Properties

Label 735.2.a.n.1.4
Level $735$
Weight $2$
Character 735.1
Self dual yes
Analytic conductor $5.869$
Analytic rank $0$
Dimension $4$
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] = [735,2,Mod(1,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, 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("735.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.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: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.334904\) of defining polynomial
Character \(\chi\) \(=\) 735.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.74912 q^{2} -1.00000 q^{3} +5.55765 q^{4} +1.00000 q^{5} -2.74912 q^{6} +9.78039 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.74912 q^{2} -1.00000 q^{3} +5.55765 q^{4} +1.00000 q^{5} -2.74912 q^{6} +9.78039 q^{8} +1.00000 q^{9} +2.74912 q^{10} -4.44549 q^{11} -5.55765 q^{12} +3.61706 q^{13} -1.00000 q^{15} +15.7721 q^{16} -4.94725 q^{17} +2.74912 q^{18} +2.74441 q^{19} +5.55765 q^{20} -12.2212 q^{22} -4.36147 q^{23} -9.78039 q^{24} +1.00000 q^{25} +9.94372 q^{26} -1.00000 q^{27} +0.660384 q^{29} -2.74912 q^{30} +1.25559 q^{31} +23.7987 q^{32} +4.44549 q^{33} -13.6006 q^{34} +5.55765 q^{36} -2.16804 q^{37} +7.54469 q^{38} -3.61706 q^{39} +9.78039 q^{40} -5.77568 q^{41} -9.11529 q^{43} -24.7064 q^{44} +1.00000 q^{45} -11.9902 q^{46} +6.48528 q^{47} -15.7721 q^{48} +2.74912 q^{50} +4.94725 q^{51} +20.1023 q^{52} +7.03127 q^{53} -2.74912 q^{54} -4.44549 q^{55} -2.74441 q^{57} +1.81547 q^{58} -13.8249 q^{59} -5.55765 q^{60} +9.41421 q^{61} +3.45178 q^{62} +33.8812 q^{64} +3.61706 q^{65} +12.2212 q^{66} +3.06608 q^{67} -27.4951 q^{68} +4.36147 q^{69} +0.277444 q^{71} +9.78039 q^{72} -10.6135 q^{73} -5.96021 q^{74} -1.00000 q^{75} +15.2524 q^{76} -9.94372 q^{78} -5.53803 q^{79} +15.7721 q^{80} +1.00000 q^{81} -15.8780 q^{82} -7.17157 q^{83} -4.94725 q^{85} -25.0590 q^{86} -0.660384 q^{87} -43.4786 q^{88} -0.828427 q^{89} +2.74912 q^{90} -24.2395 q^{92} -1.25559 q^{93} +17.8288 q^{94} +2.74441 q^{95} -23.7987 q^{96} +6.20784 q^{97} -4.44549 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{3} + 8 q^{4} + 4 q^{5} - 4 q^{6} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 4 q^{3} + 8 q^{4} + 4 q^{5} - 4 q^{6} + 12 q^{8} + 4 q^{9} + 4 q^{10} + 8 q^{11} - 8 q^{12} - 4 q^{15} + 12 q^{16} - 8 q^{17} + 4 q^{18} + 8 q^{19} + 8 q^{20} - 12 q^{24} + 4 q^{25} - 4 q^{27} + 8 q^{29} - 4 q^{30} + 8 q^{31} + 28 q^{32} - 8 q^{33} + 8 q^{34} + 8 q^{36} + 8 q^{37} - 4 q^{38} + 12 q^{40} - 8 q^{43} - 16 q^{44} + 4 q^{45} + 12 q^{46} - 8 q^{47} - 12 q^{48} + 4 q^{50} + 8 q^{51} + 32 q^{52} + 8 q^{53} - 4 q^{54} + 8 q^{55} - 8 q^{57} - 24 q^{58} - 16 q^{59} - 8 q^{60} + 32 q^{61} + 20 q^{62} + 24 q^{64} - 24 q^{68} - 8 q^{71} + 12 q^{72} - 32 q^{74} - 4 q^{75} - 8 q^{76} + 12 q^{80} + 4 q^{81} + 8 q^{82} - 40 q^{83} - 8 q^{85} - 32 q^{86} - 8 q^{87} - 40 q^{88} + 8 q^{89} + 4 q^{90} - 8 q^{92} - 8 q^{93} + 16 q^{94} + 8 q^{95} - 28 q^{96} + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.74912 1.94392 0.971960 0.235147i \(-0.0755571\pi\)
0.971960 + 0.235147i \(0.0755571\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.55765 2.77882
\(5\) 1.00000 0.447214
\(6\) −2.74912 −1.12232
\(7\) 0 0
\(8\) 9.78039 3.45789
\(9\) 1.00000 0.333333
\(10\) 2.74912 0.869347
\(11\) −4.44549 −1.34036 −0.670182 0.742197i \(-0.733784\pi\)
−0.670182 + 0.742197i \(0.733784\pi\)
\(12\) −5.55765 −1.60435
\(13\) 3.61706 1.00319 0.501596 0.865102i \(-0.332746\pi\)
0.501596 + 0.865102i \(0.332746\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 15.7721 3.94304
\(17\) −4.94725 −1.19988 −0.599942 0.800043i \(-0.704810\pi\)
−0.599942 + 0.800043i \(0.704810\pi\)
\(18\) 2.74912 0.647973
\(19\) 2.74441 0.629610 0.314805 0.949156i \(-0.398061\pi\)
0.314805 + 0.949156i \(0.398061\pi\)
\(20\) 5.55765 1.24273
\(21\) 0 0
\(22\) −12.2212 −2.60556
\(23\) −4.36147 −0.909428 −0.454714 0.890637i \(-0.650259\pi\)
−0.454714 + 0.890637i \(0.650259\pi\)
\(24\) −9.78039 −1.99641
\(25\) 1.00000 0.200000
\(26\) 9.94372 1.95012
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0.660384 0.122630 0.0613151 0.998118i \(-0.480471\pi\)
0.0613151 + 0.998118i \(0.480471\pi\)
\(30\) −2.74912 −0.501918
\(31\) 1.25559 0.225511 0.112756 0.993623i \(-0.464032\pi\)
0.112756 + 0.993623i \(0.464032\pi\)
\(32\) 23.7987 4.20706
\(33\) 4.44549 0.773860
\(34\) −13.6006 −2.33248
\(35\) 0 0
\(36\) 5.55765 0.926275
\(37\) −2.16804 −0.356424 −0.178212 0.983992i \(-0.557031\pi\)
−0.178212 + 0.983992i \(0.557031\pi\)
\(38\) 7.54469 1.22391
\(39\) −3.61706 −0.579193
\(40\) 9.78039 1.54642
\(41\) −5.77568 −0.902009 −0.451005 0.892522i \(-0.648934\pi\)
−0.451005 + 0.892522i \(0.648934\pi\)
\(42\) 0 0
\(43\) −9.11529 −1.39007 −0.695035 0.718976i \(-0.744611\pi\)
−0.695035 + 0.718976i \(0.744611\pi\)
\(44\) −24.7064 −3.72464
\(45\) 1.00000 0.149071
\(46\) −11.9902 −1.76786
\(47\) 6.48528 0.945976 0.472988 0.881069i \(-0.343175\pi\)
