Properties

Label 1045.2.a.h.1.5
Level $1045$
Weight $2$
Character 1045.1
Self dual yes
Analytic conductor $8.344$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1045,2,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11 \) 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 1.5
Root \(0.745312\) of defining polynomial
Character \(\chi\) \(=\) 1045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.745312 q^{2} -2.05058 q^{3} -1.44451 q^{4} -1.00000 q^{5} -1.52832 q^{6} -3.86782 q^{7} -2.56724 q^{8} +1.20489 q^{9} +O(q^{10})\) \(q+0.745312 q^{2} -2.05058 q^{3} -1.44451 q^{4} -1.00000 q^{5} -1.52832 q^{6} -3.86782 q^{7} -2.56724 q^{8} +1.20489 q^{9} -0.745312 q^{10} +1.00000 q^{11} +2.96208 q^{12} -2.04430 q^{13} -2.88274 q^{14} +2.05058 q^{15} +0.975626 q^{16} -3.70255 q^{17} +0.898017 q^{18} -1.00000 q^{19} +1.44451 q^{20} +7.93129 q^{21} +0.745312 q^{22} +2.09412 q^{23} +5.26433 q^{24} +1.00000 q^{25} -1.52364 q^{26} +3.68103 q^{27} +5.58711 q^{28} -1.56558 q^{29} +1.52832 q^{30} +5.38850 q^{31} +5.86162 q^{32} -2.05058 q^{33} -2.75956 q^{34} +3.86782 q^{35} -1.74047 q^{36} -7.65441 q^{37} -0.745312 q^{38} +4.19200 q^{39} +2.56724 q^{40} +2.55527 q^{41} +5.91129 q^{42} +1.43588 q^{43} -1.44451 q^{44} -1.20489 q^{45} +1.56077 q^{46} -5.29091 q^{47} -2.00060 q^{48} +7.96005 q^{49} +0.745312 q^{50} +7.59239 q^{51} +2.95301 q^{52} +2.10441 q^{53} +2.74352 q^{54} -1.00000 q^{55} +9.92961 q^{56} +2.05058 q^{57} -1.16685 q^{58} +0.643913 q^{59} -2.96208 q^{60} +14.3858 q^{61} +4.01612 q^{62} -4.66029 q^{63} +2.41748 q^{64} +2.04430 q^{65} -1.52832 q^{66} -7.38466 q^{67} +5.34837 q^{68} -4.29417 q^{69} +2.88274 q^{70} -3.12919 q^{71} -3.09323 q^{72} +6.41053 q^{73} -5.70492 q^{74} -2.05058 q^{75} +1.44451 q^{76} -3.86782 q^{77} +3.12435 q^{78} -3.06249 q^{79} -0.975626 q^{80} -11.1629 q^{81} +1.90448 q^{82} +6.28445 q^{83} -11.4568 q^{84} +3.70255 q^{85} +1.07018 q^{86} +3.21035 q^{87} -2.56724 q^{88} -1.85225 q^{89} -0.898017 q^{90} +7.90699 q^{91} -3.02498 q^{92} -11.0496 q^{93} -3.94338 q^{94} +1.00000 q^{95} -12.0197 q^{96} +6.58248 q^{97} +5.93273 q^{98} +1.20489 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9} - q^{10} + 7 q^{11} + 13 q^{12} + q^{13} + 12 q^{14} - 3 q^{15} + 3 q^{16} + q^{17} + 7 q^{18} - 7 q^{19} - 7 q^{20} + 5 q^{21} + q^{22} - 8 q^{23} + 25 q^{24} + 7 q^{25} + 12 q^{27} + 4 q^{28} + 11 q^{29} - 8 q^{30} + 7 q^{31} + 12 q^{32} + 3 q^{33} - 14 q^{34} + q^{35} + 7 q^{36} - 17 q^{37} - q^{38} + 30 q^{39} - 3 q^{40} + 17 q^{41} + 33 q^{42} - 3 q^{43} + 7 q^{44} - 2 q^{45} + 18 q^{46} + 14 q^{47} - 12 q^{48} + 6 q^{49} + q^{50} + 8 q^{51} - 17 q^{52} + 7 q^{53} - 27 q^{54} - 7 q^{55} + 36 q^{56} - 3 q^{57} - 15 q^{58} + 35 q^{59} - 13 q^{60} + 17 q^{61} + 46 q^{62} - 22 q^{63} + 5 q^{64} - q^{65} + 8 q^{66} + 4 q^{67} - 35 q^{68} - 4 q^{69} - 12 q^{70} + 10 q^{71} + 12 q^{72} + 22 q^{73} - 11 q^{74} + 3 q^{75} - 7 q^{76} - q^{77} - 41 q^{78} + 11 q^{79} - 3 q^{80} - 21 q^{81} - 14 q^{82} + 39 q^{83} + 21 q^{84} - q^{85} - 24 q^{86} - 2 q^{87} + 3 q^{88} + 18 q^{89} - 7 q^{90} - 22 q^{91} - 51 q^{92} + 10 q^{93} + 14 q^{94} + 7 q^{95} - 11 q^{96} - 4 q^{97} - 26 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.745312 0.527015 0.263508 0.964657i \(-0.415121\pi\)
0.263508 + 0.964657i \(0.415121\pi\)
\(3\) −2.05058 −1.18390 −0.591952 0.805973i \(-0.701643\pi\)
−0.591952 + 0.805973i \(0.701643\pi\)
\(4\) −1.44451 −0.722255
\(5\) −1.00000 −0.447214
\(6\) −1.52832 −0.623936
\(7\) −3.86782 −1.46190 −0.730950 0.682431i \(-0.760923\pi\)
−0.730950 + 0.682431i \(0.760923\pi\)
\(8\) −2.56724 −0.907655
\(9\) 1.20489 0.401629
\(10\) −0.745312 −0.235688
\(11\) 1.00000 0.301511
\(12\) 2.96208 0.855080
\(13\) −2.04430 −0.566987 −0.283493 0.958974i \(-0.591493\pi\)
−0.283493 + 0.958974i \(0.591493\pi\)
\(14\) −2.88274 −0.770444
\(15\) 2.05058 0.529458
\(16\) 0.975626 0.243907
\(17\) −3.70255 −0.898001 −0.449001 0.893531i \(-0.648220\pi\)
−0.449001 + 0.893531i \(0.648220\pi\)
\(18\) 0.898017 0.211665
\(19\) −1.00000 −0.229416
\(20\) 1.44451 0.323002
\(21\) 7.93129 1.73075
\(22\) 0.745312 0.158901
\(23\) 2.09412 0.436654 0.218327 0.975876i \(-0.429940\pi\)
0.218327 + 0.975876i \(0.429940\pi\)
\(24\) 5.26433 1.07458
\(25\) 1.00000 0.200000
\(26\) −1.52364 −0.298811
\(27\) 3.68103 0.708414
\(28\) 5.58711 1.05586
\(29\) −1.56558 −0.290721 −0.145361 0.989379i \(-0.546434\pi\)
−0.145361 + 0.989379i \(0.546434\pi\)
\(30\) 1.52832 0.279033
\(31\) 5.38850 0.967804 0.483902 0.875122i \(-0.339219\pi\)
0.483902 + 0.875122i \(0.339219\pi\)
\(32\) 5.86162 1.03620
\(33\) −2.05058 −0.356960
\(34\) −2.75956 −0.473261
\(35\) 3.86782 0.653781
\(36\) −1.74047 −0.290078
\(37\) −7.65441 −1.25838 −0.629188 0.777253i \(-0.716613\pi\)
−0.629188 + 0.777253i \(0.716613\pi\)
\(38\) −0.745312 −0.120906
\(39\) 4.19200 0.671258
\(40\) 2.56724 0.405916
\(41\) 2.55527 0.399067 0.199533 0.979891i \(-0.436057\pi\)
0.199533 + 0.979891i \(0.436057\pi\)
\(42\) 5.91129 0.912131
\(43\) 1.43588 0.218969 0.109485 0.993988i \(-0.465080\pi\)
0.109485 + 0.993988i \(0.465080\pi\)
\(44\) −1.44451 −0.217768
\(45\) −1.20489 −0.179614
\(46\) 1.56077 0.230124
\(47\) −5.29091 −0.771758 −0.385879 0.922549i \(-0.626102\pi\)
