Properties

Label 8034.2.a.r.1.7
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8034,2,Mod(1,8034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, 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("8034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.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: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 9x^{7} + 45x^{6} + 7x^{5} - 123x^{4} + 37x^{3} + 87x^{2} - 54x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.25914\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.25914 q^{5} -1.00000 q^{6} -0.797445 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.25914 q^{5} -1.00000 q^{6} -0.797445 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.25914 q^{10} -2.98171 q^{11} -1.00000 q^{12} +1.00000 q^{13} -0.797445 q^{14} -1.25914 q^{15} +1.00000 q^{16} +4.87471 q^{17} +1.00000 q^{18} -4.45274 q^{19} +1.25914 q^{20} +0.797445 q^{21} -2.98171 q^{22} +4.08918 q^{23} -1.00000 q^{24} -3.41456 q^{25} +1.00000 q^{26} -1.00000 q^{27} -0.797445 q^{28} -9.96702 q^{29} -1.25914 q^{30} +4.04763 q^{31} +1.00000 q^{32} +2.98171 q^{33} +4.87471 q^{34} -1.00410 q^{35} +1.00000 q^{36} -3.80662 q^{37} -4.45274 q^{38} -1.00000 q^{39} +1.25914 q^{40} -2.52399 q^{41} +0.797445 q^{42} -3.00371 q^{43} -2.98171 q^{44} +1.25914 q^{45} +4.08918 q^{46} -3.00222 q^{47} -1.00000 q^{48} -6.36408 q^{49} -3.41456 q^{50} -4.87471 q^{51} +1.00000 q^{52} -8.63309 q^{53} -1.00000 q^{54} -3.75440 q^{55} -0.797445 q^{56} +4.45274 q^{57} -9.96702 q^{58} -6.72403 q^{59} -1.25914 q^{60} +4.06686 q^{61} +4.04763 q^{62} -0.797445 q^{63} +1.00000 q^{64} +1.25914 q^{65} +2.98171 q^{66} +4.94082 q^{67} +4.87471 q^{68} -4.08918 q^{69} -1.00410 q^{70} -1.94373 q^{71} +1.00000 q^{72} -7.45502 q^{73} -3.80662 q^{74} +3.41456 q^{75} -4.45274 q^{76} +2.37775 q^{77} -1.00000 q^{78} +14.1834 q^{79} +1.25914 q^{80} +1.00000 q^{81} -2.52399 q^{82} +15.3797 q^{83} +0.797445 q^{84} +6.13796 q^{85} -3.00371 q^{86} +9.96702 q^{87} -2.98171 q^{88} +14.6866 q^{89} +1.25914 q^{90} -0.797445 q^{91} +4.08918 q^{92} -4.04763 q^{93} -3.00222 q^{94} -5.60664 q^{95} -1.00000 q^{96} -11.3927 q^{97} -6.36408 q^{98} -2.98171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} - 4 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} - 4 q^{7} + 9 q^{8} + 9 q^{9} - 4 q^{10} - 5 q^{11} - 9 q^{12} + 9 q^{13} - 4 q^{14} + 4 q^{15} + 9 q^{16} - 6 q^{17} + 9 q^{18} - 4 q^{19} - 4 q^{20} + 4 q^{21} - 5 q^{22} - 6 q^{23} - 9 q^{24} - 11 q^{25} + 9 q^{26} - 9 q^{27} - 4 q^{28} - 19 q^{29} + 4 q^{30} - 6 q^{31} + 9 q^{32} + 5 q^{33} - 6 q^{34} + 10 q^{35} + 9 q^{36} - 13 q^{37} - 4 q^{38} - 9 q^{39} - 4 q^{40} - 18 q^{41} + 4 q^{42} - 20 q^{43} - 5 q^{44} - 4 q^{45} - 6 q^{46} + 14 q^{47} - 9 q^{48} - 3 q^{49} - 11 q^{50} + 6 q^{51} + 9 q^{52} - 3 q^{53} - 9 q^{54} - 4 q^{55} - 4 q^{56} + 4 q^{57} - 19 q^{58} - 9 q^{59} + 4 q^{60} - 24 q^{61} - 6 q^{62} - 4 q^{63} + 9 q^{64} - 4 q^{65} + 5 q^{66} - 4 q^{67} - 6 q^{68} + 6 q^{69} + 10 q^{70} - 9 q^{71} + 9 q^{72} - 24 q^{73} - 13 q^{74} + 11 q^{75} - 4 q^{76} + 3 q^{77} - 9 q^{78} - 15 q^{79} - 4 q^{80} + 9 q^{81} - 18 q^{82} + 20 q^{83} + 4 q^{84} - 31 q^{85} - 20 q^{86} + 19 q^{87} - 5 q^{88} + 3 q^{89} - 4 q^{90} - 4 q^{91} - 6 q^{92} + 6 q^{93} + 14 q^{94} - 4 q^{95} - 9 q^{96} - 19 q^{97} - 3 q^{98} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.25914 0.563106 0.281553 0.959546i \(-0.409150\pi\)
0.281553 + 0.959546i \(0.409150\pi\)
\(6\) −1.00000 −0.408248
\(7\) −0.797445 −0.301406 −0.150703 0.988579i \(-0.548154\pi\)
−0.150703 + 0.988579i \(0.548154\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.25914 0.398176
\(11\) −2.98171 −0.899020 −0.449510 0.893275i \(-0.648401\pi\)
−0.449510 + 0.893275i \(0.648401\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) −0.797445 −0.213126
\(15\) −1.25914 −0.325109
\(16\) 1.00000 0.250000
\(17\) 4.87471 1.18229 0.591146 0.806565i \(-0.298676\pi\)
0.591146 + 0.806565i \(0.298676\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.45274 −1.02153 −0.510764 0.859721i \(-0.670637\pi\)
−0.510764 + 0.859721i \(0.670637\pi\)
\(20\) 1.25914 0.281553
\(21\) 0.797445 0.174017
\(22\) −2.98171 −0.635703
\(23\) 4.08918 0.852652 0.426326 0.904570i \(-0.359808\pi\)
0.426326 + 0.904570i \(0.359808\pi\)
\(24\) −1.00000 −0.204124
\(25\) −3.41456 −0.682912
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) −0.797445 −0.150703
\(29\) −9.96702 −1.85083 −0.925415 0.378955i \(-0.876283\pi\)
−0.925415 + 0.378955i \(0.876283\pi\)
\(30\) −1.25914 −0.229887
\(31\) 4.04763 0.726976 0.363488 0.931599i \(-0.381586\pi\)
0.363488 + 0.931599i \(0.381586\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.98171 0.519049
\(34\) 4.87471 0.836006
\(35\) −1.00410 −0.169723
\(36\) 1.00000 0.166667
\(37\) −3.80662 −0.625804 −0.312902 0.949785i \(-0.601301\pi\)
−0.312902 + 0.949785i \(0.601301\pi\)
\(38\) −4.45274 −0.722330
\(39\) −1.00000 −0.160128
\(40\) 1.25914 0.199088
\(41\) −2.52399 −0.394181 −0.197091 0.980385i \(-0.563149\pi\)
−0.197091 + 0.980385i \(0.563149\pi\)
\(42\) 0.797445 0.123048
\(43\) −3.00371 −0.458061 −0.229031 0.973419i \(-0.573556\pi\)
−0.229031 + 0.973419i \(0.573556\pi\)
\(44\) −2.98171 −0.449510
\(45\) 1.25914 0.187702