0.472988 + 0.881069i \(0.343175\pi\)
\(48\) −15.7721 −2.27651
\(49\) 0 0
\(50\) 2.74912 0.388784
\(51\) 4.94725 0.692754
\(52\) 20.1023 2.78769
\(53\) 7.03127 0.965820 0.482910 0.875670i \(-0.339580\pi\)
0.482910 + 0.875670i \(0.339580\pi\)
\(54\) −2.74912 −0.374108
\(55\) −4.44549 −0.599429
\(56\) 0 0
\(57\) −2.74441 −0.363505
\(58\) 1.81547 0.238383
\(59\) −13.8249 −1.79985 −0.899924 0.436046i \(-0.856378\pi\)
−0.899924 + 0.436046i \(0.856378\pi\)
\(60\) −5.55765 −0.717489
\(61\) 9.41421 1.20537 0.602683 0.797981i \(-0.294098\pi\)
0.602683 + 0.797981i \(0.294098\pi\)
\(62\) 3.45178 0.438376
\(63\) 0 0
\(64\) 33.8812 4.23514
\(65\) 3.61706 0.448641
\(66\) 12.2212 1.50432
\(67\) 3.06608 0.374581 0.187290 0.982305i \(-0.440029\pi\)
0.187290 + 0.982305i \(0.440029\pi\)
\(68\) −27.4951 −3.33427
\(69\) 4.36147 0.525059
\(70\) 0 0
\(71\) 0.277444 0.0329265 0.0164632 0.999864i \(-0.494759\pi\)
0.0164632 + 0.999864i \(0.494759\pi\)
\(72\) 9.78039 1.15263
\(73\) −10.6135 −1.24222 −0.621110 0.783724i \(-0.713318\pi\)
−0.621110 + 0.783724i \(0.713318\pi\)
\(74\) −5.96021 −0.692860
\(75\) −1.00000 −0.115470
\(76\) 15.2524 1.74957
\(77\) 0 0
\(78\) −9.94372 −1.12590
\(79\) −5.53803 −0.623077 −0.311539 0.950234i \(-0.600844\pi\)
−0.311539 + 0.950234i \(0.600844\pi\)
\(80\) 15.7721 1.76338
\(81\) 1.00000 0.111111
\(82\) −15.8780 −1.75343
\(83\) −7.17157 −0.787182 −0.393591 0.919286i \(-0.628767\pi\)
−0.393591 + 0.919286i \(0.628767\pi\)
\(84\) 0 0
\(85\) −4.94725 −0.536605
\(86\) −25.0590 −2.70218
\(87\) −0.660384 −0.0708006
\(88\) −43.4786 −4.63483
\(89\) −0.828427 −0.0878131 −0.0439065 0.999036i \(-0.513980\pi\)
−0.0439065 + 0.999036i \(0.513980\pi\)
\(90\) 2.74912 0.289782
\(91\) 0 0
\(92\) −24.2395 −2.52714
\(93\) −1.25559 −0.130199
\(94\) 17.8288 1.83890
\(95\) 2.74441 0.281570
\(96\) −23.7987 −2.42895
\(97\) 6.20784 0.630310 0.315155 0.949040i \(-0.397943\pi\)
0.315155 + 0.949040i \(0.397943\pi\)
\(98\) 0 0
\(99\) −4.44549 −0.446788
\(100\) 5.55765 0.555765
\(101\) 4.11882 0.409838 0.204919 0.978779i \(-0.434307\pi\)
0.204919 + 0.978779i \(0.434307\pi\)
\(102\) 13.6006 1.34666
\(103\) −8.16804 −0.804821 −0.402411 0.915459i \(-0.631828\pi\)
−0.402411 + 0.915459i \(0.631828\pi\)
\(104\) 35.3763 3.46893
\(105\) 0 0
\(106\) 19.3298 1.87748
\(107\) −8.52951 −0.824579 −0.412289 0.911053i \(-0.635271\pi\)
−0.412289 + 0.911053i \(0.635271\pi\)
\(108\) −5.55765 −0.534785
\(109\) −3.65685 −0.350263 −0.175132 0.984545i \(-0.556035\pi\)
−0.175132 + 0.984545i \(0.556035\pi\)
\(110\) −12.2212 −1.16524
\(111\) 2.16804 0.205782
\(112\) 0 0
\(113\) 12.6085 1.18611 0.593056 0.805161i \(-0.297921\pi\)
0.593056 + 0.805161i \(0.297921\pi\)
\(114\) −7.54469 −0.706625
\(115\) −4.36147 −0.406709
\(116\) 3.67018 0.340768
\(117\) 3.61706 0.334397
\(118\) −38.0063 −3.49876
\(119\) 0 0
\(120\) −9.78039 −0.892823
\(121\) 8.76235 0.796577
\(122\) 25.8808 2.34314
\(123\) 5.77568 0.520775
\(124\) 6.97815 0.626656
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −0.0562783 −0.00499389 −0.00249695 0.999997i \(-0.500795\pi\)
−0.00249695 + 0.999997i \(0.500795\pi\)
\(128\) 45.5459 4.02572
\(129\) 9.11529 0.802557
\(130\) 9.94372 0.872122
\(131\) 19.6631 1.71797 0.858987 0.511997i \(-0.171094\pi\)
0.858987 + 0.511997i \(0.171094\pi\)
\(132\) 24.7064 2.15042
\(133\) 0 0
\(134\) 8.42900 0.728155
\(135\) −1.00000 −0.0860663
\(136\) −48.3861 −4.14907
\(137\) −18.7507 −1.60198 −0.800989 0.598679i \(-0.795693\pi\)
−0.800989 + 0.598679i \(0.795693\pi\)
\(138\) 11.9902 1.02067
\(139\) −6.14657 −0.521345 −0.260673 0.965427i \(-0.583944\pi\)
−0.260673 + 0.965427i \(0.583944\pi\)
\(140\) 0 0
\(141\) −6.48528 −0.546159
\(142\) 0.762725 0.0640065
\(143\) −16.0796 −1.34464
\(144\) 15.7721 1.31435
\(145\) 0.660384 0.0548419
\(146\) −29.1778 −2.41478
\(147\) 0 0
\(148\) −12.0492 −0.990440
\(149\) 20.2306 1.65735 0.828677 0.559727i \(-0.189094\pi\)
0.828677 + 0.559727i \(0.189094\pi\)
\(150\) −2.74912 −0.224465
\(151\) 21.1082 1.71776 0.858882 0.512174i \(-0.171160\pi\)
0.858882 + 0.512174i \(0.171160\pi\)
\(152\) 26.8414 2.17712
\(153\) −4.94725 −0.399962
\(154\) 0 0
\(155\) 1.25559 0.100852
\(156\) −20.1023 −1.60948
\(157\) 1.96021 0.156441 0.0782207 0.996936i \(-0.475076\pi\)
0.0782207 + 0.996936i \(0.475076\pi\)
\(158\) −15.2247 −1.21121
\(159\) −7.03127 −0.557616
\(160\) 23.7987 1.88145
\(161\) 0 0
\(162\) 2.74912 0.215991
\(163\) −8.18137 −0.640814 −0.320407 0.947280i \(-0.603820\pi\)
−0.320407 + 0.947280i \(0.603820\pi\)
\(164\) −32.0992 −2.50653
\(165\) 4.44549 0.346081
\(166\) −19.7155 −1.53022
\(167\) −3.76235 −0.291139 −0.145570 0.989348i \(-0.546502\pi\)
−0.145570 + 0.989348i \(0.546502\pi\)
\(168\) 0 0
\(169\) 0.0831193 0.00639379
\(170\) −13.6006 −1.04312
\(171\) 2.74441 0.209870
\(172\) −50.6596 −3.86276
\(173\) −15.7061 −1.19411 −0.597055 0.802200i \(-0.703663\pi\)
−0.597055 + 0.802200i \(0.703663\pi\)
\(174\) −1.81547 −0.137631
\(175\) 0 0
\(176\) −70.1149 −5.28511
\(177\) 13.8249 1.03914
\(178\) −2.27744 −0.170702
