Properties

Label 483.2.a.j.1.2
Level $483$
Weight $2$
Character 483.1
Self dual yes
Analytic conductor $3.857$
Analytic rank $0$
Dimension $4$
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] = [483,2,Mod(1,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, 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("483.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 2 \) 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 \(-0.329727\) of defining polynomial
Character \(\chi\) \(=\) 483.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.329727 q^{2} -1.00000 q^{3} -1.89128 q^{4} -2.73589 q^{5} +0.329727 q^{6} -1.00000 q^{7} +1.28306 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.329727 q^{2} -1.00000 q^{3} -1.89128 q^{4} -2.73589 q^{5} +0.329727 q^{6} -1.00000 q^{7} +1.28306 q^{8} +1.00000 q^{9} +0.902098 q^{10} -2.50407 q^{11} +1.89128 q^{12} +1.48511 q^{13} +0.329727 q^{14} +2.73589 q^{15} +3.35950 q^{16} +0.902098 q^{17} -0.329727 q^{18} +2.50407 q^{19} +5.17434 q^{20} +1.00000 q^{21} +0.825659 q^{22} +1.00000 q^{23} -1.28306 q^{24} +2.48511 q^{25} -0.489682 q^{26} -1.00000 q^{27} +1.89128 q^{28} +6.68466 q^{29} -0.902098 q^{30} +1.09790 q^{31} -3.67384 q^{32} +2.50407 q^{33} -0.297446 q^{34} +2.73589 q^{35} -1.89128 q^{36} +9.91023 q^{37} -0.825659 q^{38} -1.48511 q^{39} -3.51032 q^{40} -2.30826 q^{41} -0.329727 q^{42} +5.83380 q^{43} +4.73589 q^{44} -2.73589 q^{45} -0.329727 q^{46} -6.74671 q^{47} -3.35950 q^{48} +1.00000 q^{49} -0.819409 q^{50} -0.902098 q^{51} -2.80877 q^{52} -4.91648 q^{53} +0.329727 q^{54} +6.85086 q^{55} -1.28306 q^{56} -2.50407 q^{57} -2.20411 q^{58} -7.32973 q^{59} -5.17434 q^{60} -1.00625 q^{61} -0.362008 q^{62} -1.00000 q^{63} -5.50764 q^{64} -4.06311 q^{65} -0.825659 q^{66} +11.2929 q^{67} -1.70612 q^{68} -1.00000 q^{69} -0.902098 q^{70} -0.362008 q^{71} +1.28306 q^{72} -5.34411 q^{73} -3.26767 q^{74} -2.48511 q^{75} -4.73589 q^{76} +2.50407 q^{77} +0.489682 q^{78} +10.4672 q^{79} -9.19123 q^{80} +1.00000 q^{81} +0.761098 q^{82} +7.59946 q^{83} -1.89128 q^{84} -2.46805 q^{85} -1.92356 q^{86} -6.68466 q^{87} -3.21287 q^{88} +10.6245 q^{89} +0.902098 q^{90} -1.48511 q^{91} -1.89128 q^{92} -1.09790 q^{93} +2.22457 q^{94} -6.85086 q^{95} +3.67384 q^{96} +13.9102 q^{97} -0.329727 q^{98} -2.50407 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} + 5 q^{5} - 2 q^{6} - 4 q^{7} + 9 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} + 5 q^{5} - 2 q^{6} - 4 q^{7} + 9 q^{8} + 4 q^{9} + 2 q^{10} + q^{11} - 4 q^{12} + 7 q^{13} - 2 q^{14} - 5 q^{15} + 8 q^{16} + 2 q^{17} + 2 q^{18} - q^{19} + 13 q^{20} + 4 q^{21} + 11 q^{22} + 4 q^{23} - 9 q^{24} + 11 q^{25} - 19 q^{26} - 4 q^{27} - 4 q^{28} + 2 q^{29} - 2 q^{30} + 6 q^{31} + 20 q^{32} - q^{33} + 23 q^{34} - 5 q^{35} + 4 q^{36} + 16 q^{37} - 11 q^{38} - 7 q^{39} + 3 q^{40} + 5 q^{41} + 2 q^{42} + 9 q^{43} + 3 q^{44} + 5 q^{45} + 2 q^{46} - 21 q^{47} - 8 q^{48} + 4 q^{49} - 17 q^{50} - 2 q^{51} - 24 q^{52} + 10 q^{53} - 2 q^{54} + 17 q^{55} - 9 q^{56} + q^{57} + q^{58} - 26 q^{59} - 13 q^{60} + 2 q^{61} - 19 q^{62} - 4 q^{63} + 27 q^{64} + 26 q^{65} - 11 q^{66} + 5 q^{67} + 7 q^{68} - 4 q^{69} - 2 q^{70} - 19 q^{71} + 9 q^{72} + 10 q^{73} + 9 q^{74} - 11 q^{75} - 3 q^{76} - q^{77} + 19 q^{78} - 6 q^{79} - 24 q^{80} + 4 q^{81} - 31 q^{82} - 2 q^{83} + 4 q^{84} - 7 q^{85} - 17 q^{86} - 2 q^{87} - 20 q^{88} - 17 q^{89} + 2 q^{90} - 7 q^{91} + 4 q^{92} - 6 q^{93} - 44 q^{94} - 17 q^{95} - 20 q^{96} + 32 q^{97} + 2 q^{98} + q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.329727 −0.233152 −0.116576 0.993182i \(-0.537192\pi\)
−0.116576 + 0.993182i \(0.537192\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.89128 −0.945640
\(5\) −2.73589 −1.22353 −0.611764 0.791040i \(-0.709540\pi\)
−0.611764 + 0.791040i \(0.709540\pi\)
\(6\) 0.329727 0.134611
\(7\) −1.00000 −0.377964
\(8\) 1.28306 0.453630
\(9\) 1.00000 0.333333
\(10\) 0.902098 0.285269
\(11\) −2.50407 −0.755005 −0.377502 0.926009i \(-0.623217\pi\)
−0.377502 + 0.926009i \(0.623217\pi\)
\(12\) 1.89128 0.545966
\(13\) 1.48511 0.411896 0.205948 0.978563i \(-0.433972\pi\)
0.205948 + 0.978563i \(0.433972\pi\)
\(14\) 0.329727 0.0881233
\(15\) 2.73589 0.706405
\(16\) 3.35950 0.839875
\(17\) 0.902098 0.218791 0.109396 0.993998i \(-0.465108\pi\)
0.109396 + 0.993998i \(0.465108\pi\)
\(18\) −0.329727 −0.0777174
\(19\) 2.50407 0.574473 0.287236 0.957860i \(-0.407264\pi\)
0.287236 + 0.957860i \(0.407264\pi\)
\(20\) 5.17434 1.15702
\(21\) 1.00000 0.218218
\(22\) 0.825659 0.176031
\(23\) 1.00000 0.208514
\(24\) −1.28306 −0.261904
\(25\) 2.48511 0.497023
\(26\) −0.489682 −0.0960346
\(27\) −1.00000 −0.192450
\(28\) 1.89128 0.357418
\(29\) 6.68466 1.24131 0.620655 0.784084i \(-0.286867\pi\)
0.620655 + 0.784084i \(0.286867\pi\)
\(30\) −0.902098 −0.164700
\(31\) 1.09790 0.197189 0.0985945 0.995128i \(-0.468565\pi\)
0.0985945 + 0.995128i \(0.468565\pi\)
\(32\) −3.67384 −0.649449
\(33\) 2.50407 0.435902
\(34\) −0.297446 −0.0510116
\(35\) 2.73589 0.462450
\(36\) −1.89128 −0.315213
\(37\) 9.91023 1.62923 0.814616 0.580000i \(-0.196948\pi\)
0.814616 + 0.580000i \(0.196948\pi\)
\(38\) −0.825659 −0.133940
\(39\) −1.48511 −0.237808
\(40\) −3.51032 −0.555030
\(41\) −2.30826 −0.360490 −0.180245 0.983622i \(-0.557689\pi\)
−0.180245 + 0.983622i \(0.557689\pi\)
