Properties

Label 3549.2.a.j.1.2
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3549,2,Mod(1,3549)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, 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("3549.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.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: \(28.3389076774\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.445042 q^{2} +1.00000 q^{3} -1.80194 q^{4} +1.00000 q^{5} -0.445042 q^{6} -1.00000 q^{7} +1.69202 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.445042 q^{2} +1.00000 q^{3} -1.80194 q^{4} +1.00000 q^{5} -0.445042 q^{6} -1.00000 q^{7} +1.69202 q^{8} +1.00000 q^{9} -0.445042 q^{10} +0.445042 q^{11} -1.80194 q^{12} +0.445042 q^{14} +1.00000 q^{15} +2.85086 q^{16} -0.753020 q^{17} -0.445042 q^{18} -3.40581 q^{19} -1.80194 q^{20} -1.00000 q^{21} -0.198062 q^{22} -2.64310 q^{23} +1.69202 q^{24} -4.00000 q^{25} +1.00000 q^{27} +1.80194 q^{28} -2.13706 q^{29} -0.445042 q^{30} +1.24698 q^{31} -4.65279 q^{32} +0.445042 q^{33} +0.335126 q^{34} -1.00000 q^{35} -1.80194 q^{36} +8.20775 q^{37} +1.51573 q^{38} +1.69202 q^{40} +1.51573 q^{41} +0.445042 q^{42} -8.00000 q^{43} -0.801938 q^{44} +1.00000 q^{45} +1.17629 q^{46} -8.07069 q^{47} +2.85086 q^{48} +1.00000 q^{49} +1.78017 q^{50} -0.753020 q^{51} +4.16421 q^{53} -0.445042 q^{54} +0.445042 q^{55} -1.69202 q^{56} -3.40581 q^{57} +0.951083 q^{58} -10.3840 q^{59} -1.80194 q^{60} -2.14914 q^{61} -0.554958 q^{62} -1.00000 q^{63} -3.63102 q^{64} -0.198062 q^{66} +0.740939 q^{67} +1.35690 q^{68} -2.64310 q^{69} +0.445042 q^{70} +1.66487 q^{71} +1.69202 q^{72} +5.74094 q^{73} -3.65279 q^{74} -4.00000 q^{75} +6.13706 q^{76} -0.445042 q^{77} +0.719169 q^{79} +2.85086 q^{80} +1.00000 q^{81} -0.674563 q^{82} -0.131687 q^{83} +1.80194 q^{84} -0.753020 q^{85} +3.56033 q^{86} -2.13706 q^{87} +0.753020 q^{88} +15.5918 q^{89} -0.445042 q^{90} +4.76271 q^{92} +1.24698 q^{93} +3.59179 q^{94} -3.40581 q^{95} -4.65279 q^{96} +10.8726 q^{97} -0.445042 q^{98} +0.445042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{3} - q^{4} + 3 q^{5} - q^{6} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 3 q^{3} - q^{4} + 3 q^{5} - q^{6} - 3 q^{7} + 3 q^{9} - q^{10} + q^{11} - q^{12} + q^{14} + 3 q^{15} - 5 q^{16} - 7 q^{17} - q^{18} + 3 q^{19} - q^{20} - 3 q^{21} - 5 q^{22} - 12 q^{23} - 12 q^{25} + 3 q^{27} + q^{28} - q^{29} - q^{30} - q^{31} + 4 q^{32} + q^{33} - 3 q^{35} - q^{36} + 7 q^{37} - 8 q^{38} - 8 q^{41} + q^{42} - 24 q^{43} + 2 q^{44} + 3 q^{45} + 11 q^{46} - 12 q^{47} - 5 q^{48} + 3 q^{49} + 4 q^{50} - 7 q^{51} + q^{53} - q^{54} + q^{55} + 3 q^{57} + 12 q^{58} - 21 q^{59} - q^{60} - 20 q^{61} - 2 q^{62} - 3 q^{63} + 4 q^{64} - 5 q^{66} - 12 q^{67} - 12 q^{69} + q^{70} + 6 q^{71} + 3 q^{73} + 7 q^{74} - 12 q^{75} + 13 q^{76} - q^{77} - 9 q^{79} - 5 q^{80} + 3 q^{81} + 19 q^{82} + 2 q^{83} + q^{84} - 7 q^{85} + 8 q^{86} - q^{87} + 7 q^{88} + 19 q^{89} - q^{90} - 3 q^{92} - q^{93} - 17 q^{94} + 3 q^{95} + 4 q^{96} + 16 q^{97} - q^{98} + q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.445042 −0.314692 −0.157346 0.987544i \(-0.550294\pi\)
−0.157346 + 0.987544i \(0.550294\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.80194 −0.900969
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −0.445042 −0.181688
\(7\) −1.00000 −0.377964
\(8\) 1.69202 0.598220
\(9\) 1.00000 0.333333
\(10\) −0.445042 −0.140735
\(11\) 0.445042 0.134185 0.0670926 0.997747i \(-0.478628\pi\)
0.0670926 + 0.997747i \(0.478628\pi\)
\(12\) −1.80194 −0.520175
\(13\) 0 0
\(14\) 0.445042 0.118942
\(15\) 1.00000 0.258199
\(16\) 2.85086 0.712714
\(17\) −0.753020 −0.182634 −0.0913171 0.995822i \(-0.529108\pi\)
−0.0913171 + 0.995822i \(0.529108\pi\)
\(18\) −0.445042 −0.104897
\(19\) −3.40581 −0.781347 −0.390674 0.920529i \(-0.627758\pi\)
−0.390674 + 0.920529i \(0.627758\pi\)
\(20\) −1.80194 −0.402926
\(21\) −1.00000 −0.218218
\(22\) −0.198062 −0.0422270
\(23\) −2.64310 −0.551125 −0.275563 0.961283i \(-0.588864\pi\)
−0.275563 + 0.961283i \(0.588864\pi\)
\(24\) 1.69202 0.345382
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.80194 0.340534
\(29\) −2.13706 −0.396843 −0.198421 0.980117i \(-0.563581\pi\)
−0.198421 + 0.980117i \(0.563581\pi\)
\(30\) −0.445042 −0.0812532
\(31\) 1.24698 0.223964 0.111982 0.993710i \(-0.464280\pi\)
0.111982 + 0.993710i \(0.464280\pi\)
\(32\) −4.65279 −0.822505
\(33\) 0.445042 0.0774718
\(34\) 0.335126 0.0574736
\(35\) −1.00000 −0.169031
\(36\) −1.80194 −0.300323
\(37\) 8.20775 1.34935 0.674673 0.738117i \(-0.264285\pi\)
0.674673 + 0.738117i \(0.264285\pi\)
\(38\) 1.51573 0.245884
\(39\) 0 0
\(40\) 1.69202 0.267532
\(41\) 1.51573 0.236717 0.118359 0.992971i \(-0.462237\pi\)
0.118359 + 0.992971i \(0.462237\pi\)
\(42\) 0.445042 0.0686715
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −0.801938 −0.120897
\(45\) 1.00000 0.149071
\(46\) 1.17629 0.173435
\(47\) −8.07069 −1.17723 −0.588615 0.808413i \(-0.700327\pi\)
−0.588615 + 0.808413i \(0.700327\pi\)
\(48\) 2.85086 0.411485
\(49\) 1.00000 0.142857
\(50\) 1.78017 0.251754
\(51\) −0.753020 −0.105444
\(52\) 0 0