\(46\) 4.08918 0.602916
\(47\) −3.00222 −0.437919 −0.218959 0.975734i \(-0.570266\pi\)
−0.218959 + 0.975734i \(0.570266\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.36408 −0.909155
\(50\) −3.41456 −0.482891
\(51\) −4.87471 −0.682596
\(52\) 1.00000 0.138675
\(53\) −8.63309 −1.18585 −0.592923 0.805259i \(-0.702026\pi\)
−0.592923 + 0.805259i \(0.702026\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.75440 −0.506243
\(56\) −0.797445 −0.106563
\(57\) 4.45274 0.589780
\(58\) −9.96702 −1.30873
\(59\) −6.72403 −0.875394 −0.437697 0.899123i \(-0.644206\pi\)
−0.437697 + 0.899123i \(0.644206\pi\)
\(60\) −1.25914 −0.162555
\(61\) 4.06686 0.520708 0.260354 0.965513i \(-0.416161\pi\)
0.260354 + 0.965513i \(0.416161\pi\)
\(62\) 4.04763 0.514049
\(63\) −0.797445 −0.100469
\(64\) 1.00000 0.125000
\(65\) 1.25914 0.156177
\(66\) 2.98171 0.367023
\(67\) 4.94082 0.603617 0.301808 0.953369i \(-0.402410\pi\)
0.301808 + 0.953369i \(0.402410\pi\)
\(68\) 4.87471 0.591146
\(69\) −4.08918 −0.492279
\(70\) −1.00410 −0.120013
\(71\) −1.94373 −0.230679 −0.115339 0.993326i \(-0.536796\pi\)
−0.115339 + 0.993326i \(0.536796\pi\)
\(72\) 1.00000 0.117851
\(73\) −7.45502 −0.872544 −0.436272 0.899815i \(-0.643702\pi\)
−0.436272 + 0.899815i \(0.643702\pi\)
\(74\) −3.80662 −0.442510
\(75\) 3.41456 0.394279
\(76\) −4.45274 −0.510764
\(77\) 2.37775 0.270970
\(78\) −1.00000 −0.113228
\(79\) 14.1834 1.59576 0.797878 0.602819i \(-0.205956\pi\)
0.797878 + 0.602819i \(0.205956\pi\)
\(80\) 1.25914 0.140776
\(81\) 1.00000 0.111111
\(82\) −2.52399 −0.278728
\(83\) 15.3797 1.68814 0.844068 0.536236i \(-0.180154\pi\)
0.844068 + 0.536236i \(0.180154\pi\)
\(84\) 0.797445 0.0870083
\(85\) 6.13796 0.665755
\(86\) −3.00371 −0.323898
\(87\) 9.96702 1.06858
\(88\) −2.98171 −0.317851
\(89\) 14.6866 1.55678 0.778389 0.627783i \(-0.216037\pi\)
0.778389 + 0.627783i \(0.216037\pi\)
\(90\) 1.25914 0.132725
\(91\) −0.797445 −0.0835949
\(92\) 4.08918 0.426326
\(93\) −4.04763 −0.419720
\(94\) −3.00222 −0.309655
\(95\) −5.60664 −0.575229
\(96\) −1.00000 −0.102062
\(97\) −11.3927 −1.15675 −0.578376 0.815770i \(-0.696313\pi\)
−0.578376 + 0.815770i \(0.696313\pi\)
\(98\) −6.36408 −0.642869
\(99\) −2.98171 −0.299673
\(100\) −3.41456 −0.341456
\(101\) −5.48924 −0.546200 −0.273100 0.961986i \(-0.588049\pi\)
−0.273100 + 0.961986i \(0.588049\pi\)
\(102\) −4.87471 −0.482668
\(103\) −1.00000 −0.0985329
\(104\) 1.00000 0.0980581
\(105\) 1.00410 0.0979898
\(106\) −8.63309 −0.838520
\(107\) 3.92781 0.379716 0.189858 0.981812i \(-0.439197\pi\)
0.189858 + 0.981812i \(0.439197\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 11.9866 1.14810 0.574052 0.818819i \(-0.305371\pi\)
0.574052 + 0.818819i \(0.305371\pi\)
\(110\) −3.75440 −0.357968
\(111\) 3.80662 0.361308
\(112\) −0.797445 −0.0753514
\(113\) −6.53982 −0.615215 −0.307607 0.951513i \(-0.599528\pi\)
−0.307607 + 0.951513i \(0.599528\pi\)
\(114\) 4.45274 0.417037
\(115\) 5.14886 0.480134
\(116\) −9.96702 −0.925415
\(117\) 1.00000 0.0924500
\(118\) −6.72403 −0.618997
\(119\) −3.88731 −0.356349
\(120\) −1.25914 −0.114944
\(121\) −2.10940 −0.191764
\(122\) 4.06686 0.368196
\(123\) 2.52399 0.227581
\(124\) 4.04763 0.363488
\(125\) −10.5951 −0.947658
\(126\) −0.797445 −0.0710420
\(127\) 11.0198 0.977845 0.488923 0.872327i \(-0.337390\pi\)
0.488923 + 0.872327i \(0.337390\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.00371 0.264462
\(130\) 1.25914 0.110434
\(131\) −11.0610 −0.966402 −0.483201 0.875509i \(-0.660526\pi\)
−0.483201 + 0.875509i \(0.660526\pi\)
\(132\) 2.98171 0.259525
\(133\) 3.55081 0.307895
\(134\) 4.94082 0.426822
\(135\) −1.25914 −0.108370
\(136\) 4.87471 0.418003
\(137\) −21.6808 −1.85232 −0.926159 0.377133i \(-0.876910\pi\)
−0.926159 + 0.377133i \(0.876910\pi\)
\(138\) −4.08918 −0.348094
\(139\) −4.01573 −0.340609 −0.170305 0.985391i \(-0.554475\pi\)
−0.170305 + 0.985391i \(0.554475\pi\)
\(140\) −1.00410 −0.0848617
\(141\) 3.00222 0.252833
\(142\) −1.94373 −0.163114
\(143\) −2.98171 −0.249343
\(144\) 1.00000 0.0833333
\(145\) −12.5499 −1.04221
\(146\) −7.45502 −0.616982
\(147\) 6.36408 0.524901
\(148\) −3.80662 −0.312902
\(149\) −18.0807 −1.48123 −0.740615 0.671930i \(-0.765466\pi\)
−0.740615 + 0.671930i \(0.765466\pi\)
\(150\) 3.41456 0.278798
\(151\) −9.37983 −0.763320 −0.381660 0.924303i \(-0.624647\pi\)
−0.381660 + 0.924303i \(0.624647\pi\)
\(152\) −4.45274 −0.361165
\(153\) 4.87471 0.394097
\(154\) 2.37775 0.191605
\(155\) 5.09654 0.409364
\(156\) −1.00000 −0.0800641
\(157\) −18.1379 −1.44756 −0.723780 0.690030i \(-0.757597\pi\)
−0.723780 + 0.690030i \(0.757597\pi\)
\(158\) 14.1834 1.12837
\(159\) 8.63309 0.684649
\(160\) 1.25914 0.0995440
\(161\) −3.26089 −0.256994
\(162\) 1.00000 0.0785674
\(163\) −15.9568 −1.24983 −0.624915 0.780693i \(-0.714866\pi\)
−0.624915 + 0.780693i \(0.714866\pi\)
\(164\) −2.52399 −0.197091
\(165\) 3.75440 0.292280
\(166\) 15.3797 1.19369
\(167\) −21.4308 −1.65836 −0.829182 0.558979i \(-0.811193\pi\)
−0.829182 + 0.558979i \(0.811193\pi\)
\(168\) 0.797445 0.0615242
\(169\) 1.00000 0.0769231
\(170\) 6.13796 0.470760
\(171\) −4.45274 −0.340509
\(172\) −3.00371 −0.229031
\(173\) −22.1020 −1.68038 −0.840192 0.542288i \(-0.817558\pi\)
−0.840192 + 0.542288i \(0.817558\pi\)
\(174\) 9.96702 0.755598