\(179\) −1.28097 −0.0957444 −0.0478722 0.998853i \(-0.515244\pi\)
−0.0478722 + 0.998853i \(0.515244\pi\)
\(180\) 5.55765 0.414243
\(181\) 16.5599 1.23089 0.615443 0.788181i \(-0.288977\pi\)
0.615443 + 0.788181i \(0.288977\pi\)
\(182\) 0 0
\(183\) −9.41421 −0.695919
\(184\) −42.6568 −3.14470
\(185\) −2.16804 −0.159398
\(186\) −3.45178 −0.253097
\(187\) 21.9929 1.60828
\(188\) 36.0429 2.62870
\(189\) 0 0
\(190\) 7.54469 0.547350
\(191\) 8.76272 0.634049 0.317024 0.948417i \(-0.397316\pi\)
0.317024 + 0.948417i \(0.397316\pi\)
\(192\) −33.8812 −2.44516
\(193\) 26.4986 1.90741 0.953706 0.300741i \(-0.0972340\pi\)
0.953706 + 0.300741i \(0.0972340\pi\)
\(194\) 17.0661 1.22527
\(195\) −3.61706 −0.259023
\(196\) 0 0
\(197\) 4.11439 0.293138 0.146569 0.989200i \(-0.453177\pi\)
0.146569 + 0.989200i \(0.453177\pi\)
\(198\) −12.2212 −0.868520
\(199\) −7.63538 −0.541258 −0.270629 0.962684i \(-0.587232\pi\)
−0.270629 + 0.962684i \(0.587232\pi\)
\(200\) 9.78039 0.691578
\(201\) −3.06608 −0.216264
\(202\) 11.3231 0.796693
\(203\) 0 0
\(204\) 27.4951 1.92504
\(205\) −5.77568 −0.403391
\(206\) −22.4549 −1.56451
\(207\) −4.36147 −0.303143
\(208\) 57.0488 3.95562
\(209\) −12.2002 −0.843907
\(210\) 0 0
\(211\) −0.317238 −0.0218396 −0.0109198 0.999940i \(-0.503476\pi\)
−0.0109198 + 0.999940i \(0.503476\pi\)
\(212\) 39.0773 2.68384
\(213\) −0.277444 −0.0190101
\(214\) −23.4486 −1.60291
\(215\) −9.11529 −0.621658
\(216\) −9.78039 −0.665471
\(217\) 0 0
\(218\) −10.0531 −0.680883
\(219\) 10.6135 0.717196
\(220\) −24.7064 −1.66571
\(221\) −17.8945 −1.20371
\(222\) 5.96021 0.400023
\(223\) 29.0590 1.94594 0.972968 0.230941i \(-0.0741803\pi\)
0.972968 + 0.230941i \(0.0741803\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 34.6624 2.30571
\(227\) −20.8910 −1.38658 −0.693291 0.720657i \(-0.743840\pi\)
−0.693291 + 0.720657i \(0.743840\pi\)
\(228\) −15.2524 −1.01012
\(229\) 4.92893 0.325713 0.162857 0.986650i \(-0.447929\pi\)
0.162857 + 0.986650i \(0.447929\pi\)
\(230\) −11.9902 −0.790609
\(231\) 0 0
\(232\) 6.45881 0.424042
\(233\) −1.09382 −0.0716585 −0.0358292 0.999358i \(-0.511407\pi\)
−0.0358292 + 0.999358i \(0.511407\pi\)
\(234\) 9.94372 0.650041
\(235\) 6.48528 0.423053
\(236\) −76.8339 −5.00146
\(237\) 5.53803 0.359734
\(238\) 0 0
\(239\) 25.6725 1.66062 0.830309 0.557303i \(-0.188164\pi\)
0.830309 + 0.557303i \(0.188164\pi\)
\(240\) −15.7721 −1.01809
\(241\) 24.7468 1.59408 0.797040 0.603927i \(-0.206398\pi\)
0.797040 + 0.603927i \(0.206398\pi\)
\(242\) 24.0887 1.54848
\(243\) −1.00000 −0.0641500
\(244\) 52.3209 3.34950
\(245\) 0 0
\(246\) 15.8780 1.01235
\(247\) 9.92668 0.631619
\(248\) 12.2802 0.779794
\(249\) 7.17157 0.454480
\(250\) 2.74912 0.173869
\(251\) 0.405692 0.0256070 0.0128035 0.999918i \(-0.495924\pi\)
0.0128035 + 0.999918i \(0.495924\pi\)
\(252\) 0 0
\(253\) 19.3888 1.21897
\(254\) −0.154716 −0.00970772
\(255\) 4.94725 0.309809
\(256\) 57.4486 3.59054
\(257\) −10.8910 −0.679360 −0.339680 0.940541i \(-0.610319\pi\)
−0.339680 + 0.940541i \(0.610319\pi\)
\(258\) 25.0590 1.56011
\(259\) 0 0
\(260\) 20.1023 1.24669
\(261\) 0.660384 0.0408767
\(262\) 54.0562 3.33961
\(263\) 8.76716 0.540606 0.270303 0.962775i \(-0.412876\pi\)
0.270303 + 0.962775i \(0.412876\pi\)
\(264\) 43.4786 2.67592
\(265\) 7.03127 0.431928
\(266\) 0 0
\(267\) 0.828427 0.0506989
\(268\) 17.0402 1.04089
\(269\) 22.8347 1.39226 0.696128 0.717918i \(-0.254905\pi\)
0.696128 + 0.717918i \(0.254905\pi\)
\(270\) −2.74912 −0.167306
\(271\) 10.6748 0.648448 0.324224 0.945980i \(-0.394897\pi\)
0.324224 + 0.945980i \(0.394897\pi\)
\(272\) −78.0288 −4.73119
\(273\) 0 0
\(274\) −51.5478 −3.11412
\(275\) −4.44549 −0.268073
\(276\) 24.2395 1.45905
\(277\) −12.4361 −0.747211 −0.373605 0.927588i \(-0.621879\pi\)
−0.373605 + 0.927588i \(0.621879\pi\)
\(278\) −16.8976 −1.01345
\(279\) 1.25559 0.0751705
\(280\) 0 0
\(281\) 12.9169 0.770556 0.385278 0.922800i \(-0.374105\pi\)
0.385278 + 0.922800i \(0.374105\pi\)
\(282\) −17.8288 −1.06169
\(283\) 25.6139 1.52259 0.761294 0.648407i \(-0.224564\pi\)
0.761294 + 0.648407i \(0.224564\pi\)
\(284\) 1.54193 0.0914969
\(285\) −2.74441 −0.162565
\(286\) −44.2047 −2.61388
\(287\) 0 0
\(288\) 23.7987 1.40235
\(289\) 7.47530 0.439723
\(290\) 1.81547 0.106608
\(291\) −6.20784 −0.363910
\(292\) −58.9863 −3.45191
\(293\) −1.68276 −0.0983080 −0.0491540 0.998791i \(-0.515653\pi\)
−0.0491540 + 0.998791i \(0.515653\pi\)
\(294\) 0 0
\(295\) −13.8249 −0.804917
\(296\) −21.2043 −1.23248
\(297\) 4.44549 0.257953
\(298\) 55.6163 3.22176
\(299\) −15.7757 −0.912331
\(300\) −5.55765 −0.320871
\(301\) 0 0
\(302\) 58.0290 3.33919
\(303\) −4.11882 −0.236620
\(304\) 43.2852 2.48257
\(305\) 9.41421 0.539056
\(306\) −13.6006 −0.777493
\(307\) −7.31371 −0.417415 −0.208708 0.977978i \(-0.566926\pi\)
−0.208708 + 0.977978i \(0.566926\pi\)
\(308\) 0 0
\(309\) 8.16804 0.464664
\(310\) 3.45178 0.196048
\(311\) 11.2019 0.635204 0.317602 0.948224i \(-0.397122\pi\)
0.317602 + 0.948224i \(0.397122\pi\)
\(312\) −35.3763 −2.00279