−0.385879 + 0.922549i \(0.626102\pi\)
\(48\) −2.00060 −0.288762
\(49\) 7.96005 1.13715
\(50\) 0.745312 0.105403
\(51\) 7.59239 1.06315
\(52\) 2.95301 0.409509
\(53\) 2.10441 0.289063 0.144531 0.989500i \(-0.453833\pi\)
0.144531 + 0.989500i \(0.453833\pi\)
\(54\) 2.74352 0.373345
\(55\) −1.00000 −0.134840
\(56\) 9.92961 1.32690
\(57\) 2.05058 0.271606
\(58\) −1.16685 −0.153214
\(59\) 0.643913 0.0838303 0.0419152 0.999121i \(-0.486654\pi\)
0.0419152 + 0.999121i \(0.486654\pi\)
\(60\) −2.96208 −0.382403
\(61\) 14.3858 1.84191 0.920957 0.389664i \(-0.127409\pi\)
0.920957 + 0.389664i \(0.127409\pi\)
\(62\) 4.01612 0.510047
\(63\) −4.66029 −0.587141
\(64\) 2.41748 0.302186
\(65\) 2.04430 0.253564
\(66\) −1.52832 −0.188124
\(67\) −7.38466 −0.902179 −0.451090 0.892479i \(-0.648965\pi\)
−0.451090 + 0.892479i \(0.648965\pi\)
\(68\) 5.34837 0.648586
\(69\) −4.29417 −0.516957
\(70\) 2.88274 0.344553
\(71\) −3.12919 −0.371366 −0.185683 0.982610i \(-0.559450\pi\)
−0.185683 + 0.982610i \(0.559450\pi\)
\(72\) −3.09323 −0.364540
\(73\) 6.41053 0.750296 0.375148 0.926965i \(-0.377592\pi\)
0.375148 + 0.926965i \(0.377592\pi\)
\(74\) −5.70492 −0.663184
\(75\) −2.05058 −0.236781
\(76\) 1.44451 0.165697
\(77\) −3.86782 −0.440779
\(78\) 3.12435 0.353763
\(79\) −3.06249 −0.344557 −0.172278 0.985048i \(-0.555113\pi\)
−0.172278 + 0.985048i \(0.555113\pi\)
\(80\) −0.975626 −0.109078
\(81\) −11.1629 −1.24032
\(82\) 1.90448 0.210314
\(83\) 6.28445 0.689808 0.344904 0.938638i \(-0.387911\pi\)
0.344904 + 0.938638i \(0.387911\pi\)
\(84\) −11.4568 −1.25004
\(85\) 3.70255 0.401598
\(86\) 1.07018 0.115400
\(87\) 3.21035 0.344186
\(88\) −2.56724 −0.273668
\(89\) −1.85225 −0.196338 −0.0981691 0.995170i \(-0.531299\pi\)
−0.0981691 + 0.995170i \(0.531299\pi\)
\(90\) −0.898017 −0.0946593
\(91\) 7.90699 0.828878
\(92\) −3.02498 −0.315376
\(93\) −11.0496 −1.14579
\(94\) −3.94338 −0.406729
\(95\) 1.00000 0.102598
\(96\) −12.0197 −1.22676
\(97\) 6.58248 0.668349 0.334175 0.942511i \(-0.391542\pi\)
0.334175 + 0.942511i \(0.391542\pi\)
\(98\) 5.93273 0.599296
\(99\) 1.20489 0.121096
\(100\) −1.44451 −0.144451
\(101\) −13.7356 −1.36674 −0.683372 0.730070i \(-0.739487\pi\)
−0.683372 + 0.730070i \(0.739487\pi\)
\(102\) 5.65870 0.560295
\(103\) −6.59626 −0.649949 −0.324975 0.945723i \(-0.605356\pi\)
−0.324975 + 0.945723i \(0.605356\pi\)
\(104\) 5.24820 0.514628
\(105\) −7.93129 −0.774014
\(106\) 1.56844 0.152341
\(107\) 7.49964 0.725018 0.362509 0.931980i \(-0.381920\pi\)
0.362509 + 0.931980i \(0.381920\pi\)
\(108\) −5.31728 −0.511655
\(109\) −12.0403 −1.15325 −0.576624 0.817010i \(-0.695630\pi\)
−0.576624 + 0.817010i \(0.695630\pi\)
\(110\) −0.745312 −0.0710628
\(111\) 15.6960 1.48980
\(112\) −3.77355 −0.356567
\(113\) −11.8491 −1.11467 −0.557333 0.830289i \(-0.688175\pi\)
−0.557333 + 0.830289i \(0.688175\pi\)
\(114\) 1.52832 0.143141
\(115\) −2.09412 −0.195278
\(116\) 2.26150 0.209975
\(117\) −2.46315 −0.227718
\(118\) 0.479916 0.0441799
\(119\) 14.3208 1.31279
\(120\) −5.26433 −0.480565
\(121\) 1.00000 0.0909091
\(122\) 10.7219 0.970717
\(123\) −5.23980 −0.472457
\(124\) −7.78374 −0.699001
\(125\) −1.00000 −0.0894427
\(126\) −3.47337 −0.309432
\(127\) 12.8805 1.14296 0.571479 0.820617i \(-0.306370\pi\)
0.571479 + 0.820617i \(0.306370\pi\)
\(128\) −9.92145 −0.876941
\(129\) −2.94439 −0.259239
\(130\) 1.52364 0.133632
\(131\) 14.3038 1.24973 0.624865 0.780733i \(-0.285154\pi\)
0.624865 + 0.780733i \(0.285154\pi\)
\(132\) 2.96208 0.257816
\(133\) 3.86782 0.335383
\(134\) −5.50388 −0.475462
\(135\) −3.68103 −0.316812
\(136\) 9.50533 0.815075
\(137\) −15.1224 −1.29200 −0.645998 0.763339i \(-0.723558\pi\)
−0.645998 + 0.763339i \(0.723558\pi\)
\(138\) −3.20050 −0.272444
\(139\) 9.76738 0.828458 0.414229 0.910173i \(-0.364051\pi\)
0.414229 + 0.910173i \(0.364051\pi\)
\(140\) −5.58711 −0.472197
\(141\) 10.8494 0.913688
\(142\) −2.33222 −0.195716
\(143\) −2.04430 −0.170953
\(144\) 1.17552 0.0979599
\(145\) 1.56558 0.130014
\(146\) 4.77785 0.395418
\(147\) −16.3227 −1.34628
\(148\) 11.0569 0.908869
\(149\) −14.1604 −1.16007 −0.580034 0.814592i \(-0.696961\pi\)
−0.580034 + 0.814592i \(0.696961\pi\)
\(150\) −1.52832 −0.124787
\(151\) 18.0003 1.46484 0.732420 0.680853i \(-0.238391\pi\)
0.732420 + 0.680853i \(0.238391\pi\)
\(152\) 2.56724 0.208230
\(153\) −4.46116 −0.360663
\(154\) −2.88274 −0.232298
\(155\) −5.38850 −0.432815
\(156\) −6.05539 −0.484819
\(157\) 2.90903 0.232166 0.116083 0.993240i \(-0.462966\pi\)
0.116083 + 0.993240i \(0.462966\pi\)
\(158\) −2.28251 −0.181587
\(159\) −4.31526 −0.342223
\(160\) −5.86162 −0.463402
\(161\) −8.09969 −0.638345
\(162\) −8.31985 −0.653669
\(163\) 9.73587 0.762572 0.381286 0.924457i \(-0.375481\pi\)
0.381286 + 0.924457i \(0.375481\pi\)
\(164\) −3.69112 −0.288228
\(165\) 2.05058 0.159638
\(166\) 4.68388 0.363539
\(167\) 22.9362 1.77486 0.887430 0.460943i \(-0.152489\pi\)
0.887430 + 0.460943i \(0.152489\pi\)
\(168\) −20.3615 −1.57092
\(169\) −8.82084 −0.678526
\(170\) 2.75956 0.211649
\(171\) −1.20489 −0.0921399
\(172\) −2.07414 −0.158152
\(173\) −5.44966 −0.414330 −0.207165 0.978306i \(-0.566424\pi\)
−0.207165 + 0.978306i \(0.566424\pi\)
\(174\) 2.39271 0.181391
\(175\) −3.86782 −0.292380
\(176\) 0.975626 0.0735406
\(177\) −1.32040 −0.0992471
