648 0.984094i \(-0.556849\pi\)
−0.177648 + 0.984094i \(0.556849\pi\)
\(788\) 10.0308 0.357332
\(789\) −16.0047 −0.569782
\(790\) −34.1496 −1.21499
\(791\) 16.7829 0.596731
\(792\) −7.16099 −0.254455
\(793\) −8.82508 −0.313388
\(794\) −15.7069 −0.557418
\(795\) 12.9619 0.459713
\(796\) −3.87399 −0.137310
\(797\) −6.13728 −0.217394 −0.108697 0.994075i \(-0.534668\pi\)
−0.108697 + 0.994075i \(0.534668\pi\)
\(798\) 13.3993 0.474331
\(799\) −9.88577 −0.349733
\(800\) 11.0645 0.391191
\(801\) −1.67033 −0.0590181
\(802\) −17.2187 −0.608012
\(803\) −18.7243 −0.660766
\(804\) −14.9766 −0.528184
\(805\) −3.57337 −0.125945
\(806\) 4.35361 0.153349
\(807\) −8.58719 −0.302283
\(808\) −1.46762 −0.0516305
\(809\) 28.6740 1.00812 0.504062 0.863668i \(-0.331838\pi\)
0.504062 + 0.863668i \(0.331838\pi\)
\(810\) 4.69147 0.164841
\(811\) 10.4813 0.368048 0.184024 0.982922i \(-0.441088\pi\)
0.184024 + 0.982922i \(0.441088\pi\)
\(812\) 1.69536 0.0594954
\(813\) 5.11668 0.179450
\(814\) −31.1096 −1.09039
\(815\) −3.26676 −0.114430
\(816\) 3.77149 0.132029
\(817\) 39.2990 1.37490
\(818\) 14.6889 0.513586
\(819\) −0.835524 −0.0291956
\(820\) 3.56639 0.124544
\(821\) −21.2833 −0.742793 −0.371397 0.928474i \(-0.621121\pi\)
−0.371397 + 0.928474i \(0.621121\pi\)
\(822\) −13.9177 −0.485436
\(823\) −49.6720 −1.73146 −0.865729 0.500513i \(-0.833145\pi\)
−0.865729 + 0.500513i \(0.833145\pi\)
\(824\) −21.4106 −0.745873
\(825\) 9.38682 0.326807
\(826\) −35.3128 −1.22869
\(827\) −34.0143 −1.18279 −0.591397 0.806381i \(-0.701423\pi\)
−0.591397 + 0.806381i \(0.701423\pi\)
\(828\) 1.23712 0.0429930
\(829\) 46.4099 1.61188 0.805942 0.591995i \(-0.201660\pi\)
0.805942 + 0.591995i \(0.201660\pi\)
\(830\) −18.4775 −0.641364
\(831\) 27.1069 0.940328
\(832\) 0.717608 0.0248786
\(833\) −3.90745 −0.135385
\(834\) −22.1361 −0.766512
\(835\) 43.4129 1.50236
\(836\) −35.0757 −1.21312
\(837\) −3.96881 −0.137182
\(838\) −36.7880 −1.27082
\(839\) 23.8343 0.822851 0.411426 0.911443i \(-0.365031\pi\)
0.411426 + 0.911443i \(0.365031\pi\)
\(840\) 4.90469 0.169228
\(841\) 1.00000 0.0344828
\(842\) −31.0599 −1.07040
\(843\) 27.2727 0.939322
\(844\) −2.39758 −0.0825280
\(845\) 32.9286 1.13278
\(846\) 23.3150 0.801587
\(847\) 22.2271 0.763733
\(848\) −24.5753 −0.843921
\(849\) 26.2431 0.900661
\(850\) −2.46953 −0.0847041
\(851\) −3.31418 −0.113609
\(852\) −2.73168 −0.0935859
\(853\) −44.3491 −1.51848 −0.759242 0.650809i \(-0.774430\pi\)
−0.759242 + 0.650809i \(0.774430\pi\)
\(854\) −35.6892 −1.22126
\(855\) −14.1704 −0.484619
\(856\) −10.7299 −0.366739
\(857\) −45.1503 −1.54231 −0.771153 0.636650i \(-0.780319\pi\)
−0.771153 + 0.636650i \(0.780319\pi\)
\(858\) 5.72307 0.195382
