Properties

Label 3381.2.a.bk.1.1
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $10$
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] = [3381,2,Mod(1,3381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, 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("3381.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.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: \(26.9974209234\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 6x^{8} + 36x^{7} + x^{6} - 100x^{5} + 26x^{4} + 108x^{3} - 33x^{2} - 36x + 14 \) 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 1.1
Root \(-2.23864\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.23864 q^{2} -1.00000 q^{3} +3.01150 q^{4} -2.46287 q^{5} +2.23864 q^{6} -2.26439 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.23864 q^{2} -1.00000 q^{3} +3.01150 q^{4} -2.46287 q^{5} +2.23864 q^{6} -2.26439 q^{8} +1.00000 q^{9} +5.51348 q^{10} +3.07123 q^{11} -3.01150 q^{12} -3.08168 q^{13} +2.46287 q^{15} -0.953860 q^{16} +3.92563 q^{17} -2.23864 q^{18} -2.23891 q^{19} -7.41695 q^{20} -6.87536 q^{22} -1.00000 q^{23} +2.26439 q^{24} +1.06575 q^{25} +6.89878 q^{26} -1.00000 q^{27} +4.31559 q^{29} -5.51348 q^{30} -3.47943 q^{31} +6.66412 q^{32} -3.07123 q^{33} -8.78808 q^{34} +3.01150 q^{36} -3.10069 q^{37} +5.01212 q^{38} +3.08168 q^{39} +5.57690 q^{40} -8.45567 q^{41} -3.01191 q^{43} +9.24900 q^{44} -2.46287 q^{45} +2.23864 q^{46} +9.63365 q^{47} +0.953860 q^{48} -2.38582 q^{50} -3.92563 q^{51} -9.28050 q^{52} +3.32711 q^{53} +2.23864 q^{54} -7.56404 q^{55} +2.23891 q^{57} -9.66105 q^{58} -7.60292 q^{59} +7.41695 q^{60} +1.60304 q^{61} +7.78919 q^{62} -13.0108 q^{64} +7.58980 q^{65} +6.87536 q^{66} +13.7403 q^{67} +11.8221 q^{68} +1.00000 q^{69} +10.7392 q^{71} -2.26439 q^{72} +5.47869 q^{73} +6.94133 q^{74} -1.06575 q^{75} -6.74249 q^{76} -6.89878 q^{78} -9.72680 q^{79} +2.34924 q^{80} +1.00000 q^{81} +18.9292 q^{82} +2.34134 q^{83} -9.66834 q^{85} +6.74257 q^{86} -4.31559 q^{87} -6.95444 q^{88} -5.01990 q^{89} +5.51348 q^{90} -3.01150 q^{92} +3.47943 q^{93} -21.5662 q^{94} +5.51416 q^{95} -6.66412 q^{96} -5.86596 q^{97} +3.07123 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} - 10 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} + 12 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{2} - 10 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} + 12 q^{8} + 10 q^{9} - 8 q^{10} + 2 q^{11} - 8 q^{12} + 4 q^{15} + 4 q^{16} - 12 q^{17} + 4 q^{18} - 26 q^{19} - 24 q^{20} - 8 q^{22} - 10 q^{23} - 12 q^{24} - 2 q^{25} - 4 q^{26} - 10 q^{27} + 16 q^{29} + 8 q^{30} - 12 q^{31} + 8 q^{32} - 2 q^{33} - 28 q^{34} + 8 q^{36} - 8 q^{37} - 32 q^{38} - 4 q^{40} - 10 q^{41} - 4 q^{43} - 16 q^{44} - 4 q^{45} - 4 q^{46} - 2 q^{47} - 4 q^{48} - 8 q^{50} + 12 q^{51} - 24 q^{52} + 14 q^{53} - 4 q^{54} - 16 q^{55} + 26 q^{57} - 8 q^{58} - 38 q^{59} + 24 q^{60} - 14 q^{61} + 8 q^{62} + 8 q^{64} + 12 q^{65} + 8 q^{66} - 8 q^{68} + 10 q^{69} + 24 q^{71} + 12 q^{72} - 8 q^{73} - 8 q^{74} + 2 q^{75} - 64 q^{76} + 4 q^{78} - 16 q^{79} - 28 q^{80} + 10 q^{81} + 40 q^{82} - 28 q^{83} - 4 q^{85} + 20 q^{86} - 16 q^{87} - 68 q^{88} - 32 q^{89} - 8 q^{90} - 8 q^{92} + 12 q^{93} - 56 q^{94} + 8 q^{95} - 8 q^{96} - 4 q^{97} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.23864 −1.58296 −0.791478 0.611197i \(-0.790688\pi\)
−0.791478 + 0.611197i \(0.790688\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.01150 1.50575
\(5\) −2.46287 −1.10143 −0.550715 0.834693i \(-0.685645\pi\)
−0.550715 + 0.834693i \(0.685645\pi\)
\(6\) 2.23864 0.913920
\(7\) 0 0
\(8\) −2.26439 −0.800582
\(9\) 1.00000 0.333333
\(10\) 5.51348 1.74352
\(11\) 3.07123 0.926009 0.463005 0.886356i \(-0.346771\pi\)
0.463005 + 0.886356i \(0.346771\pi\)
\(12\) −3.01150 −0.869346
\(13\) −3.08168 −0.854705 −0.427353 0.904085i \(-0.640554\pi\)
−0.427353 + 0.904085i \(0.640554\pi\)
\(14\) 0 0
\(15\) 2.46287 0.635911
\(16\) −0.953860 −0.238465
\(17\) 3.92563 0.952106 0.476053 0.879417i \(-0.342067\pi\)
0.476053 + 0.879417i \(0.342067\pi\)
\(18\) −2.23864 −0.527652
\(19\) −2.23891 −0.513642 −0.256821 0.966459i \(-0.582675\pi\)
−0.256821 + 0.966459i \(0.582675\pi\)
\(20\) −7.41695 −1.65848
\(21\) 0 0
\(22\) −6.87536 −1.46583
\(23\) −1.00000 −0.208514
\(24\) 2.26439 0.462216
\(25\) 1.06575 0.213149
\(26\) 6.89878 1.35296
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.31559 0.801385 0.400693 0.916213i \(-0.368770\pi\)
0.400693 + 0.916213i \(0.368770\pi\)
\(30\) −5.51348 −1.00662
\(31\) −3.47943 −0.624925 −0.312462 0.949930i \(-0.601154\pi\)
−0.312462 + 0.949930i \(0.601154\pi\)
\(32\) 6.66412 1.17806
\(33\) −3.07123 −0.534632
\(34\) −8.78808 −1.50714
\(35\) 0 0
\(36\) 3.01150 0.501917
\(37\) −3.10069 −0.509751 −0.254875 0.966974i \(-0.582034\pi\)
−0.254875 + 0.966974i \(0.582034\pi\)
\(38\) 5.01212 0.813073
\(39\) 3.08168 0.493464
\(40\) 5.57690 0.881785
\(41\) −8.45567 −1.32055 −0.660277 0.751022i \(-0.729561\pi\)