\(178\) −1.38051 −0.103473
\(179\) 17.7565 1.32718 0.663590 0.748096i \(-0.269032\pi\)
0.663590 + 0.748096i \(0.269032\pi\)
\(180\) 1.74047 0.129727
\(181\) −22.0588 −1.63962 −0.819810 0.572636i \(-0.805921\pi\)
−0.819810 + 0.572636i \(0.805921\pi\)
\(182\) 5.89318 0.436831
\(183\) −29.4993 −2.18065
\(184\) −5.37610 −0.396331
\(185\) 7.65441 0.562763
\(186\) −8.23538 −0.603847
\(187\) −3.70255 −0.270758
\(188\) 7.64277 0.557406
\(189\) −14.2376 −1.03563
\(190\) 0.745312 0.0540706
\(191\) 12.6095 0.912390 0.456195 0.889880i \(-0.349212\pi\)
0.456195 + 0.889880i \(0.349212\pi\)
\(192\) −4.95725 −0.357759
\(193\) −8.37178 −0.602614 −0.301307 0.953527i \(-0.597423\pi\)
−0.301307 + 0.953527i \(0.597423\pi\)
\(194\) 4.90600 0.352230
\(195\) −4.19200 −0.300196
\(196\) −11.4984 −0.821312
\(197\) 9.63861 0.686722 0.343361 0.939203i \(-0.388435\pi\)
0.343361 + 0.939203i \(0.388435\pi\)
\(198\) 0.898017 0.0638193
\(199\) −17.1167 −1.21337 −0.606684 0.794943i \(-0.707501\pi\)
−0.606684 + 0.794943i \(0.707501\pi\)
\(200\) −2.56724 −0.181531
\(201\) 15.1428 1.06809
\(202\) −10.2373 −0.720295
\(203\) 6.05539 0.425005
\(204\) −10.9673 −0.767863
\(205\) −2.55527 −0.178468
\(206\) −4.91628 −0.342533
\(207\) 2.52318 0.175373
\(208\) −1.99447 −0.138292
\(209\) −1.00000 −0.0691714
\(210\) −5.91129 −0.407918
\(211\) −0.310300 −0.0213619 −0.0106810 0.999943i \(-0.503400\pi\)
−0.0106810 + 0.999943i \(0.503400\pi\)
\(212\) −3.03984 −0.208777
\(213\) 6.41666 0.439662
\(214\) 5.58958 0.382096
\(215\) −1.43588 −0.0979261
\(216\) −9.45007 −0.642996
\(217\) −20.8418 −1.41483
\(218\) −8.97375 −0.607779
\(219\) −13.1453 −0.888279
\(220\) 1.44451 0.0973888
\(221\) 7.56913 0.509155
\(222\) 11.6984 0.785146
\(223\) −3.81396 −0.255402 −0.127701 0.991813i \(-0.540760\pi\)
−0.127701 + 0.991813i \(0.540760\pi\)
\(224\) −22.6717 −1.51482
\(225\) 1.20489 0.0803257
\(226\) −8.83126 −0.587447
\(227\) −17.7004 −1.17482 −0.587410 0.809290i \(-0.699852\pi\)
−0.587410 + 0.809290i \(0.699852\pi\)
\(228\) −2.96208 −0.196169
\(229\) 27.1634 1.79501 0.897505 0.441004i \(-0.145378\pi\)
0.897505 + 0.441004i \(0.145378\pi\)
\(230\) −1.56077 −0.102914
\(231\) 7.93129 0.521840
\(232\) 4.01921 0.263874
\(233\) 0.550837 0.0360865 0.0180433 0.999837i \(-0.494256\pi\)
0.0180433 + 0.999837i \(0.494256\pi\)
\(234\) −1.83581 −0.120011
\(235\) 5.29091 0.345141
\(236\) −0.930138 −0.0605469
\(237\) 6.27988 0.407922
\(238\) 10.6735 0.691859
\(239\) −5.37952 −0.347972 −0.173986 0.984748i \(-0.555665\pi\)
−0.173986 + 0.984748i \(0.555665\pi\)
\(240\) 2.00060 0.129138
\(241\) 22.5371 1.45174 0.725872 0.687830i \(-0.241437\pi\)
0.725872 + 0.687830i \(0.241437\pi\)
\(242\) 0.745312 0.0479105
\(243\) 11.8474 0.760009
\(244\) −20.7804 −1.33033
\(245\) −7.96005 −0.508549
\(246\) −3.90529 −0.248992
\(247\) 2.04430 0.130076
\(248\) −13.8336 −0.878432
\(249\) −12.8868 −0.816666
\(250\) −0.745312 −0.0471377
\(251\) 7.58349 0.478665 0.239333 0.970938i \(-0.423071\pi\)
0.239333 + 0.970938i \(0.423071\pi\)
\(252\) 6.73183 0.424065
\(253\) 2.09412 0.131656
\(254\) 9.59998 0.602356
\(255\) −7.59239 −0.475454
\(256\) −12.2296 −0.764347
\(257\) 13.6317 0.850320 0.425160 0.905118i \(-0.360218\pi\)
0.425160 + 0.905118i \(0.360218\pi\)
\(258\) −2.19449 −0.136623
\(259\) 29.6059 1.83962
\(260\) −2.95301 −0.183138
\(261\) −1.88635 −0.116762
\(262\) 10.6608 0.658627
\(263\) −4.03027 −0.248517 −0.124259 0.992250i \(-0.539655\pi\)
−0.124259 + 0.992250i \(0.539655\pi\)
\(264\) 5.26433 0.323997
\(265\) −2.10441 −0.129273
\(266\) 2.88274 0.176752
\(267\) 3.79819 0.232446
\(268\) 10.6672 0.651603
\(269\) 26.6319 1.62378 0.811889 0.583812i \(-0.198439\pi\)
0.811889 + 0.583812i \(0.198439\pi\)
\(270\) −2.74352 −0.166965
\(271\) 22.3349 1.35675 0.678375 0.734716i \(-0.262685\pi\)
0.678375 + 0.734716i \(0.262685\pi\)
\(272\) −3.61231 −0.219028
\(273\) −16.2139 −0.981311
\(274\) −11.2709 −0.680901
\(275\) 1.00000 0.0603023
\(276\) 6.20296 0.373374
\(277\) −22.6811 −1.36277 −0.681387 0.731923i \(-0.738623\pi\)
−0.681387 + 0.731923i \(0.738623\pi\)
\(278\) 7.27975 0.436610
\(279\) 6.49253 0.388698
\(280\) −9.92961 −0.593408
\(281\) 22.0886 1.31770 0.658848 0.752276i \(-0.271044\pi\)
0.658848 + 0.752276i \(0.271044\pi\)
\(282\) 8.08622 0.481528
\(283\) −17.4483 −1.03719 −0.518597 0.855019i \(-0.673546\pi\)
−0.518597 + 0.855019i \(0.673546\pi\)
\(284\) 4.52014 0.268221
\(285\) −2.05058 −0.121466
\(286\) −1.52364 −0.0900948
\(287\) −9.88335 −0.583396
\(288\) 7.06258 0.416167
\(289\) −3.29109 −0.193594
\(290\) 1.16685 0.0685196
\(291\) −13.4979 −0.791261
\(292\) −9.26007 −0.541905
\(293\) 17.0451 0.995785 0.497892 0.867239i \(-0.334107\pi\)
0.497892 + 0.867239i \(0.334107\pi\)
\(294\) −12.1655 −0.709509
\(295\) −0.643913 −0.0374901
\(296\) 19.6507 1.14217
\(297\) 3.68103 0.213595
\(298\) −10.5539 −0.611374
\(299\) −4.28101 −0.247577
\(300\) 2.96208 0.171016
\(301\) −5.55372 −0.320111
\(302\) 13.4158 0.771993
\(303\) 28.1660 1.61809
\(304\) −0.975626 −0.0559560
\(305\) −14.3858 −0.823729
\(306\) −3.32495 −0.190075
\(307\) −9.52713 −0.543742 −0.271871 0.962334i \(-0.587642\pi\)
−0.271871 + 0.962334i \(0.587642\pi\)
\(308\) 5.58711 0.318355
\(309\) 13.5262 0.769477
\(310\) −4.01612 −0.228100