\(313\) −31.4857 −1.77967 −0.889837 0.456278i \(-0.849182\pi\)
−0.889837 + 0.456278i \(0.849182\pi\)
\(314\) 5.38883 0.304110
\(315\) 0 0
\(316\) −30.7784 −1.73142
\(317\) 8.45048 0.474626 0.237313 0.971433i \(-0.423733\pi\)
0.237313 + 0.971433i \(0.423733\pi\)
\(318\) −19.3298 −1.08396
\(319\) −2.93573 −0.164369
\(320\) 33.8812 1.89401
\(321\) 8.52951 0.476071
\(322\) 0 0
\(323\) −13.5773 −0.755459
\(324\) 5.55765 0.308758
\(325\) 3.61706 0.200638
\(326\) −22.4915 −1.24569
\(327\) 3.65685 0.202225
\(328\) −56.4884 −3.11905
\(329\) 0 0
\(330\) 12.2212 0.672753
\(331\) 16.6533 0.915349 0.457675 0.889120i \(-0.348682\pi\)
0.457675 + 0.889120i \(0.348682\pi\)
\(332\) −39.8571 −2.18744
\(333\) −2.16804 −0.118808
\(334\) −10.3431 −0.565952
\(335\) 3.06608 0.167518
\(336\) 0 0
\(337\) 31.9178 1.73867 0.869337 0.494220i \(-0.164546\pi\)
0.869337 + 0.494220i \(0.164546\pi\)
\(338\) 0.228505 0.0124290
\(339\) −12.6085 −0.684802
\(340\) −27.4951 −1.49113
\(341\) −5.58173 −0.302268
\(342\) 7.54469 0.407970
\(343\) 0 0
\(344\) −89.1511 −4.80671
\(345\) 4.36147 0.234813
\(346\) −43.1778 −2.32126
\(347\) −7.91282 −0.424783 −0.212391 0.977185i \(-0.568125\pi\)
−0.212391 + 0.977185i \(0.568125\pi\)
\(348\) −3.67018 −0.196742
\(349\) 25.1525 1.34638 0.673190 0.739469i \(-0.264924\pi\)
0.673190 + 0.739469i \(0.264924\pi\)
\(350\) 0 0
\(351\) −3.61706 −0.193064
\(352\) −105.797 −5.63899
\(353\) −20.5667 −1.09465 −0.547327 0.836919i \(-0.684355\pi\)
−0.547327 + 0.836919i \(0.684355\pi\)
\(354\) 38.0063 2.02001
\(355\) 0.277444 0.0147252
\(356\) −4.60411 −0.244017
\(357\) 0 0
\(358\) −3.52155 −0.186120
\(359\) −14.1023 −0.744293 −0.372147 0.928174i \(-0.621378\pi\)
−0.372147 + 0.928174i \(0.621378\pi\)
\(360\) 9.78039 0.515472
\(361\) −11.4682 −0.603591
\(362\) 45.5251 2.39274
\(363\) −8.76235 −0.459904
\(364\) 0 0
\(365\) −10.6135 −0.555538
\(366\) −25.8808 −1.35281
\(367\) 0.386844 0.0201931 0.0100965 0.999949i \(-0.496786\pi\)
0.0100965 + 0.999949i \(0.496786\pi\)
\(368\) −68.7897 −3.58591
\(369\) −5.77568 −0.300670
\(370\) −5.96021 −0.309856
\(371\) 0 0
\(372\) −6.97815 −0.361800
\(373\) −10.6345 −0.550632 −0.275316 0.961354i \(-0.588783\pi\)
−0.275316 + 0.961354i \(0.588783\pi\)
\(374\) 60.4612 3.12637
\(375\) −1.00000 −0.0516398
\(376\) 63.4286 3.27108
\(377\) 2.38865 0.123022
\(378\) 0 0
\(379\) −26.4290 −1.35757 −0.678783 0.734339i \(-0.737492\pi\)
−0.678783 + 0.734339i \(0.737492\pi\)
\(380\) 15.2524 0.782434
\(381\) 0.0562783 0.00288322
\(382\) 24.0898 1.23254
\(383\) 5.14567 0.262931 0.131466 0.991321i \(-0.458032\pi\)
0.131466 + 0.991321i \(0.458032\pi\)
\(384\) −45.5459 −2.32425
\(385\) 0 0
\(386\) 72.8478 3.70785
\(387\) −9.11529 −0.463356
\(388\) 34.5010 1.75152
\(389\) −12.9080 −0.654462 −0.327231 0.944944i \(-0.606116\pi\)
−0.327231 + 0.944944i \(0.606116\pi\)
\(390\) −9.94372 −0.503520
\(391\) 21.5773 1.09121
\(392\) 0 0
\(393\) −19.6631 −0.991873
\(394\) 11.3109 0.569837
\(395\) −5.53803 −0.278649
\(396\) −24.7064 −1.24155
\(397\) −22.7627 −1.14243 −0.571214 0.820801i \(-0.693527\pi\)
−0.571214 + 0.820801i \(0.693527\pi\)
\(398\) −20.9906 −1.05216
\(399\) 0 0
\(400\) 15.7721 0.788607
\(401\) −9.85080 −0.491926 −0.245963 0.969279i \(-0.579104\pi\)
−0.245963 + 0.969279i \(0.579104\pi\)
\(402\) −8.42900 −0.420400
\(403\) 4.54156 0.226231
\(404\) 22.8910 1.13887
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 9.63801 0.477738
\(408\) 48.3861 2.39547
\(409\) −14.6742 −0.725594 −0.362797 0.931868i \(-0.618178\pi\)
−0.362797 + 0.931868i \(0.618178\pi\)
\(410\) −15.8780 −0.784159
\(411\) 18.7507 0.924903
\(412\) −45.3951 −2.23646
\(413\) 0 0
\(414\) −11.9902 −0.589285
\(415\) −7.17157 −0.352039
\(416\) 86.0813 4.22049
\(417\) 6.14657 0.300999
\(418\) −33.5398 −1.64049
\(419\) −9.89450 −0.483378 −0.241689 0.970354i \(-0.577701\pi\)
−0.241689 + 0.970354i \(0.577701\pi\)
\(420\) 0 0
\(421\) 35.4388 1.72718 0.863591 0.504193i \(-0.168210\pi\)
0.863591 + 0.504193i \(0.168210\pi\)
\(422\) −0.872125 −0.0424544
\(423\) 6.48528 0.315325
\(424\) 68.7686 3.33970
\(425\) −4.94725 −0.239977
\(426\) −0.762725 −0.0369541
\(427\) 0 0
\(428\) −47.4040 −2.29136
\(429\) 16.0796 0.776330
\(430\) −25.0590 −1.20845
\(431\) −16.6931 −0.804079 −0.402040 0.915622i \(-0.631699\pi\)
−0.402040 + 0.915622i \(0.631699\pi\)
\(432\) −15.7721 −0.758838
\(433\) −13.0363 −0.626483 −0.313241 0.949674i \(-0.601415\pi\)
−0.313241 + 0.949674i \(0.601415\pi\)
\(434\) 0 0
\(435\) −0.660384 −0.0316630
\(436\) −20.3235 −0.973319
\(437\) −11.9696 −0.572585
\(438\) 29.1778 1.39417
\(439\) 22.7114 1.08396 0.541979 0.840392i \(-0.317675\pi\)
0.541979 + 0.840392i \(0.317675\pi\)
\(440\) −43.4786 −2.07276
\(441\) 0 0
\(442\) −49.1941 −2.33992
\(443\) 8.24891 0.391917 0.195959 0.980612i \(-0.437218\pi\)
0.195959 + 0.980612i \(0.437218\pi\)
\(444\) 12.0492 0.571831
\(445\) −0.828427 −0.0392712
\(446\) 79.8867 3.78274
\(447\) −20.2306 −0.956874
\(448\) 0 0
\(449\) 24.5212 1.15723 0.578613 0.815602i \(-0.303594\pi\)