\(42\) −0.329727 −0.0508780
\(43\) 5.83380 0.889645 0.444823 0.895619i \(-0.353267\pi\)
0.444823 + 0.895619i \(0.353267\pi\)
\(44\) 4.73589 0.713963
\(45\) −2.73589 −0.407843
\(46\) −0.329727 −0.0486156
\(47\) −6.74671 −0.984109 −0.492055 0.870564i \(-0.663754\pi\)
−0.492055 + 0.870564i \(0.663754\pi\)
\(48\) −3.35950 −0.484902
\(49\) 1.00000 0.142857
\(50\) −0.819409 −0.115882
\(51\) −0.902098 −0.126319
\(52\) −2.80877 −0.389506
\(53\) −4.91648 −0.675331 −0.337666 0.941266i \(-0.609637\pi\)
−0.337666 + 0.941266i \(0.609637\pi\)
\(54\) 0.329727 0.0448702
\(55\) 6.85086 0.923770
\(56\) −1.28306 −0.171456
\(57\) −2.50407 −0.331672
\(58\) −2.20411 −0.289414
\(59\) −7.32973 −0.954249 −0.477125 0.878836i \(-0.658321\pi\)
−0.477125 + 0.878836i \(0.658321\pi\)
\(60\) −5.17434 −0.668005
\(61\) −1.00625 −0.128837 −0.0644185 0.997923i \(-0.520519\pi\)
−0.0644185 + 0.997923i \(0.520519\pi\)
\(62\) −0.362008 −0.0459751
\(63\) −1.00000 −0.125988
\(64\) −5.50764 −0.688454
\(65\) −4.06311 −0.503967
\(66\) −0.825659 −0.101632
\(67\) 11.2929 1.37964 0.689822 0.723979i \(-0.257689\pi\)
0.689822 + 0.723979i \(0.257689\pi\)
\(68\) −1.70612 −0.206898
\(69\) −1.00000 −0.120386
\(70\) −0.902098 −0.107821
\(71\) −0.362008 −0.0429624 −0.0214812 0.999769i \(-0.506838\pi\)
−0.0214812 + 0.999769i \(0.506838\pi\)
\(72\) 1.28306 0.151210
\(73\) −5.34411 −0.625481 −0.312741 0.949839i \(-0.601247\pi\)
−0.312741 + 0.949839i \(0.601247\pi\)
\(74\) −3.26767 −0.379859
\(75\) −2.48511 −0.286956
\(76\) −4.73589 −0.543244
\(77\) 2.50407 0.285365
\(78\) 0.489682 0.0554456
\(79\) 10.4672 1.17765 0.588827 0.808259i \(-0.299590\pi\)
0.588827 + 0.808259i \(0.299590\pi\)
\(80\) −9.19123 −1.02761
\(81\) 1.00000 0.111111
\(82\) 0.761098 0.0840492
\(83\) 7.59946 0.834149 0.417075 0.908872i \(-0.363055\pi\)
0.417075 + 0.908872i \(0.363055\pi\)
\(84\) −1.89128 −0.206356
\(85\) −2.46805 −0.267697
\(86\) −1.92356 −0.207423
\(87\) −6.68466 −0.716671
\(88\) −3.21287 −0.342493
\(89\) 10.6245 1.12619 0.563097 0.826391i \(-0.309610\pi\)
0.563097 + 0.826391i \(0.309610\pi\)
\(90\) 0.902098 0.0950895
\(91\) −1.48511 −0.155682
\(92\) −1.89128 −0.197180
\(93\) −1.09790 −0.113847
\(94\) 2.22457 0.229447
\(95\) −6.85086 −0.702884
\(96\) 3.67384 0.374960
\(97\) 13.9102 1.41237 0.706185 0.708027i \(-0.250415\pi\)
0.706185 + 0.708027i \(0.250415\pi\)
\(98\) −0.329727 −0.0333075
\(99\) −2.50407 −0.251668
\(100\) −4.70005 −0.470005
\(101\) 1.81502 0.180601 0.0903004 0.995915i \(-0.471217\pi\)
0.0903004 + 0.995915i \(0.471217\pi\)
\(102\) 0.297446 0.0294516
\(103\) −16.6605 −1.64161 −0.820805 0.571209i \(-0.806475\pi\)
−0.820805 + 0.571209i \(0.806475\pi\)
\(104\) 1.90549 0.186849
\(105\) −2.73589 −0.266996
\(106\) 1.62110 0.157455
\(107\) 1.68092 0.162500 0.0812502 0.996694i \(-0.474109\pi\)
0.0812502 + 0.996694i \(0.474109\pi\)
\(108\) 1.89128 0.181989
\(109\) 11.4333 1.09511 0.547554 0.836771i \(-0.315559\pi\)
0.547554 + 0.836771i \(0.315559\pi\)
\(110\) −2.25892 −0.215379
\(111\) −9.91023 −0.940638
\(112\) −3.35950 −0.317443
\(113\) 3.14914 0.296246 0.148123 0.988969i \(-0.452677\pi\)
0.148123 + 0.988969i \(0.452677\pi\)
\(114\) 0.825659 0.0773301
\(115\) −2.73589 −0.255123
\(116\) −12.6426 −1.17383
\(117\) 1.48511 0.137299
\(118\) 2.41681 0.222485
\(119\) −0.902098 −0.0826952
\(120\) 3.51032 0.320447
\(121\) −4.72964 −0.429968
\(122\) 0.331788 0.0300387
\(123\) 2.30826 0.208129
\(124\) −2.07644 −0.186470
\(125\) 6.88046 0.615407
\(126\) 0.329727 0.0293744
\(127\) 0.449266 0.0398659 0.0199329 0.999801i \(-0.493655\pi\)
0.0199329 + 0.999801i \(0.493655\pi\)
\(128\) 9.16370 0.809964
\(129\) −5.83380 −0.513637
\(130\) 1.33972 0.117501
\(131\) −19.9759 −1.74530 −0.872649 0.488347i \(-0.837600\pi\)
−0.872649 + 0.488347i \(0.837600\pi\)
\(132\) −4.73589 −0.412207
\(133\) −2.50407 −0.217130
\(134\) −3.72357 −0.321667
\(135\) 2.73589 0.235468
\(136\) 1.15745 0.0992503
\(137\) 0.270356 0.0230981 0.0115490 0.999933i \(-0.496324\pi\)
0.0115490 + 0.999933i \(0.496324\pi\)
\(138\) 0.329727 0.0280682
\(139\) 4.42512 0.375334 0.187667 0.982233i \(-0.439907\pi\)
0.187667 + 0.982233i \(0.439907\pi\)
\(140\) −5.17434 −0.437312
\(141\) 6.74671 0.568176
\(142\) 0.119364 0.0100168
\(143\) −3.71883 −0.310984
\(144\) 3.35950 0.279958
\(145\) −18.2885 −1.51878
\(146\) 1.76210 0.145832
\(147\) −1.00000 −0.0824786
\(148\) −18.7430 −1.54067
\(149\) −3.69530 −0.302731 −0.151365 0.988478i \(-0.548367\pi\)
−0.151365 + 0.988478i \(0.548367\pi\)
\(150\) 0.819409 0.0669045
\(151\) 8.63174 0.702441 0.351221 0.936293i \(-0.385767\pi\)
0.351221 + 0.936293i \(0.385767\pi\)
\(152\) 3.21287 0.260598
\(153\) 0.902098 0.0729303
\(154\) −0.825659 −0.0665335
\(155\) −3.00374 −0.241266
\(156\) 2.80877 0.224881
\(157\) 9.93438 0.792850 0.396425 0.918067i \(-0.370251\pi\)
0.396425 + 0.918067i \(0.370251\pi\)
\(158\) −3.45133 −0.274573
\(159\) 4.91648 0.389903
\(160\) 10.0512 0.794620
\(161\) −1.00000 −0.0788110
\(162\) −0.329727 −0.0259058
\(163\) 14.7082 1.15203 0.576017 0.817438i \(-0.304606\pi\)
0.576017 + 0.817438i \(0.304606\pi\)
\(164\) 4.36558 0.340894
\(165\) −6.85086 −0.533339
\(166\) −2.50575 −0.194484
\(167\) −5.47430 −0.423614 −0.211807 0.977312i \(-0.567935\pi\)
−0.211807 + 0.977312i \(0.567935\pi\)
\(168\) 1.28306 0.0989903
\(169\) −10.7944 −0.830341
\(170\) 0.813782 0.0624142