\(311\) −25.1706 −1.42730 −0.713648 0.700505i \(-0.752958\pi\)
−0.713648 + 0.700505i \(0.752958\pi\)
\(312\) −10.7619 −0.609270
\(313\) −8.19113 −0.462990 −0.231495 0.972836i \(-0.574362\pi\)
−0.231495 + 0.972836i \(0.574362\pi\)
\(314\) 2.16813 0.122355
\(315\) 4.66029 0.262577
\(316\) 4.42379 0.248858
\(317\) 20.9949 1.17919 0.589596 0.807699i \(-0.299287\pi\)
0.589596 + 0.807699i \(0.299287\pi\)
\(318\) −3.21622 −0.180357
\(319\) −1.56558 −0.0876557
\(320\) −2.41748 −0.135141
\(321\) −15.3786 −0.858351
\(322\) −6.03680 −0.336418
\(323\) 3.70255 0.206016
\(324\) 16.1249 0.895829
\(325\) −2.04430 −0.113397
\(326\) 7.25627 0.401887
\(327\) 24.6895 1.36533
\(328\) −6.55999 −0.362215
\(329\) 20.4643 1.12823
\(330\) 1.52832 0.0841315
\(331\) 3.28061 0.180319 0.0901594 0.995927i \(-0.471262\pi\)
0.0901594 + 0.995927i \(0.471262\pi\)
\(332\) −9.07795 −0.498217
\(333\) −9.22269 −0.505400
\(334\) 17.0947 0.935378
\(335\) 7.38466 0.403467
\(336\) 7.73797 0.422141
\(337\) −18.4493 −1.00500 −0.502499 0.864578i \(-0.667586\pi\)
−0.502499 + 0.864578i \(0.667586\pi\)
\(338\) −6.57428 −0.357594
\(339\) 24.2975 1.31966
\(340\) −5.34837 −0.290056
\(341\) 5.38850 0.291804
\(342\) −0.898017 −0.0485592
\(343\) −3.71331 −0.200500
\(344\) −3.68624 −0.198749
\(345\) 4.29417 0.231190
\(346\) −4.06170 −0.218358
\(347\) 29.8792 1.60400 0.801999 0.597325i \(-0.203770\pi\)
0.801999 + 0.597325i \(0.203770\pi\)
\(348\) −4.63738 −0.248590
\(349\) 27.5340 1.47386 0.736932 0.675967i \(-0.236274\pi\)
0.736932 + 0.675967i \(0.236274\pi\)
\(350\) −2.88274 −0.154089
\(351\) −7.52512 −0.401661
\(352\) 5.86162 0.312425
\(353\) 3.83620 0.204180 0.102090 0.994775i \(-0.467447\pi\)
0.102090 + 0.994775i \(0.467447\pi\)
\(354\) −0.984108 −0.0523047
\(355\) 3.12919 0.166080
\(356\) 2.67559 0.141806
\(357\) −29.3660 −1.55421
\(358\) 13.2341 0.699445
\(359\) 32.7643 1.72923 0.864617 0.502432i \(-0.167561\pi\)
0.864617 + 0.502432i \(0.167561\pi\)
\(360\) 3.09323 0.163027
\(361\) 1.00000 0.0526316
\(362\) −16.4407 −0.864105
\(363\) −2.05058 −0.107628
\(364\) −11.4217 −0.598661
\(365\) −6.41053 −0.335543
\(366\) −21.9862 −1.14924
\(367\) 21.0340 1.09797 0.548983 0.835833i \(-0.315015\pi\)
0.548983 + 0.835833i \(0.315015\pi\)
\(368\) 2.04308 0.106503
\(369\) 3.07881 0.160277
\(370\) 5.70492 0.296585
\(371\) −8.13948 −0.422581
\(372\) 15.9612 0.827550
\(373\) 2.07953 0.107674 0.0538371 0.998550i \(-0.482855\pi\)
0.0538371 + 0.998550i \(0.482855\pi\)
\(374\) −2.75956 −0.142693
\(375\) 2.05058 0.105892
\(376\) 13.5830 0.700490
\(377\) 3.20052 0.164835
\(378\) −10.6114 −0.545793
\(379\) 30.7796 1.58104 0.790520 0.612436i \(-0.209810\pi\)
0.790520 + 0.612436i \(0.209810\pi\)
\(380\) −1.44451 −0.0741018
\(381\) −26.4125 −1.35315
\(382\) 9.39800 0.480843
\(383\) −0.0147152 −0.000751912 0 −0.000375956 1.00000i \(-0.500120\pi\)
−0.000375956 1.00000i \(0.500120\pi\)
\(384\) 20.3448 1.03821
\(385\) 3.86782 0.197122
\(386\) −6.23959 −0.317587
\(387\) 1.73007 0.0879444
\(388\) −9.50845 −0.482718
\(389\) −22.9201 −1.16209 −0.581047 0.813870i \(-0.697357\pi\)
−0.581047 + 0.813870i \(0.697357\pi\)
\(390\) −3.12435 −0.158208
\(391\) −7.75359 −0.392116
\(392\) −20.4353 −1.03214
\(393\) −29.3311 −1.47956
\(394\) 7.18377 0.361913
\(395\) 3.06249 0.154090
\(396\) −1.74047 −0.0874619
\(397\) −17.6494 −0.885800 −0.442900 0.896571i \(-0.646050\pi\)
−0.442900 + 0.896571i \(0.646050\pi\)
\(398\) −12.7573 −0.639463
\(399\) −7.93129 −0.397061
\(400\) 0.975626 0.0487813
\(401\) −18.7621 −0.936934 −0.468467 0.883481i \(-0.655193\pi\)
−0.468467 + 0.883481i \(0.655193\pi\)
\(402\) 11.2861 0.562902
\(403\) −11.0157 −0.548732
\(404\) 19.8412 0.987137
\(405\) 11.1629 0.554689
\(406\) 4.51316 0.223984
\(407\) −7.65441 −0.379415
\(408\) −19.4915 −0.964971
\(409\) −15.7447 −0.778524 −0.389262 0.921127i \(-0.627270\pi\)
−0.389262 + 0.921127i \(0.627270\pi\)
\(410\) −1.90448 −0.0940555
\(411\) 31.0097 1.52960
\(412\) 9.52837 0.469429
\(413\) −2.49054 −0.122552
\(414\) 1.88055 0.0924242
\(415\) −6.28445 −0.308491
\(416\) −11.9829 −0.587510
\(417\) −20.0288 −0.980815
\(418\) −0.745312 −0.0364544
\(419\) 25.7072 1.25588 0.627938 0.778263i \(-0.283899\pi\)
0.627938 + 0.778263i \(0.283899\pi\)
\(420\) 11.4568 0.559036
\(421\) −17.2694 −0.841657 −0.420828 0.907140i \(-0.638261\pi\)
−0.420828 + 0.907140i \(0.638261\pi\)
\(422\) −0.231271 −0.0112581
\(423\) −6.37494 −0.309960
\(424\) −5.40252 −0.262369
\(425\) −3.70255 −0.179600
\(426\) 4.78241 0.231709
\(427\) −55.6418 −2.69269
\(428\) −10.8333 −0.523647
\(429\) 4.19200 0.202392
\(430\) −1.07018 −0.0516086
\(431\) −2.81774 −0.135726 −0.0678628 0.997695i \(-0.521618\pi\)
−0.0678628 + 0.997695i \(0.521618\pi\)
\(432\) 3.59131 0.172787
\(433\) 8.85731 0.425655 0.212828 0.977090i \(-0.431733\pi\)
0.212828 + 0.977090i \(0.431733\pi\)
\(434\) −15.5336 −0.745638
\(435\) −3.21035 −0.153925
\(436\) 17.3923 0.832939
\(437\) −2.09412 −0.100175
\(438\) −9.79737 −0.468137
\(439\) −9.71132 −0.463496 −0.231748 0.972776i \(-0.574444\pi\)
−0.231748 + 0.972776i \(0.574444\pi\)
\(440\) 2.56724 0.122388
\(441\) 9.59096 0.456712
\(442\) 5.64137 0.268332
\(443\) 28.3968 1.34917 0.674586 0.738196i \(-0.264322\pi\)
0.674586 + 0.738196i \(0.264322\pi\)
\(444\) −22.6730 −1.07601