\(175\) 2.72292 0.205833
\(176\) −2.98171 −0.224755
\(177\) 6.72403 0.505409
\(178\) 14.6866 1.10081
\(179\) 8.00632 0.598421 0.299210 0.954187i \(-0.403277\pi\)
0.299210 + 0.954187i \(0.403277\pi\)
\(180\) 1.25914 0.0938510
\(181\) −21.8618 −1.62497 −0.812487 0.582979i \(-0.801887\pi\)
−0.812487 + 0.582979i \(0.801887\pi\)
\(182\) −0.797445 −0.0591105
\(183\) −4.06686 −0.300631
\(184\) 4.08918 0.301458
\(185\) −4.79308 −0.352394
\(186\) −4.04763 −0.296787
\(187\) −14.5350 −1.06290
\(188\) −3.00222 −0.218959
\(189\) 0.797445 0.0580056
\(190\) −5.60664 −0.406748
\(191\) 22.9946 1.66383 0.831915 0.554904i \(-0.187245\pi\)
0.831915 + 0.554904i \(0.187245\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 12.0887 0.870165 0.435083 0.900391i \(-0.356719\pi\)
0.435083 + 0.900391i \(0.356719\pi\)
\(194\) −11.3927 −0.817948
\(195\) −1.25914 −0.0901691
\(196\) −6.36408 −0.454577
\(197\) −0.828221 −0.0590083 −0.0295042 0.999565i \(-0.509393\pi\)
−0.0295042 + 0.999565i \(0.509393\pi\)
\(198\) −2.98171 −0.211901
\(199\) −16.9393 −1.20080 −0.600399 0.799701i \(-0.704991\pi\)
−0.600399 + 0.799701i \(0.704991\pi\)
\(200\) −3.41456 −0.241446
\(201\) −4.94082 −0.348498
\(202\) −5.48924 −0.386222
\(203\) 7.94815 0.557851
\(204\) −4.87471 −0.341298
\(205\) −3.17807 −0.221966
\(206\) −1.00000 −0.0696733
\(207\) 4.08918 0.284217
\(208\) 1.00000 0.0693375
\(209\) 13.2768 0.918374
\(210\) 1.00410 0.0692893
\(211\) 26.7360 1.84058 0.920291 0.391235i \(-0.127952\pi\)
0.920291 + 0.391235i \(0.127952\pi\)
\(212\) −8.63309 −0.592923
\(213\) 1.94373 0.133182
\(214\) 3.92781 0.268500
\(215\) −3.78210 −0.257937
\(216\) −1.00000 −0.0680414
\(217\) −3.22776 −0.219115
\(218\) 11.9866 0.811832
\(219\) 7.45502 0.503764
\(220\) −3.75440 −0.253122
\(221\) 4.87471 0.327909
\(222\) 3.80662 0.255483
\(223\) 16.4842 1.10387 0.551933 0.833889i \(-0.313891\pi\)
0.551933 + 0.833889i \(0.313891\pi\)
\(224\) −0.797445 −0.0532815
\(225\) −3.41456 −0.227637
\(226\) −6.53982 −0.435023
\(227\) 4.26467 0.283056 0.141528 0.989934i \(-0.454798\pi\)
0.141528 + 0.989934i \(0.454798\pi\)
\(228\) 4.45274 0.294890
\(229\) 21.0701 1.39235 0.696177 0.717870i \(-0.254883\pi\)
0.696177 + 0.717870i \(0.254883\pi\)
\(230\) 5.14886 0.339506
\(231\) −2.37775 −0.156444
\(232\) −9.96702 −0.654367
\(233\) 13.3803 0.876572 0.438286 0.898836i \(-0.355586\pi\)
0.438286 + 0.898836i \(0.355586\pi\)
\(234\) 1.00000 0.0653720
\(235\) −3.78023 −0.246595
\(236\) −6.72403 −0.437697
\(237\) −14.1834 −0.921310
\(238\) −3.88731 −0.251977
\(239\) 29.5612 1.91215 0.956076 0.293118i \(-0.0946930\pi\)
0.956076 + 0.293118i \(0.0946930\pi\)
\(240\) −1.25914 −0.0812773
\(241\) −2.85139 −0.183674 −0.0918371 0.995774i \(-0.529274\pi\)
−0.0918371 + 0.995774i \(0.529274\pi\)
\(242\) −2.10940 −0.135597
\(243\) −1.00000 −0.0641500
\(244\) 4.06686 0.260354
\(245\) −8.01329 −0.511950
\(246\) 2.52399 0.160924
\(247\) −4.45274 −0.283321
\(248\) 4.04763 0.257025
\(249\) −15.3797 −0.974646
\(250\) −10.5951 −0.670095
\(251\) −17.9435 −1.13258 −0.566292 0.824205i \(-0.691622\pi\)
−0.566292 + 0.824205i \(0.691622\pi\)
\(252\) −0.797445 −0.0502343
\(253\) −12.1927 −0.766551
\(254\) 11.0198 0.691441
\(255\) −6.13796 −0.384374
\(256\) 1.00000 0.0625000
\(257\) −4.24943 −0.265072 −0.132536 0.991178i \(-0.542312\pi\)
−0.132536 + 0.991178i \(0.542312\pi\)
\(258\) 3.00371 0.187003
\(259\) 3.03557 0.188621
\(260\) 1.25914 0.0780887
\(261\) −9.96702 −0.616943
\(262\) −11.0610 −0.683350
\(263\) 13.4098 0.826881 0.413440 0.910531i \(-0.364327\pi\)
0.413440 + 0.910531i \(0.364327\pi\)
\(264\) 2.98171 0.183512
\(265\) −10.8703 −0.667757
\(266\) 3.55081 0.217714
\(267\) −14.6866 −0.898806
\(268\) 4.94082 0.301808
\(269\) 26.4982 1.61562 0.807811 0.589442i \(-0.200652\pi\)
0.807811 + 0.589442i \(0.200652\pi\)
\(270\) −1.25914 −0.0766290
\(271\) −13.3519 −0.811073 −0.405537 0.914079i \(-0.632915\pi\)
−0.405537 + 0.914079i \(0.632915\pi\)
\(272\) 4.87471 0.295573
\(273\) 0.797445 0.0482635
\(274\) −21.6808 −1.30979
\(275\) 10.1812 0.613951
\(276\) −4.08918 −0.246140
\(277\) −5.91529 −0.355415 −0.177708 0.984083i \(-0.556868\pi\)
−0.177708 + 0.984083i \(0.556868\pi\)
\(278\) −4.01573 −0.240847
\(279\) 4.04763 0.242325
\(280\) −1.00410 −0.0600063
\(281\) 8.20337 0.489372 0.244686 0.969602i \(-0.421315\pi\)
0.244686 + 0.969602i \(0.421315\pi\)
\(282\) 3.00222 0.178780
\(283\) −18.8189 −1.11867 −0.559333 0.828943i \(-0.688943\pi\)
−0.559333 + 0.828943i \(0.688943\pi\)
\(284\) −1.94373 −0.115339
\(285\) 5.60664 0.332108
\(286\) −2.98171 −0.176312
\(287\) 2.01274 0.118808
\(288\) 1.00000 0.0589256
\(289\) 6.76281 0.397813
\(290\) −12.5499 −0.736956
\(291\) 11.3927 0.667851
\(292\) −7.45502 −0.436272
\(293\) −33.5092 −1.95763 −0.978815 0.204746i \(-0.934363\pi\)
−0.978815 + 0.204746i \(0.934363\pi\)
\(294\) 6.36408 0.371161
\(295\) −8.46651 −0.492940
\(296\) −3.80662 −0.221255
\(297\) 2.98171 0.173016
\(298\) −18.0807 −1.04739
\(299\) 4.08918 0.236483
\(300\) 3.41456 0.197140
\(301\) 2.39529 0.138062
\(302\) −9.37983 −0.539748
\(303\) 5.48924 0.315349
\(304\) −4.45274 −0.255382
\(305\) 5.12075 0.293214
\(306\) 4.87471 0.278669
\(307\) 4.39459 0.250812 0.125406 0.992105i \(-0.459977\pi\)
0.125406 + 0.992105i \(0.459977\pi\)