\(53\) 4.16421 0.571998 0.285999 0.958230i \(-0.407675\pi\)
0.285999 + 0.958230i \(0.407675\pi\)
\(54\) −0.445042 −0.0605625
\(55\) 0.445042 0.0600094
\(56\) −1.69202 −0.226106
\(57\) −3.40581 −0.451111
\(58\) 0.951083 0.124883
\(59\) −10.3840 −1.35189 −0.675944 0.736953i \(-0.736264\pi\)
−0.675944 + 0.736953i \(0.736264\pi\)
\(60\) −1.80194 −0.232629
\(61\) −2.14914 −0.275170 −0.137585 0.990490i \(-0.543934\pi\)
−0.137585 + 0.990490i \(0.543934\pi\)
\(62\) −0.554958 −0.0704798
\(63\) −1.00000 −0.125988
\(64\) −3.63102 −0.453878
\(65\) 0 0
\(66\) −0.198062 −0.0243798
\(67\) 0.740939 0.0905201 0.0452600 0.998975i \(-0.485588\pi\)
0.0452600 + 0.998975i \(0.485588\pi\)
\(68\) 1.35690 0.164548
\(69\) −2.64310 −0.318192
\(70\) 0.445042 0.0531927
\(71\) 1.66487 0.197584 0.0987921 0.995108i \(-0.468502\pi\)
0.0987921 + 0.995108i \(0.468502\pi\)
\(72\) 1.69202 0.199407
\(73\) 5.74094 0.671926 0.335963 0.941875i \(-0.390938\pi\)
0.335963 + 0.941875i \(0.390938\pi\)
\(74\) −3.65279 −0.424629
\(75\) −4.00000 −0.461880
\(76\) 6.13706 0.703969
\(77\) −0.445042 −0.0507172
\(78\) 0 0
\(79\) 0.719169 0.0809128 0.0404564 0.999181i \(-0.487119\pi\)
0.0404564 + 0.999181i \(0.487119\pi\)
\(80\) 2.85086 0.318735
\(81\) 1.00000 0.111111
\(82\) −0.674563 −0.0744930
\(83\) −0.131687 −0.0144545 −0.00722724 0.999974i \(-0.502301\pi\)
−0.00722724 + 0.999974i \(0.502301\pi\)
\(84\) 1.80194 0.196608
\(85\) −0.753020 −0.0816765
\(86\) 3.56033 0.383921
\(87\) −2.13706 −0.229117
\(88\) 0.753020 0.0802722
\(89\) 15.5918 1.65273 0.826363 0.563137i \(-0.190406\pi\)
0.826363 + 0.563137i \(0.190406\pi\)
\(90\) −0.445042 −0.0469115
\(91\) 0 0
\(92\) 4.76271 0.496547
\(93\) 1.24698 0.129306
\(94\) 3.59179 0.370465
\(95\) −3.40581 −0.349429
\(96\) −4.65279 −0.474874
\(97\) 10.8726 1.10395 0.551974 0.833861i \(-0.313875\pi\)
0.551974 + 0.833861i \(0.313875\pi\)
\(98\) −0.445042 −0.0449560
\(99\) 0.445042 0.0447284
\(100\) 7.20775 0.720775
\(101\) −17.0858 −1.70010 −0.850048 0.526705i \(-0.823427\pi\)
−0.850048 + 0.526705i \(0.823427\pi\)
\(102\) 0.335126 0.0331824
\(103\) −7.11529 −0.701091 −0.350545 0.936546i \(-0.614004\pi\)
−0.350545 + 0.936546i \(0.614004\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) −1.85325 −0.180003
\(107\) −13.3545 −1.29103 −0.645514 0.763748i \(-0.723357\pi\)
−0.645514 + 0.763748i \(0.723357\pi\)
\(108\) −1.80194 −0.173392
\(109\) −3.93900 −0.377288 −0.188644 0.982046i \(-0.560409\pi\)
−0.188644 + 0.982046i \(0.560409\pi\)
\(110\) −0.198062 −0.0188845
\(111\) 8.20775 0.779045
\(112\) −2.85086 −0.269380
\(113\) −12.5797 −1.18340 −0.591700 0.806158i \(-0.701543\pi\)
−0.591700 + 0.806158i \(0.701543\pi\)
\(114\) 1.51573 0.141961
\(115\) −2.64310 −0.246471
\(116\) 3.85086 0.357543
\(117\) 0 0
\(118\) 4.62133 0.425428
\(119\) 0.753020 0.0690293
\(120\) 1.69202 0.154460
\(121\) −10.8019 −0.981994
\(122\) 0.956459 0.0865938
\(123\) 1.51573 0.136669
\(124\) −2.24698 −0.201785
\(125\) −9.00000 −0.804984
\(126\) 0.445042 0.0396475
\(127\) −11.3991 −1.01151 −0.505754 0.862678i \(-0.668786\pi\)
−0.505754 + 0.862678i \(0.668786\pi\)
\(128\) 10.9215 0.965337
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 2.65279 0.231776 0.115888 0.993262i \(-0.463029\pi\)
0.115888 + 0.993262i \(0.463029\pi\)
\(132\) −0.801938 −0.0697997
\(133\) 3.40581 0.295321
\(134\) −0.329749 −0.0284860
\(135\) 1.00000 0.0860663
\(136\) −1.27413 −0.109255
\(137\) −19.9855 −1.70748 −0.853739 0.520701i \(-0.825671\pi\)
−0.853739 + 0.520701i \(0.825671\pi\)
\(138\) 1.17629 0.100133
\(139\) −5.54527 −0.470344 −0.235172 0.971954i \(-0.575565\pi\)
−0.235172 + 0.971954i \(0.575565\pi\)
\(140\) 1.80194 0.152292
\(141\) −8.07069 −0.679675
\(142\) −0.740939 −0.0621782
\(143\) 0 0
\(144\) 2.85086 0.237571
\(145\) −2.13706 −0.177473
\(146\) −2.55496 −0.211450
\(147\) 1.00000 0.0824786
\(148\) −14.7899 −1.21572
\(149\) 17.6015 1.44197 0.720985 0.692951i \(-0.243690\pi\)
0.720985 + 0.692951i \(0.243690\pi\)
\(150\) 1.78017 0.145350
\(151\) −3.41550 −0.277950 −0.138975 0.990296i \(-0.544381\pi\)
−0.138975 + 0.990296i \(0.544381\pi\)
\(152\) −5.76271 −0.467417
\(153\) −0.753020 −0.0608781
\(154\) 0.198062 0.0159603
\(155\) 1.24698 0.100160
\(156\) 0 0
\(157\) −17.1196 −1.36629 −0.683147 0.730281i \(-0.739389\pi\)
−0.683147 + 0.730281i \(0.739389\pi\)
\(158\) −0.320060 −0.0254626
\(159\) 4.16421 0.330243
\(160\) −4.65279 −0.367836
\(161\) 2.64310 0.208306
\(162\) −0.445042 −0.0349658
\(163\) 3.59419 0.281518 0.140759 0.990044i \(-0.455046\pi\)
0.140759 + 0.990044i \(0.455046\pi\)
\(164\) −2.73125 −0.213275
\(165\) 0.445042 0.0346465
\(166\) 0.0586060 0.00454871
\(167\) 8.47219 0.655598 0.327799 0.944748i \(-0.393693\pi\)
0.327799 + 0.944748i \(0.393693\pi\)
\(168\) −1.69202 −0.130542
\(169\) 0 0
\(170\) 0.335126 0.0257030
\(171\) −3.40581 −0.260449
\(172\) 14.4155 1.09917
\(173\) 19.2500 1.46355 0.731774 0.681548i \(-0.238693\pi\)
0.731774 + 0.681548i \(0.238693\pi\)
\(174\) 0.951083 0.0721014
\(175\) 4.00000 0.302372
\(176\) 1.26875 0.0956356
\(177\) −10.3840 −0.780512
\(178\) −6.93900 −0.520100
\(179\) −5.63773 −0.421384 −0.210692 0.977553i \(-0.567572\pi\)