\(445\) 1.85225 0.0878051
\(446\) −2.84259 −0.134601
\(447\) 29.0371 1.37341
\(448\) −9.35040 −0.441765
\(449\) 34.9628 1.65000 0.824998 0.565136i \(-0.191176\pi\)
0.824998 + 0.565136i \(0.191176\pi\)
\(450\) 0.898017 0.0423329
\(451\) 2.55527 0.120323
\(452\) 17.1161 0.805073
\(453\) −36.9110 −1.73423
\(454\) −13.1924 −0.619148
\(455\) −7.90699 −0.370685
\(456\) −5.26433 −0.246525
\(457\) 38.6808 1.80941 0.904705 0.426039i \(-0.140091\pi\)
0.904705 + 0.426039i \(0.140091\pi\)
\(458\) 20.2452 0.945998
\(459\) −13.6292 −0.636157
\(460\) 3.02498 0.141040
\(461\) −19.6046 −0.913078 −0.456539 0.889703i \(-0.650911\pi\)
−0.456539 + 0.889703i \(0.650911\pi\)
\(462\) 5.91129 0.275018
\(463\) −3.18393 −0.147970 −0.0739850 0.997259i \(-0.523572\pi\)
−0.0739850 + 0.997259i \(0.523572\pi\)
\(464\) −1.52742 −0.0709088
\(465\) 11.0496 0.512411
\(466\) 0.410545 0.0190182
\(467\) 28.7840 1.33197 0.665983 0.745967i \(-0.268012\pi\)
0.665983 + 0.745967i \(0.268012\pi\)
\(468\) 3.55804 0.164470
\(469\) 28.5625 1.31890
\(470\) 3.94338 0.181895
\(471\) −5.96520 −0.274862
\(472\) −1.65308 −0.0760890
\(473\) 1.43588 0.0660217
\(474\) 4.68047 0.214981
\(475\) −1.00000 −0.0458831
\(476\) −20.6866 −0.948167
\(477\) 2.53557 0.116096
\(478\) −4.00942 −0.183387
\(479\) 9.57656 0.437564 0.218782 0.975774i \(-0.429792\pi\)
0.218782 + 0.975774i \(0.429792\pi\)
\(480\) 12.0197 0.548623
\(481\) 15.6479 0.713483
\(482\) 16.7972 0.765091
\(483\) 16.6091 0.755739
\(484\) −1.44451 −0.0656595
\(485\) −6.58248 −0.298895
\(486\) 8.82999 0.400537
\(487\) −32.8807 −1.48997 −0.744984 0.667083i \(-0.767543\pi\)
−0.744984 + 0.667083i \(0.767543\pi\)
\(488\) −36.9318 −1.67182
\(489\) −19.9642 −0.902812
\(490\) −5.93273 −0.268013
\(491\) 4.43763 0.200267 0.100134 0.994974i \(-0.468073\pi\)
0.100134 + 0.994974i \(0.468073\pi\)
\(492\) 7.56894 0.341234
\(493\) 5.79665 0.261068
\(494\) 1.52364 0.0685519
\(495\) −1.20489 −0.0541556
\(496\) 5.25716 0.236054
\(497\) 12.1031 0.542900
\(498\) −9.60468 −0.430396
\(499\) −18.3904 −0.823266 −0.411633 0.911350i \(-0.635041\pi\)
−0.411633 + 0.911350i \(0.635041\pi\)
\(500\) 1.44451 0.0646004
\(501\) −47.0326 −2.10126
\(502\) 5.65207 0.252264
\(503\) −35.0565 −1.56309 −0.781545 0.623849i \(-0.785568\pi\)
−0.781545 + 0.623849i \(0.785568\pi\)
\(504\) 11.9641 0.532921
\(505\) 13.7356 0.611226
\(506\) 1.56077 0.0693849
\(507\) 18.0879 0.803310
\(508\) −18.6060 −0.825507
\(509\) 29.8890 1.32481 0.662403 0.749147i \(-0.269537\pi\)
0.662403 + 0.749147i \(0.269537\pi\)
\(510\) −5.65870 −0.250572
\(511\) −24.7948 −1.09686
\(512\) 10.7281 0.474118
\(513\) −3.68103 −0.162521
\(514\) 10.1598 0.448132
\(515\) 6.59626 0.290666
\(516\) 4.25319 0.187236
\(517\) −5.29091 −0.232694
\(518\) 22.0656 0.969508
\(519\) 11.1750 0.490527
\(520\) −5.24820 −0.230149
\(521\) 3.35923 0.147170 0.0735852 0.997289i \(-0.476556\pi\)
0.0735852 + 0.997289i \(0.476556\pi\)
\(522\) −1.40592 −0.0615353
\(523\) 41.7591 1.82600 0.913000 0.407961i \(-0.133760\pi\)
0.913000 + 0.407961i \(0.133760\pi\)
\(524\) −20.6620 −0.902623
\(525\) 7.93129 0.346150
\(526\) −3.00381 −0.130972
\(527\) −19.9512 −0.869089
\(528\) −2.00060 −0.0870650
\(529\) −18.6147 −0.809333
\(530\) −1.56844 −0.0681288
\(531\) 0.775842 0.0336687
\(532\) −5.58711 −0.242232
\(533\) −5.22375 −0.226266
\(534\) 2.83084 0.122502
\(535\) −7.49964 −0.324238
\(536\) 18.9582 0.818867
\(537\) −36.4111 −1.57125
\(538\) 19.8491 0.855756
\(539\) 7.96005 0.342864
\(540\) 5.31728 0.228819
\(541\) −10.2418 −0.440331 −0.220165 0.975463i \(-0.570660\pi\)
−0.220165 + 0.975463i \(0.570660\pi\)
\(542\) 16.6465 0.715028
\(543\) 45.2334 1.94115
\(544\) −21.7030 −0.930507
\(545\) 12.0403 0.515748
\(546\) −12.0844 −0.517166
\(547\) −12.8348 −0.548779 −0.274389 0.961619i \(-0.588476\pi\)
−0.274389 + 0.961619i \(0.588476\pi\)
\(548\) 21.8445 0.933150
\(549\) 17.3333 0.739766
\(550\) 0.745312 0.0317802
\(551\) 1.56558 0.0666960
\(552\) 11.0241 0.469218
\(553\) 11.8452 0.503707
\(554\) −16.9045 −0.718203
\(555\) −15.6960 −0.666258
\(556\) −14.1091 −0.598358
\(557\) 28.3802 1.20251 0.601254 0.799058i \(-0.294668\pi\)
0.601254 + 0.799058i \(0.294668\pi\)
\(558\) 4.83896 0.204850
\(559\) −2.93536 −0.124153
\(560\) 3.77355 0.159462
\(561\) 7.59239 0.320551
\(562\) 16.4629 0.694446
\(563\) 18.5823 0.783149 0.391575 0.920146i \(-0.371930\pi\)
0.391575 + 0.920146i \(0.371930\pi\)
\(564\) −15.6721 −0.659915
\(565\) 11.8491 0.498494
\(566\) −13.0044 −0.546618
\(567\) 43.1761 1.81323
\(568\) 8.03336 0.337072
\(569\) 5.80939 0.243542 0.121771 0.992558i \(-0.461143\pi\)
0.121771 + 0.992558i \(0.461143\pi\)
\(570\) −1.52832 −0.0640145
\(571\) −25.5528 −1.06935 −0.534675 0.845058i \(-0.679566\pi\)
−0.534675 + 0.845058i \(0.679566\pi\)
\(572\) 2.95301 0.123472
\(573\) −25.8568 −1.08018
\(574\) −7.36618 −0.307459
\(575\) 2.09412 0.0873309
\(576\) 2.91279 0.121366
\(577\) −1.44616 −0.0602043 −0.0301021 0.999547i \(-0.509583\pi\)
−0.0301021 + 0.999547i \(0.509583\pi\)
\(578\) −2.45289 −0.102027
\(579\) 17.1670 0.713437
\(580\) −2.26150 −0.0939035
\(581\) −24.3071 −1.00843
\(582\) −10.0602 −0.417007
\(583\) 2.10441 0.0871557
\(584\) −16.4573 −0.681010
\(585\) 2.46315 0.101839
\(586\) 12.7039 0.524794
\(587\) −5.15981 −0.212968 −0.106484 0.994314i \(-0.533959\pi\)