\(308\) 2.37775 0.135485
\(309\) 1.00000 0.0568880
\(310\) 5.09654 0.289464
\(311\) −10.2447 −0.580925 −0.290462 0.956886i \(-0.593809\pi\)
−0.290462 + 0.956886i \(0.593809\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −25.1202 −1.41988 −0.709939 0.704264i \(-0.751277\pi\)
−0.709939 + 0.704264i \(0.751277\pi\)
\(314\) −18.1379 −1.02358
\(315\) −1.00410 −0.0565745
\(316\) 14.1834 0.797878
\(317\) −5.81992 −0.326879 −0.163440 0.986553i \(-0.552259\pi\)
−0.163440 + 0.986553i \(0.552259\pi\)
\(318\) 8.63309 0.484120
\(319\) 29.7188 1.66393
\(320\) 1.25914 0.0703882
\(321\) −3.92781 −0.219229
\(322\) −3.26089 −0.181722
\(323\) −21.7058 −1.20774
\(324\) 1.00000 0.0555556
\(325\) −3.41456 −0.189406
\(326\) −15.9568 −0.883763
\(327\) −11.9866 −0.662858
\(328\) −2.52399 −0.139364
\(329\) 2.39411 0.131991
\(330\) 3.75440 0.206673
\(331\) −20.4154 −1.12213 −0.561067 0.827771i \(-0.689609\pi\)
−0.561067 + 0.827771i \(0.689609\pi\)
\(332\) 15.3797 0.844068
\(333\) −3.80662 −0.208601
\(334\) −21.4308 −1.17264
\(335\) 6.22120 0.339900
\(336\) 0.797445 0.0435042
\(337\) −13.7105 −0.746859 −0.373430 0.927658i \(-0.621818\pi\)
−0.373430 + 0.927658i \(0.621818\pi\)
\(338\) 1.00000 0.0543928
\(339\) 6.53982 0.355194
\(340\) 6.13796 0.332878
\(341\) −12.0689 −0.653565
\(342\) −4.45274 −0.240777
\(343\) 10.6571 0.575430
\(344\) −3.00371 −0.161949
\(345\) −5.14886 −0.277205
\(346\) −22.1020 −1.18821
\(347\) 28.3176 1.52017 0.760084 0.649825i \(-0.225158\pi\)
0.760084 + 0.649825i \(0.225158\pi\)
\(348\) 9.96702 0.534289
\(349\) −2.68702 −0.143833 −0.0719164 0.997411i \(-0.522911\pi\)
−0.0719164 + 0.997411i \(0.522911\pi\)
\(350\) 2.72292 0.145546
\(351\) −1.00000 −0.0533761
\(352\) −2.98171 −0.158926
\(353\) 31.9501 1.70053 0.850266 0.526354i \(-0.176441\pi\)
0.850266 + 0.526354i \(0.176441\pi\)
\(354\) 6.72403 0.357378
\(355\) −2.44744 −0.129896
\(356\) 14.6866 0.778389
\(357\) 3.88731 0.205738
\(358\) 8.00632 0.423147
\(359\) −23.3446 −1.23208 −0.616040 0.787715i \(-0.711264\pi\)
−0.616040 + 0.787715i \(0.711264\pi\)
\(360\) 1.25914 0.0663627
\(361\) 0.826888 0.0435204
\(362\) −21.8618 −1.14903
\(363\) 2.10940 0.110715
\(364\) −0.797445 −0.0417975
\(365\) −9.38694 −0.491335
\(366\) −4.06686 −0.212578
\(367\) −1.63073 −0.0851236 −0.0425618 0.999094i \(-0.513552\pi\)
−0.0425618 + 0.999094i \(0.513552\pi\)
\(368\) 4.08918 0.213163
\(369\) −2.52399 −0.131394
\(370\) −4.79308 −0.249180
\(371\) 6.88441 0.357421
\(372\) −4.04763 −0.209860
\(373\) −9.90305 −0.512761 −0.256380 0.966576i \(-0.582530\pi\)
−0.256380 + 0.966576i \(0.582530\pi\)
\(374\) −14.5350 −0.751586
\(375\) 10.5951 0.547130
\(376\) −3.00222 −0.154828
\(377\) −9.96702 −0.513328
\(378\) 0.797445 0.0410161
\(379\) 7.74486 0.397827 0.198913 0.980017i \(-0.436259\pi\)
0.198913 + 0.980017i \(0.436259\pi\)
\(380\) −5.60664 −0.287614
\(381\) −11.0198 −0.564559
\(382\) 22.9946 1.17650
\(383\) 37.7230 1.92756 0.963779 0.266703i \(-0.0859342\pi\)
0.963779 + 0.266703i \(0.0859342\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 2.99393 0.152585
\(386\) 12.0887 0.615300
\(387\) −3.00371 −0.152687
\(388\) −11.3927 −0.578376
\(389\) −18.3085 −0.928276 −0.464138 0.885763i \(-0.653636\pi\)
−0.464138 + 0.885763i \(0.653636\pi\)
\(390\) −1.25914 −0.0637592
\(391\) 19.9336 1.00808
\(392\) −6.36408 −0.321435
\(393\) 11.0610 0.557953
\(394\) −0.828221 −0.0417252
\(395\) 17.8589 0.898580
\(396\) −2.98171 −0.149837
\(397\) −22.2253 −1.11546 −0.557728 0.830023i \(-0.688327\pi\)
−0.557728 + 0.830023i \(0.688327\pi\)
\(398\) −16.9393 −0.849092
\(399\) −3.55081 −0.177763
\(400\) −3.41456 −0.170728
\(401\) 10.6908 0.533875 0.266937 0.963714i \(-0.413988\pi\)
0.266937 + 0.963714i \(0.413988\pi\)
\(402\) −4.94082 −0.246426
\(403\) 4.04763 0.201627
\(404\) −5.48924 −0.273100
\(405\) 1.25914 0.0625673
\(406\) 7.94815 0.394460
\(407\) 11.3502 0.562610
\(408\) −4.87471 −0.241334
\(409\) 5.36398 0.265232 0.132616 0.991168i \(-0.457662\pi\)
0.132616 + 0.991168i \(0.457662\pi\)
\(410\) −3.17807 −0.156954
\(411\) 21.6808 1.06944
\(412\) −1.00000 −0.0492665
\(413\) 5.36204 0.263849
\(414\) 4.08918 0.200972
\(415\) 19.3652 0.950599
\(416\) 1.00000 0.0490290
\(417\) 4.01573 0.196651
\(418\) 13.2768 0.649389
\(419\) −24.4431 −1.19412 −0.597061 0.802196i \(-0.703665\pi\)
−0.597061 + 0.802196i \(0.703665\pi\)
\(420\) 1.00410 0.0489949
\(421\) 15.5098 0.755900 0.377950 0.925826i \(-0.376629\pi\)
0.377950 + 0.925826i \(0.376629\pi\)
\(422\) 26.7360 1.30149
\(423\) −3.00222 −0.145973
\(424\) −8.63309 −0.419260
\(425\) −16.6450 −0.807400
\(426\) 1.94373 0.0941741
\(427\) −3.24309 −0.156944
\(428\) 3.92781 0.189858
\(429\) 2.98171 0.143958
\(430\) −3.78210 −0.182389
\(431\) −21.4370 −1.03258 −0.516292 0.856413i \(-0.672688\pi\)
−0.516292 + 0.856413i \(0.672688\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −6.20622 −0.298252 −0.149126 0.988818i \(-0.547646\pi\)
−0.149126 + 0.988818i \(0.547646\pi\)
\(434\) −3.22776 −0.154937
\(435\) 12.5499 0.601722
\(436\) 11.9866 0.574052
\(437\) −18.2080 −0.871009
\(438\) 7.45502 0.356215
\(439\) 11.1121 0.530350 0.265175 0.964200i \(-0.414570\pi\)
0.265175 + 0.964200i \(0.414570\pi\)
\(440\) −3.75440 −0.178984
\(441\) −6.36408 −0.303052
\(442\) 4.87471 0.231866