−0.660277 + 0.751022i \(0.729561\pi\)
\(42\) 0 0
\(43\) −3.01191 −0.459312 −0.229656 0.973272i \(-0.573760\pi\)
−0.229656 + 0.973272i \(0.573760\pi\)
\(44\) 9.24900 1.39434
\(45\) −2.46287 −0.367144
\(46\) 2.23864 0.330069
\(47\) 9.63365 1.40521 0.702606 0.711579i \(-0.252020\pi\)
0.702606 + 0.711579i \(0.252020\pi\)
\(48\) 0.953860 0.137678
\(49\) 0 0
\(50\) −2.38582 −0.337406
\(51\) −3.92563 −0.549699
\(52\) −9.28050 −1.28697
\(53\) 3.32711 0.457014 0.228507 0.973542i \(-0.426616\pi\)
0.228507 + 0.973542i \(0.426616\pi\)
\(54\) 2.23864 0.304640
\(55\) −7.56404 −1.01993
\(56\) 0 0
\(57\) 2.23891 0.296551
\(58\) −9.66105 −1.26856
\(59\) −7.60292 −0.989816 −0.494908 0.868945i \(-0.664798\pi\)
−0.494908 + 0.868945i \(0.664798\pi\)
\(60\) 7.41695 0.957524
\(61\) 1.60304 0.205249 0.102624 0.994720i \(-0.467276\pi\)
0.102624 + 0.994720i \(0.467276\pi\)
\(62\) 7.78919 0.989229
\(63\) 0 0
\(64\) −13.0108 −1.62635
\(65\) 7.58980 0.941398
\(66\) 6.87536 0.846299
\(67\) 13.7403 1.67865 0.839325 0.543631i \(-0.182951\pi\)
0.839325 + 0.543631i \(0.182951\pi\)
\(68\) 11.8221 1.43363
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 10.7392 1.27451 0.637256 0.770652i \(-0.280070\pi\)
0.637256 + 0.770652i \(0.280070\pi\)
\(72\) −2.26439 −0.266861
\(73\) 5.47869 0.641232 0.320616 0.947209i \(-0.396110\pi\)
0.320616 + 0.947209i \(0.396110\pi\)
\(74\) 6.94133 0.806913
\(75\) −1.06575 −0.123062
\(76\) −6.74249 −0.773417
\(77\) 0 0
\(78\) −6.89878 −0.781132
\(79\) −9.72680 −1.09435 −0.547175 0.837018i \(-0.684297\pi\)
−0.547175 + 0.837018i \(0.684297\pi\)
\(80\) 2.34924 0.262653
\(81\) 1.00000 0.111111
\(82\) 18.9292 2.09038
\(83\) 2.34134 0.256995 0.128498 0.991710i \(-0.458985\pi\)
0.128498 + 0.991710i \(0.458985\pi\)
\(84\) 0 0
\(85\) −9.66834 −1.04868
\(86\) 6.74257 0.727070
\(87\) −4.31559 −0.462680
\(88\) −6.95444 −0.741346
\(89\) −5.01990 −0.532108 −0.266054 0.963958i \(-0.585720\pi\)
−0.266054 + 0.963958i \(0.585720\pi\)
\(90\) 5.51348 0.581172
\(91\) 0 0
\(92\) −3.01150 −0.313971
\(93\) 3.47943 0.360800
\(94\) −21.5662 −2.22439
\(95\) 5.51416 0.565741
\(96\) −6.66412 −0.680154
\(97\) −5.86596 −0.595598 −0.297799 0.954629i \(-0.596253\pi\)
−0.297799 + 0.954629i \(0.596253\pi\)
\(98\) 0 0
\(99\) 3.07123 0.308670
\(100\) 3.20950 0.320950
\(101\) −0.968892 −0.0964084 −0.0482042 0.998838i \(-0.515350\pi\)
−0.0482042 + 0.998838i \(0.515350\pi\)
\(102\) 8.78808 0.870149
\(103\) 6.40172 0.630781 0.315390 0.948962i \(-0.397865\pi\)
0.315390 + 0.948962i \(0.397865\pi\)
\(104\) 6.97812 0.684261
\(105\) 0 0
\(106\) −7.44820 −0.723434
\(107\) 16.1158 1.55798 0.778988 0.627039i \(-0.215733\pi\)
0.778988 + 0.627039i \(0.215733\pi\)
\(108\) −3.01150 −0.289782
\(109\) 6.45532 0.618308 0.309154 0.951012i \(-0.399954\pi\)
0.309154 + 0.951012i \(0.399954\pi\)
\(110\) 16.9332 1.61451
\(111\) 3.10069 0.294305
\(112\) 0 0
\(113\) 8.50261 0.799858 0.399929 0.916546i \(-0.369035\pi\)
0.399929 + 0.916546i \(0.369035\pi\)
\(114\) −5.01212 −0.469428
\(115\) 2.46287 0.229664
\(116\) 12.9964 1.20669
\(117\) −3.08168 −0.284902
\(118\) 17.0202 1.56684
\(119\) 0 0
\(120\) −5.57690 −0.509099
\(121\) −1.56757 −0.142507
\(122\) −3.58863 −0.324900
\(123\) 8.45567 0.762422
\(124\) −10.4783 −0.940981
\(125\) 9.68957 0.866661
\(126\) 0 0
\(127\) −9.69548 −0.860334 −0.430167 0.902749i \(-0.641545\pi\)
−0.430167 + 0.902749i \(0.641545\pi\)
\(128\) 15.7983 1.39639
\(129\) 3.01191 0.265184
\(130\) −16.9908 −1.49019
\(131\) −18.2934 −1.59830 −0.799150 0.601132i \(-0.794717\pi\)
−0.799150 + 0.601132i \(0.794717\pi\)
\(132\) −9.24900 −0.805022
\(133\) 0 0
\(134\) −30.7596 −2.65723
\(135\) 2.46287 0.211970
\(136\) −8.88916 −0.762239
\(137\) 4.69464 0.401090 0.200545 0.979685i \(-0.435729\pi\)
0.200545 + 0.979685i \(0.435729\pi\)
\(138\) −2.23864 −0.190566
\(139\) 9.69649 0.822446 0.411223 0.911535i \(-0.365102\pi\)
0.411223 + 0.911535i \(0.365102\pi\)
\(140\) 0 0
\(141\) −9.63365 −0.811299
\(142\) −24.0412 −2.01750
\(143\) −9.46454 −0.791465
\(144\) −0.953860 −0.0794883
\(145\) −10.6288 −0.882670
\(146\) −12.2648 −1.01504
\(147\) 0 0
\(148\) −9.33774 −0.767558
\(149\) −10.3636 −0.849017 −0.424509 0.905424i \(-0.639553\pi\)
−0.424509 + 0.905424i \(0.639553\pi\)
\(150\) 2.38582 0.194802
\(151\) 10.8567 0.883503 0.441752 0.897137i \(-0.354357\pi\)
0.441752 + 0.897137i \(0.354357\pi\)
\(152\) 5.06976 0.411212
\(153\) 3.92563 0.317369
\(154\) 0 0
\(155\) 8.56941 0.688311
\(156\) 9.28050 0.743034
\(157\) 2.52233 0.201304 0.100652 0.994922i \(-0.467907\pi\)
0.100652 + 0.994922i \(0.467907\pi\)
\(158\) 21.7748 1.73231
\(159\) −3.32711 −0.263857
\(160\) −16.4129 −1.29755
\(161\) 0 0
\(162\) −2.23864 −0.175884
\(163\) −14.4762 −1.13386 −0.566932 0.823765i \(-0.691870\pi\)
−0.566932 + 0.823765i \(0.691870\pi\)
\(164\) −25.4643 −1.98842
\(165\) 7.56404 0.588860
\(166\) −5.24141 −0.406812
\(167\) −15.6448 −1.21063 −0.605314 0.795987i \(-0.706953\pi\)
−0.605314 + 0.795987i \(0.706953\pi\)
\(168\) 0 0
\(169\) −3.50323 −0.269479