−0.210692 + 0.977553i \(0.567572\pi\)
\(180\) −1.80194 −0.134309
\(181\) −14.0368 −1.04335 −0.521675 0.853144i \(-0.674693\pi\)
−0.521675 + 0.853144i \(0.674693\pi\)
\(182\) 0 0
\(183\) −2.14914 −0.158869
\(184\) −4.47219 −0.329694
\(185\) 8.20775 0.603446
\(186\) −0.554958 −0.0406915
\(187\) −0.335126 −0.0245068
\(188\) 14.5429 1.06065
\(189\) −1.00000 −0.0727393
\(190\) 1.51573 0.109963
\(191\) −19.4252 −1.40556 −0.702779 0.711409i \(-0.748058\pi\)
−0.702779 + 0.711409i \(0.748058\pi\)
\(192\) −3.63102 −0.262046
\(193\) 6.49827 0.467756 0.233878 0.972266i \(-0.424858\pi\)
0.233878 + 0.972266i \(0.424858\pi\)
\(194\) −4.83877 −0.347404
\(195\) 0 0
\(196\) −1.80194 −0.128710
\(197\) −13.8998 −0.990318 −0.495159 0.868802i \(-0.664890\pi\)
−0.495159 + 0.868802i \(0.664890\pi\)
\(198\) −0.198062 −0.0140757
\(199\) −0.984935 −0.0698202 −0.0349101 0.999390i \(-0.511114\pi\)
−0.0349101 + 0.999390i \(0.511114\pi\)
\(200\) −6.76809 −0.478576
\(201\) 0.740939 0.0522618
\(202\) 7.60388 0.535007
\(203\) 2.13706 0.149992
\(204\) 1.35690 0.0950017
\(205\) 1.51573 0.105863
\(206\) 3.16660 0.220628
\(207\) −2.64310 −0.183708
\(208\) 0 0
\(209\) −1.51573 −0.104845
\(210\) 0.445042 0.0307108
\(211\) 1.44935 0.0997776 0.0498888 0.998755i \(-0.484113\pi\)
0.0498888 + 0.998755i \(0.484113\pi\)
\(212\) −7.50365 −0.515353
\(213\) 1.66487 0.114075
\(214\) 5.94331 0.406277
\(215\) −8.00000 −0.545595
\(216\) 1.69202 0.115127
\(217\) −1.24698 −0.0846505
\(218\) 1.75302 0.118730
\(219\) 5.74094 0.387937
\(220\) −0.801938 −0.0540666
\(221\) 0 0
\(222\) −3.65279 −0.245159
\(223\) −13.7832 −0.922988 −0.461494 0.887143i \(-0.652686\pi\)
−0.461494 + 0.887143i \(0.652686\pi\)
\(224\) 4.65279 0.310878
\(225\) −4.00000 −0.266667
\(226\) 5.59850 0.372407
\(227\) 0.259061 0.0171945 0.00859725 0.999963i \(-0.497263\pi\)
0.00859725 + 0.999963i \(0.497263\pi\)
\(228\) 6.13706 0.406437
\(229\) −24.7265 −1.63397 −0.816985 0.576658i \(-0.804356\pi\)
−0.816985 + 0.576658i \(0.804356\pi\)
\(230\) 1.17629 0.0775624
\(231\) −0.445042 −0.0292816
\(232\) −3.61596 −0.237399
\(233\) 13.5961 0.890711 0.445355 0.895354i \(-0.353077\pi\)
0.445355 + 0.895354i \(0.353077\pi\)
\(234\) 0 0
\(235\) −8.07069 −0.526474
\(236\) 18.7114 1.21801
\(237\) 0.719169 0.0467150
\(238\) −0.335126 −0.0217230
\(239\) −16.9269 −1.09491 −0.547456 0.836835i \(-0.684404\pi\)
−0.547456 + 0.836835i \(0.684404\pi\)
\(240\) 2.85086 0.184022
\(241\) −7.91484 −0.509840 −0.254920 0.966962i \(-0.582049\pi\)
−0.254920 + 0.966962i \(0.582049\pi\)
\(242\) 4.80731 0.309026
\(243\) 1.00000 0.0641500
\(244\) 3.87263 0.247919
\(245\) 1.00000 0.0638877
\(246\) −0.674563 −0.0430086
\(247\) 0 0
\(248\) 2.10992 0.133980
\(249\) −0.131687 −0.00834529
\(250\) 4.00538 0.253322
\(251\) 13.7845 0.870069 0.435034 0.900414i \(-0.356736\pi\)
0.435034 + 0.900414i \(0.356736\pi\)
\(252\) 1.80194 0.113511
\(253\) −1.17629 −0.0739528
\(254\) 5.07308 0.318313
\(255\) −0.753020 −0.0471560
\(256\) 2.40150 0.150094
\(257\) 1.56571 0.0976664 0.0488332 0.998807i \(-0.484450\pi\)
0.0488332 + 0.998807i \(0.484450\pi\)
\(258\) 3.56033 0.221657
\(259\) −8.20775 −0.510005
\(260\) 0 0
\(261\) −2.13706 −0.132281
\(262\) −1.18060 −0.0729380
\(263\) 16.1371 0.995054 0.497527 0.867449i \(-0.334242\pi\)
0.497527 + 0.867449i \(0.334242\pi\)
\(264\) 0.753020 0.0463452
\(265\) 4.16421 0.255805
\(266\) −1.51573 −0.0929353
\(267\) 15.5918 0.954202
\(268\) −1.33513 −0.0815558
\(269\) −28.7318 −1.75181 −0.875906 0.482482i \(-0.839735\pi\)
−0.875906 + 0.482482i \(0.839735\pi\)
\(270\) −0.445042 −0.0270844
\(271\) 1.18359 0.0718978 0.0359489 0.999354i \(-0.488555\pi\)
0.0359489 + 0.999354i \(0.488555\pi\)
\(272\) −2.14675 −0.130166
\(273\) 0 0
\(274\) 8.89440 0.537330
\(275\) −1.78017 −0.107348
\(276\) 4.76271 0.286681
\(277\) −25.8866 −1.55538 −0.777688 0.628650i \(-0.783608\pi\)
−0.777688 + 0.628650i \(0.783608\pi\)
\(278\) 2.46788 0.148013
\(279\) 1.24698 0.0746547
\(280\) −1.69202 −0.101118
\(281\) 18.3793 1.09641 0.548207 0.836343i \(-0.315310\pi\)
0.548207 + 0.836343i \(0.315310\pi\)
\(282\) 3.59179 0.213888
\(283\) 6.02954 0.358419 0.179209 0.983811i \(-0.442646\pi\)
0.179209 + 0.983811i \(0.442646\pi\)
\(284\) −3.00000 −0.178017
\(285\) −3.40581 −0.201743
\(286\) 0 0
\(287\) −1.51573 −0.0894707
\(288\) −4.65279 −0.274168
\(289\) −16.4330 −0.966645
\(290\) 0.951083 0.0558495
\(291\) 10.8726 0.637365
\(292\) −10.3448 −0.605384
\(293\) −23.3575 −1.36456 −0.682279 0.731091i \(-0.739011\pi\)
−0.682279 + 0.731091i \(0.739011\pi\)
\(294\) −0.445042 −0.0259554
\(295\) −10.3840 −0.604582
\(296\) 13.8877 0.807206
\(297\) 0.445042 0.0258239
\(298\) −7.83340 −0.453776
\(299\) 0 0
\(300\) 7.20775 0.416140
\(301\) 8.00000 0.461112
\(302\) 1.52004 0.0874686
\(303\) −17.0858 −0.981551
\(304\) −9.70948 −0.556877
\(305\) −2.14914 −0.123060
\(306\) 0.335126 0.0191579
\(307\) −4.14782 −0.236728 −0.118364 0.992970i \(-0.537765\pi\)
−0.118364 + 0.992970i \(0.537765\pi\)
\(308\) 0.801938 0.0456946
\(309\) −7.11529 −0.404775
\(310\) −0.554958 −0.0315195
\(311\) 23.9584 1.35856 0.679278 0.733882i \(-0.262293\pi\)