−0.106484 + 0.994314i \(0.533959\pi\)
\(588\) 23.5784 0.972355
\(589\) −5.38850 −0.222029
\(590\) −0.479916 −0.0197578
\(591\) −19.7648 −0.813013
\(592\) −7.46784 −0.306926
\(593\) 3.99385 0.164008 0.0820038 0.996632i \(-0.473868\pi\)
0.0820038 + 0.996632i \(0.473868\pi\)
\(594\) 2.74352 0.112568
\(595\) −14.3208 −0.587096
\(596\) 20.4549 0.837864
\(597\) 35.0991 1.43651
\(598\) −3.19069 −0.130477
\(599\) −20.1155 −0.821899 −0.410949 0.911658i \(-0.634803\pi\)
−0.410949 + 0.911658i \(0.634803\pi\)
\(600\) 5.26433 0.214915
\(601\) 34.2267 1.39614 0.698069 0.716031i \(-0.254043\pi\)
0.698069 + 0.716031i \(0.254043\pi\)
\(602\) −4.13926 −0.168704
\(603\) −8.89767 −0.362341
\(604\) −26.0015 −1.05799
\(605\) −1.00000 −0.0406558
\(606\) 20.9925 0.852760
\(607\) 10.0999 0.409943 0.204971 0.978768i \(-0.434290\pi\)
0.204971 + 0.978768i \(0.434290\pi\)
\(608\) −5.86162 −0.237720
\(609\) −12.4171 −0.503165
\(610\) −10.7219 −0.434118
\(611\) 10.8162 0.437577
\(612\) 6.44418 0.260491
\(613\) 0.860528 0.0347564 0.0173782 0.999849i \(-0.494468\pi\)
0.0173782 + 0.999849i \(0.494468\pi\)
\(614\) −7.10069 −0.286561
\(615\) 5.23980 0.211289
\(616\) 9.92961 0.400075
\(617\) −40.8378 −1.64407 −0.822033 0.569439i \(-0.807160\pi\)
−0.822033 + 0.569439i \(0.807160\pi\)
\(618\) 10.0812 0.405527
\(619\) −12.7328 −0.511774 −0.255887 0.966707i \(-0.582367\pi\)
−0.255887 + 0.966707i \(0.582367\pi\)
\(620\) 7.78374 0.312603
\(621\) 7.70852 0.309332
\(622\) −18.7600 −0.752207
\(623\) 7.16418 0.287027
\(624\) 4.08983 0.163724
\(625\) 1.00000 0.0400000
\(626\) −6.10495 −0.244003
\(627\) 2.05058 0.0818924
\(628\) −4.20212 −0.167683
\(629\) 28.3409 1.13002
\(630\) 3.47337 0.138382
\(631\) −27.2662 −1.08545 −0.542725 0.839911i \(-0.682607\pi\)
−0.542725 + 0.839911i \(0.682607\pi\)
\(632\) 7.86213 0.312739
\(633\) 0.636296 0.0252905
\(634\) 15.6478 0.621452
\(635\) −12.8805 −0.511146
\(636\) 6.23344 0.247172
\(637\) −16.2727 −0.644749
\(638\) −1.16685 −0.0461959
\(639\) −3.77031 −0.149151
\(640\) 9.92145 0.392180
\(641\) −16.6825 −0.658920 −0.329460 0.944170i \(-0.606867\pi\)
−0.329460 + 0.944170i \(0.606867\pi\)
\(642\) −11.4619 −0.452364
\(643\) −19.6733 −0.775839 −0.387919 0.921693i \(-0.626806\pi\)
−0.387919 + 0.921693i \(0.626806\pi\)
\(644\) 11.7001 0.461048
\(645\) 2.94439 0.115935
\(646\) 2.75956 0.108573
\(647\) 28.9630 1.13865 0.569326 0.822112i \(-0.307204\pi\)
0.569326 + 0.822112i \(0.307204\pi\)
\(648\) 28.6578 1.12579
\(649\) 0.643913 0.0252758
\(650\) −1.52364 −0.0597621
\(651\) 42.7378 1.67502
\(652\) −14.0636 −0.550771
\(653\) −2.45120 −0.0959230 −0.0479615 0.998849i \(-0.515272\pi\)
−0.0479615 + 0.998849i \(0.515272\pi\)
\(654\) 18.4014 0.719552
\(655\) −14.3038 −0.558896
\(656\) 2.49299 0.0973350
\(657\) 7.72396 0.301340
\(658\) 15.2523 0.594596
\(659\) 4.60105 0.179231 0.0896157 0.995976i \(-0.471436\pi\)
0.0896157 + 0.995976i \(0.471436\pi\)
\(660\) −2.96208 −0.115299
\(661\) 1.55553 0.0605029 0.0302515 0.999542i \(-0.490369\pi\)
0.0302515 + 0.999542i \(0.490369\pi\)
\(662\) 2.44508 0.0950308
\(663\) −15.5211 −0.602790
\(664\) −16.1337 −0.626108
\(665\) −3.86782 −0.149988
\(666\) −6.87378 −0.266354
\(667\) −3.27851 −0.126945
\(668\) −33.1316 −1.28190
\(669\) 7.82084 0.302371
\(670\) 5.50388 0.212633
\(671\) 14.3858 0.555358
\(672\) 46.4902 1.79340
\(673\) 40.7350 1.57022 0.785110 0.619356i \(-0.212606\pi\)
0.785110 + 0.619356i \(0.212606\pi\)
\(674\) −13.7505 −0.529650
\(675\) 3.68103 0.141683
\(676\) 12.7418 0.490069
\(677\) 11.5539 0.444051 0.222025 0.975041i \(-0.428733\pi\)
0.222025 + 0.975041i \(0.428733\pi\)
\(678\) 18.1092 0.695480
\(679\) −25.4599 −0.977059
\(680\) −9.50533 −0.364513
\(681\) 36.2962 1.39087
\(682\) 4.01612 0.153785
\(683\) 18.1027 0.692679 0.346339 0.938109i \(-0.387425\pi\)
0.346339 + 0.938109i \(0.387425\pi\)
\(684\) 1.74047 0.0665485
\(685\) 15.1224 0.577798
\(686\) −2.76758 −0.105667
\(687\) −55.7008 −2.12512
\(688\) 1.40088 0.0534081
\(689\) −4.30204 −0.163895
\(690\) 3.20050 0.121841
\(691\) −6.04864 −0.230101 −0.115051 0.993360i \(-0.536703\pi\)
−0.115051 + 0.993360i \(0.536703\pi\)
\(692\) 7.87208 0.299252
\(693\) −4.66029 −0.177030
\(694\) 22.2693 0.845332
\(695\) −9.76738 −0.370498
\(696\) −8.24173 −0.312402
\(697\) −9.46104 −0.358363
\(698\) 20.5215 0.776749
\(699\) −1.12954 −0.0427230
\(700\) 5.58711 0.211173
\(701\) 31.8527 1.20306 0.601530 0.798850i \(-0.294558\pi\)
0.601530 + 0.798850i \(0.294558\pi\)
\(702\) −5.60857 −0.211682
\(703\) 7.65441 0.288691
\(704\) 2.41748 0.0911124
\(705\) −10.8494 −0.408614
\(706\) 2.85917 0.107606
\(707\) 53.1269 1.99804
\(708\) 1.90733 0.0716817
\(709\) −38.1780 −1.43381 −0.716903 0.697173i \(-0.754441\pi\)
−0.716903 + 0.697173i \(0.754441\pi\)
\(710\) 2.33222 0.0875268
\(711\) −3.68995 −0.138384
\(712\) 4.75516 0.178207
\(713\) 11.2842 0.422596
\(714\) −21.8869 −0.819095
\(715\) 2.04430 0.0764525
\(716\) −25.6494 −0.958563
\(717\) 11.0311 0.411965
\(718\) 24.4196 0.911333
\(719\) −28.3038 −1.05555 −0.527777 0.849383i \(-0.676974\pi\)
−0.527777 + 0.849383i \(0.676974\pi\)
\(720\) −1.17552 −0.0438090
\(721\) 25.5132 0.950160
\(722\) 0.745312 0.0277377
\(723\) −46.2142 −1.71872
\(724\) 31.8642 1.18422
\(725\) −1.56558 −0.0581442