\(170\) 21.6439 1.66001
\(171\) −2.23891 −0.171214
\(172\) −9.07037 −0.691609
\(173\) 21.9908 1.67193 0.835963 0.548785i \(-0.184909\pi\)
0.835963 + 0.548785i \(0.184909\pi\)
\(174\) 9.66105 0.732402
\(175\) 0 0
\(176\) −2.92952 −0.220821
\(177\) 7.60292 0.571471
\(178\) 11.2377 0.842304
\(179\) −18.7258 −1.39963 −0.699815 0.714324i \(-0.746734\pi\)
−0.699815 + 0.714324i \(0.746734\pi\)
\(180\) −7.41695 −0.552827
\(181\) 4.88885 0.363386 0.181693 0.983355i \(-0.441842\pi\)
0.181693 + 0.983355i \(0.441842\pi\)
\(182\) 0 0
\(183\) −1.60304 −0.118500
\(184\) 2.26439 0.166933
\(185\) 7.63661 0.561455
\(186\) −7.78919 −0.571131
\(187\) 12.0565 0.881659
\(188\) 29.0117 2.11590
\(189\) 0 0
\(190\) −12.3442 −0.895543
\(191\) −13.3658 −0.967118 −0.483559 0.875312i \(-0.660656\pi\)
−0.483559 + 0.875312i \(0.660656\pi\)
\(192\) 13.0108 0.938976
\(193\) −11.0215 −0.793344 −0.396672 0.917960i \(-0.629835\pi\)
−0.396672 + 0.917960i \(0.629835\pi\)
\(194\) 13.1318 0.942806
\(195\) −7.58980 −0.543517
\(196\) 0 0
\(197\) 16.6336 1.18510 0.592549 0.805535i \(-0.298122\pi\)
0.592549 + 0.805535i \(0.298122\pi\)
\(198\) −6.87536 −0.488611
\(199\) 14.7498 1.04559 0.522794 0.852459i \(-0.324890\pi\)
0.522794 + 0.852459i \(0.324890\pi\)
\(200\) −2.41326 −0.170644
\(201\) −13.7403 −0.969169
\(202\) 2.16900 0.152610
\(203\) 0 0
\(204\) −11.8221 −0.827709
\(205\) 20.8252 1.45450
\(206\) −14.3311 −0.998498
\(207\) −1.00000 −0.0695048
\(208\) 2.93949 0.203817
\(209\) −6.87620 −0.475637
\(210\) 0 0
\(211\) 7.38157 0.508168 0.254084 0.967182i \(-0.418226\pi\)
0.254084 + 0.967182i \(0.418226\pi\)
\(212\) 10.0196 0.688150
\(213\) −10.7392 −0.735839
\(214\) −36.0775 −2.46621
\(215\) 7.41795 0.505900
\(216\) 2.26439 0.154072
\(217\) 0 0
\(218\) −14.4511 −0.978754
\(219\) −5.47869 −0.370216
\(220\) −22.7791 −1.53577
\(221\) −12.0976 −0.813770
\(222\) −6.94133 −0.465872
\(223\) 25.6512 1.71773 0.858865 0.512202i \(-0.171170\pi\)
0.858865 + 0.512202i \(0.171170\pi\)
\(224\) 0 0
\(225\) 1.06575 0.0710498
\(226\) −19.0343 −1.26614
\(227\) −9.34980 −0.620568 −0.310284 0.950644i \(-0.600424\pi\)
−0.310284 + 0.950644i \(0.600424\pi\)
\(228\) 6.74249 0.446532
\(229\) −18.0873 −1.19524 −0.597621 0.801779i \(-0.703887\pi\)
−0.597621 + 0.801779i \(0.703887\pi\)
\(230\) −5.51348 −0.363548
\(231\) 0 0
\(232\) −9.77217 −0.641574
\(233\) −17.8642 −1.17033 −0.585163 0.810916i \(-0.698969\pi\)
−0.585163 + 0.810916i \(0.698969\pi\)
\(234\) 6.89878 0.450987
\(235\) −23.7265 −1.54774
\(236\) −22.8962 −1.49042
\(237\) 9.72680 0.631823
\(238\) 0 0
\(239\) 15.3433 0.992475 0.496237 0.868187i \(-0.334715\pi\)
0.496237 + 0.868187i \(0.334715\pi\)
\(240\) −2.34924 −0.151643
\(241\) 3.96319 0.255292 0.127646 0.991820i \(-0.459258\pi\)
0.127646 + 0.991820i \(0.459258\pi\)
\(242\) 3.50923 0.225582
\(243\) −1.00000 −0.0641500
\(244\) 4.82757 0.309053
\(245\) 0 0
\(246\) −18.9292 −1.20688
\(247\) 6.89962 0.439012
\(248\) 7.87879 0.500303
\(249\) −2.34134 −0.148376
\(250\) −21.6914 −1.37189
\(251\) 5.68753 0.358994 0.179497 0.983759i \(-0.442553\pi\)
0.179497 + 0.983759i \(0.442553\pi\)
\(252\) 0 0
\(253\) −3.07123 −0.193086
\(254\) 21.7047 1.36187
\(255\) 9.66834 0.605455
\(256\) −9.34505 −0.584066
\(257\) 12.2060 0.761387 0.380693 0.924701i \(-0.375685\pi\)
0.380693 + 0.924701i \(0.375685\pi\)
\(258\) −6.74257 −0.419774
\(259\) 0 0
\(260\) 22.8567 1.41751
\(261\) 4.31559 0.267128
\(262\) 40.9522 2.53004
\(263\) −10.6450 −0.656397 −0.328199 0.944609i \(-0.606442\pi\)
−0.328199 + 0.944609i \(0.606442\pi\)
\(264\) 6.95444 0.428016
\(265\) −8.19426 −0.503369
\(266\) 0 0
\(267\) 5.01990 0.307213
\(268\) 41.3790 2.52763
\(269\) −24.8992 −1.51813 −0.759066 0.651014i \(-0.774344\pi\)
−0.759066 + 0.651014i \(0.774344\pi\)
\(270\) −5.51348 −0.335540
\(271\) 6.57668 0.399505 0.199752 0.979846i \(-0.435986\pi\)
0.199752 + 0.979846i \(0.435986\pi\)
\(272\) −3.74451 −0.227044
\(273\) 0 0
\(274\) −10.5096 −0.634908
\(275\) 3.27315 0.197378
\(276\) 3.01150 0.181271
\(277\) −23.2474 −1.39680 −0.698400 0.715708i \(-0.746104\pi\)
−0.698400 + 0.715708i \(0.746104\pi\)
\(278\) −21.7069 −1.30190
\(279\) −3.47943 −0.208308
\(280\) 0 0
\(281\) 10.7437 0.640913 0.320456 0.947263i \(-0.396164\pi\)
0.320456 + 0.947263i \(0.396164\pi\)
\(282\) 21.5662 1.28425
\(283\) −24.5170 −1.45739 −0.728694 0.684840i \(-0.759872\pi\)
−0.728694 + 0.684840i \(0.759872\pi\)
\(284\) 32.3412 1.91910
\(285\) −5.51416 −0.326631
\(286\) 21.1877 1.25285
\(287\) 0 0
\(288\) 6.66412 0.392687
\(289\) −1.58939 −0.0934937
\(290\) 23.7939 1.39723
\(291\) 5.86596 0.343869
\(292\) 16.4991 0.965536
\(293\) −33.2732 −1.94384 −0.971920 0.235311i \(-0.924389\pi\)
−0.971920 + 0.235311i \(0.924389\pi\)
\(294\) 0 0
\(295\) 18.7250 1.09021
\(296\) 7.02117 0.408097
\(297\) −3.07123 −0.178211
\(298\) 23.2003 1.34396
\(299\) 3.08168 0.178218
\(300\) −3.20950 −0.185301
\(301\) 0 0
\(302\) −24.3042 −1.39855
\(303\) 0.968892 0.0556614
\(304\) 2.13561 0.122486