0.578613 + 0.815602i \(0.303594\pi\)
\(450\) 2.74912 0.129595
\(451\) 25.6757 1.20902
\(452\) 70.0738 3.29599
\(453\) −21.1082 −0.991751
\(454\) −57.4317 −2.69541
\(455\) 0 0
\(456\) −26.8414 −1.25696
\(457\) −28.3468 −1.32601 −0.663004 0.748616i \(-0.730719\pi\)
−0.663004 + 0.748616i \(0.730719\pi\)
\(458\) 13.5502 0.633160
\(459\) 4.94725 0.230918
\(460\) −24.2395 −1.13017
\(461\) 38.3727 1.78720 0.893598 0.448868i \(-0.148173\pi\)
0.893598 + 0.448868i \(0.148173\pi\)
\(462\) 0 0
\(463\) 2.38530 0.110855 0.0554273 0.998463i \(-0.482348\pi\)
0.0554273 + 0.998463i \(0.482348\pi\)
\(464\) 10.4157 0.483536
\(465\) −1.25559 −0.0582268
\(466\) −3.00704 −0.139298
\(467\) −19.8079 −0.916598 −0.458299 0.888798i \(-0.651541\pi\)
−0.458299 + 0.888798i \(0.651541\pi\)
\(468\) 20.1023 0.929231
\(469\) 0 0
\(470\) 17.8288 0.822381
\(471\) −1.96021 −0.0903215
\(472\) −135.213 −6.22368
\(473\) 40.5219 1.86320
\(474\) 15.2247 0.699293
\(475\) 2.74441 0.125922
\(476\) 0 0
\(477\) 7.03127 0.321940
\(478\) 70.5768 3.22811
\(479\) −3.43959 −0.157159 −0.0785795 0.996908i \(-0.525038\pi\)
−0.0785795 + 0.996908i \(0.525038\pi\)
\(480\) −23.7987 −1.08626
\(481\) −7.84194 −0.357562
\(482\) 68.0318 3.09876
\(483\) 0 0
\(484\) 48.6981 2.21355
\(485\) 6.20784 0.281883
\(486\) −2.74912 −0.124703
\(487\) −7.12235 −0.322745 −0.161372 0.986894i \(-0.551592\pi\)
−0.161372 + 0.986894i \(0.551592\pi\)
\(488\) 92.0747 4.16803
\(489\) 8.18137 0.369974
\(490\) 0 0
\(491\) −31.6896 −1.43013 −0.715066 0.699057i \(-0.753603\pi\)
−0.715066 + 0.699057i \(0.753603\pi\)
\(492\) 32.0992 1.44714
\(493\) −3.26709 −0.147142
\(494\) 27.2896 1.22782
\(495\) −4.44549 −0.199810
\(496\) 19.8034 0.889200
\(497\) 0 0
\(498\) 19.7155 0.883473
\(499\) −39.0027 −1.74600 −0.873001 0.487718i \(-0.837829\pi\)
−0.873001 + 0.487718i \(0.837829\pi\)
\(500\) 5.55765 0.248546
\(501\) 3.76235 0.168089
\(502\) 1.11529 0.0497780
\(503\) −18.5308 −0.826247 −0.413123 0.910675i \(-0.635562\pi\)
−0.413123 + 0.910675i \(0.635562\pi\)
\(504\) 0 0
\(505\) 4.11882 0.183285
\(506\) 53.3022 2.36957
\(507\) −0.0831193 −0.00369146
\(508\) −0.312775 −0.0138771
\(509\) −39.8053 −1.76434 −0.882169 0.470934i \(-0.843917\pi\)
−0.882169 + 0.470934i \(0.843917\pi\)
\(510\) 13.6006 0.602244
\(511\) 0 0
\(512\) 66.8412 2.95399
\(513\) −2.74441 −0.121168
\(514\) −29.9406 −1.32062
\(515\) −8.16804 −0.359927
\(516\) 50.6596 2.23016
\(517\) −28.8302 −1.26795
\(518\) 0 0
\(519\) 15.7061 0.689420
\(520\) 35.3763 1.55135
\(521\) 26.3594 1.15483 0.577413 0.816452i \(-0.304062\pi\)
0.577413 + 0.816452i \(0.304062\pi\)
\(522\) 1.81547 0.0794611
\(523\) −24.3727 −1.06574 −0.532872 0.846196i \(-0.678887\pi\)
−0.532872 + 0.846196i \(0.678887\pi\)
\(524\) 109.281 4.77395
\(525\) 0 0
\(526\) 24.1019 1.05089
\(527\) −6.21174 −0.270588
\(528\) 70.1149 3.05136
\(529\) −3.97762 −0.172940
\(530\) 19.3298 0.839633
\(531\) −13.8249 −0.599949
\(532\) 0 0
\(533\) −20.8910 −0.904888
\(534\) 2.27744 0.0985546
\(535\) −8.52951 −0.368763
\(536\) 29.9874 1.29526
\(537\) 1.28097 0.0552781
\(538\) 62.7753 2.70643
\(539\) 0 0
\(540\) −5.55765 −0.239163
\(541\) 24.3627 1.04744 0.523718 0.851892i \(-0.324545\pi\)
0.523718 + 0.851892i \(0.324545\pi\)
\(542\) 29.3463 1.26053
\(543\) −16.5599 −0.710652
\(544\) −117.738 −5.04798
\(545\) −3.65685 −0.156642
\(546\) 0 0
\(547\) 10.1421 0.433646 0.216823 0.976211i \(-0.430430\pi\)
0.216823 + 0.976211i \(0.430430\pi\)
\(548\) −104.210 −4.45161
\(549\) 9.41421 0.401789
\(550\) −12.2212 −0.521112
\(551\) 1.81236 0.0772092
\(552\) 42.6568 1.81560
\(553\) 0 0
\(554\) −34.1882 −1.45252
\(555\) 2.16804 0.0920283
\(556\) −34.1605 −1.44873
\(557\) −14.5560 −0.616756 −0.308378 0.951264i \(-0.599786\pi\)
−0.308378 + 0.951264i \(0.599786\pi\)
\(558\) 3.45178 0.146125
\(559\) −32.9706 −1.39451
\(560\) 0 0
\(561\) −21.9929 −0.928543
\(562\) 35.5100 1.49790
\(563\) −36.5845 −1.54185 −0.770926 0.636925i \(-0.780206\pi\)
−0.770926 + 0.636925i \(0.780206\pi\)
\(564\) −36.0429 −1.51768
\(565\) 12.6085 0.530445
\(566\) 70.4156 2.95979
\(567\) 0 0
\(568\) 2.71351 0.113856
\(569\) −3.64687 −0.152885 −0.0764424 0.997074i \(-0.524356\pi\)
−0.0764424 + 0.997074i \(0.524356\pi\)
\(570\) −7.54469 −0.316012
\(571\) −38.5933 −1.61508 −0.807540 0.589812i \(-0.799202\pi\)
−0.807540 + 0.589812i \(0.799202\pi\)
\(572\) −89.3647 −3.73653
\(573\) −8.76272 −0.366068
\(574\) 0 0
\(575\) −4.36147 −0.181886
\(576\) 33.8812 1.41171
\(577\) 14.0523 0.585006 0.292503 0.956265i \(-0.405512\pi\)
0.292503 + 0.956265i \(0.405512\pi\)
\(578\) 20.5505 0.854787
\(579\) −26.4986 −1.10124
\(580\) 3.67018 0.152396
\(581\) 0 0
\(582\) −17.0661 −0.707412
\(583\) −31.2574 −1.29455
\(584\) −103.804 −4.29546
\(585\) 3.61706 0.149547
\(586\) −4.62611 −0.191103
\(587\) −27.0402 −1.11607 −0.558034 0.829818i \(-0.688444\pi\)
−0.558034 + 0.829818i \(0.688444\pi\)
\(588\) 0 0
\(589\) 3.44586 0.141984
\(590\) −38.0063 −1.56469
\(591\) −4.11439 −0.169243
\(592\) −34.1947 −1.40539
\(593\) 21.6764 0.890145 0.445073 0.895495i \(-0.353178\pi\)