\(726\) −1.52832 −0.0567214
\(727\) −31.1881 −1.15670 −0.578352 0.815787i \(-0.696304\pi\)
−0.578352 + 0.815787i \(0.696304\pi\)
\(728\) −20.2991 −0.752335
\(729\) 9.19472 0.340545
\(730\) −4.77785 −0.176836
\(731\) −5.31642 −0.196635
\(732\) 42.6120 1.57498
\(733\) 4.62188 0.170713 0.0853565 0.996350i \(-0.472797\pi\)
0.0853565 + 0.996350i \(0.472797\pi\)
\(734\) 15.6769 0.578645
\(735\) 16.3227 0.602073
\(736\) 12.2749 0.452460
\(737\) −7.38466 −0.272017
\(738\) 2.29468 0.0844683
\(739\) −5.87519 −0.216122 −0.108061 0.994144i \(-0.534464\pi\)
−0.108061 + 0.994144i \(0.534464\pi\)
\(740\) −11.0569 −0.406458
\(741\) −4.19200 −0.153997
\(742\) −6.06646 −0.222707
\(743\) −19.5669 −0.717839 −0.358919 0.933369i \(-0.616855\pi\)
−0.358919 + 0.933369i \(0.616855\pi\)
\(744\) 28.3668 1.03998
\(745\) 14.1604 0.518798
\(746\) 1.54990 0.0567460
\(747\) 7.57205 0.277047
\(748\) 5.34837 0.195556
\(749\) −29.0073 −1.05990
\(750\) 1.52832 0.0558065
\(751\) −17.1912 −0.627314 −0.313657 0.949536i \(-0.601554\pi\)
−0.313657 + 0.949536i \(0.601554\pi\)
\(752\) −5.16195 −0.188237
\(753\) −15.5506 −0.566694
\(754\) 2.38538 0.0868706
\(755\) −18.0003 −0.655096
\(756\) 20.5663 0.747989
\(757\) −54.9157 −1.99595 −0.997973 0.0636322i \(-0.979732\pi\)
−0.997973 + 0.0636322i \(0.979732\pi\)
\(758\) 22.9404 0.833233
\(759\) −4.29417 −0.155868
\(760\) −2.56724 −0.0931234
\(761\) −2.42585 −0.0879372 −0.0439686 0.999033i \(-0.514000\pi\)
−0.0439686 + 0.999033i \(0.514000\pi\)
\(762\) −19.6855 −0.713132
\(763\) 46.5696 1.68593
\(764\) −18.2145 −0.658978
\(765\) 4.46116 0.161293
\(766\) −0.0109674 −0.000396269 0
\(767\) −1.31635 −0.0475307
\(768\) 25.0777 0.904913
\(769\) −4.51504 −0.162816 −0.0814082 0.996681i \(-0.525942\pi\)
−0.0814082 + 0.996681i \(0.525942\pi\)
\(770\) 2.88274 0.103887
\(771\) −27.9528 −1.00670
\(772\) 12.0931 0.435241
\(773\) −4.60263 −0.165545 −0.0827726 0.996568i \(-0.526378\pi\)
−0.0827726 + 0.996568i \(0.526378\pi\)
\(774\) 1.28944 0.0463480
\(775\) 5.38850 0.193561
\(776\) −16.8988 −0.606630
\(777\) −60.7093 −2.17793
\(778\) −17.0826 −0.612441
\(779\) −2.55527 −0.0915522
\(780\) 6.05539 0.216818
\(781\) −3.12919 −0.111971
\(782\) −5.77885 −0.206651
\(783\) −5.76295 −0.205951
\(784\) 7.76604 0.277358
\(785\) −2.90903 −0.103828
\(786\) −21.8608 −0.779751
\(787\) 0.615078 0.0219252 0.0109626 0.999940i \(-0.496510\pi\)
0.0109626 + 0.999940i \(0.496510\pi\)
\(788\) −13.9231 −0.495988
\(789\) 8.26440 0.294220
\(790\) 2.28251 0.0812081
\(791\) 45.8301 1.62953
\(792\) −3.09323 −0.109913
\(793\) −29.4089 −1.04434
\(794\) −13.1544 −0.466831
\(795\) 4.31526 0.153047
\(796\) 24.7252 0.876360
\(797\) 54.0423 1.91428 0.957139 0.289630i \(-0.0935322\pi\)
0.957139 + 0.289630i \(0.0935322\pi\)
\(798\) −5.91129 −0.209257
\(799\) 19.5899 0.693040
\(800\) 5.86162 0.207239
\(801\) −2.23175 −0.0788550
\(802\) −13.9836 −0.493778
\(803\) 6.41053 0.226223
\(804\) −21.8740 −0.771436
\(805\) 8.09969 0.285476
\(806\) −8.21015 −0.289190
\(807\) −54.6110 −1.92240
\(808\) 35.2625 1.24053
\(809\) −14.6049 −0.513480 −0.256740 0.966481i \(-0.582648\pi\)
−0.256740 + 0.966481i \(0.582648\pi\)
\(810\) 8.31985 0.292330
\(811\) 18.1014 0.635626 0.317813 0.948153i \(-0.397052\pi\)
0.317813 + 0.948153i \(0.397052\pi\)
\(812\) −8.74707 −0.306962
\(813\) −45.7996 −1.60626
\(814\) −5.70492 −0.199958
\(815\) −9.73587 −0.341033
\(816\) 7.40733 0.259309
\(817\) −1.43588 −0.0502350
\(818\) −11.7347 −0.410294
\(819\) 9.52702 0.332901
\(820\) 3.69112 0.128899
\(821\) 17.5804 0.613562 0.306781 0.951780i \(-0.400748\pi\)
0.306781 + 0.951780i \(0.400748\pi\)
\(822\) 23.1120 0.806122
\(823\) −42.5561 −1.48341 −0.741706 0.670725i \(-0.765983\pi\)
−0.741706 + 0.670725i \(0.765983\pi\)
\(824\) 16.9342 0.589930
\(825\) −2.05058 −0.0713921
\(826\) −1.85623 −0.0645866
\(827\) 48.0006 1.66914 0.834572 0.550899i \(-0.185715\pi\)
0.834572 + 0.550899i \(0.185715\pi\)
\(828\) −3.64475 −0.126664
\(829\) 34.1185 1.18498 0.592492 0.805576i \(-0.298144\pi\)
0.592492 + 0.805576i \(0.298144\pi\)
\(830\) −4.68388 −0.162580
\(831\) 46.5094 1.61339
\(832\) −4.94206 −0.171335
\(833\) −29.4725 −1.02116
\(834\) −14.9277 −0.516905
\(835\) −22.9362 −0.793741
\(836\) 1.44451 0.0499594
\(837\) 19.8352 0.685606
\(838\) 19.1599 0.661866
\(839\) −13.4842 −0.465528 −0.232764 0.972533i \(-0.574777\pi\)
−0.232764 + 0.972533i \(0.574777\pi\)
\(840\) 20.3615 0.702538
\(841\) −26.5490 −0.915481
\(842\) −12.8711 −0.443566
\(843\) −45.2945 −1.56003
\(844\) 0.448231 0.0154288
\(845\) 8.82084 0.303446
\(846\) −4.75132 −0.163354
\(847\) −3.86782 −0.132900
\(848\) 2.05312 0.0705043
\(849\) 35.7792 1.22794
\(850\) −2.75956 −0.0946521
\(851\) −16.0293 −0.549476
\(852\) −9.26892 −0.317548
\(853\) 7.34151 0.251369 0.125684 0.992070i \(-0.459887\pi\)
0.125684 + 0.992070i \(0.459887\pi\)
\(854\) −41.4705 −1.41909
\(855\) 1.20489 0.0412062
\(856\) −19.2533 −0.658066
\(857\) −25.9177 −0.885331 −0.442666 0.896687i \(-0.645967\pi\)
−0.442666 + 0.896687i \(0.645967\pi\)
\(858\) 3.12435 0.106664
\(859\) −12.6379 −0.431200 −0.215600 0.976482i \(-0.569171\pi\)
−0.215600 + 0.976482i \(0.569171\pi\)
\(860\) 2.07414 0.0707276
\(861\) 20.2666 0.690684
\(862\) −2.10009 −0.0715295