\(305\) −3.94809 −0.226067
\(306\) −8.78808 −0.502381
\(307\) −20.2345 −1.15485 −0.577423 0.816445i \(-0.695942\pi\)
−0.577423 + 0.816445i \(0.695942\pi\)
\(308\) 0 0
\(309\) −6.40172 −0.364181
\(310\) −19.1838 −1.08957
\(311\) −22.8603 −1.29629 −0.648145 0.761517i \(-0.724455\pi\)
−0.648145 + 0.761517i \(0.724455\pi\)
\(312\) −6.97812 −0.395059
\(313\) −23.6714 −1.33798 −0.668992 0.743269i \(-0.733274\pi\)
−0.668992 + 0.743269i \(0.733274\pi\)
\(314\) −5.64659 −0.318655
\(315\) 0 0
\(316\) −29.2923 −1.64782
\(317\) 12.4283 0.698041 0.349021 0.937115i \(-0.386514\pi\)
0.349021 + 0.937115i \(0.386514\pi\)
\(318\) 7.44820 0.417675
\(319\) 13.2542 0.742090
\(320\) 32.0441 1.79132
\(321\) −16.1158 −0.899497
\(322\) 0 0
\(323\) −8.78915 −0.489042
\(324\) 3.01150 0.167306
\(325\) −3.28429 −0.182180
\(326\) 32.4070 1.79486
\(327\) −6.45532 −0.356980
\(328\) 19.1469 1.05721
\(329\) 0 0
\(330\) −16.9332 −0.932139
\(331\) −7.30561 −0.401553 −0.200776 0.979637i \(-0.564346\pi\)
−0.200776 + 0.979637i \(0.564346\pi\)
\(332\) 7.05094 0.386971
\(333\) −3.10069 −0.169917
\(334\) 35.0230 1.91637
\(335\) −33.8407 −1.84892
\(336\) 0 0
\(337\) 31.1798 1.69847 0.849235 0.528015i \(-0.177063\pi\)
0.849235 + 0.528015i \(0.177063\pi\)
\(338\) 7.84246 0.426574
\(339\) −8.50261 −0.461798
\(340\) −29.1162 −1.57905
\(341\) −10.6861 −0.578686
\(342\) 5.01212 0.271024
\(343\) 0 0
\(344\) 6.82013 0.367717
\(345\) −2.46287 −0.132597
\(346\) −49.2294 −2.64659
\(347\) −6.98372 −0.374906 −0.187453 0.982274i \(-0.560023\pi\)
−0.187453 + 0.982274i \(0.560023\pi\)
\(348\) −12.9964 −0.696681
\(349\) −12.5064 −0.669454 −0.334727 0.942315i \(-0.608644\pi\)
−0.334727 + 0.942315i \(0.608644\pi\)
\(350\) 0 0
\(351\) 3.08168 0.164488
\(352\) 20.4670 1.09090
\(353\) 19.9458 1.06161 0.530803 0.847495i \(-0.321890\pi\)
0.530803 + 0.847495i \(0.321890\pi\)
\(354\) −17.0202 −0.904613
\(355\) −26.4494 −1.40379
\(356\) −15.1174 −0.801222
\(357\) 0 0
\(358\) 41.9203 2.21555
\(359\) 31.2088 1.64714 0.823570 0.567215i \(-0.191979\pi\)
0.823570 + 0.567215i \(0.191979\pi\)
\(360\) 5.57690 0.293928
\(361\) −13.9873 −0.736172
\(362\) −10.9444 −0.575224
\(363\) 1.56757 0.0822763
\(364\) 0 0
\(365\) −13.4933 −0.706273
\(366\) 3.58863 0.187581
\(367\) −15.0953 −0.787966 −0.393983 0.919118i \(-0.628903\pi\)
−0.393983 + 0.919118i \(0.628903\pi\)
\(368\) 0.953860 0.0497234
\(369\) −8.45567 −0.440184
\(370\) −17.0956 −0.888759
\(371\) 0 0
\(372\) 10.4783 0.543276
\(373\) −27.2574 −1.41134 −0.705668 0.708543i \(-0.749353\pi\)
−0.705668 + 0.708543i \(0.749353\pi\)
\(374\) −26.9902 −1.39563
\(375\) −9.68957 −0.500367
\(376\) −21.8143 −1.12499
\(377\) −13.2993 −0.684948
\(378\) 0 0
\(379\) −28.6079 −1.46949 −0.734745 0.678344i \(-0.762698\pi\)
−0.734745 + 0.678344i \(0.762698\pi\)
\(380\) 16.6059 0.851865
\(381\) 9.69548 0.496714
\(382\) 29.9213 1.53091
\(383\) −18.3899 −0.939682 −0.469841 0.882751i \(-0.655689\pi\)
−0.469841 + 0.882751i \(0.655689\pi\)
\(384\) −15.7983 −0.806205
\(385\) 0 0
\(386\) 24.6731 1.25583
\(387\) −3.01191 −0.153104
\(388\) −17.6654 −0.896823
\(389\) −32.8984 −1.66801 −0.834007 0.551753i \(-0.813959\pi\)
−0.834007 + 0.551753i \(0.813959\pi\)
\(390\) 16.9908 0.860363
\(391\) −3.92563 −0.198528
\(392\) 0 0
\(393\) 18.2934 0.922779
\(394\) −37.2367 −1.87596
\(395\) 23.9559 1.20535
\(396\) 9.24900 0.464780
\(397\) −17.3815 −0.872352 −0.436176 0.899861i \(-0.643668\pi\)
−0.436176 + 0.899861i \(0.643668\pi\)
\(398\) −33.0195 −1.65512
\(399\) 0 0
\(400\) −1.01657 −0.0508287
\(401\) −23.2849 −1.16279 −0.581396 0.813621i \(-0.697493\pi\)
−0.581396 + 0.813621i \(0.697493\pi\)
\(402\) 30.7596 1.53415
\(403\) 10.7225 0.534126
\(404\) −2.91782 −0.145167
\(405\) −2.46287 −0.122381
\(406\) 0 0
\(407\) −9.52293 −0.472034
\(408\) 8.88916 0.440079
\(409\) −0.712227 −0.0352174 −0.0176087 0.999845i \(-0.505605\pi\)
−0.0176087 + 0.999845i \(0.505605\pi\)
\(410\) −46.6202 −2.30241
\(411\) −4.69464 −0.231569
\(412\) 19.2788 0.949799
\(413\) 0 0
\(414\) 2.23864 0.110023
\(415\) −5.76642 −0.283062
\(416\) −20.5367 −1.00690
\(417\) −9.69649 −0.474839
\(418\) 15.3933 0.752913
\(419\) 5.77916 0.282330 0.141165 0.989986i \(-0.454915\pi\)
0.141165 + 0.989986i \(0.454915\pi\)
\(420\) 0 0
\(421\) −14.8487 −0.723680 −0.361840 0.932240i \(-0.617851\pi\)
−0.361840 + 0.932240i \(0.617851\pi\)
\(422\) −16.5247 −0.804408
\(423\) 9.63365 0.468404
\(424\) −7.53387 −0.365877
\(425\) 4.18373 0.202941
\(426\) 24.0412 1.16480
\(427\) 0 0
\(428\) 48.5328 2.34592
\(429\) 9.46454 0.456953
\(430\) −16.6061 −0.800818
\(431\) 4.44128 0.213929 0.106964 0.994263i \(-0.465887\pi\)
0.106964 + 0.994263i \(0.465887\pi\)
\(432\) 0.953860 0.0458926
\(433\) 8.07973 0.388287 0.194143 0.980973i \(-0.437807\pi\)
0.194143 + 0.980973i \(0.437807\pi\)
\(434\) 0 0
\(435\) 10.6288 0.509610
\(436\) 19.4402 0.931018
\(437\) 2.23891 0.107102
\(438\) 12.2648 0.586035
\(439\) −11.5837 −0.552860 −0.276430 0.961034i \(-0.589151\pi\)
−0.276430 + 0.961034i \(0.589151\pi\)