0.445073 + 0.895495i \(0.353178\pi\)
\(594\) 12.2212 0.501440
\(595\) 0 0
\(596\) 112.434 4.60550
\(597\) 7.63538 0.312495
\(598\) −43.3692 −1.77350
\(599\) 25.9898 1.06191 0.530957 0.847399i \(-0.321833\pi\)
0.530957 + 0.847399i \(0.321833\pi\)
\(600\) −9.78039 −0.399283
\(601\) 12.0746 0.492533 0.246267 0.969202i \(-0.420796\pi\)
0.246267 + 0.969202i \(0.420796\pi\)
\(602\) 0 0
\(603\) 3.06608 0.124860
\(604\) 117.312 4.77336
\(605\) 8.76235 0.356240
\(606\) −11.3231 −0.459971
\(607\) 41.7390 1.69413 0.847067 0.531486i \(-0.178366\pi\)
0.847067 + 0.531486i \(0.178366\pi\)
\(608\) 65.3133 2.64880
\(609\) 0 0
\(610\) 25.8808 1.04788
\(611\) 23.4576 0.948995
\(612\) −27.4951 −1.11142
\(613\) −11.8016 −0.476662 −0.238331 0.971184i \(-0.576600\pi\)
−0.238331 + 0.971184i \(0.576600\pi\)
\(614\) −20.1062 −0.811422
\(615\) 5.77568 0.232898
\(616\) 0 0
\(617\) 12.2913 0.494829 0.247415 0.968910i \(-0.420419\pi\)
0.247415 + 0.968910i \(0.420419\pi\)
\(618\) 22.4549 0.903269
\(619\) 24.3772 0.979801 0.489900 0.871778i \(-0.337033\pi\)
0.489900 + 0.871778i \(0.337033\pi\)
\(620\) 6.97815 0.280249
\(621\) 4.36147 0.175020
\(622\) 30.7955 1.23479
\(623\) 0 0
\(624\) −57.0488 −2.28378
\(625\) 1.00000 0.0400000
\(626\) −86.5578 −3.45954
\(627\) 12.2002 0.487230
\(628\) 10.8941 0.434723
\(629\) 10.7259 0.427668
\(630\) 0 0
\(631\) −43.3200 −1.72454 −0.862271 0.506448i \(-0.830958\pi\)
−0.862271 + 0.506448i \(0.830958\pi\)
\(632\) −54.1641 −2.15453
\(633\) 0.317238 0.0126091
\(634\) 23.2314 0.922635
\(635\) −0.0562783 −0.00223334
\(636\) −39.0773 −1.54952
\(637\) 0 0
\(638\) −8.07066 −0.319521
\(639\) 0.277444 0.0109755
\(640\) 45.5459 1.80036
\(641\) 18.5408 0.732316 0.366158 0.930553i \(-0.380673\pi\)
0.366158 + 0.930553i \(0.380673\pi\)
\(642\) 23.4486 0.925443
\(643\) 8.55489 0.337372 0.168686 0.985670i \(-0.446048\pi\)
0.168686 + 0.985670i \(0.446048\pi\)
\(644\) 0 0
\(645\) 9.11529 0.358914
\(646\) −37.3255 −1.46855
\(647\) 43.0886 1.69399 0.846994 0.531603i \(-0.178410\pi\)
0.846994 + 0.531603i \(0.178410\pi\)
\(648\) 9.78039 0.384210
\(649\) 61.4584 2.41245
\(650\) 9.94372 0.390025
\(651\) 0 0
\(652\) −45.4692 −1.78071
\(653\) −14.0277 −0.548948 −0.274474 0.961595i \(-0.588504\pi\)
−0.274474 + 0.961595i \(0.588504\pi\)
\(654\) 10.0531 0.393108
\(655\) 19.6631 0.768302
\(656\) −91.0949 −3.55666
\(657\) −10.6135 −0.414073
\(658\) 0 0
\(659\) −19.1488 −0.745932 −0.372966 0.927845i \(-0.621659\pi\)
−0.372966 + 0.927845i \(0.621659\pi\)
\(660\) 24.7064 0.961697
\(661\) −26.5717 −1.03352 −0.516759 0.856131i \(-0.672862\pi\)
−0.516759 + 0.856131i \(0.672862\pi\)
\(662\) 45.7819 1.77937
\(663\) 17.8945 0.694965
\(664\) −70.1408 −2.72199
\(665\) 0 0
\(666\) −5.96021 −0.230953
\(667\) −2.88024 −0.111523
\(668\) −20.9098 −0.809025
\(669\) −29.0590 −1.12349
\(670\) 8.42900 0.325641
\(671\) −41.8508 −1.61563
\(672\) 0 0
\(673\) 46.6900 1.79977 0.899883 0.436132i \(-0.143652\pi\)
0.899883 + 0.436132i \(0.143652\pi\)
\(674\) 87.7458 3.37984
\(675\) −1.00000 −0.0384900
\(676\) 0.461948 0.0177672
\(677\) 41.6953 1.60248 0.801240 0.598343i \(-0.204174\pi\)
0.801240 + 0.598343i \(0.204174\pi\)
\(678\) −34.6624 −1.33120
\(679\) 0 0
\(680\) −48.3861 −1.85552
\(681\) 20.8910 0.800544
\(682\) −15.3448 −0.587584
\(683\) −21.5430 −0.824321 −0.412160 0.911111i \(-0.635226\pi\)
−0.412160 + 0.911111i \(0.635226\pi\)
\(684\) 15.2524 0.583192
\(685\) −18.7507 −0.716426
\(686\) 0 0
\(687\) −4.92893 −0.188050
\(688\) −143.768 −5.48110
\(689\) 25.4325 0.968902
\(690\) 11.9902 0.456458
\(691\) 22.9224 0.872010 0.436005 0.899944i \(-0.356393\pi\)
0.436005 + 0.899944i \(0.356393\pi\)
\(692\) −87.2888 −3.31822
\(693\) 0 0
\(694\) −21.7533 −0.825743
\(695\) −6.14657 −0.233153
\(696\) −6.45881 −0.244821
\(697\) 28.5737 1.08231
\(698\) 69.1471 2.61726
\(699\) 1.09382 0.0413720
\(700\) 0 0
\(701\) −51.4317 −1.94255 −0.971275 0.237960i \(-0.923521\pi\)
−0.971275 + 0.237960i \(0.923521\pi\)
\(702\) −9.94372 −0.375302
\(703\) −5.94999 −0.224408
\(704\) −150.618 −5.67664
\(705\) −6.48528 −0.244250
\(706\) −56.5402 −2.12792
\(707\) 0 0
\(708\) 76.8339 2.88759
\(709\) −34.4682 −1.29448 −0.647241 0.762286i \(-0.724077\pi\)
−0.647241 + 0.762286i \(0.724077\pi\)
\(710\) 0.762725 0.0286246
\(711\) −5.53803 −0.207692
\(712\) −8.10234 −0.303648
\(713\) −5.47623 −0.205086
\(714\) 0 0
\(715\) −16.0796 −0.601343
\(716\) −7.11920 −0.266057
\(717\) −25.6725 −0.958759
\(718\) −38.7690 −1.44685
\(719\) 42.0751 1.56914 0.784568 0.620043i \(-0.212885\pi\)
0.784568 + 0.620043i \(0.212885\pi\)
\(720\) 15.7721 0.587793
\(721\) 0 0
\(722\) −31.5275 −1.17333
\(723\) −24.7468 −0.920342
\(724\) 92.0340 3.42042
\(725\) 0.660384 0.0245260
\(726\) −24.0887 −0.894017
\(727\) 36.6722 1.36010 0.680048 0.733168i \(-0.261959\pi\)
0.680048 + 0.733168i \(0.261959\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −29.1778 −1.07992
\(731\) 45.0957 1.66792
\(732\) −52.3209 −1.93384
\(733\) −46.3696 −1.71270 −0.856350 0.516397i \(-0.827273\pi\)