\(171\) 2.50407 0.191491
\(172\) −11.0333 −0.841284
\(173\) 18.1727 1.38165 0.690823 0.723024i \(-0.257248\pi\)
0.690823 + 0.723024i \(0.257248\pi\)
\(174\) 2.20411 0.167093
\(175\) −2.48511 −0.187857
\(176\) −8.41242 −0.634110
\(177\) 7.32973 0.550936
\(178\) −3.50318 −0.262575
\(179\) 10.1860 0.761341 0.380670 0.924711i \(-0.375693\pi\)
0.380670 + 0.924711i \(0.375693\pi\)
\(180\) 5.17434 0.385673
\(181\) 13.5186 1.00483 0.502416 0.864626i \(-0.332445\pi\)
0.502416 + 0.864626i \(0.332445\pi\)
\(182\) 0.489682 0.0362977
\(183\) 1.00625 0.0743841
\(184\) 1.28306 0.0945885
\(185\) −27.1133 −1.99341
\(186\) 0.362008 0.0265437
\(187\) −2.25892 −0.165188
\(188\) 12.7599 0.930613
\(189\) 1.00000 0.0727393
\(190\) 2.25892 0.163879
\(191\) −0.631742 −0.0457113 −0.0228556 0.999739i \(-0.507276\pi\)
−0.0228556 + 0.999739i \(0.507276\pi\)
\(192\) 5.50764 0.397479
\(193\) 1.36826 0.0984893 0.0492447 0.998787i \(-0.484319\pi\)
0.0492447 + 0.998787i \(0.484319\pi\)
\(194\) −4.58658 −0.329297
\(195\) 4.06311 0.290966
\(196\) −1.89128 −0.135091
\(197\) 24.7611 1.76416 0.882078 0.471104i \(-0.156144\pi\)
0.882078 + 0.471104i \(0.156144\pi\)
\(198\) 0.825659 0.0586770
\(199\) 1.44470 0.102412 0.0512059 0.998688i \(-0.483694\pi\)
0.0512059 + 0.998688i \(0.483694\pi\)
\(200\) 3.18855 0.225465
\(201\) −11.2929 −0.796538
\(202\) −0.598460 −0.0421075
\(203\) −6.68466 −0.469171
\(204\) 1.70612 0.119452
\(205\) 6.31517 0.441070
\(206\) 5.49342 0.382745
\(207\) 1.00000 0.0695048
\(208\) 4.98924 0.345941
\(209\) −6.27036 −0.433730
\(210\) 0.902098 0.0622507
\(211\) −23.0990 −1.59020 −0.795099 0.606480i \(-0.792581\pi\)
−0.795099 + 0.606480i \(0.792581\pi\)
\(212\) 9.29845 0.638620
\(213\) 0.362008 0.0248044
\(214\) −0.554244 −0.0378873
\(215\) −15.9606 −1.08851
\(216\) −1.28306 −0.0873012
\(217\) −1.09790 −0.0745304
\(218\) −3.76986 −0.255327
\(219\) 5.34411 0.361122
\(220\) −12.9569 −0.873554
\(221\) 1.33972 0.0901192
\(222\) 3.26767 0.219312
\(223\) −18.6497 −1.24888 −0.624438 0.781074i \(-0.714672\pi\)
−0.624438 + 0.781074i \(0.714672\pi\)
\(224\) 3.67384 0.245469
\(225\) 2.48511 0.165674
\(226\) −1.03836 −0.0690704
\(227\) 20.6334 1.36949 0.684744 0.728783i \(-0.259914\pi\)
0.684744 + 0.728783i \(0.259914\pi\)
\(228\) 4.73589 0.313642
\(229\) 14.9544 0.988214 0.494107 0.869401i \(-0.335495\pi\)
0.494107 + 0.869401i \(0.335495\pi\)
\(230\) 0.902098 0.0594826
\(231\) −2.50407 −0.164756
\(232\) 8.57682 0.563096
\(233\) 28.1052 1.84123 0.920617 0.390467i \(-0.127687\pi\)
0.920617 + 0.390467i \(0.127687\pi\)
\(234\) −0.489682 −0.0320115
\(235\) 18.4583 1.20409
\(236\) 13.8626 0.902376
\(237\) −10.4672 −0.679919
\(238\) 0.297446 0.0192806
\(239\) 25.8536 1.67233 0.836166 0.548476i \(-0.184792\pi\)
0.836166 + 0.548476i \(0.184792\pi\)
\(240\) 9.19123 0.593292
\(241\) −14.8311 −0.955356 −0.477678 0.878535i \(-0.658521\pi\)
−0.477678 + 0.878535i \(0.658521\pi\)
\(242\) 1.55949 0.100248
\(243\) −1.00000 −0.0641500
\(244\) 1.90310 0.121833
\(245\) −2.73589 −0.174790
\(246\) −0.761098 −0.0485258
\(247\) 3.71883 0.236623
\(248\) 1.40867 0.0894509
\(249\) −7.59946 −0.481596
\(250\) −2.26868 −0.143484
\(251\) −16.2841 −1.02784 −0.513922 0.857837i \(-0.671808\pi\)
−0.513922 + 0.857837i \(0.671808\pi\)
\(252\) 1.89128 0.119139
\(253\) −2.50407 −0.157429
\(254\) −0.148135 −0.00929482
\(255\) 2.46805 0.154555
\(256\) 7.99375 0.499609
\(257\) 22.9446 1.43125 0.715623 0.698486i \(-0.246143\pi\)
0.715623 + 0.698486i \(0.246143\pi\)
\(258\) 1.92356 0.119756
\(259\) −9.91023 −0.615792
\(260\) 7.68448 0.476571
\(261\) 6.68466 0.413770
\(262\) 6.58658 0.406920
\(263\) −0.776932 −0.0479077 −0.0239538 0.999713i \(-0.507625\pi\)
−0.0239538 + 0.999713i \(0.507625\pi\)
\(264\) 3.21287 0.197739
\(265\) 13.4510 0.826287
\(266\) 0.825659 0.0506244
\(267\) −10.6245 −0.650208
\(268\) −21.3580 −1.30465
\(269\) 3.05937 0.186533 0.0932666 0.995641i \(-0.470269\pi\)
0.0932666 + 0.995641i \(0.470269\pi\)
\(270\) −0.902098 −0.0549000
\(271\) −16.0631 −0.975765 −0.487882 0.872909i \(-0.662230\pi\)
−0.487882 + 0.872909i \(0.662230\pi\)
\(272\) 3.03060 0.183757
\(273\) 1.48511 0.0898832
\(274\) −0.0891438 −0.00538537
\(275\) −6.22289 −0.375255
\(276\) 1.89128 0.113842
\(277\) −11.6103 −0.697594 −0.348797 0.937198i \(-0.613410\pi\)
−0.348797 + 0.937198i \(0.613410\pi\)
\(278\) −1.45908 −0.0875100
\(279\) 1.09790 0.0657296
\(280\) 3.51032 0.209782
\(281\) 9.89753 0.590437 0.295219 0.955430i \(-0.404608\pi\)
0.295219 + 0.955430i \(0.404608\pi\)
\(282\) −2.22457 −0.132471
\(283\) 13.9994 0.832177 0.416088 0.909324i \(-0.363401\pi\)
0.416088 + 0.909324i \(0.363401\pi\)
\(284\) 0.684658 0.0406270
\(285\) 6.85086 0.405810
\(286\) 1.22620 0.0725066
\(287\) 2.30826 0.136253
\(288\) −3.67384 −0.216483
\(289\) −16.1862 −0.952130
\(290\) 6.03022 0.354107
\(291\) −13.9102 −0.815432
\(292\) 10.1072 0.591480
\(293\) −8.14038 −0.475566 −0.237783 0.971318i \(-0.576421\pi\)
−0.237783 + 0.971318i \(0.576421\pi\)
\(294\) 0.329727 0.0192301
\(295\) 20.0534 1.16755
\(296\) 12.7154 0.739070
\(297\) 2.50407 0.145301
\(298\) 1.21844 0.0705824
\(299\) 1.48511 0.0858863
\(300\) 4.70005 0.271357
\(301\) −5.83380 −0.336254
\(302\) −2.84612 −0.163776
\(303\) −1.81502 −0.104270
\(304\) 8.41242 0.482485
\(305\) 2.75299 0.157636
\(306\) −0.297446 −0.0170039