\(440\) 17.1279 0.816541
\(441\) 0 0
\(442\) 27.0821 1.28816
\(443\) −36.6133 −1.73955 −0.869775 0.493448i \(-0.835736\pi\)
−0.869775 + 0.493448i \(0.835736\pi\)
\(444\) 9.33774 0.443150
\(445\) 12.3634 0.586080
\(446\) −57.4237 −2.71909
\(447\) 10.3636 0.490180
\(448\) 0 0
\(449\) 8.72363 0.411694 0.205847 0.978584i \(-0.434005\pi\)
0.205847 + 0.978584i \(0.434005\pi\)
\(450\) −2.38582 −0.112469
\(451\) −25.9693 −1.22284
\(452\) 25.6056 1.20439
\(453\) −10.8567 −0.510091
\(454\) 20.9308 0.982332
\(455\) 0 0
\(456\) −5.06976 −0.237413
\(457\) 22.5857 1.05651 0.528256 0.849085i \(-0.322846\pi\)
0.528256 + 0.849085i \(0.322846\pi\)
\(458\) 40.4909 1.89202
\(459\) −3.92563 −0.183233
\(460\) 7.41695 0.345817
\(461\) −29.1463 −1.35748 −0.678740 0.734379i \(-0.737474\pi\)
−0.678740 + 0.734379i \(0.737474\pi\)
\(462\) 0 0
\(463\) −0.0153471 −0.000713240 0 −0.000356620 1.00000i \(-0.500114\pi\)
−0.000356620 1.00000i \(0.500114\pi\)
\(464\) −4.11647 −0.191102
\(465\) −8.56941 −0.397397
\(466\) 39.9916 1.85257
\(467\) −16.3186 −0.755135 −0.377568 0.925982i \(-0.623239\pi\)
−0.377568 + 0.925982i \(0.623239\pi\)
\(468\) −9.28050 −0.428991
\(469\) 0 0
\(470\) 53.1149 2.45001
\(471\) −2.52233 −0.116223
\(472\) 17.2160 0.792429
\(473\) −9.25025 −0.425327
\(474\) −21.7748 −1.00015
\(475\) −2.38611 −0.109482
\(476\) 0 0
\(477\) 3.32711 0.152338
\(478\) −34.3481 −1.57104
\(479\) −19.2029 −0.877403 −0.438701 0.898633i \(-0.644561\pi\)
−0.438701 + 0.898633i \(0.644561\pi\)
\(480\) 16.4129 0.749143
\(481\) 9.55535 0.435687
\(482\) −8.87216 −0.404116
\(483\) 0 0
\(484\) −4.72075 −0.214580
\(485\) 14.4471 0.656010
\(486\) 2.23864 0.101547
\(487\) 38.5000 1.74460 0.872300 0.488971i \(-0.162628\pi\)
0.872300 + 0.488971i \(0.162628\pi\)
\(488\) −3.62991 −0.164318
\(489\) 14.4762 0.654637
\(490\) 0 0
\(491\) 5.99057 0.270351 0.135175 0.990822i \(-0.456840\pi\)
0.135175 + 0.990822i \(0.456840\pi\)
\(492\) 25.4643 1.14802
\(493\) 16.9414 0.763004
\(494\) −15.4458 −0.694937
\(495\) −7.56404 −0.339978
\(496\) 3.31889 0.149023
\(497\) 0 0
\(498\) 5.24141 0.234873
\(499\) −12.3152 −0.551302 −0.275651 0.961258i \(-0.588893\pi\)
−0.275651 + 0.961258i \(0.588893\pi\)
\(500\) 29.1802 1.30498
\(501\) 15.6448 0.698957
\(502\) −12.7323 −0.568271
\(503\) 12.8732 0.573990 0.286995 0.957932i \(-0.407344\pi\)
0.286995 + 0.957932i \(0.407344\pi\)
\(504\) 0 0
\(505\) 2.38626 0.106187
\(506\) 6.87536 0.305647
\(507\) 3.50323 0.155584
\(508\) −29.1979 −1.29545
\(509\) −6.75440 −0.299383 −0.149692 0.988733i \(-0.547828\pi\)
−0.149692 + 0.988733i \(0.547828\pi\)
\(510\) −21.6439 −0.958409
\(511\) 0 0
\(512\) −10.6765 −0.471837
\(513\) 2.23891 0.0988504
\(514\) −27.3247 −1.20524
\(515\) −15.7666 −0.694761
\(516\) 9.07037 0.399301
\(517\) 29.5871 1.30124
\(518\) 0 0
\(519\) −21.9908 −0.965288
\(520\) −17.1862 −0.753666
\(521\) 13.1416 0.575745 0.287872 0.957669i \(-0.407052\pi\)
0.287872 + 0.957669i \(0.407052\pi\)
\(522\) −9.66105 −0.422853
\(523\) −29.6346 −1.29583 −0.647915 0.761713i \(-0.724359\pi\)
−0.647915 + 0.761713i \(0.724359\pi\)
\(524\) −55.0905 −2.40664
\(525\) 0 0
\(526\) 23.8302 1.03905
\(527\) −13.6590 −0.594995
\(528\) 2.92952 0.127491
\(529\) 1.00000 0.0434783
\(530\) 18.3440 0.796812
\(531\) −7.60292 −0.329939
\(532\) 0 0
\(533\) 26.0577 1.12868
\(534\) −11.2377 −0.486304
\(535\) −39.6912 −1.71600
\(536\) −31.1134 −1.34390
\(537\) 18.7258 0.808077
\(538\) 55.7403 2.40314
\(539\) 0 0
\(540\) 7.41695 0.319175
\(541\) −10.9740 −0.471809 −0.235904 0.971776i \(-0.575805\pi\)
−0.235904 + 0.971776i \(0.575805\pi\)
\(542\) −14.7228 −0.632398
\(543\) −4.88885 −0.209801
\(544\) 26.1609 1.12164
\(545\) −15.8986 −0.681023
\(546\) 0 0
\(547\) 12.8543 0.549609 0.274804 0.961500i \(-0.411387\pi\)
0.274804 + 0.961500i \(0.411387\pi\)
\(548\) 14.1379 0.603941
\(549\) 1.60304 0.0684162
\(550\) −7.32740 −0.312441
\(551\) −9.66223 −0.411625
\(552\) −2.26439 −0.0963787
\(553\) 0 0
\(554\) 52.0425 2.21107
\(555\) −7.63661 −0.324156
\(556\) 29.2010 1.23840
\(557\) −27.4905 −1.16481 −0.582405 0.812899i \(-0.697888\pi\)
−0.582405 + 0.812899i \(0.697888\pi\)
\(558\) 7.78919 0.329743
\(559\) 9.28175 0.392576
\(560\) 0 0
\(561\) −12.0565 −0.509026
\(562\) −24.0512 −1.01454
\(563\) 39.8365 1.67891 0.839455 0.543429i \(-0.182874\pi\)
0.839455 + 0.543429i \(0.182874\pi\)
\(564\) −29.0117 −1.22161
\(565\) −20.9408 −0.880988
\(566\) 54.8848 2.30698
\(567\) 0 0
\(568\) −24.3178 −1.02035
\(569\) 23.4654 0.983720 0.491860 0.870674i \(-0.336317\pi\)
0.491860 + 0.870674i \(0.336317\pi\)
\(570\) 12.3442 0.517042
\(571\) −0.958267 −0.0401022 −0.0200511 0.999799i \(-0.506383\pi\)
−0.0200511 + 0.999799i \(0.506383\pi\)
\(572\) −28.5025 −1.19175
\(573\) 13.3658 0.558366
\(574\) 0 0
\(575\) −1.06575 −0.0444447
\(576\) −13.0108 −0.542118
\(577\) −36.0588 −1.50115 −0.750574 0.660786i \(-0.770223\pi\)
−0.750574 + 0.660786i \(0.770223\pi\)
\(578\) 3.55808 0.147996
\(579\) 11.0215 0.458037
\(580\) −32.0085 −1.32908