0.679278 + 0.733882i \(0.262293\pi\)
\(312\) 0 0
\(313\) −16.6920 −0.943489 −0.471744 0.881735i \(-0.656376\pi\)
−0.471744 + 0.881735i \(0.656376\pi\)
\(314\) 7.61894 0.429962
\(315\) −1.00000 −0.0563436
\(316\) −1.29590 −0.0728999
\(317\) 9.70948 0.545339 0.272669 0.962108i \(-0.412093\pi\)
0.272669 + 0.962108i \(0.412093\pi\)
\(318\) −1.85325 −0.103925
\(319\) −0.951083 −0.0532504
\(320\) −3.63102 −0.202980
\(321\) −13.3545 −0.745376
\(322\) −1.17629 −0.0655522
\(323\) 2.56465 0.142701
\(324\) −1.80194 −0.100108
\(325\) 0 0
\(326\) −1.59956 −0.0885916
\(327\) −3.93900 −0.217827
\(328\) 2.56465 0.141609
\(329\) 8.07069 0.444951
\(330\) −0.198062 −0.0109030
\(331\) 13.3207 0.732169 0.366085 0.930582i \(-0.380698\pi\)
0.366085 + 0.930582i \(0.380698\pi\)
\(332\) 0.237291 0.0130230
\(333\) 8.20775 0.449782
\(334\) −3.77048 −0.206311
\(335\) 0.740939 0.0404818
\(336\) −2.85086 −0.155527
\(337\) 26.8974 1.46519 0.732597 0.680663i \(-0.238308\pi\)
0.732597 + 0.680663i \(0.238308\pi\)
\(338\) 0 0
\(339\) −12.5797 −0.683236
\(340\) 1.35690 0.0735880
\(341\) 0.554958 0.0300527
\(342\) 1.51573 0.0819613
\(343\) −1.00000 −0.0539949
\(344\) −13.5362 −0.729821
\(345\) −2.64310 −0.142300
\(346\) −8.56704 −0.460567
\(347\) 18.1860 0.976275 0.488137 0.872767i \(-0.337677\pi\)
0.488137 + 0.872767i \(0.337677\pi\)
\(348\) 3.85086 0.206427
\(349\) 21.5623 1.15420 0.577100 0.816673i \(-0.304184\pi\)
0.577100 + 0.816673i \(0.304184\pi\)
\(350\) −1.78017 −0.0951540
\(351\) 0 0
\(352\) −2.07069 −0.110368
\(353\) −0.997607 −0.0530973 −0.0265486 0.999648i \(-0.508452\pi\)
−0.0265486 + 0.999648i \(0.508452\pi\)
\(354\) 4.62133 0.245621
\(355\) 1.66487 0.0883624
\(356\) −28.0954 −1.48906
\(357\) 0.753020 0.0398541
\(358\) 2.50902 0.132606
\(359\) 14.4983 0.765189 0.382595 0.923916i \(-0.375031\pi\)
0.382595 + 0.923916i \(0.375031\pi\)
\(360\) 1.69202 0.0891774
\(361\) −7.40044 −0.389497
\(362\) 6.24698 0.328334
\(363\) −10.8019 −0.566955
\(364\) 0 0
\(365\) 5.74094 0.300494
\(366\) 0.956459 0.0499949
\(367\) −8.33752 −0.435215 −0.217607 0.976036i \(-0.569825\pi\)
−0.217607 + 0.976036i \(0.569825\pi\)
\(368\) −7.53511 −0.392795
\(369\) 1.51573 0.0789057
\(370\) −3.65279 −0.189900
\(371\) −4.16421 −0.216195
\(372\) −2.24698 −0.116500
\(373\) 35.8974 1.85870 0.929348 0.369205i \(-0.120370\pi\)
0.929348 + 0.369205i \(0.120370\pi\)
\(374\) 0.149145 0.00771210
\(375\) −9.00000 −0.464758
\(376\) −13.6558 −0.704243
\(377\) 0 0
\(378\) 0.445042 0.0228905
\(379\) −34.6088 −1.77773 −0.888867 0.458166i \(-0.848507\pi\)
−0.888867 + 0.458166i \(0.848507\pi\)
\(380\) 6.13706 0.314825
\(381\) −11.3991 −0.583994
\(382\) 8.64502 0.442318
\(383\) −9.31037 −0.475738 −0.237869 0.971297i \(-0.576449\pi\)
−0.237869 + 0.971297i \(0.576449\pi\)
\(384\) 10.9215 0.557338
\(385\) −0.445042 −0.0226814
\(386\) −2.89200 −0.147199
\(387\) −8.00000 −0.406663
\(388\) −19.5918 −0.994623
\(389\) 38.5448 1.95430 0.977149 0.212554i \(-0.0681781\pi\)
0.977149 + 0.212554i \(0.0681781\pi\)
\(390\) 0 0
\(391\) 1.99031 0.100654
\(392\) 1.69202 0.0854600
\(393\) 2.65279 0.133816
\(394\) 6.18598 0.311645
\(395\) 0.719169 0.0361853
\(396\) −0.801938 −0.0402989
\(397\) 3.67696 0.184541 0.0922706 0.995734i \(-0.470588\pi\)
0.0922706 + 0.995734i \(0.470588\pi\)
\(398\) 0.438337 0.0219719
\(399\) 3.40581 0.170504
\(400\) −11.4034 −0.570171
\(401\) 23.9530 1.19616 0.598078 0.801438i \(-0.295931\pi\)
0.598078 + 0.801438i \(0.295931\pi\)
\(402\) −0.329749 −0.0164464
\(403\) 0 0
\(404\) 30.7875 1.53173
\(405\) 1.00000 0.0496904
\(406\) −0.951083 −0.0472014
\(407\) 3.65279 0.181062
\(408\) −1.27413 −0.0630787
\(409\) −28.0344 −1.38621 −0.693107 0.720835i \(-0.743759\pi\)
−0.693107 + 0.720835i \(0.743759\pi\)
\(410\) −0.674563 −0.0333143
\(411\) −19.9855 −0.985813
\(412\) 12.8213 0.631661
\(413\) 10.3840 0.510965
\(414\) 1.17629 0.0578116
\(415\) −0.131687 −0.00646424
\(416\) 0 0
\(417\) −5.54527 −0.271553
\(418\) 0.674563 0.0329940
\(419\) −27.5918 −1.34795 −0.673974 0.738755i \(-0.735414\pi\)
−0.673974 + 0.738755i \(0.735414\pi\)
\(420\) 1.80194 0.0879256
\(421\) −17.6746 −0.861405 −0.430703 0.902494i \(-0.641734\pi\)
−0.430703 + 0.902494i \(0.641734\pi\)
\(422\) −0.645023 −0.0313992
\(423\) −8.07069 −0.392410
\(424\) 7.04593 0.342181
\(425\) 3.01208 0.146107
\(426\) −0.740939 −0.0358986
\(427\) 2.14914 0.104004
\(428\) 24.0640 1.16318
\(429\) 0 0
\(430\) 3.56033 0.171695
\(431\) 7.59717 0.365943 0.182971 0.983118i \(-0.441428\pi\)
0.182971 + 0.983118i \(0.441428\pi\)
\(432\) 2.85086 0.137162
\(433\) 19.3927 0.931952 0.465976 0.884797i \(-0.345703\pi\)
0.465976 + 0.884797i \(0.345703\pi\)
\(434\) 0.554958 0.0266388
\(435\) −2.13706 −0.102464
\(436\) 7.09783 0.339925
\(437\) 9.00192 0.430620
\(438\) −2.55496 −0.122081
\(439\) −13.9554 −0.666055 −0.333027 0.942917i \(-0.608070\pi\)
−0.333027 + 0.942917i \(0.608070\pi\)
\(440\) 0.753020 0.0358988
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 24.2271 1.15107 0.575533 0.817779i \(-0.304795\pi\)
0.575533 + 0.817779i \(0.304795\pi\)
\(444\) −14.7899 −0.701896
\(445\) 15.5918 0.739122
\(446\) 6.13408 0.290457