\(443\) −8.79189 −0.417715 −0.208858 0.977946i \(-0.566975\pi\)
−0.208858 + 0.977946i \(0.566975\pi\)
\(444\) 3.80662 0.180654
\(445\) 18.4925 0.876630
\(446\) 16.4842 0.780550
\(447\) 18.0807 0.855188
\(448\) −0.797445 −0.0376757
\(449\) −14.1718 −0.668811 −0.334405 0.942429i \(-0.608535\pi\)
−0.334405 + 0.942429i \(0.608535\pi\)
\(450\) −3.41456 −0.160964
\(451\) 7.52581 0.354377
\(452\) −6.53982 −0.307607
\(453\) 9.37983 0.440703
\(454\) 4.26467 0.200151
\(455\) −1.00410 −0.0470728
\(456\) 4.45274 0.208519
\(457\) 0.0240313 0.00112414 0.000562069 1.00000i \(-0.499821\pi\)
0.000562069 1.00000i \(0.499821\pi\)
\(458\) 21.0701 0.984543
\(459\) −4.87471 −0.227532
\(460\) 5.14886 0.240067
\(461\) −9.22105 −0.429467 −0.214734 0.976673i \(-0.568888\pi\)
−0.214734 + 0.976673i \(0.568888\pi\)
\(462\) −2.37775 −0.110623
\(463\) −4.11875 −0.191415 −0.0957073 0.995410i \(-0.530511\pi\)
−0.0957073 + 0.995410i \(0.530511\pi\)
\(464\) −9.96702 −0.462708
\(465\) −5.09654 −0.236347
\(466\) 13.3803 0.619830
\(467\) 11.3460 0.525032 0.262516 0.964928i \(-0.415448\pi\)
0.262516 + 0.964928i \(0.415448\pi\)
\(468\) 1.00000 0.0462250
\(469\) −3.94003 −0.181934
\(470\) −3.78023 −0.174369
\(471\) 18.1379 0.835750
\(472\) −6.72403 −0.309498
\(473\) 8.95619 0.411806
\(474\) −14.1834 −0.651465
\(475\) 15.2041 0.697614
\(476\) −3.88731 −0.178175
\(477\) −8.63309 −0.395282
\(478\) 29.5612 1.35210
\(479\) −18.0330 −0.823949 −0.411975 0.911195i \(-0.635161\pi\)
−0.411975 + 0.911195i \(0.635161\pi\)
\(480\) −1.25914 −0.0574718
\(481\) −3.80662 −0.173567
\(482\) −2.85139 −0.129877
\(483\) 3.26089 0.148376
\(484\) −2.10940 −0.0958818
\(485\) −14.3450 −0.651374
\(486\) −1.00000 −0.0453609
\(487\) −41.4619 −1.87882 −0.939409 0.342799i \(-0.888625\pi\)
−0.939409 + 0.342799i \(0.888625\pi\)
\(488\) 4.06686 0.184098
\(489\) 15.9568 0.721590
\(490\) −8.01329 −0.362004
\(491\) −34.9707 −1.57820 −0.789102 0.614262i \(-0.789454\pi\)
−0.789102 + 0.614262i \(0.789454\pi\)
\(492\) 2.52399 0.113790
\(493\) −48.5864 −2.18822
\(494\) −4.45274 −0.200338
\(495\) −3.75440 −0.168748
\(496\) 4.04763 0.181744
\(497\) 1.55002 0.0695278
\(498\) −15.3797 −0.689179
\(499\) −18.4859 −0.827542 −0.413771 0.910381i \(-0.635789\pi\)
−0.413771 + 0.910381i \(0.635789\pi\)
\(500\) −10.5951 −0.473829
\(501\) 21.4308 0.957457
\(502\) −17.9435 −0.800857
\(503\) −35.5052 −1.58310 −0.791550 0.611104i \(-0.790726\pi\)
−0.791550 + 0.611104i \(0.790726\pi\)
\(504\) −0.797445 −0.0355210
\(505\) −6.91174 −0.307568
\(506\) −12.1927 −0.542034
\(507\) −1.00000 −0.0444116
\(508\) 11.0198 0.488923
\(509\) 1.89530 0.0840078 0.0420039 0.999117i \(-0.486626\pi\)
0.0420039 + 0.999117i \(0.486626\pi\)
\(510\) −6.13796 −0.271793
\(511\) 5.94497 0.262990
\(512\) 1.00000 0.0441942
\(513\) 4.45274 0.196593
\(514\) −4.24943 −0.187434
\(515\) −1.25914 −0.0554845
\(516\) 3.00371 0.132231
\(517\) 8.95176 0.393698
\(518\) 3.03557 0.133375
\(519\) 22.1020 0.970171
\(520\) 1.25914 0.0552171
\(521\) 27.6583 1.21173 0.605865 0.795567i \(-0.292827\pi\)
0.605865 + 0.795567i \(0.292827\pi\)
\(522\) −9.96702 −0.436245
\(523\) −14.4338 −0.631147 −0.315573 0.948901i \(-0.602197\pi\)
−0.315573 + 0.948901i \(0.602197\pi\)
\(524\) −11.0610 −0.483201
\(525\) −2.72292 −0.118838
\(526\) 13.4098 0.584693
\(527\) 19.7310 0.859497
\(528\) 2.98171 0.129762
\(529\) −6.27863 −0.272984
\(530\) −10.8703 −0.472176
\(531\) −6.72403 −0.291798
\(532\) 3.55081 0.153947
\(533\) −2.52399 −0.109326
\(534\) −14.6866 −0.635552
\(535\) 4.94568 0.213820
\(536\) 4.94082 0.213411
\(537\) −8.00632 −0.345498
\(538\) 26.4982 1.14242
\(539\) 18.9759 0.817348
\(540\) −1.25914 −0.0541849
\(541\) 12.3984 0.533050 0.266525 0.963828i \(-0.414125\pi\)
0.266525 + 0.963828i \(0.414125\pi\)
\(542\) −13.3519 −0.573515
\(543\) 21.8618 0.938179
\(544\) 4.87471 0.209002
\(545\) 15.0928 0.646504
\(546\) 0.797445 0.0341275
\(547\) 7.76399 0.331964 0.165982 0.986129i \(-0.446921\pi\)
0.165982 + 0.986129i \(0.446921\pi\)
\(548\) −21.6808 −0.926159
\(549\) 4.06686 0.173569
\(550\) 10.1812 0.434129
\(551\) 44.3806 1.89068
\(552\) −4.08918 −0.174047
\(553\) −11.3105 −0.480970
\(554\) −5.91529 −0.251317
\(555\) 4.79308 0.203455
\(556\) −4.01573 −0.170305
\(557\) 15.2003 0.644059 0.322029 0.946730i \(-0.395635\pi\)
0.322029 + 0.946730i \(0.395635\pi\)
\(558\) 4.04763 0.171350
\(559\) −3.00371 −0.127043
\(560\) −1.00410 −0.0424308
\(561\) 14.5350 0.613667
\(562\) 8.20337 0.346038
\(563\) 28.7671 1.21239 0.606195 0.795316i \(-0.292695\pi\)
0.606195 + 0.795316i \(0.292695\pi\)
\(564\) 3.00222 0.126416
\(565\) −8.23457 −0.346431
\(566\) −18.8189 −0.791017
\(567\) −0.797445 −0.0334895
\(568\) −1.94373 −0.0815572
\(569\) −13.8644 −0.581226 −0.290613 0.956841i \(-0.593859\pi\)
−0.290613 + 0.956841i \(0.593859\pi\)
\(570\) 5.60664 0.234836
\(571\) 4.34900 0.182000 0.0909999 0.995851i \(-0.470994\pi\)
0.0909999 + 0.995851i \(0.470994\pi\)
\(572\) −2.98171 −0.124672
\(573\) −22.9946 −0.960612
\(574\) 2.01274 0.0840103
\(575\) −13.9627 −0.582286
\(576\) 1.00000 0.0416667
\(577\) −26.1842 −1.09006 −0.545030 0.838416i \(-0.683482\pi\)
−0.545030 + 0.838416i \(0.683482\pi\)
\(578\) 6.76281 0.281296
\(579\) −12.0887 −0.502390
\(580\) −12.5499 −0.521107
\(581\) −12.2644 −0.508814
\(582\) 11.3927 0.472242
\(583\) 25.7414 1.06610