\(581\) 0 0
\(582\) −13.1318 −0.544329
\(583\) 10.2183 0.423199
\(584\) −12.4059 −0.513359
\(585\) 7.58980 0.313799
\(586\) 74.4866 3.07701
\(587\) 19.5627 0.807439 0.403720 0.914883i \(-0.367717\pi\)
0.403720 + 0.914883i \(0.367717\pi\)
\(588\) 0 0
\(589\) 7.79015 0.320987
\(590\) −41.9186 −1.72576
\(591\) −16.6336 −0.684216
\(592\) 2.95763 0.121558
\(593\) 0.346580 0.0142324 0.00711618 0.999975i \(-0.497735\pi\)
0.00711618 + 0.999975i \(0.497735\pi\)
\(594\) 6.87536 0.282100
\(595\) 0 0
\(596\) −31.2099 −1.27841
\(597\) −14.7498 −0.603670
\(598\) −6.89878 −0.282112
\(599\) −13.3827 −0.546804 −0.273402 0.961900i \(-0.588149\pi\)
−0.273402 + 0.961900i \(0.588149\pi\)
\(600\) 2.41326 0.0985211
\(601\) 19.1384 0.780670 0.390335 0.920673i \(-0.372359\pi\)
0.390335 + 0.920673i \(0.372359\pi\)
\(602\) 0 0
\(603\) 13.7403 0.559550
\(604\) 32.6949 1.33034
\(605\) 3.86074 0.156961
\(606\) −2.16900 −0.0881096
\(607\) −26.2427 −1.06516 −0.532579 0.846381i \(-0.678777\pi\)
−0.532579 + 0.846381i \(0.678777\pi\)
\(608\) −14.9204 −0.605102
\(609\) 0 0
\(610\) 8.83835 0.357854
\(611\) −29.6878 −1.20104
\(612\) 11.8221 0.477878
\(613\) −29.5838 −1.19488 −0.597440 0.801914i \(-0.703815\pi\)
−0.597440 + 0.801914i \(0.703815\pi\)
\(614\) 45.2978 1.82807
\(615\) −20.8252 −0.839755
\(616\) 0 0
\(617\) −42.6171 −1.71570 −0.857851 0.513899i \(-0.828201\pi\)
−0.857851 + 0.513899i \(0.828201\pi\)
\(618\) 14.3311 0.576483
\(619\) −30.0740 −1.20878 −0.604389 0.796689i \(-0.706583\pi\)
−0.604389 + 0.796689i \(0.706583\pi\)
\(620\) 25.8068 1.03643
\(621\) 1.00000 0.0401286
\(622\) 51.1760 2.05197
\(623\) 0 0
\(624\) −2.93949 −0.117674
\(625\) −29.1929 −1.16772
\(626\) 52.9916 2.11797
\(627\) 6.87620 0.274609
\(628\) 7.59600 0.303114
\(629\) −12.1722 −0.485337
\(630\) 0 0
\(631\) 39.5441 1.57423 0.787113 0.616809i \(-0.211575\pi\)
0.787113 + 0.616809i \(0.211575\pi\)
\(632\) 22.0252 0.876117
\(633\) −7.38157 −0.293391
\(634\) −27.8224 −1.10497
\(635\) 23.8787 0.947599
\(636\) −10.0196 −0.397303
\(637\) 0 0
\(638\) −29.6713 −1.17470
\(639\) 10.7392 0.424837
\(640\) −38.9093 −1.53802
\(641\) 35.9145 1.41854 0.709269 0.704938i \(-0.249025\pi\)
0.709269 + 0.704938i \(0.249025\pi\)
\(642\) 36.0775 1.42387
\(643\) 16.2300 0.640049 0.320025 0.947409i \(-0.396309\pi\)
0.320025 + 0.947409i \(0.396309\pi\)
\(644\) 0 0
\(645\) −7.41795 −0.292081
\(646\) 19.6757 0.774131
\(647\) 38.0710 1.49673 0.748364 0.663289i \(-0.230840\pi\)
0.748364 + 0.663289i \(0.230840\pi\)
\(648\) −2.26439 −0.0889535
\(649\) −23.3503 −0.916579
\(650\) 7.35235 0.288383
\(651\) 0 0
\(652\) −43.5951 −1.70732
\(653\) −8.45161 −0.330737 −0.165368 0.986232i \(-0.552881\pi\)
−0.165368 + 0.986232i \(0.552881\pi\)
\(654\) 14.4511 0.565084
\(655\) 45.0543 1.76042
\(656\) 8.06552 0.314906
\(657\) 5.47869 0.213744
\(658\) 0 0
\(659\) 46.3402 1.80516 0.902579 0.430523i \(-0.141671\pi\)
0.902579 + 0.430523i \(0.141671\pi\)
\(660\) 22.7791 0.886676
\(661\) −14.1551 −0.550569 −0.275285 0.961363i \(-0.588772\pi\)
−0.275285 + 0.961363i \(0.588772\pi\)
\(662\) 16.3546 0.635641
\(663\) 12.0976 0.469830
\(664\) −5.30170 −0.205746
\(665\) 0 0
\(666\) 6.94133 0.268971
\(667\) −4.31559 −0.167100
\(668\) −47.1142 −1.82290
\(669\) −25.6512 −0.991732
\(670\) 75.7571 2.92675
\(671\) 4.92331 0.190062
\(672\) 0 0
\(673\) −41.1357 −1.58567 −0.792833 0.609439i \(-0.791395\pi\)
−0.792833 + 0.609439i \(0.791395\pi\)
\(674\) −69.8002 −2.68860
\(675\) −1.06575 −0.0410206
\(676\) −10.5500 −0.405768
\(677\) 40.5057 1.55676 0.778380 0.627793i \(-0.216042\pi\)
0.778380 + 0.627793i \(0.216042\pi\)
\(678\) 19.0343 0.731006
\(679\) 0 0
\(680\) 21.8929 0.839553
\(681\) 9.34980 0.358285
\(682\) 23.9224 0.916035
\(683\) −15.1819 −0.580920 −0.290460 0.956887i \(-0.593808\pi\)
−0.290460 + 0.956887i \(0.593808\pi\)
\(684\) −6.74249 −0.257806
\(685\) −11.5623 −0.441773
\(686\) 0 0
\(687\) 18.0873 0.690073
\(688\) 2.87294 0.109530
\(689\) −10.2531 −0.390612
\(690\) 5.51348 0.209895
\(691\) −35.8668 −1.36444 −0.682219 0.731148i \(-0.738985\pi\)
−0.682219 + 0.731148i \(0.738985\pi\)
\(692\) 66.2252 2.51751
\(693\) 0 0
\(694\) 15.6340 0.593460
\(695\) −23.8812 −0.905867
\(696\) 9.77217 0.370413
\(697\) −33.1939 −1.25731
\(698\) 27.9974 1.05972
\(699\) 17.8642 0.675688
\(700\) 0 0
\(701\) −47.2118 −1.78316 −0.891582 0.452860i \(-0.850404\pi\)
−0.891582 + 0.452860i \(0.850404\pi\)
\(702\) −6.89878 −0.260377
\(703\) 6.94218 0.261829
\(704\) −39.9592 −1.50602
\(705\) 23.7265 0.893590
\(706\) −44.6514 −1.68048
\(707\) 0 0
\(708\) 22.8962 0.860492
\(709\) 44.0818 1.65553 0.827764 0.561076i \(-0.189612\pi\)
0.827764 + 0.561076i \(0.189612\pi\)
\(710\) 59.2105 2.22213
\(711\) −9.72680 −0.364783
\(712\) 11.3670 0.425996
\(713\) 3.47943 0.130306
\(714\) 0 0
\(715\) 23.3100 0.871744
\(716\) −56.3927 −2.10750
\(717\) −15.3433 −0.573006
\(718\) −69.8653 −2.60735
\(719\) −36.8733 −1.37514 −0.687570 0.726118i \(-0.741323\pi\)
−0.687570 + 0.726118i \(0.741323\pi\)