\(859\) −14.2349 −0.485687 −0.242844 0.970065i \(-0.578080\pi\)
−0.242844 + 0.970065i \(0.578080\pi\)
\(860\) −23.3275 −0.795462
\(861\) 1.51508 0.0516338
\(862\) 5.33139 0.181588
\(863\) −19.9095 −0.677727 −0.338864 0.940835i \(-0.610043\pi\)
−0.338864 + 0.940835i \(0.610043\pi\)
\(864\) −6.14970 −0.209217
\(865\) 21.4280 0.728573
\(866\) −39.3882 −1.33846
\(867\) 16.4180 0.557585
\(868\) 6.72855 0.228382
\(869\) 37.9766 1.28827
\(870\) −4.69147 −0.159056
\(871\) −7.38093 −0.250093
\(872\) −2.55278 −0.0864480
\(873\) −18.5166 −0.626693
\(874\) −9.77765 −0.330734
\(875\) 11.4376 0.386663
\(876\) −4.43996 −0.150013
\(877\) −13.7071 −0.462857 −0.231429 0.972852i \(-0.574340\pi\)
−0.231429 + 0.972852i \(0.574340\pi\)
\(878\) 52.9555 1.78716
\(879\) −24.4461 −0.824545
\(880\) −67.2553 −2.26718
\(881\) −12.2371 −0.412278 −0.206139 0.978523i \(-0.566090\pi\)
−0.206139 + 0.978523i \(0.566090\pi\)
\(882\) 9.21549 0.310302
\(883\) −42.7294 −1.43796 −0.718979 0.695032i \(-0.755390\pi\)
−0.718979 + 0.695032i \(0.755390\pi\)
\(884\) −0.575410 −0.0193531
\(885\) 37.3450 1.25534
\(886\) 25.0434 0.841349
\(887\) −42.5741 −1.42950 −0.714750 0.699380i \(-0.753459\pi\)
−0.714750 + 0.699380i \(0.753459\pi\)
\(888\) 4.54894 0.152652
\(889\) 10.3263 0.346333
\(890\) −7.83628 −0.262673
\(891\) −5.21722 −0.174783
\(892\) 29.1080 0.974607
\(893\) −70.4224 −2.35660
\(894\) −24.4072 −0.816297
\(895\) 47.7368 1.59566
\(896\) −13.9531 −0.466140
\(897\) 0.609692 0.0203570
\(898\) −31.1684 −1.04010
\(899\) 3.96881 0.132367
\(900\) 2.22583 0.0741944
\(901\) 3.79224 0.126338
\(902\) −10.3778 −0.345543
\(903\) −9.91005 −0.329786
\(904\) 16.8094 0.559073
\(905\) 46.4481 1.54399
\(906\) −14.0726 −0.467532
\(907\) −41.7581 −1.38656 −0.693278 0.720671i \(-0.743834\pi\)
−0.693278 + 0.720671i \(0.743834\pi\)
\(908\) 20.0996 0.667030
\(909\) −1.06925 −0.0354647
\(910\) −3.91983 −0.129941
\(911\) 15.8964 0.526671 0.263335 0.964704i \(-0.415177\pi\)
0.263335 + 0.964704i \(0.415177\pi\)
\(912\) 26.8666 0.889643
\(913\) 20.5482 0.680047
\(914\) 24.3876 0.806671
\(915\) 37.7431 1.24775
\(916\) −16.7061 −0.551986
\(917\) −5.17906 −0.171028
\(918\) 1.37257 0.0453015
\(919\) −15.4175 −0.508576 −0.254288 0.967129i \(-0.581841\pi\)
−0.254288 + 0.967129i \(0.581841\pi\)
\(920\) −3.57901 −0.117996
\(921\) −21.4171 −0.705718
\(922\) 24.4041 0.803706
\(923\) −1.34626 −0.0443126
\(924\) 8.84506 0.290981
\(925\) −5.96287 −0.196058
\(926\) 48.6461 1.59861
\(927\) −15.5989 −0.512336
\(928\) 6.14970 0.201874
\(929\) −9.48569 −0.311215 −0.155608 0.987819i \(-0.549734\pi\)
−0.155608 + 0.987819i \(0.549734\pi\)
\(930\) −18.6195 −0.610558
\(931\) −27.8351 −0.912260
\(932\) −15.3615 −0.503182