\(584\) −7.45502 −0.308491
\(585\) 1.25914 0.0520592
\(586\) −33.5092 −1.38425
\(587\) −13.8575 −0.571960 −0.285980 0.958236i \(-0.592319\pi\)
−0.285980 + 0.958236i \(0.592319\pi\)
\(588\) 6.36408 0.262450
\(589\) −18.0230 −0.742626
\(590\) −8.46651 −0.348561
\(591\) 0.828221 0.0340685
\(592\) −3.80662 −0.156451
\(593\) 28.9779 1.18998 0.594989 0.803734i \(-0.297156\pi\)
0.594989 + 0.803734i \(0.297156\pi\)
\(594\) 2.98171 0.122341
\(595\) −4.89468 −0.200662
\(596\) −18.0807 −0.740615
\(597\) 16.9393 0.693281
\(598\) 4.08918 0.167219
\(599\) 36.9924 1.51147 0.755734 0.654879i \(-0.227281\pi\)
0.755734 + 0.654879i \(0.227281\pi\)
\(600\) 3.41456 0.139399
\(601\) 1.03333 0.0421504 0.0210752 0.999778i \(-0.493291\pi\)
0.0210752 + 0.999778i \(0.493291\pi\)
\(602\) 2.39529 0.0976247
\(603\) 4.94082 0.201206
\(604\) −9.37983 −0.381660
\(605\) −2.65604 −0.107983
\(606\) 5.48924 0.222985
\(607\) 22.0381 0.894499 0.447249 0.894409i \(-0.352404\pi\)
0.447249 + 0.894409i \(0.352404\pi\)
\(608\) −4.45274 −0.180582
\(609\) −7.94815 −0.322075
\(610\) 5.12075 0.207333
\(611\) −3.00222 −0.121457
\(612\) 4.87471 0.197049
\(613\) −16.8807 −0.681804 −0.340902 0.940099i \(-0.610733\pi\)
−0.340902 + 0.940099i \(0.610733\pi\)
\(614\) 4.39459 0.177351
\(615\) 3.17807 0.128152
\(616\) 2.37775 0.0958023
\(617\) 27.3160 1.09970 0.549850 0.835263i \(-0.314685\pi\)
0.549850 + 0.835263i \(0.314685\pi\)
\(618\) 1.00000 0.0402259
\(619\) 34.1151 1.37120 0.685600 0.727978i \(-0.259540\pi\)
0.685600 + 0.727978i \(0.259540\pi\)
\(620\) 5.09654 0.204682
\(621\) −4.08918 −0.164093
\(622\) −10.2447 −0.410776
\(623\) −11.7118 −0.469222
\(624\) −1.00000 −0.0400320
\(625\) 3.73200 0.149280
\(626\) −25.1202 −1.00400
\(627\) −13.2768 −0.530224
\(628\) −18.1379 −0.723780
\(629\) −18.5562 −0.739883
\(630\) −1.00410 −0.0400042
\(631\) 13.4519 0.535513 0.267756 0.963487i \(-0.413718\pi\)
0.267756 + 0.963487i \(0.413718\pi\)
\(632\) 14.1834 0.564185
\(633\) −26.7360 −1.06266
\(634\) −5.81992 −0.231138
\(635\) 13.8755 0.550630
\(636\) 8.63309 0.342324
\(637\) −6.36408 −0.252154
\(638\) 29.7188 1.17658
\(639\) −1.94373 −0.0768928
\(640\) 1.25914 0.0497720
\(641\) −15.5587 −0.614533 −0.307266 0.951624i \(-0.599414\pi\)
−0.307266 + 0.951624i \(0.599414\pi\)
\(642\) −3.92781 −0.155018
\(643\) −7.19979 −0.283932 −0.141966 0.989872i \(-0.545342\pi\)
−0.141966 + 0.989872i \(0.545342\pi\)
\(644\) −3.26089 −0.128497
\(645\) 3.78210 0.148920
\(646\) −21.7058 −0.854004
\(647\) −14.9448 −0.587539 −0.293769 0.955876i \(-0.594910\pi\)
−0.293769 + 0.955876i \(0.594910\pi\)
\(648\) 1.00000 0.0392837
\(649\) 20.0491 0.786996
\(650\) −3.41456 −0.133930
\(651\) 3.22776 0.126506
\(652\) −15.9568 −0.624915
\(653\) −33.1012 −1.29535 −0.647675 0.761917i \(-0.724259\pi\)
−0.647675 + 0.761917i \(0.724259\pi\)
\(654\) −11.9866 −0.468712
\(655\) −13.9274 −0.544187
\(656\) −2.52399 −0.0985453
\(657\) −7.45502 −0.290848
\(658\) 2.39411 0.0933319
\(659\) −32.9257 −1.28260 −0.641302 0.767289i \(-0.721605\pi\)
−0.641302 + 0.767289i \(0.721605\pi\)
\(660\) 3.75440 0.146140
\(661\) 11.5643 0.449798 0.224899 0.974382i \(-0.427795\pi\)
0.224899 + 0.974382i \(0.427795\pi\)
\(662\) −20.4154 −0.793468
\(663\) −4.87471 −0.189318
\(664\) 15.3797 0.596846
\(665\) 4.47098 0.173377
\(666\) −3.80662 −0.147503
\(667\) −40.7569 −1.57811
\(668\) −21.4308 −0.829182
\(669\) −16.4842 −0.637317
\(670\) 6.22120 0.240346
\(671\) −12.1262 −0.468126
\(672\) 0.797445 0.0307621
\(673\) 2.06251 0.0795038 0.0397519 0.999210i \(-0.487343\pi\)
0.0397519 + 0.999210i \(0.487343\pi\)
\(674\) −13.7105 −0.528109
\(675\) 3.41456 0.131426
\(676\) 1.00000 0.0384615
\(677\) −14.5098 −0.557657 −0.278829 0.960341i \(-0.589946\pi\)
−0.278829 + 0.960341i \(0.589946\pi\)
\(678\) 6.53982 0.251160
\(679\) 9.08504 0.348652
\(680\) 6.13796 0.235380
\(681\) −4.26467 −0.163422
\(682\) −12.0689 −0.462140
\(683\) −2.22822 −0.0852604 −0.0426302 0.999091i \(-0.513574\pi\)
−0.0426302 + 0.999091i \(0.513574\pi\)
\(684\) −4.45274 −0.170255
\(685\) −27.2993 −1.04305
\(686\) 10.6571 0.406891
\(687\) −21.0701 −0.803876
\(688\) −3.00371 −0.114515
\(689\) −8.63309 −0.328895
\(690\) −5.14886 −0.196014
\(691\) 14.0702 0.535256 0.267628 0.963522i \(-0.413760\pi\)
0.267628 + 0.963522i \(0.413760\pi\)
\(692\) −22.1020 −0.840192
\(693\) 2.37775 0.0903232
\(694\) 28.3176 1.07492
\(695\) −5.05637 −0.191799
\(696\) 9.96702 0.377799
\(697\) −12.3037 −0.466037
\(698\) −2.68702 −0.101705
\(699\) −13.3803 −0.506089
\(700\) 2.72292 0.102917
\(701\) −44.1774 −1.66856 −0.834280 0.551342i \(-0.814116\pi\)
−0.834280 + 0.551342i \(0.814116\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 16.9499 0.639277
\(704\) −2.98171 −0.112377
\(705\) 3.78023 0.142372
\(706\) 31.9501 1.20246
\(707\) 4.37737 0.164628
\(708\) 6.72403 0.252704
\(709\) 38.7468 1.45517 0.727583 0.686019i \(-0.240643\pi\)
0.727583 + 0.686019i \(0.240643\pi\)
\(710\) −2.44744 −0.0918507
\(711\) 14.1834 0.531919
\(712\) 14.6866 0.550404
\(713\) 16.5515 0.619857
\(714\) 3.88731 0.145479
\(715\) −3.75440 −0.140407
\(716\) 8.00632 0.299210
\(717\) −29.5612 −1.10398
\(718\) −23.3446 −0.871213
\(719\) 4.72624 0.176259 0.0881295 0.996109i \(-0.471911\pi\)
0.0881295 + 0.996109i \(0.471911\pi\)
\(720\) 1.25914 0.0469255