\(307\) 13.5095 0.771027 0.385514 0.922702i \(-0.374024\pi\)
0.385514 + 0.922702i \(0.374024\pi\)
\(308\) −4.73589 −0.269853
\(309\) 16.6605 0.947783
\(310\) 0.990415 0.0562518
\(311\) −24.3035 −1.37813 −0.689063 0.724701i \(-0.741978\pi\)
−0.689063 + 0.724701i \(0.741978\pi\)
\(312\) −1.90549 −0.107877
\(313\) −33.7836 −1.90956 −0.954782 0.297308i \(-0.903911\pi\)
−0.954782 + 0.297308i \(0.903911\pi\)
\(314\) −3.27563 −0.184855
\(315\) 2.73589 0.154150
\(316\) −19.7964 −1.11364
\(317\) 32.1594 1.80625 0.903126 0.429376i \(-0.141267\pi\)
0.903126 + 0.429376i \(0.141267\pi\)
\(318\) −1.62110 −0.0909067
\(319\) −16.7388 −0.937195
\(320\) 15.0683 0.842344
\(321\) −1.68092 −0.0938196
\(322\) 0.329727 0.0183750
\(323\) 2.25892 0.125689
\(324\) −1.89128 −0.105071
\(325\) 3.69068 0.204722
\(326\) −4.84969 −0.268599
\(327\) −11.4333 −0.632261
\(328\) −2.96164 −0.163529
\(329\) 6.74671 0.371958
\(330\) 2.25892 0.124349
\(331\) −4.84004 −0.266033 −0.133016 0.991114i \(-0.542466\pi\)
−0.133016 + 0.991114i \(0.542466\pi\)
\(332\) −14.3727 −0.788805
\(333\) 9.91023 0.543077
\(334\) 1.80502 0.0987665
\(335\) −30.8961 −1.68803
\(336\) 3.35950 0.183276
\(337\) −2.59195 −0.141192 −0.0705962 0.997505i \(-0.522490\pi\)
−0.0705962 + 0.997505i \(0.522490\pi\)
\(338\) 3.55922 0.193596
\(339\) −3.14914 −0.171038
\(340\) 4.66776 0.253145
\(341\) −2.74922 −0.148879
\(342\) −0.825659 −0.0446465
\(343\) −1.00000 −0.0539949
\(344\) 7.48511 0.403570
\(345\) 2.73589 0.147296
\(346\) −5.99204 −0.322134
\(347\) −15.9390 −0.855651 −0.427825 0.903861i \(-0.640720\pi\)
−0.427825 + 0.903861i \(0.640720\pi\)
\(348\) 12.6426 0.677712
\(349\) −8.02603 −0.429624 −0.214812 0.976655i \(-0.568914\pi\)
−0.214812 + 0.976655i \(0.568914\pi\)
\(350\) 0.819409 0.0437993
\(351\) −1.48511 −0.0792695
\(352\) 9.19954 0.490337
\(353\) 10.7871 0.574141 0.287070 0.957909i \(-0.407319\pi\)
0.287070 + 0.957909i \(0.407319\pi\)
\(354\) −2.41681 −0.128452
\(355\) 0.990415 0.0525658
\(356\) −20.0939 −1.06497
\(357\) 0.902098 0.0477441
\(358\) −3.35862 −0.177508
\(359\) −2.17891 −0.114998 −0.0574992 0.998346i \(-0.518313\pi\)
−0.0574992 + 0.998346i \(0.518313\pi\)
\(360\) −3.51032 −0.185010
\(361\) −12.7296 −0.669981
\(362\) −4.45746 −0.234279
\(363\) 4.72964 0.248242
\(364\) 2.80877 0.147219
\(365\) 14.6209 0.765294
\(366\) −0.331788 −0.0173428
\(367\) 27.0388 1.41141 0.705707 0.708504i \(-0.250630\pi\)
0.705707 + 0.708504i \(0.250630\pi\)
\(368\) 3.35950 0.175126
\(369\) −2.30826 −0.120163
\(370\) 8.94001 0.464769
\(371\) 4.91648 0.255251
\(372\) 2.07644 0.107658
\(373\) 16.4189 0.850140 0.425070 0.905161i \(-0.360249\pi\)
0.425070 + 0.905161i \(0.360249\pi\)
\(374\) 0.744826 0.0385140
\(375\) −6.88046 −0.355306
\(376\) −8.65644 −0.446422
\(377\) 9.92748 0.511291
\(378\) −0.329727 −0.0169593
\(379\) −34.6399 −1.77933 −0.889667 0.456610i \(-0.849064\pi\)
−0.889667 + 0.456610i \(0.849064\pi\)
\(380\) 12.9569 0.664675
\(381\) −0.449266 −0.0230166
\(382\) 0.208303 0.0106577
\(383\) −10.6722 −0.545322 −0.272661 0.962110i \(-0.587904\pi\)
−0.272661 + 0.962110i \(0.587904\pi\)
\(384\) −9.16370 −0.467633
\(385\) −6.85086 −0.349152
\(386\) −0.451152 −0.0229630
\(387\) 5.83380 0.296548
\(388\) −26.3081 −1.33559
\(389\) 0.0302201 0.00153222 0.000766110 1.00000i \(-0.499756\pi\)
0.000766110 1.00000i \(0.499756\pi\)
\(390\) −1.33972 −0.0678393
\(391\) 0.902098 0.0456211
\(392\) 1.28306 0.0648044
\(393\) 19.9759 1.00765
\(394\) −8.16441 −0.411317
\(395\) −28.6372 −1.44089
\(396\) 4.73589 0.237988
\(397\) 20.4749 1.02761 0.513803 0.857908i \(-0.328236\pi\)
0.513803 + 0.857908i \(0.328236\pi\)
\(398\) −0.476356 −0.0238776
\(399\) 2.50407 0.125360
\(400\) 8.34874 0.417437
\(401\) −36.4304 −1.81925 −0.909623 0.415435i \(-0.863629\pi\)
−0.909623 + 0.415435i \(0.863629\pi\)
\(402\) 3.72357 0.185715
\(403\) 1.63051 0.0812214
\(404\) −3.43270 −0.170783
\(405\) −2.73589 −0.135948
\(406\) 2.20411 0.109388
\(407\) −24.8159 −1.23008
\(408\) −1.15745 −0.0573022
\(409\) 37.4054 1.84958 0.924790 0.380479i \(-0.124241\pi\)
0.924790 + 0.380479i \(0.124241\pi\)
\(410\) −2.08228 −0.102837
\(411\) −0.270356 −0.0133357
\(412\) 31.5097 1.55237
\(413\) 7.32973 0.360672
\(414\) −0.329727 −0.0162052
\(415\) −20.7913 −1.02061
\(416\) −5.45607 −0.267506
\(417\) −4.42512 −0.216699
\(418\) 2.06751 0.101125
\(419\) −37.2017 −1.81742 −0.908710 0.417428i \(-0.862932\pi\)
−0.908710 + 0.417428i \(0.862932\pi\)
\(420\) 5.17434 0.252482
\(421\) 27.6326 1.34673 0.673366 0.739309i \(-0.264848\pi\)
0.673366 + 0.739309i \(0.264848\pi\)
\(422\) 7.61636 0.370758
\(423\) −6.74671 −0.328036
\(424\) −6.30815 −0.306351
\(425\) 2.24182 0.108744
\(426\) −0.119364 −0.00578320
\(427\) 1.00625 0.0486958
\(428\) −3.17908 −0.153667
\(429\) 3.71883 0.179547
\(430\) 5.26266 0.253788
\(431\) 35.0676 1.68915 0.844573 0.535441i \(-0.179855\pi\)
0.844573 + 0.535441i \(0.179855\pi\)
\(432\) −3.35950 −0.161634
\(433\) −1.00708 −0.0483970 −0.0241985 0.999707i \(-0.507703\pi\)
−0.0241985 + 0.999707i \(0.507703\pi\)
\(434\) 0.362008 0.0173769
\(435\) 18.2885 0.876867
\(436\) −21.6235 −1.03558
\(437\) 2.50407 0.119786
\(438\) −1.76210 −0.0841964
\(439\) −4.04649 −0.193129 −0.0965643 0.995327i \(-0.530785\pi\)
−0.0965643 + 0.995327i \(0.530785\pi\)
\(440\) 8.79007 0.419050
\(441\) 1.00000 0.0476190
\(442\) −0.441742 −0.0210115