\(863\) 37.4073 1.27336 0.636679 0.771129i \(-0.280307\pi\)
0.636679 + 0.771129i \(0.280307\pi\)
\(864\) 21.5768 0.734057
\(865\) 5.44966 0.185294
\(866\) 6.60146 0.224327
\(867\) 6.74866 0.229196
\(868\) 30.1061 1.02187
\(869\) −3.06249 −0.103888
\(870\) −2.39271 −0.0811206
\(871\) 15.0964 0.511524
\(872\) 30.9102 1.04675
\(873\) 7.93113 0.268428
\(874\) −1.56077 −0.0527940
\(875\) 3.86782 0.130756
\(876\) 18.9885 0.641563
\(877\) −49.5616 −1.67358 −0.836788 0.547527i \(-0.815569\pi\)
−0.836788 + 0.547527i \(0.815569\pi\)
\(878\) −7.23796 −0.244269
\(879\) −34.9523 −1.17891
\(880\) −0.975626 −0.0328883
\(881\) −19.9617 −0.672527 −0.336263 0.941768i \(-0.609163\pi\)
−0.336263 + 0.941768i \(0.609163\pi\)
\(882\) 7.14826 0.240694
\(883\) 26.4622 0.890524 0.445262 0.895400i \(-0.353111\pi\)
0.445262 + 0.895400i \(0.353111\pi\)
\(884\) −10.9337 −0.367739
\(885\) 1.32040 0.0443846
\(886\) 21.1645 0.711035
\(887\) −0.489272 −0.0164281 −0.00821407 0.999966i \(-0.502615\pi\)
−0.00821407 + 0.999966i \(0.502615\pi\)
\(888\) −40.2953 −1.35222
\(889\) −49.8194 −1.67089
\(890\) 1.38051 0.0462746
\(891\) −11.1629 −0.373971
\(892\) 5.50930 0.184465
\(893\) 5.29091 0.177053
\(894\) 21.6417 0.723808
\(895\) −17.7565 −0.593533
\(896\) 38.3744 1.28200
\(897\) 8.77856 0.293108
\(898\) 26.0582 0.869574
\(899\) −8.43614 −0.281361
\(900\) −1.74047 −0.0580156
\(901\) −7.79169 −0.259579
\(902\) 1.90448 0.0634122
\(903\) 11.3884 0.378981
\(904\) 30.4194 1.01173
\(905\) 22.0588 0.733260
\(906\) −27.5102 −0.913966
\(907\) 48.9032 1.62381 0.811903 0.583793i \(-0.198432\pi\)
0.811903 + 0.583793i \(0.198432\pi\)
\(908\) 25.5685 0.848519
\(909\) −16.5498 −0.548924
\(910\) −5.89318 −0.195357
\(911\) −7.54945 −0.250125 −0.125062 0.992149i \(-0.539913\pi\)
−0.125062 + 0.992149i \(0.539913\pi\)
\(912\) 2.00060 0.0662465
\(913\) 6.28445 0.207985
\(914\) 28.8292 0.953587
\(915\) 29.4993 0.975216
\(916\) −39.2378 −1.29645
\(917\) −55.3246 −1.82698
\(918\) −10.1580 −0.335264
\(919\) −31.0368 −1.02381 −0.511905 0.859042i \(-0.671060\pi\)
−0.511905 + 0.859042i \(0.671060\pi\)
\(920\) 5.37610 0.177245
\(921\) 19.5362 0.643738
\(922\) −14.6116 −0.481206
\(923\) 6.39700 0.210560
\(924\) −11.4568 −0.376902
\(925\) −7.65441 −0.251675
\(926\) −2.37303 −0.0779825
\(927\) −7.94775 −0.261038
\(928\) −9.17683 −0.301244
\(929\) 3.46660 0.113735 0.0568677 0.998382i \(-0.481889\pi\)
0.0568677 + 0.998382i \(0.481889\pi\)
\(930\) 8.23538 0.270049
\(931\) −7.96005 −0.260880
\(932\) −0.795689 −0.0260637
\(933\) 51.6144 1.68978
\(934\) 21.4531 0.701966
\(935\) 3.70255 0.121086
\(936\) 6.32348 0.206689
\(937\) −26.4876 −0.865313 −0.432657 0.901559i \(-0.642424\pi\)
−0.432657 + 0.901559i \(0.642424\pi\)
\(938\) 21.2880 0.695078
\(939\) 16.7966 0.548136
\(940\) −7.64277 −0.249280
\(941\) −48.9435 −1.59551 −0.797756 0.602981i \(-0.793980\pi\)
−0.797756 + 0.602981i \(0.793980\pi\)
\(942\) −4.44594 −0.144856
\(943\) 5.35105 0.174254
\(944\) 0.628218 0.0204468
\(945\) 14.2376 0.463148
\(946\) 1.07018 0.0347945
\(947\) 14.6896 0.477346 0.238673 0.971100i \(-0.423288\pi\)
0.238673 + 0.971100i \(0.423288\pi\)
\(948\) −9.07135 −0.294624
\(949\) −13.1050 −0.425408
\(950\) −0.745312 −0.0241811
\(951\) −43.0518 −1.39605
\(952\) −36.7649 −1.19156
\(953\) 13.8224 0.447752 0.223876 0.974618i \(-0.428129\pi\)
0.223876 + 0.974618i \(0.428129\pi\)
\(954\) 1.88979 0.0611844
\(955\) −12.6095 −0.408033
\(956\) 7.77076 0.251324
\(957\) 3.21035 0.103776
\(958\) 7.13753 0.230603
\(959\) 58.4908 1.88877
\(960\) 4.95725 0.159995
\(961\) −1.96404 −0.0633561
\(962\) 11.6626 0.376017
\(963\) 9.03621 0.291188
\(964\) −32.5551 −1.04853
\(965\) 8.37178 0.269497
\(966\) 12.3789 0.398286
\(967\) −48.0516 −1.54523 −0.772617 0.634872i \(-0.781053\pi\)
−0.772617 + 0.634872i \(0.781053\pi\)
\(968\) −2.56724 −0.0825141
\(969\) −7.59239 −0.243903
\(970\) −4.90600 −0.157522
\(971\) −14.9180 −0.478740 −0.239370 0.970928i \(-0.576941\pi\)
−0.239370 + 0.970928i \(0.576941\pi\)
\(972\) −17.1136 −0.548920
\(973\) −37.7785 −1.21112
\(974\) −24.5064 −0.785236
\(975\) 4.19200 0.134252
\(976\) 14.0352 0.449255
\(977\) −13.7541 −0.440032 −0.220016 0.975496i \(-0.570611\pi\)
−0.220016 + 0.975496i \(0.570611\pi\)
\(978\) −14.8796 −0.475796
\(979\) −1.85225 −0.0591982
\(980\) 11.4984 0.367302
\(981\) −14.5071 −0.463177
\(982\) 3.30742 0.105544
\(983\) 34.7768 1.10921 0.554604 0.832115i \(-0.312870\pi\)
0.554604 + 0.832115i \(0.312870\pi\)
\(984\) 13.4518 0.428828
\(985\) −9.63861 −0.307112
\(986\) 4.32031 0.137587
\(987\) −41.9637 −1.33572
\(988\) −2.95301 −0.0939478
\(989\) 3.00690 0.0956139
\(990\) −0.898017 −0.0285408
\(991\) −58.8419 −1.86917 −0.934587 0.355734i \(-0.884231\pi\)
−0.934587 + 0.355734i \(0.884231\pi\)
\(992\) 31.5853 1.00284
\(993\) −6.72717 −0.213480
\(994\) 9.02062 0.286117
\(995\) 17.1167 0.542634
\(996\) 18.6151 0.589841
\(997\) 9.48200 0.300298 0.150149 0.988663i \(-0.452025\pi\)
0.150149 + 0.988663i \(0.452025\pi\)
\(998\) −13.7066 −0.433874
\(999\) −28.1761 −0.891452
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1045.2.a.h.1.5 7
3.2 odd 2 9405.2.a.bd.1.3 7
5.4 even 2 5225.2.a.m.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.h.1.5 7 1.1 even 1 trivial
5225.2.a.m.1.3 7 5.4 even 2
9405.2.a.bd.1.3 7 3.2 odd 2