\(721\) 0.797445 0.0296984
\(722\) 0.826888 0.0307736
\(723\) 2.85139 0.106044
\(724\) −21.8618 −0.812487
\(725\) 34.0330 1.26395
\(726\) 2.10940 0.0782872
\(727\) 49.5432 1.83746 0.918729 0.394890i \(-0.129217\pi\)
0.918729 + 0.394890i \(0.129217\pi\)
\(728\) −0.797445 −0.0295553
\(729\) 1.00000 0.0370370
\(730\) −9.38694 −0.347426
\(731\) −14.6422 −0.541562
\(732\) −4.06686 −0.150315
\(733\) 42.5643 1.57215 0.786075 0.618131i \(-0.212110\pi\)
0.786075 + 0.618131i \(0.212110\pi\)
\(734\) −1.63073 −0.0601915
\(735\) 8.01329 0.295575
\(736\) 4.08918 0.150729
\(737\) −14.7321 −0.542664
\(738\) −2.52399 −0.0929094
\(739\) 8.83116 0.324859 0.162430 0.986720i \(-0.448067\pi\)
0.162430 + 0.986720i \(0.448067\pi\)
\(740\) −4.79308 −0.176197
\(741\) 4.45274 0.163575
\(742\) 6.88441 0.252735
\(743\) 18.9037 0.693509 0.346754 0.937956i \(-0.387284\pi\)
0.346754 + 0.937956i \(0.387284\pi\)
\(744\) −4.04763 −0.148393
\(745\) −22.7662 −0.834089
\(746\) −9.90305 −0.362577
\(747\) 15.3797 0.562712
\(748\) −14.5350 −0.531452
\(749\) −3.13221 −0.114449
\(750\) 10.5951 0.386880
\(751\) −25.3504 −0.925049 −0.462525 0.886606i \(-0.653056\pi\)
−0.462525 + 0.886606i \(0.653056\pi\)
\(752\) −3.00222 −0.109480
\(753\) 17.9435 0.653897
\(754\) −9.96702 −0.362978
\(755\) −11.8105 −0.429830
\(756\) 0.797445 0.0290028
\(757\) −47.3667 −1.72157 −0.860785 0.508968i \(-0.830027\pi\)
−0.860785 + 0.508968i \(0.830027\pi\)
\(758\) 7.74486 0.281306
\(759\) 12.1927 0.442569
\(760\) −5.60664 −0.203374
\(761\) 42.7967 1.55138 0.775689 0.631115i \(-0.217402\pi\)
0.775689 + 0.631115i \(0.217402\pi\)
\(762\) −11.0198 −0.399204
\(763\) −9.55862 −0.346045
\(764\) 22.9946 0.831915
\(765\) 6.13796 0.221918
\(766\) 37.7230 1.36299
\(767\) −6.72403 −0.242791
\(768\) −1.00000 −0.0360844
\(769\) −24.5633 −0.885777 −0.442888 0.896577i \(-0.646046\pi\)
−0.442888 + 0.896577i \(0.646046\pi\)
\(770\) 2.99393 0.107894
\(771\) 4.24943 0.153039
\(772\) 12.0887 0.435083
\(773\) 4.82029 0.173374 0.0866869 0.996236i \(-0.472372\pi\)
0.0866869 + 0.996236i \(0.472372\pi\)
\(774\) −3.00371 −0.107966
\(775\) −13.8209 −0.496460
\(776\) −11.3927 −0.408974
\(777\) −3.03557 −0.108900
\(778\) −18.3085 −0.656390
\(779\) 11.2387 0.402667
\(780\) −1.25914 −0.0450846
\(781\) 5.79565 0.207385
\(782\) 19.9336 0.712823
\(783\) 9.96702 0.356192
\(784\) −6.36408 −0.227289
\(785\) −22.8382 −0.815130
\(786\) 11.0610 0.394532
\(787\) −10.9917 −0.391812 −0.195906 0.980623i \(-0.562765\pi\)
−0.195906 + 0.980623i \(0.562765\pi\)
\(788\) −0.828221 −0.0295042
\(789\) −13.4098 −0.477400
\(790\) 17.8589 0.635392
\(791\) 5.21515 0.185429
\(792\) −2.98171 −0.105950
\(793\) 4.06686 0.144418
\(794\) −22.2253 −0.788747
\(795\) 10.8703 0.385530
\(796\) −16.9393 −0.600399
\(797\) 19.7528 0.699680 0.349840 0.936809i \(-0.386236\pi\)
0.349840 + 0.936809i \(0.386236\pi\)
\(798\) −3.55081 −0.125697
\(799\) −14.6350 −0.517748
\(800\) −3.41456 −0.120723
\(801\) 14.6866 0.518926
\(802\) 10.6908 0.377506
\(803\) 22.2287 0.784435
\(804\) −4.94082 −0.174249
\(805\) −4.10593 −0.144715
\(806\) 4.04763 0.142572
\(807\) −26.4982 −0.932779
\(808\) −5.48924 −0.193111
\(809\) 5.92615 0.208352 0.104176 0.994559i \(-0.466779\pi\)
0.104176 + 0.994559i \(0.466779\pi\)
\(810\) 1.25914 0.0442418
\(811\) −11.0575 −0.388281 −0.194140 0.980974i \(-0.562192\pi\)
−0.194140 + 0.980974i \(0.562192\pi\)
\(812\) 7.94815 0.278925
\(813\) 13.3519 0.468273
\(814\) 11.3502 0.397825
\(815\) −20.0919 −0.703787
\(816\) −4.87471 −0.170649
\(817\) 13.3747 0.467922
\(818\) 5.36398 0.187547
\(819\) −0.797445 −0.0278650
\(820\) −3.17807 −0.110983
\(821\) 43.2722 1.51021 0.755106 0.655603i \(-0.227586\pi\)
0.755106 + 0.655603i \(0.227586\pi\)
\(822\) 21.6808 0.756206
\(823\) 2.19947 0.0766686 0.0383343 0.999265i \(-0.487795\pi\)
0.0383343 + 0.999265i \(0.487795\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −10.1812 −0.354465
\(826\) 5.36204 0.186569
\(827\) 28.5423 0.992512 0.496256 0.868176i \(-0.334708\pi\)
0.496256 + 0.868176i \(0.334708\pi\)
\(828\) 4.08918 0.142109
\(829\) 27.4658 0.953926 0.476963 0.878923i \(-0.341738\pi\)
0.476963 + 0.878923i \(0.341738\pi\)
\(830\) 19.3652 0.672175
\(831\) 5.91529 0.205199
\(832\) 1.00000 0.0346688
\(833\) −31.0231 −1.07489
\(834\) 4.01573 0.139053
\(835\) −26.9844 −0.933834
\(836\) 13.2768 0.459187
\(837\) −4.04763 −0.139907
\(838\) −24.4431 −0.844372
\(839\) 44.7604 1.54530 0.772651 0.634831i \(-0.218930\pi\)
0.772651 + 0.634831i \(0.218930\pi\)
\(840\) 1.00410 0.0346446
\(841\) 70.3416 2.42557
\(842\) 15.5098 0.534502
\(843\) −8.20337 −0.282539
\(844\) 26.7360 0.920291
\(845\) 1.25914 0.0433158
\(846\) −3.00222 −0.103218
\(847\) 1.68213 0.0577986
\(848\) −8.63309 −0.296462
\(849\) 18.8189 0.645862
\(850\) −16.6450 −0.570918
\(851\) −15.5659 −0.533593
\(852\) 1.94373 0.0665912
\(853\) 5.23001 0.179072 0.0895361 0.995984i \(-0.471462\pi\)
0.0895361 + 0.995984i \(0.471462\pi\)
\(854\) −3.24309 −0.110976
\(855\) −5.60664 −0.191743
\(856\) 3.92781 0.134250
\(857\) 50.2733 1.71730 0.858652 0.512559i \(-0.171302\pi\)
0.858652 + 0.512559i \(0.171302\pi\)
\(858\) 2.98171 0.101794
\(859\) 8.37204 0.285650 0.142825 0.989748i \(-0.454381\pi\)
0.142825 + 0.989748i \(0.454381\pi\)
\(860\) −3.78210 −0.128968