\(443\) −29.3947 −1.39659 −0.698293 0.715812i \(-0.746057\pi\)
−0.698293 + 0.715812i \(0.746057\pi\)
\(444\) 18.7430 0.889505
\(445\) −29.0675 −1.37793
\(446\) 6.14931 0.291178
\(447\) 3.69530 0.174782
\(448\) 5.50764 0.260211
\(449\) −4.87627 −0.230126 −0.115063 0.993358i \(-0.536707\pi\)
−0.115063 + 0.993358i \(0.536707\pi\)
\(450\) −0.819409 −0.0386273
\(451\) 5.78005 0.272172
\(452\) −5.95590 −0.280142
\(453\) −8.63174 −0.405555
\(454\) −6.80340 −0.319299
\(455\) 4.06311 0.190482
\(456\) −3.21287 −0.150456
\(457\) −35.5212 −1.66161 −0.830806 0.556562i \(-0.812120\pi\)
−0.830806 + 0.556562i \(0.812120\pi\)
\(458\) −4.93087 −0.230404
\(459\) −0.902098 −0.0421064
\(460\) 5.17434 0.241255
\(461\) −18.9442 −0.882319 −0.441160 0.897429i \(-0.645433\pi\)
−0.441160 + 0.897429i \(0.645433\pi\)
\(462\) 0.825659 0.0384131
\(463\) 14.1933 0.659618 0.329809 0.944048i \(-0.393016\pi\)
0.329809 + 0.944048i \(0.393016\pi\)
\(464\) 22.4571 1.04255
\(465\) 3.00374 0.139295
\(466\) −9.26705 −0.429288
\(467\) −31.0355 −1.43615 −0.718075 0.695966i \(-0.754976\pi\)
−0.718075 + 0.695966i \(0.754976\pi\)
\(468\) −2.80877 −0.129835
\(469\) −11.2929 −0.521457
\(470\) −6.08620 −0.280735
\(471\) −9.93438 −0.457752
\(472\) −9.40449 −0.432877
\(473\) −14.6082 −0.671687
\(474\) 3.45133 0.158525
\(475\) 6.22289 0.285526
\(476\) 1.70612 0.0781999
\(477\) −4.91648 −0.225110
\(478\) −8.52465 −0.389908
\(479\) 10.5214 0.480735 0.240367 0.970682i \(-0.422732\pi\)
0.240367 + 0.970682i \(0.422732\pi\)
\(480\) −10.0512 −0.458774
\(481\) 14.7178 0.671075
\(482\) 4.89022 0.222743
\(483\) 1.00000 0.0455016
\(484\) 8.94508 0.406595
\(485\) −38.0569 −1.72808
\(486\) 0.329727 0.0149567
\(487\) −31.7378 −1.43818 −0.719089 0.694918i \(-0.755441\pi\)
−0.719089 + 0.694918i \(0.755441\pi\)
\(488\) −1.29108 −0.0584444
\(489\) −14.7082 −0.665127
\(490\) 0.902098 0.0407527
\(491\) −31.8788 −1.43867 −0.719336 0.694662i \(-0.755554\pi\)
−0.719336 + 0.694662i \(0.755554\pi\)
\(492\) −4.36558 −0.196815
\(493\) 6.03022 0.271587
\(494\) −1.22620 −0.0551692
\(495\) 6.85086 0.307923
\(496\) 3.68840 0.165614
\(497\) 0.362008 0.0162383
\(498\) 2.50575 0.112285
\(499\) −39.4279 −1.76503 −0.882517 0.470280i \(-0.844153\pi\)
−0.882517 + 0.470280i \(0.844153\pi\)
\(500\) −13.0129 −0.581954
\(501\) 5.47430 0.244573
\(502\) 5.36932 0.239644
\(503\) 24.4589 1.09057 0.545284 0.838251i \(-0.316422\pi\)
0.545284 + 0.838251i \(0.316422\pi\)
\(504\) −1.28306 −0.0571521
\(505\) −4.96569 −0.220970
\(506\) 0.825659 0.0367050
\(507\) 10.7944 0.479398
\(508\) −0.849687 −0.0376988
\(509\) 20.3927 0.903889 0.451944 0.892046i \(-0.350731\pi\)
0.451944 + 0.892046i \(0.350731\pi\)
\(510\) −0.813782 −0.0360349
\(511\) 5.34411 0.236410
\(512\) −20.9632 −0.926449
\(513\) −2.50407 −0.110557
\(514\) −7.56547 −0.333699
\(515\) 45.5814 2.00856
\(516\) 11.0333 0.485716
\(517\) 16.8942 0.743007
\(518\) 3.26767 0.143573
\(519\) −18.1727 −0.797694
\(520\) −5.21322 −0.228615
\(521\) 26.3058 1.15248 0.576238 0.817282i \(-0.304520\pi\)
0.576238 + 0.817282i \(0.304520\pi\)
\(522\) −2.20411 −0.0964714
\(523\) −40.4256 −1.76769 −0.883843 0.467783i \(-0.845053\pi\)
−0.883843 + 0.467783i \(0.845053\pi\)
\(524\) 37.7799 1.65042
\(525\) 2.48511 0.108459
\(526\) 0.256176 0.0111698
\(527\) 0.990415 0.0431432
\(528\) 8.41242 0.366103
\(529\) 1.00000 0.0434783
\(530\) −4.43515 −0.192651
\(531\) −7.32973 −0.318083
\(532\) 4.73589 0.205327
\(533\) −3.42804 −0.148485
\(534\) 3.50318 0.151598
\(535\) −4.59881 −0.198824
\(536\) 14.4894 0.625849
\(537\) −10.1860 −0.439560
\(538\) −1.00876 −0.0434906
\(539\) −2.50407 −0.107858
\(540\) −5.17434 −0.222668
\(541\) 0.360122 0.0154828 0.00774142 0.999970i \(-0.497536\pi\)
0.00774142 + 0.999970i \(0.497536\pi\)
\(542\) 5.29644 0.227502
\(543\) −13.5186 −0.580140
\(544\) −3.31417 −0.142094
\(545\) −31.2802 −1.33990
\(546\) −0.489682 −0.0209565
\(547\) 20.2023 0.863789 0.431894 0.901924i \(-0.357845\pi\)
0.431894 + 0.901924i \(0.357845\pi\)
\(548\) −0.511319 −0.0218425
\(549\) −1.00625 −0.0429457
\(550\) 2.05186 0.0874915
\(551\) 16.7388 0.713099
\(552\) −1.28306 −0.0546107
\(553\) −10.4672 −0.445111
\(554\) 3.82822 0.162646
\(555\) 27.1133 1.15090
\(556\) −8.36914 −0.354931
\(557\) 9.71086 0.411463 0.205731 0.978609i \(-0.434043\pi\)
0.205731 + 0.978609i \(0.434043\pi\)
\(558\) −0.362008 −0.0153250
\(559\) 8.66385 0.366442
\(560\) 9.19123 0.388401
\(561\) 2.25892 0.0953715
\(562\) −3.26348 −0.137662
\(563\) 27.2789 1.14967 0.574835 0.818269i \(-0.305066\pi\)
0.574835 + 0.818269i \(0.305066\pi\)
\(564\) −12.7599 −0.537290
\(565\) −8.61570 −0.362465
\(566\) −4.61598 −0.194024
\(567\) −1.00000 −0.0419961
\(568\) −0.464478 −0.0194891
\(569\) 19.6132 0.822229 0.411115 0.911584i \(-0.365140\pi\)
0.411115 + 0.911584i \(0.365140\pi\)
\(570\) −2.25892 −0.0946156
\(571\) 21.1859 0.886601 0.443301 0.896373i \(-0.353807\pi\)
0.443301 + 0.896373i \(0.353807\pi\)
\(572\) 7.03334 0.294079
\(573\) 0.631742 0.0263914
\(574\) −0.761098 −0.0317676
\(575\) 2.48511 0.103636
\(576\) −5.50764 −0.229485
\(577\) −12.4079 −0.516547 −0.258273 0.966072i \(-0.583154\pi\)
−0.258273 + 0.966072i \(0.583154\pi\)
\(578\) 5.33704 0.221991
\(579\) −1.36826 −0.0568629
\(580\) 34.5887 1.43622
\(581\) −7.59946 −0.315279
\(582\) 4.58658 0.190120
\(583\) 12.3112 0.509878
\(584\) −6.85682 −0.283737