\(447\) 17.6015 0.832522
\(448\) 3.63102 0.171550
\(449\) −6.01447 −0.283841 −0.141920 0.989878i \(-0.545328\pi\)
−0.141920 + 0.989878i \(0.545328\pi\)
\(450\) 1.78017 0.0839179
\(451\) 0.674563 0.0317639
\(452\) 22.6679 1.06621
\(453\) −3.41550 −0.160474
\(454\) −0.115293 −0.00541097
\(455\) 0 0
\(456\) −5.76271 −0.269864
\(457\) −6.64848 −0.311003 −0.155501 0.987836i \(-0.549699\pi\)
−0.155501 + 0.987836i \(0.549699\pi\)
\(458\) 11.0043 0.514198
\(459\) −0.753020 −0.0351480
\(460\) 4.76271 0.222062
\(461\) 13.5386 0.630554 0.315277 0.949000i \(-0.397903\pi\)
0.315277 + 0.949000i \(0.397903\pi\)
\(462\) 0.198062 0.00921469
\(463\) −0.907542 −0.0421771 −0.0210885 0.999778i \(-0.506713\pi\)
−0.0210885 + 0.999778i \(0.506713\pi\)
\(464\) −6.09246 −0.282835
\(465\) 1.24698 0.0578273
\(466\) −6.05084 −0.280300
\(467\) −21.6189 −1.00041 −0.500203 0.865908i \(-0.666741\pi\)
−0.500203 + 0.865908i \(0.666741\pi\)
\(468\) 0 0
\(469\) −0.740939 −0.0342134
\(470\) 3.59179 0.165677
\(471\) −17.1196 −0.788830
\(472\) −17.5700 −0.808726
\(473\) −3.56033 −0.163704
\(474\) −0.320060 −0.0147008
\(475\) 13.6233 0.625078
\(476\) −1.35690 −0.0621932
\(477\) 4.16421 0.190666
\(478\) 7.53319 0.344560
\(479\) −27.4064 −1.25223 −0.626115 0.779730i \(-0.715356\pi\)
−0.626115 + 0.779730i \(0.715356\pi\)
\(480\) −4.65279 −0.212370
\(481\) 0 0
\(482\) 3.52243 0.160442
\(483\) 2.64310 0.120265
\(484\) 19.4644 0.884746
\(485\) 10.8726 0.493700
\(486\) −0.445042 −0.0201875
\(487\) −26.1317 −1.18414 −0.592070 0.805887i \(-0.701689\pi\)
−0.592070 + 0.805887i \(0.701689\pi\)
\(488\) −3.63640 −0.164612
\(489\) 3.59419 0.162535
\(490\) −0.445042 −0.0201049
\(491\) 17.6364 0.795920 0.397960 0.917403i \(-0.369718\pi\)
0.397960 + 0.917403i \(0.369718\pi\)
\(492\) −2.73125 −0.123134
\(493\) 1.60925 0.0724771
\(494\) 0 0
\(495\) 0.445042 0.0200031
\(496\) 3.55496 0.159622
\(497\) −1.66487 −0.0746798
\(498\) 0.0586060 0.00262620
\(499\) 16.6843 0.746890 0.373445 0.927652i \(-0.378177\pi\)
0.373445 + 0.927652i \(0.378177\pi\)
\(500\) 16.2174 0.725266
\(501\) 8.47219 0.378509
\(502\) −6.13467 −0.273804
\(503\) 41.6373 1.85651 0.928257 0.371940i \(-0.121307\pi\)
0.928257 + 0.371940i \(0.121307\pi\)
\(504\) −1.69202 −0.0753686
\(505\) −17.0858 −0.760306
\(506\) 0.523499 0.0232724
\(507\) 0 0
\(508\) 20.5405 0.911337
\(509\) 16.7356 0.741791 0.370895 0.928675i \(-0.379051\pi\)
0.370895 + 0.928675i \(0.379051\pi\)
\(510\) 0.335126 0.0148396
\(511\) −5.74094 −0.253964
\(512\) −22.9119 −1.01257
\(513\) −3.40581 −0.150370
\(514\) −0.696807 −0.0307349
\(515\) −7.11529 −0.313537
\(516\) 14.4155 0.634607
\(517\) −3.59179 −0.157967
\(518\) 3.65279 0.160495
\(519\) 19.2500 0.844980
\(520\) 0 0
\(521\) 14.9801 0.656292 0.328146 0.944627i \(-0.393576\pi\)
0.328146 + 0.944627i \(0.393576\pi\)
\(522\) 0.951083 0.0416278
\(523\) −43.9754 −1.92291 −0.961454 0.274965i \(-0.911334\pi\)
−0.961454 + 0.274965i \(0.911334\pi\)
\(524\) −4.78017 −0.208823
\(525\) 4.00000 0.174574
\(526\) −7.18167 −0.313136
\(527\) −0.939001 −0.0409035
\(528\) 1.26875 0.0552153
\(529\) −16.0140 −0.696261
\(530\) −1.85325 −0.0804999
\(531\) −10.3840 −0.450629
\(532\) −6.13706 −0.266075
\(533\) 0 0
\(534\) −6.93900 −0.300280
\(535\) −13.3545 −0.577366
\(536\) 1.25368 0.0541509
\(537\) −5.63773 −0.243286
\(538\) 12.7869 0.551281
\(539\) 0.445042 0.0191693
\(540\) −1.80194 −0.0775431
\(541\) 16.6485 0.715774 0.357887 0.933765i \(-0.383497\pi\)
0.357887 + 0.933765i \(0.383497\pi\)
\(542\) −0.526746 −0.0226257
\(543\) −14.0368 −0.602378
\(544\) 3.50365 0.150218
\(545\) −3.93900 −0.168728
\(546\) 0 0
\(547\) −44.8592 −1.91804 −0.959021 0.283336i \(-0.908559\pi\)
−0.959021 + 0.283336i \(0.908559\pi\)
\(548\) 36.0127 1.53839
\(549\) −2.14914 −0.0917233
\(550\) 0.792249 0.0337816
\(551\) 7.27844 0.310072
\(552\) −4.47219 −0.190349
\(553\) −0.719169 −0.0305822
\(554\) 11.5206 0.489465
\(555\) 8.20775 0.348400
\(556\) 9.99223 0.423765
\(557\) −2.23596 −0.0947408 −0.0473704 0.998877i \(-0.515084\pi\)
−0.0473704 + 0.998877i \(0.515084\pi\)
\(558\) −0.554958 −0.0234933
\(559\) 0 0
\(560\) −2.85086 −0.120471
\(561\) −0.335126 −0.0141490
\(562\) −8.17954 −0.345033
\(563\) 17.0019 0.716545 0.358273 0.933617i \(-0.383366\pi\)
0.358273 + 0.933617i \(0.383366\pi\)
\(564\) 14.5429 0.612366
\(565\) −12.5797 −0.529232
\(566\) −2.68340 −0.112792
\(567\) −1.00000 −0.0419961
\(568\) 2.81700 0.118199
\(569\) 1.81940 0.0762731 0.0381365 0.999273i \(-0.487858\pi\)
0.0381365 + 0.999273i \(0.487858\pi\)
\(570\) 1.51573 0.0634869
\(571\) 31.3924 1.31373 0.656866 0.754008i \(-0.271882\pi\)
0.656866 + 0.754008i \(0.271882\pi\)
\(572\) 0 0
\(573\) −19.4252 −0.811499
\(574\) 0.674563 0.0281557
\(575\) 10.5724 0.440900
\(576\) −3.63102 −0.151293
\(577\) 14.4752 0.602609 0.301305 0.953528i \(-0.402578\pi\)
0.301305 + 0.953528i \(0.402578\pi\)
\(578\) 7.31336 0.304195
\(579\) 6.49827 0.270059
\(580\) 3.85086 0.159898
\(581\) 0.131687 0.00546328
\(582\) −4.83877 −0.200574
\(583\) 1.85325 0.0767537
\(584\) 9.71379 0.401960
\(585\) 0 0
\(586\) 10.3951 0.429416
\(587\) 16.9892 0.701221 0.350611 0.936521i \(-0.385974\pi\)