−0.856350 + 0.516397i \(0.827273\pi\)
\(734\) 1.06348 0.0392538
\(735\) 0 0
\(736\) −103.797 −3.82602
\(737\) −13.6302 −0.502075
\(738\) −15.8780 −0.584478
\(739\) 1.35846 0.0499719 0.0249859 0.999688i \(-0.492046\pi\)
0.0249859 + 0.999688i \(0.492046\pi\)
\(740\) −12.0492 −0.442938
\(741\) −9.92668 −0.364666
\(742\) 0 0
\(743\) −26.2748 −0.963930 −0.481965 0.876191i \(-0.660077\pi\)
−0.481965 + 0.876191i \(0.660077\pi\)
\(744\) −12.2802 −0.450214
\(745\) 20.2306 0.741191
\(746\) −29.2354 −1.07039
\(747\) −7.17157 −0.262394
\(748\) 122.229 4.46914
\(749\) 0 0
\(750\) −2.74912 −0.100384
\(751\) −25.2271 −0.920548 −0.460274 0.887777i \(-0.652249\pi\)
−0.460274 + 0.887777i \(0.652249\pi\)
\(752\) 102.287 3.73002
\(753\) −0.405692 −0.0147842
\(754\) 6.56668 0.239144
\(755\) 21.1082 0.768207
\(756\) 0 0
\(757\) 6.02331 0.218921 0.109460 0.993991i \(-0.465088\pi\)
0.109460 + 0.993991i \(0.465088\pi\)
\(758\) −72.6564 −2.63900
\(759\) −19.3888 −0.703770
\(760\) 26.8414 0.973638
\(761\) −30.8214 −1.11727 −0.558637 0.829412i \(-0.688675\pi\)
−0.558637 + 0.829412i \(0.688675\pi\)
\(762\) 0.154716 0.00560476
\(763\) 0 0
\(764\) 48.7001 1.76191
\(765\) −4.94725 −0.178868
\(766\) 14.1460 0.511117
\(767\) −50.0055 −1.80559
\(768\) −57.4486 −2.07300
\(769\) −31.8736 −1.14939 −0.574695 0.818367i \(-0.694879\pi\)
−0.574695 + 0.818367i \(0.694879\pi\)
\(770\) 0 0
\(771\) 10.8910 0.392229
\(772\) 147.270 5.30036
\(773\) 17.0912 0.614727 0.307364 0.951592i \(-0.400553\pi\)
0.307364 + 0.951592i \(0.400553\pi\)
\(774\) −25.0590 −0.900728
\(775\) 1.25559 0.0451023
\(776\) 60.7151 2.17954
\(777\) 0 0
\(778\) −35.4857 −1.27222
\(779\) −15.8508 −0.567914
\(780\) −20.1023 −0.719779
\(781\) −1.23337 −0.0441335
\(782\) 59.3184 2.12122
\(783\) −0.660384 −0.0236002
\(784\) 0 0
\(785\) 1.96021 0.0699627
\(786\) −54.0562 −1.92812
\(787\) −28.1617 −1.00386 −0.501929 0.864909i \(-0.667376\pi\)
−0.501929 + 0.864909i \(0.667376\pi\)
\(788\) 22.8663 0.814580
\(789\) −8.76716 −0.312119
\(790\) −15.2247 −0.541670
\(791\) 0 0
\(792\) −43.4786 −1.54494
\(793\) 34.0518 1.20921
\(794\) −62.5774 −2.22079
\(795\) −7.03127 −0.249374
\(796\) −42.4347 −1.50406
\(797\) 9.60058 0.340070 0.170035 0.985438i \(-0.445612\pi\)
0.170035 + 0.985438i \(0.445612\pi\)
\(798\) 0 0
\(799\) −32.0843 −1.13506
\(800\) 23.7987 0.841411
\(801\) −0.828427 −0.0292710
\(802\) −27.0810 −0.956264
\(803\) 47.1823 1.66503
\(804\) −17.0402 −0.600960
\(805\) 0 0
\(806\) 12.4853 0.439775
\(807\) −22.8347 −0.803819
\(808\) 40.2837 1.41718
\(809\) −21.0331 −0.739485 −0.369742 0.929134i \(-0.620554\pi\)
−0.369742 + 0.929134i \(0.620554\pi\)
\(810\) 2.74912 0.0965941
\(811\) 0.770313 0.0270493 0.0135247 0.999909i \(-0.495695\pi\)
0.0135247 + 0.999909i \(0.495695\pi\)
\(812\) 0 0
\(813\) −10.6748 −0.374382
\(814\) 26.4960 0.928685
\(815\) −8.18137 −0.286581
\(816\) 78.0288 2.73155
\(817\) −25.0161 −0.875201
\(818\) −40.3412 −1.41050
\(819\) 0 0
\(820\) −32.0992 −1.12095
\(821\) −16.4782 −0.575094 −0.287547 0.957767i \(-0.592840\pi\)
−0.287547 + 0.957767i \(0.592840\pi\)
\(822\) 51.5478 1.79794
\(823\) −16.0563 −0.559687 −0.279843 0.960046i \(-0.590283\pi\)
−0.279843 + 0.960046i \(0.590283\pi\)
\(824\) −79.8867 −2.78298
\(825\) 4.44549 0.154772
\(826\) 0 0
\(827\) −1.11736 −0.0388545 −0.0194272 0.999811i \(-0.506184\pi\)
−0.0194272 + 0.999811i \(0.506184\pi\)
\(828\) −24.2395 −0.842380
\(829\) −52.5724 −1.82592 −0.912958 0.408054i \(-0.866207\pi\)
−0.912958 + 0.408054i \(0.866207\pi\)
\(830\) −19.7155 −0.684335
\(831\) 12.4361 0.431402
\(832\) 122.550 4.24866
\(833\) 0 0
\(834\) 16.8976 0.585117
\(835\) −3.76235 −0.130202
\(836\) −67.8045 −2.34507
\(837\) −1.25559 −0.0433997
\(838\) −27.2012 −0.939648
\(839\) 8.65667 0.298861 0.149431 0.988772i \(-0.452256\pi\)
0.149431 + 0.988772i \(0.452256\pi\)
\(840\) 0 0
\(841\) −28.5639 −0.984962
\(842\) 97.4254 3.35750
\(843\) −12.9169 −0.444881
\(844\) −1.76310 −0.0606884
\(845\) 0.0831193 0.00285939
\(846\) 17.8288 0.612967
\(847\) 0 0
\(848\) 110.898 3.80826
\(849\) −25.6139 −0.879067
\(850\) −13.6006 −0.466496
\(851\) 9.45584 0.324142
\(852\) −1.54193 −0.0528258
\(853\) −21.9361 −0.751078 −0.375539 0.926807i \(-0.622542\pi\)
−0.375539 + 0.926807i \(0.622542\pi\)
\(854\) 0 0
\(855\) 2.74441 0.0938567
\(856\) −83.4219 −2.85130
\(857\) 25.1253 0.858263 0.429132 0.903242i \(-0.358820\pi\)
0.429132 + 0.903242i \(0.358820\pi\)
\(858\) 44.2047 1.50912
\(859\) 8.09183 0.276090 0.138045 0.990426i \(-0.455918\pi\)
0.138045 + 0.990426i \(0.455918\pi\)
\(860\) −50.6596 −1.72748
\(861\) 0 0
\(862\) −45.8913 −1.56307
\(863\) 23.0674 0.785222 0.392611 0.919705i \(-0.371572\pi\)
0.392611 + 0.919705i \(0.371572\pi\)
\(864\) −23.7987 −0.809649
\(865\) −15.7061 −0.534023
\(866\) −35.8382 −1.21783
\(867\) −7.47530 −0.253874
\(868\) 0 0
\(869\) 24.6192 0.835150
\(870\) −1.81547 −0.0615503
\(871\) 11.0902 0.375776
\(872\) −35.7655 −1.21117
\(873\) 6.20784 0.210103
\(874\) −32.9059 −1.11306
\(875\) 0 0