\(585\) −4.06311 −0.167989
\(586\) 2.68410 0.110879
\(587\) −9.35493 −0.386119 −0.193060 0.981187i \(-0.561841\pi\)
−0.193060 + 0.981187i \(0.561841\pi\)
\(588\) 1.89128 0.0779951
\(589\) 2.74922 0.113280
\(590\) −6.61214 −0.272217
\(591\) −24.7611 −1.01854
\(592\) 33.2934 1.36835
\(593\) −16.7836 −0.689218 −0.344609 0.938746i \(-0.611989\pi\)
−0.344609 + 0.938746i \(0.611989\pi\)
\(594\) −0.825659 −0.0338772
\(595\) 2.46805 0.101180
\(596\) 6.98885 0.286275
\(597\) −1.44470 −0.0591275
\(598\) −0.489682 −0.0200246
\(599\) −16.0512 −0.655836 −0.327918 0.944706i \(-0.606347\pi\)
−0.327918 + 0.944706i \(0.606347\pi\)
\(600\) −3.18855 −0.130172
\(601\) −24.4364 −0.996781 −0.498390 0.866953i \(-0.666075\pi\)
−0.498390 + 0.866953i \(0.666075\pi\)
\(602\) 1.92356 0.0783985
\(603\) 11.2929 0.459882
\(604\) −16.3250 −0.664257
\(605\) 12.9398 0.526078
\(606\) 0.598460 0.0243108
\(607\) 28.2329 1.14594 0.572969 0.819577i \(-0.305792\pi\)
0.572969 + 0.819577i \(0.305792\pi\)
\(608\) −9.19954 −0.373091
\(609\) 6.68466 0.270876
\(610\) −0.907736 −0.0367532
\(611\) −10.0196 −0.405351
\(612\) −1.70612 −0.0689658
\(613\) 17.9873 0.726500 0.363250 0.931692i \(-0.381667\pi\)
0.363250 + 0.931692i \(0.381667\pi\)
\(614\) −4.45445 −0.179767
\(615\) −6.31517 −0.254652
\(616\) 3.21287 0.129450
\(617\) 12.9998 0.523353 0.261677 0.965156i \(-0.415725\pi\)
0.261677 + 0.965156i \(0.415725\pi\)
\(618\) −5.49342 −0.220978
\(619\) −1.26266 −0.0507505 −0.0253752 0.999678i \(-0.508078\pi\)
−0.0253752 + 0.999678i \(0.508078\pi\)
\(620\) 5.68092 0.228151
\(621\) −1.00000 −0.0401286
\(622\) 8.01353 0.321313
\(623\) −10.6245 −0.425661
\(624\) −4.98924 −0.199729
\(625\) −31.2498 −1.24999
\(626\) 11.1394 0.445219
\(627\) 6.27036 0.250414
\(628\) −18.7887 −0.749750
\(629\) 8.94001 0.356461
\(630\) −0.902098 −0.0359405
\(631\) −24.4568 −0.973611 −0.486806 0.873510i \(-0.661838\pi\)
−0.486806 + 0.873510i \(0.661838\pi\)
\(632\) 13.4301 0.534220
\(633\) 23.0990 0.918101
\(634\) −10.6038 −0.421132
\(635\) −1.22914 −0.0487771
\(636\) −9.29845 −0.368707
\(637\) 1.48511 0.0588423
\(638\) 5.51925 0.218509
\(639\) −0.362008 −0.0143208
\(640\) −25.0709 −0.991014
\(641\) −1.29037 −0.0509665 −0.0254833 0.999675i \(-0.508112\pi\)
−0.0254833 + 0.999675i \(0.508112\pi\)
\(642\) 0.554244 0.0218743
\(643\) 34.3110 1.35309 0.676547 0.736399i \(-0.263476\pi\)
0.676547 + 0.736399i \(0.263476\pi\)
\(644\) 1.89128 0.0745269
\(645\) 15.9606 0.628450
\(646\) −0.744826 −0.0293048
\(647\) −2.62092 −0.103039 −0.0515196 0.998672i \(-0.516406\pi\)
−0.0515196 + 0.998672i \(0.516406\pi\)
\(648\) 1.28306 0.0504034
\(649\) 18.3541 0.720463
\(650\) −1.21692 −0.0477314
\(651\) 1.09790 0.0430302
\(652\) −27.8173 −1.08941
\(653\) 7.53697 0.294944 0.147472 0.989066i \(-0.452886\pi\)
0.147472 + 0.989066i \(0.452886\pi\)
\(654\) 3.76986 0.147413
\(655\) 54.6518 2.13542
\(656\) −7.75462 −0.302767
\(657\) −5.34411 −0.208494
\(658\) −2.22457 −0.0867229
\(659\) 22.8213 0.888990 0.444495 0.895781i \(-0.353383\pi\)
0.444495 + 0.895781i \(0.353383\pi\)
\(660\) 12.9569 0.504347
\(661\) 27.3668 1.06445 0.532223 0.846604i \(-0.321357\pi\)
0.532223 + 0.846604i \(0.321357\pi\)
\(662\) 1.59589 0.0620262
\(663\) −1.33972 −0.0520304
\(664\) 9.75057 0.378396
\(665\) 6.85086 0.265665
\(666\) −3.26767 −0.126620
\(667\) 6.68466 0.258831
\(668\) 10.3534 0.400586
\(669\) 18.6497 0.721039
\(670\) 10.1873 0.393569
\(671\) 2.51972 0.0972726
\(672\) −3.67384 −0.141721
\(673\) 22.9790 0.885774 0.442887 0.896577i \(-0.353954\pi\)
0.442887 + 0.896577i \(0.353954\pi\)
\(674\) 0.854636 0.0329193
\(675\) −2.48511 −0.0956521
\(676\) 20.4153 0.785204
\(677\) −8.80814 −0.338524 −0.169262 0.985571i \(-0.554138\pi\)
−0.169262 + 0.985571i \(0.554138\pi\)
\(678\) 1.03836 0.0398778
\(679\) −13.9102 −0.533826
\(680\) −3.16665 −0.121436
\(681\) −20.6334 −0.790674
\(682\) 0.906493 0.0347114
\(683\) 34.4889 1.31968 0.659840 0.751406i \(-0.270624\pi\)
0.659840 + 0.751406i \(0.270624\pi\)
\(684\) −4.73589 −0.181081
\(685\) −0.739666 −0.0282612
\(686\) 0.329727 0.0125890
\(687\) −14.9544 −0.570546
\(688\) 19.5986 0.747191
\(689\) −7.30154 −0.278166
\(690\) −0.902098 −0.0343423
\(691\) 37.6110 1.43079 0.715395 0.698721i \(-0.246247\pi\)
0.715395 + 0.698721i \(0.246247\pi\)
\(692\) −34.3697 −1.30654
\(693\) 2.50407 0.0951217
\(694\) 5.25552 0.199497
\(695\) −12.1067 −0.459232
\(696\) −8.57682 −0.325104
\(697\) −2.08228 −0.0788721
\(698\) 2.64640 0.100168
\(699\) −28.1052 −1.06304
\(700\) 4.70005 0.177645
\(701\) −47.5348 −1.79536 −0.897682 0.440644i \(-0.854750\pi\)
−0.897682 + 0.440644i \(0.854750\pi\)
\(702\) 0.489682 0.0184819
\(703\) 24.8159 0.935949
\(704\) 13.7915 0.519786
\(705\) −18.4583 −0.695179
\(706\) −3.55681 −0.133862
\(707\) −1.81502 −0.0682607
\(708\) −13.8626 −0.520987
\(709\) −41.1332 −1.54479 −0.772395 0.635143i \(-0.780941\pi\)
−0.772395 + 0.635143i \(0.780941\pi\)
\(710\) −0.326567 −0.0122558
\(711\) 10.4672 0.392551
\(712\) 13.6319 0.510876
\(713\) 1.09790 0.0411167
\(714\) −0.297446 −0.0111317
\(715\) 10.1743 0.380498
\(716\) −19.2647 −0.719954
\(717\) −25.8536 −0.965522
\(718\) 0.718446 0.0268122
\(719\) 8.25857 0.307993 0.153996 0.988071i \(-0.450786\pi\)
0.153996 + 0.988071i \(0.450786\pi\)
\(720\) −9.19123 −0.342537
\(721\) 16.6605 0.620470
\(722\) 4.19731 0.156208
\(723\) 14.8311 0.551575