0.350611 + 0.936521i \(0.385974\pi\)
\(588\) −1.80194 −0.0743107
\(589\) −4.24698 −0.174994
\(590\) 4.62133 0.190257
\(591\) −13.8998 −0.571760
\(592\) 23.3991 0.961697
\(593\) 16.1666 0.663883 0.331941 0.943300i \(-0.392296\pi\)
0.331941 + 0.943300i \(0.392296\pi\)
\(594\) −0.198062 −0.00812659
\(595\) 0.753020 0.0308708
\(596\) −31.7168 −1.29917
\(597\) −0.984935 −0.0403107
\(598\) 0 0
\(599\) 0.997607 0.0407611 0.0203806 0.999792i \(-0.493512\pi\)
0.0203806 + 0.999792i \(0.493512\pi\)
\(600\) −6.76809 −0.276306
\(601\) −11.3297 −0.462150 −0.231075 0.972936i \(-0.574224\pi\)
−0.231075 + 0.972936i \(0.574224\pi\)
\(602\) −3.56033 −0.145108
\(603\) 0.740939 0.0301734
\(604\) 6.15452 0.250424
\(605\) −10.8019 −0.439161
\(606\) 7.60388 0.308886
\(607\) 32.1027 1.30301 0.651505 0.758644i \(-0.274138\pi\)
0.651505 + 0.758644i \(0.274138\pi\)
\(608\) 15.8465 0.642662
\(609\) 2.13706 0.0865982
\(610\) 0.956459 0.0387259
\(611\) 0 0
\(612\) 1.35690 0.0548493
\(613\) 14.5972 0.589574 0.294787 0.955563i \(-0.404751\pi\)
0.294787 + 0.955563i \(0.404751\pi\)
\(614\) 1.84595 0.0744966
\(615\) 1.51573 0.0611201
\(616\) −0.753020 −0.0303401
\(617\) −34.2717 −1.37973 −0.689864 0.723939i \(-0.742330\pi\)
−0.689864 + 0.723939i \(0.742330\pi\)
\(618\) 3.16660 0.127379
\(619\) 13.9202 0.559501 0.279750 0.960073i \(-0.409748\pi\)
0.279750 + 0.960073i \(0.409748\pi\)
\(620\) −2.24698 −0.0902409
\(621\) −2.64310 −0.106064
\(622\) −10.6625 −0.427527
\(623\) −15.5918 −0.624672
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 7.42865 0.296909
\(627\) −1.51573 −0.0605324
\(628\) 30.8485 1.23099
\(629\) −6.18060 −0.246437
\(630\) 0.445042 0.0177309
\(631\) 36.4292 1.45023 0.725113 0.688630i \(-0.241788\pi\)
0.725113 + 0.688630i \(0.241788\pi\)
\(632\) 1.21685 0.0484036
\(633\) 1.44935 0.0576066
\(634\) −4.32113 −0.171614
\(635\) −11.3991 −0.452360
\(636\) −7.50365 −0.297539
\(637\) 0 0
\(638\) 0.423272 0.0167575
\(639\) 1.66487 0.0658614
\(640\) 10.9215 0.431712
\(641\) 14.6756 0.579652 0.289826 0.957079i \(-0.406402\pi\)
0.289826 + 0.957079i \(0.406402\pi\)
\(642\) 5.94331 0.234564
\(643\) −18.7966 −0.741264 −0.370632 0.928780i \(-0.620859\pi\)
−0.370632 + 0.928780i \(0.620859\pi\)
\(644\) −4.76271 −0.187677
\(645\) −8.00000 −0.315000
\(646\) −1.14138 −0.0449068
\(647\) −19.7198 −0.775264 −0.387632 0.921814i \(-0.626707\pi\)
−0.387632 + 0.921814i \(0.626707\pi\)
\(648\) 1.69202 0.0664689
\(649\) −4.62133 −0.181403
\(650\) 0 0
\(651\) −1.24698 −0.0488730
\(652\) −6.47650 −0.253639
\(653\) 6.61058 0.258692 0.129346 0.991600i \(-0.458712\pi\)
0.129346 + 0.991600i \(0.458712\pi\)
\(654\) 1.75302 0.0685485
\(655\) 2.65279 0.103653
\(656\) 4.32113 0.168712
\(657\) 5.74094 0.223975
\(658\) −3.59179 −0.140023
\(659\) −26.9812 −1.05104 −0.525519 0.850782i \(-0.676129\pi\)
−0.525519 + 0.850782i \(0.676129\pi\)
\(660\) −0.801938 −0.0312154
\(661\) −25.6353 −0.997099 −0.498549 0.866861i \(-0.666134\pi\)
−0.498549 + 0.866861i \(0.666134\pi\)
\(662\) −5.92825 −0.230408
\(663\) 0 0
\(664\) −0.222816 −0.00864695
\(665\) 3.40581 0.132072
\(666\) −3.65279 −0.141543
\(667\) 5.64848 0.218710
\(668\) −15.2664 −0.590673
\(669\) −13.7832 −0.532887
\(670\) −0.329749 −0.0127393
\(671\) −0.956459 −0.0369237
\(672\) 4.65279 0.179485
\(673\) −34.8122 −1.34191 −0.670956 0.741497i \(-0.734116\pi\)
−0.670956 + 0.741497i \(0.734116\pi\)
\(674\) −11.9705 −0.461085
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 19.9782 0.767826 0.383913 0.923369i \(-0.374576\pi\)
0.383913 + 0.923369i \(0.374576\pi\)
\(678\) 5.59850 0.215009
\(679\) −10.8726 −0.417253
\(680\) −1.27413 −0.0488605
\(681\) 0.259061 0.00992725
\(682\) −0.246980 −0.00945734
\(683\) −31.5340 −1.20662 −0.603308 0.797508i \(-0.706151\pi\)
−0.603308 + 0.797508i \(0.706151\pi\)
\(684\) 6.13706 0.234656
\(685\) −19.9855 −0.763608
\(686\) 0.445042 0.0169918
\(687\) −24.7265 −0.943373
\(688\) −22.8068 −0.869503
\(689\) 0 0
\(690\) 1.17629 0.0447807
\(691\) 32.4805 1.23562 0.617809 0.786328i \(-0.288020\pi\)
0.617809 + 0.786328i \(0.288020\pi\)
\(692\) −34.6872 −1.31861
\(693\) −0.445042 −0.0169057
\(694\) −8.09352 −0.307226
\(695\) −5.54527 −0.210344
\(696\) −3.61596 −0.137062
\(697\) −1.14138 −0.0432327
\(698\) −9.59611 −0.363218
\(699\) 13.5961 0.514252
\(700\) −7.20775 −0.272427
\(701\) 33.7904 1.27625 0.638124 0.769934i \(-0.279711\pi\)
0.638124 + 0.769934i \(0.279711\pi\)
\(702\) 0 0
\(703\) −27.9541 −1.05431
\(704\) −1.61596 −0.0609037
\(705\) −8.07069 −0.303960
\(706\) 0.443977 0.0167093
\(707\) 17.0858 0.642576
\(708\) 18.7114 0.703217
\(709\) −27.4499 −1.03090 −0.515452 0.856918i \(-0.672376\pi\)
−0.515452 + 0.856918i \(0.672376\pi\)
\(710\) −0.740939 −0.0278069
\(711\) 0.719169 0.0269709
\(712\) 26.3817 0.988694
\(713\) −3.29590 −0.123432
\(714\) −0.335126 −0.0125418
\(715\) 0 0
\(716\) 10.1588 0.379653
\(717\) −16.9269 −0.632147
\(718\) −6.45234 −0.240799
\(719\) 34.0984 1.27166 0.635828 0.771830i \(-0.280659\pi\)
0.635828 + 0.771830i \(0.280659\pi\)
\(720\) 2.85086 0.106245
\(721\) 7.11529 0.264987
\(722\) 3.29350 0.122572
\(723\) −7.91484 −0.294356