\(933\) 3.00084 0.0982432
\(934\) −50.8984 −1.66545
\(935\) 10.3782 0.339404
\(936\) −0.836844 −0.0273531
\(937\) 42.0624 1.37412 0.687060 0.726601i \(-0.258901\pi\)
0.687060 + 0.726601i \(0.258901\pi\)
\(938\) −29.8490 −0.974603
\(939\) −28.8668 −0.942034
\(940\) 41.8021 1.36343
\(941\) 9.80889 0.319761 0.159880 0.987136i \(-0.448889\pi\)
0.159880 + 0.987136i \(0.448889\pi\)
\(942\) −6.06626 −0.197649
\(943\) −1.10557 −0.0360023
\(944\) −70.8046 −2.30449
\(945\) 3.57337 0.116242
\(946\) 67.8806 2.20699
\(947\) 26.0436 0.846303 0.423152 0.906059i \(-0.360924\pi\)
0.423152 + 0.906059i \(0.360924\pi\)
\(948\) 9.00513 0.292473
\(949\) −2.18815 −0.0710303
\(950\) −17.5919 −0.570758
\(951\) −7.14197 −0.231594
\(952\) 1.43495 0.0465071
\(953\) 1.85061 0.0599470 0.0299735 0.999551i \(-0.490458\pi\)
0.0299735 + 0.999551i \(0.490458\pi\)
\(954\) −8.94377 −0.289565
\(955\) 69.2135 2.23970
\(956\) 7.59033 0.245489
\(957\) 5.21722 0.168649
\(958\) −2.20159 −0.0711301
\(959\) −10.6008 −0.342316
\(960\) −3.06907 −0.0990537
\(961\) −15.2486 −0.491890
\(962\) −3.63551 −0.117214
\(963\) −7.81735 −0.251911
\(964\) −26.1723 −0.842953
\(965\) −58.6507 −1.88803
\(966\) 2.46563 0.0793304
\(967\) −54.5680 −1.75479 −0.877395 0.479769i \(-0.840721\pi\)
−0.877395 + 0.479769i \(0.840721\pi\)
\(968\) 22.2622 0.715535
\(969\) −4.14581 −0.133183
\(970\) −86.8701 −2.78923
\(971\) −18.6434 −0.598295 −0.299148 0.954207i \(-0.596702\pi\)
−0.299148 + 0.954207i \(0.596702\pi\)
\(972\) −1.23712 −0.0396807
\(973\) −16.8605 −0.540524
\(974\) 51.8438 1.66118
\(975\) 1.09696 0.0351308
\(976\) −71.5594 −2.29056
\(977\) −3.74771 −0.119900 −0.0599499 0.998201i \(-0.519094\pi\)
−0.0599499 + 0.998201i \(0.519094\pi\)
\(978\) 2.25407 0.0720774
\(979\) 8.71446 0.278515
\(980\) 16.5227 0.527798
\(981\) −1.85985 −0.0593806
\(982\) 51.8333 1.65407
\(983\) 56.5888 1.80490 0.902451 0.430792i \(-0.141766\pi\)
0.902451 + 0.430792i \(0.141766\pi\)
\(984\) 1.51747 0.0483752
\(985\) −21.1422 −0.673647
\(986\) −1.37257 −0.0437115
\(987\) 17.7584 0.565258
\(988\) −4.09900 −0.130407
\(989\) 7.23148 0.229948
\(990\) −24.4764 −0.777911
\(991\) −37.9225 −1.20465 −0.602324 0.798251i \(-0.705759\pi\)
−0.602324 + 0.798251i \(0.705759\pi\)
\(992\) 24.4070 0.774922
\(993\) −2.92127 −0.0927037
\(994\) −5.44435 −0.172684
\(995\) 8.16534 0.258859
\(996\) 4.87246 0.154390
\(997\) −44.4838 −1.40882 −0.704408 0.709795i \(-0.748788\pi\)
−0.704408 + 0.709795i \(0.748788\pi\)
\(998\) 15.5053 0.490812
\(999\) 3.31418 0.104856
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.h.1.2 5
3.2 odd 2 6003.2.a.h.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.h.1.2 5 1.1 even 1 trivial
6003.2.a.h.1.4 5 3.2 odd 2