\(724\) −25.5675 −0.950209
\(725\) 16.6121 0.616959
\(726\) −1.55949 −0.0578782
\(727\) 0.985235 0.0365403 0.0182702 0.999833i \(-0.494184\pi\)
0.0182702 + 0.999833i \(0.494184\pi\)
\(728\) −1.90549 −0.0706222
\(729\) 1.00000 0.0370370
\(730\) −4.82092 −0.178430
\(731\) 5.26266 0.194646
\(732\) −1.90310 −0.0703406
\(733\) 27.7847 1.02625 0.513125 0.858314i \(-0.328488\pi\)
0.513125 + 0.858314i \(0.328488\pi\)
\(734\) −8.91542 −0.329074
\(735\) 2.73589 0.100915
\(736\) −3.67384 −0.135420
\(737\) −28.2781 −1.04164
\(738\) 0.761098 0.0280164
\(739\) 30.0540 1.10555 0.552777 0.833329i \(-0.313568\pi\)
0.552777 + 0.833329i \(0.313568\pi\)
\(740\) 51.2789 1.88505
\(741\) −3.71883 −0.136614
\(742\) −1.62110 −0.0595124
\(743\) 40.1617 1.47339 0.736695 0.676225i \(-0.236385\pi\)
0.736695 + 0.676225i \(0.236385\pi\)
\(744\) −1.40867 −0.0516445
\(745\) 10.1100 0.370400
\(746\) −5.41377 −0.198212
\(747\) 7.59946 0.278050
\(748\) 4.27224 0.156209
\(749\) −1.68092 −0.0614194
\(750\) 2.26868 0.0828403
\(751\) −2.15350 −0.0785823 −0.0392912 0.999228i \(-0.512510\pi\)
−0.0392912 + 0.999228i \(0.512510\pi\)
\(752\) −22.6656 −0.826529
\(753\) 16.2841 0.593426
\(754\) −3.27336 −0.119209
\(755\) −23.6155 −0.859457
\(756\) −1.89128 −0.0687852
\(757\) −32.0577 −1.16516 −0.582579 0.812774i \(-0.697956\pi\)
−0.582579 + 0.812774i \(0.697956\pi\)
\(758\) 11.4217 0.414856
\(759\) 2.50407 0.0908919
\(760\) −8.79007 −0.318849
\(761\) 0.669651 0.0242748 0.0121374 0.999926i \(-0.496136\pi\)
0.0121374 + 0.999926i \(0.496136\pi\)
\(762\) 0.148135 0.00536637
\(763\) −11.4333 −0.413912
\(764\) 1.19480 0.0432264
\(765\) −2.46805 −0.0892324
\(766\) 3.51890 0.127143
\(767\) −10.8855 −0.393052
\(768\) −7.99375 −0.288450
\(769\) 43.5115 1.56906 0.784532 0.620089i \(-0.212903\pi\)
0.784532 + 0.620089i \(0.212903\pi\)
\(770\) 2.25892 0.0814057
\(771\) −22.9446 −0.826331
\(772\) −2.58776 −0.0931355
\(773\) 21.0046 0.755482 0.377741 0.925911i \(-0.376701\pi\)
0.377741 + 0.925911i \(0.376701\pi\)
\(774\) −1.92356 −0.0691410
\(775\) 2.72841 0.0980074
\(776\) 17.8477 0.640694
\(777\) 9.91023 0.355528
\(778\) −0.00996438 −0.000357240 0
\(779\) −5.78005 −0.207092
\(780\) −7.68448 −0.275149
\(781\) 0.906493 0.0324369
\(782\) −0.297446 −0.0106367
\(783\) −6.68466 −0.238890
\(784\) 3.35950 0.119982
\(785\) −27.1794 −0.970075
\(786\) −6.58658 −0.234936
\(787\) 22.1975 0.791255 0.395627 0.918411i \(-0.370527\pi\)
0.395627 + 0.918411i \(0.370527\pi\)
\(788\) −46.8302 −1.66826
\(789\) 0.776932 0.0276595
\(790\) 9.44246 0.335948
\(791\) −3.14914 −0.111970
\(792\) −3.21287 −0.114164
\(793\) −1.49440 −0.0530675
\(794\) −6.75113 −0.239589
\(795\) −13.4510 −0.477057
\(796\) −2.73233 −0.0968447
\(797\) 22.8502 0.809397 0.404699 0.914450i \(-0.367376\pi\)
0.404699 + 0.914450i \(0.367376\pi\)
\(798\) −0.825659 −0.0292280
\(799\) −6.08620 −0.215314
\(800\) −9.12991 −0.322791
\(801\) 10.6245 0.375398
\(802\) 12.0121 0.424161
\(803\) 13.3820 0.472241
\(804\) 21.3580 0.753238
\(805\) 2.73589 0.0964276
\(806\) −0.537623 −0.0189370
\(807\) −3.05937 −0.107695
\(808\) 2.32878 0.0819260
\(809\) −38.0621 −1.33819 −0.669097 0.743175i \(-0.733319\pi\)
−0.669097 + 0.743175i \(0.733319\pi\)
\(810\) 0.902098 0.0316965
\(811\) 21.0208 0.738142 0.369071 0.929401i \(-0.379676\pi\)
0.369071 + 0.929401i \(0.379676\pi\)
\(812\) 12.6426 0.443667
\(813\) 16.0631 0.563358
\(814\) 8.18248 0.286796
\(815\) −40.2400 −1.40955
\(816\) −3.03060 −0.106092
\(817\) 14.6082 0.511077
\(818\) −12.3336 −0.431234
\(819\) −1.48511 −0.0518941
\(820\) −11.9437 −0.417094
\(821\) −10.4132 −0.363425 −0.181712 0.983352i \(-0.558164\pi\)
−0.181712 + 0.983352i \(0.558164\pi\)
\(822\) 0.0891438 0.00310925
\(823\) 30.0396 1.04711 0.523557 0.851991i \(-0.324605\pi\)
0.523557 + 0.851991i \(0.324605\pi\)
\(824\) −21.3765 −0.744684
\(825\) 6.22289 0.216653
\(826\) −2.41681 −0.0840916
\(827\) 28.6978 0.997920 0.498960 0.866625i \(-0.333715\pi\)
0.498960 + 0.866625i \(0.333715\pi\)
\(828\) −1.89128 −0.0657265
\(829\) 16.4384 0.570931 0.285465 0.958389i \(-0.407852\pi\)
0.285465 + 0.958389i \(0.407852\pi\)
\(830\) 6.85546 0.237957
\(831\) 11.6103 0.402756
\(832\) −8.17946 −0.283572
\(833\) 0.902098 0.0312559
\(834\) 1.45908 0.0505239
\(835\) 14.9771 0.518304
\(836\) 11.8590 0.410152
\(837\) −1.09790 −0.0379490
\(838\) 12.2664 0.423736
\(839\) −12.5437 −0.433055 −0.216528 0.976277i \(-0.569473\pi\)
−0.216528 + 0.976277i \(0.569473\pi\)
\(840\) −3.51032 −0.121117
\(841\) 15.6847 0.540850
\(842\) −9.11123 −0.313994
\(843\) −9.89753 −0.340889
\(844\) 43.6866 1.50375
\(845\) 29.5324 1.01595
\(846\) 2.22457 0.0764824
\(847\) 4.72964 0.162512
\(848\) −16.5169 −0.567194
\(849\) −13.9994 −0.480457
\(850\) −0.739188 −0.0253539
\(851\) 9.91023 0.339718
\(852\) −0.684658 −0.0234560
\(853\) −22.0613 −0.755365 −0.377683 0.925935i \(-0.623279\pi\)
−0.377683 + 0.925935i \(0.623279\pi\)
\(854\) −0.331788 −0.0113535
\(855\) −6.85086 −0.234295
\(856\) 2.15672 0.0737151
\(857\) 6.15359 0.210203 0.105101 0.994462i \(-0.466483\pi\)
0.105101 + 0.994462i \(0.466483\pi\)
\(858\) −1.22620 −0.0418617
\(859\) 23.0698 0.787131 0.393566 0.919296i \(-0.371241\pi\)
0.393566 + 0.919296i \(0.371241\pi\)
\(860\) 30.1860 1.02934
\(861\) −2.30826 −0.0786655
\(862\) −11.5627 −0.393828
\(863\) −5.22145 −0.177740 −0.0888702 0.996043i \(-0.528326\pi\)