\(724\) 25.2935 0.940026
\(725\) 8.54825 0.317474
\(726\) 4.80731 0.178416
\(727\) −42.3086 −1.56914 −0.784569 0.620042i \(-0.787116\pi\)
−0.784569 + 0.620042i \(0.787116\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −2.55496 −0.0945632
\(731\) 6.02416 0.222812
\(732\) 3.87263 0.143136
\(733\) 37.6920 1.39219 0.696093 0.717951i \(-0.254920\pi\)
0.696093 + 0.717951i \(0.254920\pi\)
\(734\) 3.71054 0.136959
\(735\) 1.00000 0.0368856
\(736\) 12.2978 0.453304
\(737\) 0.329749 0.0121465
\(738\) −0.674563 −0.0248310
\(739\) −28.9353 −1.06440 −0.532201 0.846618i \(-0.678635\pi\)
−0.532201 + 0.846618i \(0.678635\pi\)
\(740\) −14.7899 −0.543686
\(741\) 0 0
\(742\) 1.85325 0.0680349
\(743\) −52.3430 −1.92028 −0.960139 0.279521i \(-0.909824\pi\)
−0.960139 + 0.279521i \(0.909824\pi\)
\(744\) 2.10992 0.0773533
\(745\) 17.6015 0.644868
\(746\) −15.9758 −0.584917
\(747\) −0.131687 −0.00481816
\(748\) 0.603875 0.0220799
\(749\) 13.3545 0.487963
\(750\) 4.00538 0.146256
\(751\) 2.35690 0.0860044 0.0430022 0.999075i \(-0.486308\pi\)
0.0430022 + 0.999075i \(0.486308\pi\)
\(752\) −23.0084 −0.839029
\(753\) 13.7845 0.502334
\(754\) 0 0
\(755\) −3.41550 −0.124303
\(756\) 1.80194 0.0655358
\(757\) −15.2433 −0.554026 −0.277013 0.960866i \(-0.589344\pi\)
−0.277013 + 0.960866i \(0.589344\pi\)
\(758\) 15.4024 0.559439
\(759\) −1.17629 −0.0426967
\(760\) −5.76271 −0.209035
\(761\) 11.9360 0.432680 0.216340 0.976318i \(-0.430588\pi\)
0.216340 + 0.976318i \(0.430588\pi\)
\(762\) 5.07308 0.183778
\(763\) 3.93900 0.142601
\(764\) 35.0030 1.26636
\(765\) −0.753020 −0.0272255
\(766\) 4.14351 0.149711
\(767\) 0 0
\(768\) 2.40150 0.0866567
\(769\) 35.0073 1.26240 0.631198 0.775622i \(-0.282564\pi\)
0.631198 + 0.775622i \(0.282564\pi\)
\(770\) 0.198062 0.00713767
\(771\) 1.56571 0.0563877
\(772\) −11.7095 −0.421433
\(773\) 46.0489 1.65626 0.828132 0.560533i \(-0.189404\pi\)
0.828132 + 0.560533i \(0.189404\pi\)
\(774\) 3.56033 0.127974
\(775\) −4.98792 −0.179171
\(776\) 18.3967 0.660404
\(777\) −8.20775 −0.294451
\(778\) −17.1540 −0.615002
\(779\) −5.16229 −0.184958
\(780\) 0 0
\(781\) 0.740939 0.0265129
\(782\) −0.885772 −0.0316751
\(783\) −2.13706 −0.0763724
\(784\) 2.85086 0.101816
\(785\) −17.1196 −0.611025
\(786\) −1.18060 −0.0421107
\(787\) 46.4965 1.65742 0.828710 0.559678i \(-0.189075\pi\)
0.828710 + 0.559678i \(0.189075\pi\)
\(788\) 25.0465 0.892245
\(789\) 16.1371 0.574495
\(790\) −0.320060 −0.0113872
\(791\) 12.5797 0.447283
\(792\) 0.753020 0.0267574
\(793\) 0 0
\(794\) −1.63640 −0.0580736
\(795\) 4.16421 0.147689
\(796\) 1.77479 0.0629058
\(797\) 9.40389 0.333103 0.166552 0.986033i \(-0.446737\pi\)
0.166552 + 0.986033i \(0.446737\pi\)
\(798\) −1.51573 −0.0536562
\(799\) 6.07739 0.215003
\(800\) 18.6112 0.658004
\(801\) 15.5918 0.550909
\(802\) −10.6601 −0.376421
\(803\) 2.55496 0.0901625
\(804\) −1.33513 −0.0470862
\(805\) 2.64310 0.0931572
\(806\) 0 0
\(807\) −28.7318 −1.01141
\(808\) −28.9095 −1.01703
\(809\) −42.9023 −1.50836 −0.754182 0.656665i \(-0.771966\pi\)
−0.754182 + 0.656665i \(0.771966\pi\)
\(810\) −0.445042 −0.0156372
\(811\) 51.3411 1.80283 0.901415 0.432957i \(-0.142530\pi\)
0.901415 + 0.432957i \(0.142530\pi\)
\(812\) −3.85086 −0.135139
\(813\) 1.18359 0.0415102
\(814\) −1.62565 −0.0569789
\(815\) 3.59419 0.125899
\(816\) −2.14675 −0.0751514
\(817\) 27.2465 0.953235
\(818\) 12.4765 0.436231
\(819\) 0 0
\(820\) −2.73125 −0.0953794
\(821\) 29.4553 1.02800 0.513999 0.857791i \(-0.328164\pi\)
0.513999 + 0.857791i \(0.328164\pi\)
\(822\) 8.89440 0.310228
\(823\) 22.8068 0.794996 0.397498 0.917603i \(-0.369878\pi\)
0.397498 + 0.917603i \(0.369878\pi\)
\(824\) −12.0392 −0.419406
\(825\) −1.78017 −0.0619775
\(826\) −4.62133 −0.160797
\(827\) 28.5512 0.992824 0.496412 0.868087i \(-0.334651\pi\)
0.496412 + 0.868087i \(0.334651\pi\)
\(828\) 4.76271 0.165516
\(829\) 48.2887 1.67714 0.838568 0.544797i \(-0.183393\pi\)
0.838568 + 0.544797i \(0.183393\pi\)
\(830\) 0.0586060 0.00203424
\(831\) −25.8866 −0.897997
\(832\) 0 0
\(833\) −0.753020 −0.0260906
\(834\) 2.46788 0.0854556
\(835\) 8.47219 0.293192
\(836\) 2.73125 0.0944623
\(837\) 1.24698 0.0431019
\(838\) 12.2795 0.424188
\(839\) −37.9191 −1.30911 −0.654557 0.756012i \(-0.727145\pi\)
−0.654557 + 0.756012i \(0.727145\pi\)
\(840\) −1.69202 −0.0583803
\(841\) −24.4330 −0.842516
\(842\) 7.86592 0.271078
\(843\) 18.3793 0.633015
\(844\) −2.61165 −0.0898965
\(845\) 0 0
\(846\) 3.59179 0.123488
\(847\) 10.8019 0.371159
\(848\) 11.8716 0.407671
\(849\) 6.02954 0.206933
\(850\) −1.34050 −0.0459789
\(851\) −21.6939 −0.743659
\(852\) −3.00000 −0.102778
\(853\) 10.4679 0.358413 0.179207 0.983811i \(-0.442647\pi\)
0.179207 + 0.983811i \(0.442647\pi\)
\(854\) −0.956459 −0.0327294
\(855\) −3.40581 −0.116476
\(856\) −22.5961 −0.772319
\(857\) 36.7942 1.25686 0.628432 0.777864i \(-0.283697\pi\)
0.628432 + 0.777864i \(0.283697\pi\)
\(858\) 0 0
\(859\) −11.0476 −0.376939 −0.188469 0.982079i \(-0.560353\pi\)
−0.188469 + 0.982079i \(0.560353\pi\)
\(860\) 14.4155 0.491565
\(861\) −1.51573 −0.0516559
\(862\) −3.38106 −0.115159
\(863\) 48.9372 1.66584 0.832921 0.553392i \(-0.186667\pi\)