\(720\) 2.34924 0.0875509
\(721\) 0 0
\(722\) 31.3124 1.16533
\(723\) −3.96319 −0.147393
\(724\) 14.7228 0.547168
\(725\) 4.59933 0.170815
\(726\) −3.50923 −0.130240
\(727\) −1.06104 −0.0393517 −0.0196758 0.999806i \(-0.506263\pi\)
−0.0196758 + 0.999806i \(0.506263\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 30.2067 1.11800
\(731\) −11.8237 −0.437314
\(732\) −4.82757 −0.178432
\(733\) 41.2144 1.52229 0.761144 0.648583i \(-0.224638\pi\)
0.761144 + 0.648583i \(0.224638\pi\)
\(734\) 33.7928 1.24732
\(735\) 0 0
\(736\) −6.66412 −0.245643
\(737\) 42.1997 1.55444
\(738\) 18.9292 0.696793
\(739\) 2.39486 0.0880964 0.0440482 0.999029i \(-0.485974\pi\)
0.0440482 + 0.999029i \(0.485974\pi\)
\(740\) 22.9977 0.845411
\(741\) −6.89962 −0.253464
\(742\) 0 0
\(743\) 30.3260 1.11255 0.556277 0.830997i \(-0.312229\pi\)
0.556277 + 0.830997i \(0.312229\pi\)
\(744\) −7.87879 −0.288850
\(745\) 25.5242 0.935134
\(746\) 61.0195 2.23408
\(747\) 2.34134 0.0856651
\(748\) 36.3082 1.32756
\(749\) 0 0
\(750\) 21.6914 0.792059
\(751\) −36.2501 −1.32278 −0.661392 0.750041i \(-0.730034\pi\)
−0.661392 + 0.750041i \(0.730034\pi\)
\(752\) −9.18915 −0.335094
\(753\) −5.68753 −0.207265
\(754\) 29.7723 1.08424
\(755\) −26.7386 −0.973117
\(756\) 0 0
\(757\) 18.3390 0.666543 0.333272 0.942831i \(-0.391847\pi\)
0.333272 + 0.942831i \(0.391847\pi\)
\(758\) 64.0428 2.32614
\(759\) 3.07123 0.111478
\(760\) −12.4862 −0.452922
\(761\) 10.9230 0.395958 0.197979 0.980206i \(-0.436562\pi\)
0.197979 + 0.980206i \(0.436562\pi\)
\(762\) −21.7047 −0.786277
\(763\) 0 0
\(764\) −40.2512 −1.45624
\(765\) −9.66834 −0.349560
\(766\) 41.1684 1.48748
\(767\) 23.4298 0.846001
\(768\) 9.34505 0.337210
\(769\) 38.2640 1.37983 0.689917 0.723888i \(-0.257647\pi\)
0.689917 + 0.723888i \(0.257647\pi\)
\(770\) 0 0
\(771\) −12.2060 −0.439587
\(772\) −33.1912 −1.19458
\(773\) 48.7903 1.75486 0.877432 0.479701i \(-0.159255\pi\)
0.877432 + 0.479701i \(0.159255\pi\)
\(774\) 6.74257 0.242357
\(775\) −3.70820 −0.133202
\(776\) 13.2828 0.476825
\(777\) 0 0
\(778\) 73.6476 2.64039
\(779\) 18.9315 0.678291
\(780\) −22.8567 −0.818401
\(781\) 32.9826 1.18021
\(782\) 8.78808 0.314261
\(783\) −4.31559 −0.154227
\(784\) 0 0
\(785\) −6.21218 −0.221722
\(786\) −40.9522 −1.46072
\(787\) −11.3998 −0.406358 −0.203179 0.979142i \(-0.565127\pi\)
−0.203179 + 0.979142i \(0.565127\pi\)
\(788\) 50.0922 1.78446
\(789\) 10.6450 0.378971
\(790\) −53.6285 −1.90802
\(791\) 0 0
\(792\) −6.95444 −0.247115
\(793\) −4.94007 −0.175427
\(794\) 38.9109 1.38090
\(795\) 8.19426 0.290620
\(796\) 44.4191 1.57439
\(797\) 38.3621 1.35886 0.679428 0.733742i \(-0.262228\pi\)
0.679428 + 0.733742i \(0.262228\pi\)
\(798\) 0 0
\(799\) 37.8182 1.33791
\(800\) 7.10227 0.251103
\(801\) −5.01990 −0.177369
\(802\) 52.1264 1.84065
\(803\) 16.8263 0.593787
\(804\) −41.3790 −1.45933
\(805\) 0 0
\(806\) −24.0038 −0.845499
\(807\) 24.8992 0.876494
\(808\) 2.19395 0.0771828
\(809\) 5.33756 0.187659 0.0938293 0.995588i \(-0.470089\pi\)
0.0938293 + 0.995588i \(0.470089\pi\)
\(810\) 5.51348 0.193724
\(811\) −15.5586 −0.546335 −0.273168 0.961966i \(-0.588071\pi\)
−0.273168 + 0.961966i \(0.588071\pi\)
\(812\) 0 0
\(813\) −6.57668 −0.230654
\(814\) 21.3184 0.747209
\(815\) 35.6531 1.24887
\(816\) 3.74451 0.131084
\(817\) 6.74340 0.235922
\(818\) 1.59442 0.0557476
\(819\) 0 0
\(820\) 62.7153 2.19011
\(821\) 46.1313 1.60999 0.804997 0.593279i \(-0.202167\pi\)
0.804997 + 0.593279i \(0.202167\pi\)
\(822\) 10.5096 0.366564
\(823\) −0.0521219 −0.00181685 −0.000908427 1.00000i \(-0.500289\pi\)
−0.000908427 1.00000i \(0.500289\pi\)
\(824\) −14.4960 −0.504991
\(825\) −3.27315 −0.113956
\(826\) 0 0
\(827\) −46.4334 −1.61465 −0.807324 0.590109i \(-0.799085\pi\)
−0.807324 + 0.590109i \(0.799085\pi\)
\(828\) −3.01150 −0.104657
\(829\) 25.8592 0.898127 0.449063 0.893500i \(-0.351758\pi\)
0.449063 + 0.893500i \(0.351758\pi\)
\(830\) 12.9089 0.448076
\(831\) 23.2474 0.806443
\(832\) 40.0953 1.39005
\(833\) 0 0
\(834\) 21.7069 0.751650
\(835\) 38.5311 1.33342
\(836\) −20.7077 −0.716191
\(837\) 3.47943 0.120267
\(838\) −12.9374 −0.446917
\(839\) 41.4474 1.43092 0.715462 0.698652i \(-0.246216\pi\)
0.715462 + 0.698652i \(0.246216\pi\)
\(840\) 0 0
\(841\) −10.3757 −0.357782
\(842\) 33.2408 1.14555
\(843\) −10.7437 −0.370031
\(844\) 22.2296 0.765175
\(845\) 8.62801 0.296812
\(846\) −21.5662 −0.741463
\(847\) 0 0
\(848\) −3.17360 −0.108982
\(849\) 24.5170 0.841423
\(850\) −9.36587 −0.321247
\(851\) 3.10069 0.106290
\(852\) −32.3412 −1.10799
\(853\) 17.6926 0.605782 0.302891 0.953025i \(-0.402048\pi\)
0.302891 + 0.953025i \(0.402048\pi\)
\(854\) 0 0
\(855\) 5.51416 0.188580
\(856\) −36.4925 −1.24729
\(857\) 32.5992 1.11357 0.556784 0.830657i \(-0.312035\pi\)
0.556784 + 0.830657i \(0.312035\pi\)
\(858\) −21.1877 −0.723336
\(859\) 2.57341 0.0878036 0.0439018 0.999036i \(-0.486021\pi\)
0.0439018 + 0.999036i \(0.486021\pi\)
\(860\) 22.3392 0.761759
\(861\) 0 0
\(862\) −9.94242 −0.338640