−0.0888702 + 0.996043i \(0.528326\pi\)
\(864\) 3.67384 0.124987
\(865\) −49.7186 −1.69048
\(866\) 0.332061 0.0112839
\(867\) 16.1862 0.549713
\(868\) 2.07644 0.0704789
\(869\) −26.2106 −0.889135
\(870\) −6.03022 −0.204444
\(871\) 16.7712 0.568271
\(872\) 14.6696 0.496774
\(873\) 13.9102 0.470790
\(874\) −0.825659 −0.0279283
\(875\) −6.88046 −0.232602
\(876\) −10.1072 −0.341491
\(877\) 49.8716 1.68404 0.842022 0.539443i \(-0.181365\pi\)
0.842022 + 0.539443i \(0.181365\pi\)
\(878\) 1.33424 0.0450284
\(879\) 8.14038 0.274568
\(880\) 23.0155 0.775852
\(881\) −20.3054 −0.684107 −0.342053 0.939681i \(-0.611122\pi\)
−0.342053 + 0.939681i \(0.611122\pi\)
\(882\) −0.329727 −0.0111025
\(883\) −50.3500 −1.69441 −0.847206 0.531265i \(-0.821717\pi\)
−0.847206 + 0.531265i \(0.821717\pi\)
\(884\) −2.53378 −0.0852203
\(885\) −20.0534 −0.674086
\(886\) 9.69224 0.325617
\(887\) 47.2694 1.58715 0.793576 0.608471i \(-0.208217\pi\)
0.793576 + 0.608471i \(0.208217\pi\)
\(888\) −12.7154 −0.426702
\(889\) −0.449266 −0.0150679
\(890\) 9.58434 0.321268
\(891\) −2.50407 −0.0838894
\(892\) 35.2718 1.18099
\(893\) −16.8942 −0.565344
\(894\) −1.21844 −0.0407508
\(895\) −27.8679 −0.931522
\(896\) −9.16370 −0.306138
\(897\) −1.48511 −0.0495865
\(898\) 1.60784 0.0536543
\(899\) 7.33910 0.244773
\(900\) −4.70005 −0.156668
\(901\) −4.43515 −0.147756
\(902\) −1.90584 −0.0634575
\(903\) 5.83380 0.194137
\(904\) 4.04053 0.134386
\(905\) −36.9855 −1.22944
\(906\) 2.84612 0.0945560
\(907\) −5.65035 −0.187617 −0.0938084 0.995590i \(-0.529904\pi\)
−0.0938084 + 0.995590i \(0.529904\pi\)
\(908\) −39.0236 −1.29504
\(909\) 1.81502 0.0602003
\(910\) −1.33972 −0.0444112
\(911\) 18.2196 0.603641 0.301820 0.953365i \(-0.402406\pi\)
0.301820 + 0.953365i \(0.402406\pi\)
\(912\) −8.41242 −0.278563
\(913\) −19.0296 −0.629787
\(914\) 11.7123 0.387409
\(915\) −2.75299 −0.0910111
\(916\) −28.2829 −0.934495
\(917\) 19.9759 0.659661
\(918\) 0.297446 0.00981719
\(919\) 7.79561 0.257154 0.128577 0.991700i \(-0.458959\pi\)
0.128577 + 0.991700i \(0.458959\pi\)
\(920\) −3.51032 −0.115732
\(921\) −13.5095 −0.445153
\(922\) 6.24642 0.205715
\(923\) −0.537623 −0.0176961
\(924\) 4.73589 0.155799
\(925\) 24.6281 0.809766
\(926\) −4.67992 −0.153792
\(927\) −16.6605 −0.547203
\(928\) −24.5584 −0.806168
\(929\) 11.5912 0.380293 0.190147 0.981756i \(-0.439104\pi\)
0.190147 + 0.981756i \(0.439104\pi\)
\(930\) −0.990415 −0.0324770
\(931\) 2.50407 0.0820675
\(932\) −53.1548 −1.74114
\(933\) 24.3035 0.795662
\(934\) 10.2332 0.334842
\(935\) 6.18015 0.202113
\(936\) 1.90549 0.0622829
\(937\) −32.0050 −1.04556 −0.522778 0.852469i \(-0.675105\pi\)
−0.522778 + 0.852469i \(0.675105\pi\)
\(938\) 3.72357 0.121579
\(939\) 33.7836 1.10249
\(940\) −34.9098 −1.13863
\(941\) 11.1090 0.362141 0.181071 0.983470i \(-0.442044\pi\)
0.181071 + 0.983470i \(0.442044\pi\)
\(942\) 3.27563 0.106726
\(943\) −2.30826 −0.0751674
\(944\) −24.6242 −0.801450
\(945\) −2.73589 −0.0889986
\(946\) 4.81673 0.156605
\(947\) −12.1262 −0.394049 −0.197025 0.980399i \(-0.563128\pi\)
−0.197025 + 0.980399i \(0.563128\pi\)
\(948\) 19.7964 0.642959
\(949\) −7.93661 −0.257633
\(950\) −2.05186 −0.0665710
\(951\) −32.1594 −1.04284
\(952\) −1.15745 −0.0375131
\(953\) 27.6645 0.896140 0.448070 0.893999i \(-0.352112\pi\)
0.448070 + 0.893999i \(0.352112\pi\)
\(954\) 1.62110 0.0524850
\(955\) 1.72838 0.0559291
\(956\) −48.8965 −1.58142
\(957\) 16.7388 0.541090
\(958\) −3.46919 −0.112084
\(959\) −0.270356 −0.00873026
\(960\) −15.0683 −0.486327
\(961\) −29.7946 −0.961117
\(962\) −4.85287 −0.156463
\(963\) 1.68092 0.0541668
\(964\) 28.0498 0.903423
\(965\) −3.74341 −0.120505
\(966\) −0.329727 −0.0106088
\(967\) −56.1838 −1.80675 −0.903374 0.428853i \(-0.858918\pi\)
−0.903374 + 0.428853i \(0.858918\pi\)
\(968\) −6.06842 −0.195046
\(969\) −2.25892 −0.0725668
\(970\) 12.5484 0.402905
\(971\) −43.9884 −1.41166 −0.705828 0.708383i \(-0.749425\pi\)
−0.705828 + 0.708383i \(0.749425\pi\)
\(972\) 1.89128 0.0606628
\(973\) −4.42512 −0.141863
\(974\) 10.4648 0.335315
\(975\) −3.69068 −0.118196
\(976\) −3.38050 −0.108207
\(977\) −6.47804 −0.207251 −0.103625 0.994616i \(-0.533044\pi\)
−0.103625 + 0.994616i \(0.533044\pi\)
\(978\) 4.84969 0.155076
\(979\) −26.6044 −0.850282
\(980\) 5.17434 0.165288
\(981\) 11.4333 0.365036
\(982\) 10.5113 0.335430
\(983\) 37.8961 1.20870 0.604349 0.796720i \(-0.293433\pi\)
0.604349 + 0.796720i \(0.293433\pi\)
\(984\) 2.96164 0.0944138
\(985\) −67.7437 −2.15849
\(986\) −1.98833 −0.0633212
\(987\) −6.74671 −0.214750
\(988\) −7.03334 −0.223760
\(989\) 5.83380 0.185504
\(990\) −2.25892 −0.0717931
\(991\) 53.3499 1.69472 0.847358 0.531022i \(-0.178192\pi\)
0.847358 + 0.531022i \(0.178192\pi\)
\(992\) −4.03351 −0.128064
\(993\) 4.84004 0.153594
\(994\) −0.119364 −0.00378599
\(995\) −3.95254 −0.125304
\(996\) 14.3727 0.455417
\(997\) 24.8754 0.787813 0.393907 0.919150i \(-0.371123\pi\)
0.393907 + 0.919150i \(0.371123\pi\)
\(998\) 13.0004 0.411522
\(999\) −9.91023 −0.313546
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 483.2.a.j.1.2 4
3.2 odd 2 1449.2.a.o.1.3 4
4.3 odd 2 7728.2.a.ce.1.1 4
7.6 odd 2 3381.2.a.x.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.j.1.2 4 1.1 even 1 trivial
1449.2.a.o.1.3 4 3.2 odd 2
3381.2.a.x.1.2 4 7.6 odd 2
7728.2.a.ce.1.1 4 4.3 odd 2