0.832921 + 0.553392i \(0.186667\pi\)
\(864\) −4.65279 −0.158291
\(865\) 19.2500 0.654518
\(866\) −8.63055 −0.293278
\(867\) −16.4330 −0.558093
\(868\) 2.24698 0.0762675
\(869\) 0.320060 0.0108573
\(870\) 0.951083 0.0322447
\(871\) 0 0
\(872\) −6.66487 −0.225701
\(873\) 10.8726 0.367983
\(874\) −4.00623 −0.135513
\(875\) 9.00000 0.304256
\(876\) −10.3448 −0.349519
\(877\) 2.31203 0.0780716 0.0390358 0.999238i \(-0.487571\pi\)
0.0390358 + 0.999238i \(0.487571\pi\)
\(878\) 6.21073 0.209602
\(879\) −23.3575 −0.787828
\(880\) 1.26875 0.0427695
\(881\) 39.7351 1.33871 0.669355 0.742943i \(-0.266571\pi\)
0.669355 + 0.742943i \(0.266571\pi\)
\(882\) −0.445042 −0.0149853
\(883\) 53.9480 1.81549 0.907747 0.419519i \(-0.137801\pi\)
0.907747 + 0.419519i \(0.137801\pi\)
\(884\) 0 0
\(885\) −10.3840 −0.349056
\(886\) −10.7821 −0.362231
\(887\) −7.81727 −0.262478 −0.131239 0.991351i \(-0.541896\pi\)
−0.131239 + 0.991351i \(0.541896\pi\)
\(888\) 13.8877 0.466040
\(889\) 11.3991 0.382314
\(890\) −6.93900 −0.232596
\(891\) 0.445042 0.0149095
\(892\) 24.8364 0.831584
\(893\) 27.4873 0.919826
\(894\) −7.83340 −0.261988
\(895\) −5.63773 −0.188448
\(896\) −10.9215 −0.364863
\(897\) 0 0
\(898\) 2.67669 0.0893224
\(899\) −2.66487 −0.0888785
\(900\) 7.20775 0.240258
\(901\) −3.13574 −0.104466
\(902\) −0.300209 −0.00999586
\(903\) 8.00000 0.266223
\(904\) −21.2851 −0.707933
\(905\) −14.0368 −0.466600
\(906\) 1.52004 0.0505000
\(907\) 0.554958 0.0184271 0.00921354 0.999958i \(-0.497067\pi\)
0.00921354 + 0.999958i \(0.497067\pi\)
\(908\) −0.466812 −0.0154917
\(909\) −17.0858 −0.566699
\(910\) 0 0
\(911\) −9.44265 −0.312849 −0.156424 0.987690i \(-0.549997\pi\)
−0.156424 + 0.987690i \(0.549997\pi\)
\(912\) −9.70948 −0.321513
\(913\) −0.0586060 −0.00193958
\(914\) 2.95885 0.0978701
\(915\) −2.14914 −0.0710485
\(916\) 44.5555 1.47216
\(917\) −2.65279 −0.0876029
\(918\) 0.335126 0.0110608
\(919\) 22.4862 0.741751 0.370875 0.928683i \(-0.379058\pi\)
0.370875 + 0.928683i \(0.379058\pi\)
\(920\) −4.47219 −0.147444
\(921\) −4.14782 −0.136675
\(922\) −6.02523 −0.198430
\(923\) 0 0
\(924\) 0.801938 0.0263818
\(925\) −32.8310 −1.07948
\(926\) 0.403894 0.0132728
\(927\) −7.11529 −0.233697
\(928\) 9.94331 0.326405
\(929\) 29.2500 0.959660 0.479830 0.877361i \(-0.340698\pi\)
0.479830 + 0.877361i \(0.340698\pi\)
\(930\) −0.554958 −0.0181978
\(931\) −3.40581 −0.111621
\(932\) −24.4993 −0.802502
\(933\) 23.9584 0.784362
\(934\) 9.62133 0.314820
\(935\) −0.335126 −0.0109598
\(936\) 0 0
\(937\) −13.4679 −0.439976 −0.219988 0.975503i \(-0.570602\pi\)
−0.219988 + 0.975503i \(0.570602\pi\)
\(938\) 0.329749 0.0107667
\(939\) −16.6920 −0.544724
\(940\) 14.5429 0.474336
\(941\) −15.3418 −0.500129 −0.250065 0.968229i \(-0.580452\pi\)
−0.250065 + 0.968229i \(0.580452\pi\)
\(942\) 7.61894 0.248239
\(943\) −4.00623 −0.130461
\(944\) −29.6034 −0.963509
\(945\) −1.00000 −0.0325300
\(946\) 1.58450 0.0515165
\(947\) −16.2228 −0.527171 −0.263585 0.964636i \(-0.584905\pi\)
−0.263585 + 0.964636i \(0.584905\pi\)
\(948\) −1.29590 −0.0420888
\(949\) 0 0
\(950\) −6.06292 −0.196707
\(951\) 9.70948 0.314851
\(952\) 1.27413 0.0412947
\(953\) −1.85218 −0.0599981 −0.0299990 0.999550i \(-0.509550\pi\)
−0.0299990 + 0.999550i \(0.509550\pi\)
\(954\) −1.85325 −0.0600011
\(955\) −19.4252 −0.628584
\(956\) 30.5013 0.986481
\(957\) −0.951083 −0.0307441
\(958\) 12.1970 0.394067
\(959\) 19.9855 0.645366
\(960\) −3.63102 −0.117191
\(961\) −29.4450 −0.949840
\(962\) 0 0
\(963\) −13.3545 −0.430343
\(964\) 14.2620 0.459350
\(965\) 6.49827 0.209187
\(966\) −1.17629 −0.0378466
\(967\) 13.8030 0.443875 0.221937 0.975061i \(-0.428762\pi\)
0.221937 + 0.975061i \(0.428762\pi\)
\(968\) −18.2771 −0.587449
\(969\) 2.56465 0.0823883
\(970\) −4.83877 −0.155364
\(971\) 36.2814 1.16433 0.582163 0.813072i \(-0.302207\pi\)
0.582163 + 0.813072i \(0.302207\pi\)
\(972\) −1.80194 −0.0577972
\(973\) 5.54527 0.177773
\(974\) 11.6297 0.372639
\(975\) 0 0
\(976\) −6.12690 −0.196117
\(977\) −31.5472 −1.00928 −0.504642 0.863329i \(-0.668375\pi\)
−0.504642 + 0.863329i \(0.668375\pi\)
\(978\) −1.59956 −0.0511484
\(979\) 6.93900 0.221771
\(980\) −1.80194 −0.0575608
\(981\) −3.93900 −0.125763
\(982\) −7.84894 −0.250470
\(983\) 26.6547 0.850153 0.425077 0.905157i \(-0.360247\pi\)
0.425077 + 0.905157i \(0.360247\pi\)
\(984\) 2.56465 0.0817580
\(985\) −13.8998 −0.442884
\(986\) −0.716185 −0.0228080
\(987\) 8.07069 0.256893
\(988\) 0 0
\(989\) 21.1448 0.672367
\(990\) −0.198062 −0.00629483
\(991\) 1.15990 0.0368454 0.0184227 0.999830i \(-0.494136\pi\)
0.0184227 + 0.999830i \(0.494136\pi\)
\(992\) −5.80194 −0.184212
\(993\) 13.3207 0.422718
\(994\) 0.740939 0.0235012
\(995\) −0.984935 −0.0312245
\(996\) 0.237291 0.00751885
\(997\) −33.2984 −1.05457 −0.527286 0.849688i \(-0.676790\pi\)
−0.527286 + 0.849688i \(0.676790\pi\)
\(998\) −7.42519 −0.235040
\(999\) 8.20775 0.259682
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 3549.2.a.j.1.2 3
13.12 even 2 3549.2.a.p.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3549.2.a.j.1.2 3 1.1 even 1 trivial
3549.2.a.p.1.2 yes 3 13.12 even 2