\(863\) 6.05407 0.206083 0.103041 0.994677i \(-0.467143\pi\)
0.103041 + 0.994677i \(0.467143\pi\)
\(864\) −6.66412 −0.226718
\(865\) −54.1605 −1.84151
\(866\) −18.0876 −0.614641
\(867\) 1.58939 0.0539786
\(868\) 0 0
\(869\) −29.8732 −1.01338
\(870\) −23.7939 −0.806690
\(871\) −42.3434 −1.43475
\(872\) −14.6173 −0.495006
\(873\) −5.86596 −0.198533
\(874\) −5.01212 −0.169537
\(875\) 0 0
\(876\) −16.4991 −0.557453
\(877\) −10.5887 −0.357556 −0.178778 0.983889i \(-0.557214\pi\)
−0.178778 + 0.983889i \(0.557214\pi\)
\(878\) 25.9317 0.875153
\(879\) 33.2732 1.12228
\(880\) 7.21503 0.243219
\(881\) −37.2400 −1.25465 −0.627323 0.778759i \(-0.715849\pi\)
−0.627323 + 0.778759i \(0.715849\pi\)
\(882\) 0 0
\(883\) −46.3648 −1.56030 −0.780150 0.625592i \(-0.784857\pi\)
−0.780150 + 0.625592i \(0.784857\pi\)
\(884\) −36.4318 −1.22534
\(885\) −18.7250 −0.629435
\(886\) 81.9639 2.75363
\(887\) −23.4809 −0.788410 −0.394205 0.919023i \(-0.628980\pi\)
−0.394205 + 0.919023i \(0.628980\pi\)
\(888\) −7.02117 −0.235615
\(889\) 0 0
\(890\) −27.6771 −0.927739
\(891\) 3.07123 0.102890
\(892\) 77.2486 2.58647
\(893\) −21.5689 −0.721775
\(894\) −23.2003 −0.775934
\(895\) 46.1192 1.54160
\(896\) 0 0
\(897\) −3.08168 −0.102894
\(898\) −19.5291 −0.651693
\(899\) −15.0158 −0.500805
\(900\) 3.20950 0.106983
\(901\) 13.0610 0.435126
\(902\) 58.1358 1.93571
\(903\) 0 0
\(904\) −19.2532 −0.640352
\(905\) −12.0406 −0.400244
\(906\) 24.3042 0.807451
\(907\) −13.4392 −0.446241 −0.223120 0.974791i \(-0.571624\pi\)
−0.223120 + 0.974791i \(0.571624\pi\)
\(908\) −28.1569 −0.934420
\(909\) −0.968892 −0.0321361
\(910\) 0 0
\(911\) −8.86837 −0.293822 −0.146911 0.989150i \(-0.546933\pi\)
−0.146911 + 0.989150i \(0.546933\pi\)
\(912\) −2.13561 −0.0707171
\(913\) 7.19078 0.237980
\(914\) −50.5611 −1.67241
\(915\) 3.94809 0.130520
\(916\) −54.4699 −1.79974
\(917\) 0 0
\(918\) 8.78808 0.290050
\(919\) −14.9225 −0.492249 −0.246124 0.969238i \(-0.579157\pi\)
−0.246124 + 0.969238i \(0.579157\pi\)
\(920\) −5.57690 −0.183865
\(921\) 20.2345 0.666750
\(922\) 65.2481 2.14883
\(923\) −33.0949 −1.08933
\(924\) 0 0
\(925\) −3.30455 −0.108653
\(926\) 0.0343566 0.00112903
\(927\) 6.40172 0.210260
\(928\) 28.7596 0.944081
\(929\) 50.2999 1.65029 0.825143 0.564924i \(-0.191094\pi\)
0.825143 + 0.564924i \(0.191094\pi\)
\(930\) 19.1838 0.629062
\(931\) 0 0
\(932\) −53.7982 −1.76222
\(933\) 22.8603 0.748414
\(934\) 36.5315 1.19535
\(935\) −29.6937 −0.971086
\(936\) 6.97812 0.228087
\(937\) −19.5619 −0.639060 −0.319530 0.947576i \(-0.603525\pi\)
−0.319530 + 0.947576i \(0.603525\pi\)
\(938\) 0 0
\(939\) 23.6714 0.772486
\(940\) −71.4523 −2.33052
\(941\) 10.1938 0.332307 0.166154 0.986100i \(-0.446865\pi\)
0.166154 + 0.986100i \(0.446865\pi\)
\(942\) 5.64659 0.183976
\(943\) 8.45567 0.275354
\(944\) 7.25212 0.236036
\(945\) 0 0
\(946\) 20.7080 0.673274
\(947\) −0.762412 −0.0247751 −0.0123875 0.999923i \(-0.503943\pi\)
−0.0123875 + 0.999923i \(0.503943\pi\)
\(948\) 29.2923 0.951369
\(949\) −16.8836 −0.548065
\(950\) 5.34165 0.173306
\(951\) −12.4283 −0.403014
\(952\) 0 0
\(953\) 5.06240 0.163987 0.0819936 0.996633i \(-0.473871\pi\)
0.0819936 + 0.996633i \(0.473871\pi\)
\(954\) −7.44820 −0.241145
\(955\) 32.9184 1.06521
\(956\) 46.2063 1.49442
\(957\) −13.2542 −0.428446
\(958\) 42.9883 1.38889
\(959\) 0 0
\(960\) −32.0441 −1.03422
\(961\) −18.8935 −0.609469
\(962\) −21.3910 −0.689673
\(963\) 16.1158 0.519325
\(964\) 11.9352 0.384406
\(965\) 27.1445 0.873813
\(966\) 0 0
\(967\) −21.7637 −0.699872 −0.349936 0.936774i \(-0.613797\pi\)
−0.349936 + 0.936774i \(0.613797\pi\)
\(968\) 3.54960 0.114088
\(969\) 8.78915 0.282348
\(970\) −32.3419 −1.03844
\(971\) −60.0563 −1.92730 −0.963649 0.267173i \(-0.913911\pi\)
−0.963649 + 0.267173i \(0.913911\pi\)
\(972\) −3.01150 −0.0965940
\(973\) 0 0
\(974\) −86.1875 −2.76163
\(975\) 3.28429 0.105182
\(976\) −1.52908 −0.0489446
\(977\) 22.6281 0.723936 0.361968 0.932190i \(-0.382105\pi\)
0.361968 + 0.932190i \(0.382105\pi\)
\(978\) −32.4070 −1.03626
\(979\) −15.4172 −0.492737
\(980\) 0 0
\(981\) 6.45532 0.206103
\(982\) −13.4107 −0.427954
\(983\) 3.29349 0.105046 0.0525230 0.998620i \(-0.483274\pi\)
0.0525230 + 0.998620i \(0.483274\pi\)
\(984\) −19.1469 −0.610381
\(985\) −40.9665 −1.30530
\(986\) −37.9257 −1.20780
\(987\) 0 0
\(988\) 20.7782 0.661043
\(989\) 3.01191 0.0957731
\(990\) 16.9332 0.538171
\(991\) −43.6326 −1.38604 −0.693018 0.720920i \(-0.743719\pi\)
−0.693018 + 0.720920i \(0.743719\pi\)
\(992\) −23.1874 −0.736200
\(993\) 7.30561 0.231837
\(994\) 0 0
\(995\) −36.3270 −1.15164
\(996\) −7.05094 −0.223418
\(997\) −0.571973 −0.0181146 −0.00905728 0.999959i \(-0.502883\pi\)
−0.00905728 + 0.999959i \(0.502883\pi\)
\(998\) 27.5692 0.872687
\(999\) 3.10069 0.0981016
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 3381.2.a.bk.1.1 10
7.6 odd 2 3381.2.a.bl.1.1 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3381.2.a.bk.1.1 10 1.1 even 1 trivial
3381.2.a.bl.1.1 yes 10 7.6 odd 2