\(861\) −2.01274 −0.0685941
\(862\) −21.4370 −0.730146
\(863\) −52.7861 −1.79686 −0.898429 0.439118i \(-0.855291\pi\)
−0.898429 + 0.439118i \(0.855291\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −27.8296 −0.946235
\(866\) −6.20622 −0.210896
\(867\) −6.76281 −0.229677
\(868\) −3.22776 −0.109557
\(869\) −42.2908 −1.43462
\(870\) 12.5499 0.425482
\(871\) 4.94082 0.167413
\(872\) 11.9866 0.405916
\(873\) −11.3927 −0.385584
\(874\) −18.2080 −0.615896
\(875\) 8.44903 0.285629
\(876\) 7.45502 0.251882
\(877\) 55.2655 1.86618 0.933091 0.359640i \(-0.117101\pi\)
0.933091 + 0.359640i \(0.117101\pi\)
\(878\) 11.1121 0.375014
\(879\) 33.5092 1.13024
\(880\) −3.75440 −0.126561
\(881\) −13.6393 −0.459520 −0.229760 0.973247i \(-0.573794\pi\)
−0.229760 + 0.973247i \(0.573794\pi\)
\(882\) −6.36408 −0.214290
\(883\) 1.41598 0.0476515 0.0238257 0.999716i \(-0.492415\pi\)
0.0238257 + 0.999716i \(0.492415\pi\)
\(884\) 4.87471 0.163954
\(885\) 8.46651 0.284599
\(886\) −8.79189 −0.295369
\(887\) 48.1239 1.61584 0.807922 0.589290i \(-0.200592\pi\)
0.807922 + 0.589290i \(0.200592\pi\)
\(888\) 3.80662 0.127742
\(889\) −8.78764 −0.294728
\(890\) 18.4925 0.619871
\(891\) −2.98171 −0.0998911
\(892\) 16.4842 0.551933
\(893\) 13.3681 0.447347
\(894\) 18.0807 0.604709
\(895\) 10.0811 0.336974
\(896\) −0.797445 −0.0266408
\(897\) −4.08918 −0.136534
\(898\) −14.1718 −0.472921
\(899\) −40.3428 −1.34551
\(900\) −3.41456 −0.113819
\(901\) −42.0838 −1.40202
\(902\) 7.52581 0.250582
\(903\) −2.39529 −0.0797103
\(904\) −6.53982 −0.217511
\(905\) −27.5271 −0.915033
\(906\) 9.37983 0.311624
\(907\) 11.7722 0.390890 0.195445 0.980715i \(-0.437385\pi\)
0.195445 + 0.980715i \(0.437385\pi\)
\(908\) 4.26467 0.141528
\(909\) −5.48924 −0.182067
\(910\) −1.00410 −0.0332855
\(911\) −17.8337 −0.590858 −0.295429 0.955365i \(-0.595463\pi\)
−0.295429 + 0.955365i \(0.595463\pi\)
\(912\) 4.45274 0.147445
\(913\) −45.8577 −1.51767
\(914\) 0.0240313 0.000794886 0
\(915\) −5.12075 −0.169287
\(916\) 21.0701 0.696177
\(917\) 8.82052 0.291279
\(918\) −4.87471 −0.160889
\(919\) −47.9606 −1.58207 −0.791037 0.611769i \(-0.790458\pi\)
−0.791037 + 0.611769i \(0.790458\pi\)
\(920\) 5.14886 0.169753
\(921\) −4.39459 −0.144807
\(922\) −9.22105 −0.303679
\(923\) −1.94373 −0.0639787
\(924\) −2.37775 −0.0782222
\(925\) 12.9979 0.427369
\(926\) −4.11875 −0.135351
\(927\) −1.00000 −0.0328443
\(928\) −9.96702 −0.327184
\(929\) 38.1205 1.25069 0.625346 0.780347i \(-0.284958\pi\)
0.625346 + 0.780347i \(0.284958\pi\)
\(930\) −5.09654 −0.167122
\(931\) 28.3376 0.928727
\(932\) 13.3803 0.438286
\(933\) 10.2447 0.335397
\(934\) 11.3460 0.371254
\(935\) −18.3016 −0.598527
\(936\) 1.00000 0.0326860
\(937\) −5.39528 −0.176256 −0.0881280 0.996109i \(-0.528088\pi\)
−0.0881280 + 0.996109i \(0.528088\pi\)
\(938\) −3.94003 −0.128646
\(939\) 25.1202 0.819766
\(940\) −3.78023 −0.123297
\(941\) 56.5679 1.84406 0.922030 0.387118i \(-0.126530\pi\)
0.922030 + 0.387118i \(0.126530\pi\)
\(942\) 18.1379 0.590964
\(943\) −10.3210 −0.336100
\(944\) −6.72403 −0.218848
\(945\) 1.00410 0.0326633
\(946\) 8.95619 0.291191
\(947\) 25.7348 0.836269 0.418134 0.908385i \(-0.362684\pi\)
0.418134 + 0.908385i \(0.362684\pi\)
\(948\) −14.1834 −0.460655
\(949\) −7.45502 −0.242000
\(950\) 15.2041 0.493287
\(951\) 5.81992 0.188724
\(952\) −3.88731 −0.125989
\(953\) −14.0912 −0.456459 −0.228229 0.973607i \(-0.573294\pi\)
−0.228229 + 0.973607i \(0.573294\pi\)
\(954\) −8.63309 −0.279507
\(955\) 28.9535 0.936912
\(956\) 29.5612 0.956076
\(957\) −29.7188 −0.960672
\(958\) −18.0330 −0.582620
\(959\) 17.2893 0.558299
\(960\) −1.25914 −0.0406387
\(961\) −14.6167 −0.471507
\(962\) −3.80662 −0.122730
\(963\) 3.92781 0.126572
\(964\) −2.85139 −0.0918371
\(965\) 15.2214 0.489995
\(966\) 3.26089 0.104917
\(967\) 17.9570 0.577459 0.288730 0.957411i \(-0.406767\pi\)
0.288730 + 0.957411i \(0.406767\pi\)
\(968\) −2.10940 −0.0677987
\(969\) 21.7058 0.697291
\(970\) −14.3450 −0.460591
\(971\) 14.3002 0.458915 0.229457 0.973319i \(-0.426305\pi\)
0.229457 + 0.973319i \(0.426305\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 3.20232 0.102662
\(974\) −41.4619 −1.32852
\(975\) 3.41456 0.109353
\(976\) 4.06686 0.130177
\(977\) 24.7427 0.791588 0.395794 0.918339i \(-0.370469\pi\)
0.395794 + 0.918339i \(0.370469\pi\)
\(978\) 15.9568 0.510241
\(979\) −43.7912 −1.39957
\(980\) −8.01329 −0.255975
\(981\) 11.9866 0.382701
\(982\) −34.9707 −1.11596
\(983\) 5.61508 0.179093 0.0895467 0.995983i \(-0.471458\pi\)
0.0895467 + 0.995983i \(0.471458\pi\)
\(984\) 2.52399 0.0804619
\(985\) −1.04285 −0.0332279
\(986\) −48.5864 −1.54731
\(987\) −2.39411 −0.0762052
\(988\) −4.45274 −0.141661
\(989\) −12.2827 −0.390567
\(990\) −3.75440 −0.119323
\(991\) −36.2473 −1.15143 −0.575717 0.817649i \(-0.695277\pi\)
−0.575717 + 0.817649i \(0.695277\pi\)
\(992\) 4.04763 0.128512
\(993\) 20.4154 0.647864
\(994\) 1.55002 0.0491636
\(995\) −21.3290 −0.676176
\(996\) −15.3797 −0.487323
\(997\) −5.00212 −0.158419 −0.0792094 0.996858i \(-0.525240\pi\)
−0.0792094 + 0.996858i \(0.525240\pi\)
\(998\) −18.4859 −0.585161
\(999\) 3.80662 0.120436
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 8034.2.a.r.1.7 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.r.1.7 9 1.1 even 1 trivial