\(876\) 58.9863 1.99296
\(877\) −4.32351 −0.145994 −0.0729972 0.997332i \(-0.523256\pi\)
−0.0729972 + 0.997332i \(0.523256\pi\)
\(878\) 62.4364 2.10713
\(879\) 1.68276 0.0567581
\(880\) −70.1149 −2.36357
\(881\) −19.9929 −0.673579 −0.336790 0.941580i \(-0.609341\pi\)
−0.336790 + 0.941580i \(0.609341\pi\)
\(882\) 0 0
\(883\) 1.89450 0.0637551 0.0318776 0.999492i \(-0.489851\pi\)
0.0318776 + 0.999492i \(0.489851\pi\)
\(884\) −99.4513 −3.34491
\(885\) 13.8249 0.464719
\(886\) 22.6772 0.761856
\(887\) −5.22525 −0.175447 −0.0877234 0.996145i \(-0.527959\pi\)
−0.0877234 + 0.996145i \(0.527959\pi\)
\(888\) 21.2043 0.711570
\(889\) 0 0
\(890\) −2.27744 −0.0763401
\(891\) −4.44549 −0.148929
\(892\) 161.500 5.40741
\(893\) 17.7982 0.595595
\(894\) −55.6163 −1.86009
\(895\) −1.28097 −0.0428182
\(896\) 0 0
\(897\) 15.7757 0.526735
\(898\) 67.4116 2.24955
\(899\) 0.829175 0.0276545
\(900\) 5.55765 0.185255
\(901\) −34.7855 −1.15887
\(902\) 70.5855 2.35024
\(903\) 0 0
\(904\) 123.316 4.10144
\(905\) 16.5599 0.550469
\(906\) −58.0290 −1.92789
\(907\) −5.41921 −0.179942 −0.0899709 0.995944i \(-0.528677\pi\)
−0.0899709 + 0.995944i \(0.528677\pi\)
\(908\) −116.105 −3.85307
\(909\) 4.11882 0.136613
\(910\) 0 0
\(911\) 0.226686 0.00751043 0.00375522 0.999993i \(-0.498805\pi\)
0.00375522 + 0.999993i \(0.498805\pi\)
\(912\) −43.2852 −1.43332
\(913\) 31.8811 1.05511
\(914\) −77.9287 −2.57765
\(915\) −9.41421 −0.311224
\(916\) 27.3933 0.905099
\(917\) 0 0
\(918\) 13.6006 0.448886
\(919\) −8.10624 −0.267400 −0.133700 0.991022i \(-0.542686\pi\)
−0.133700 + 0.991022i \(0.542686\pi\)
\(920\) −42.6568 −1.40635
\(921\) 7.31371 0.240995
\(922\) 105.491 3.47417
\(923\) 1.00353 0.0330316
\(924\) 0 0
\(925\) −2.16804 −0.0712848
\(926\) 6.55748 0.215492
\(927\) −8.16804 −0.268274
\(928\) 15.7163 0.515912
\(929\) 2.29945 0.0754424 0.0377212 0.999288i \(-0.487990\pi\)
0.0377212 + 0.999288i \(0.487990\pi\)
\(930\) −3.45178 −0.113188
\(931\) 0 0
\(932\) −6.07906 −0.199126
\(933\) −11.2019 −0.366735
\(934\) −54.4541 −1.78179
\(935\) 21.9929 0.719246
\(936\) 35.3763 1.15631
\(937\) −16.1023 −0.526041 −0.263020 0.964790i \(-0.584719\pi\)
−0.263020 + 0.964790i \(0.584719\pi\)
\(938\) 0 0
\(939\) 31.4857 1.02750
\(940\) 36.0429 1.17559
\(941\) 13.5247 0.440893 0.220446 0.975399i \(-0.429249\pi\)
0.220446 + 0.975399i \(0.429249\pi\)
\(942\) −5.38883 −0.175578
\(943\) 25.1904 0.820313
\(944\) −218.048 −7.09687
\(945\) 0 0
\(946\) 111.400 3.62191
\(947\) 6.85381 0.222719 0.111359 0.993780i \(-0.464480\pi\)
0.111359 + 0.993780i \(0.464480\pi\)
\(948\) 30.7784 0.999637
\(949\) −38.3898 −1.24618
\(950\) 7.54469 0.244782
\(951\) −8.45048 −0.274026
\(952\) 0 0
\(953\) −44.9483 −1.45602 −0.728009 0.685568i \(-0.759554\pi\)
−0.728009 + 0.685568i \(0.759554\pi\)
\(954\) 19.3298 0.625825
\(955\) 8.76272 0.283555
\(956\) 142.679 4.61457
\(957\) 2.93573 0.0948986
\(958\) −9.45584 −0.305504
\(959\) 0 0
\(960\) −33.8812 −1.09351
\(961\) −29.4235 −0.949145
\(962\) −21.5584 −0.695071
\(963\) −8.52951 −0.274860
\(964\) 137.534 4.42967
\(965\) 26.4986 0.853020
\(966\) 0 0
\(967\) 12.4245 0.399546 0.199773 0.979842i \(-0.435979\pi\)
0.199773 + 0.979842i \(0.435979\pi\)
\(968\) 85.6992 2.75448
\(969\) 13.5773 0.436165
\(970\) 17.0661 0.547959
\(971\) 20.8158 0.668012 0.334006 0.942571i \(-0.391599\pi\)
0.334006 + 0.942571i \(0.391599\pi\)
\(972\) −5.55765 −0.178262
\(973\) 0 0
\(974\) −19.5802 −0.627390
\(975\) −3.61706 −0.115839
\(976\) 148.482 4.75281
\(977\) 42.7418 1.36743 0.683716 0.729748i \(-0.260363\pi\)
0.683716 + 0.729748i \(0.260363\pi\)
\(978\) 22.4915 0.719200
\(979\) 3.68276 0.117702
\(980\) 0 0
\(981\) −3.65685 −0.116754
\(982\) −87.1184 −2.78006
\(983\) 5.68982 0.181477 0.0907386 0.995875i \(-0.471077\pi\)
0.0907386 + 0.995875i \(0.471077\pi\)
\(984\) 56.4884 1.80078
\(985\) 4.11439 0.131095
\(986\) −8.98160 −0.286033
\(987\) 0 0
\(988\) 55.1690 1.75516
\(989\) 39.7560 1.26417
\(990\) −12.2212 −0.388414
\(991\) 59.1216 1.87806 0.939029 0.343838i \(-0.111727\pi\)
0.939029 + 0.343838i \(0.111727\pi\)
\(992\) 29.8815 0.948739
\(993\) −16.6533 −0.528477
\(994\) 0 0
\(995\) −7.63538 −0.242058
\(996\) 39.8571 1.26292
\(997\) −45.5294 −1.44193 −0.720965 0.692972i \(-0.756301\pi\)
−0.720965 + 0.692972i \(0.756301\pi\)
\(998\) −107.223 −3.39409
\(999\) 2.16804 0.0685938
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 735.2.a.n.1.4 4
3.2 odd 2 2205.2.a.bf.1.1 4
5.4 even 2 3675.2.a.bl.1.1 4
7.2 even 3 735.2.i.n.361.1 8
7.3 odd 6 735.2.i.m.226.1 8
7.4 even 3 735.2.i.n.226.1 8
7.5 odd 6 735.2.i.m.361.1 8
7.6 odd 2 735.2.a.o.1.4 yes 4
21.20 even 2 2205.2.a.bg.1.1 4
35.34 odd 2 3675.2.a.bk.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
735.2.a.n.1.4 4 1.1 even 1 trivial
735.2.a.o.1.4 yes 4 7.6 odd 2
735.2.i.m.226.1 8 7.3 odd 6
735.2.i.m.361.1 8 7.5 odd 6
735.2.i.n.226.1 8 7.4 even 3
735.2.i.n.361.1 8 7.2 even 3
2205.2.a.bf.1.1 4 3.2 odd 2
2205.2.a.bg.1.1 4 21.20 even 2
3675.2.a.bk.1.1 4 35.34 odd 2
3675.2.a.bl.1.1 4 5.4 even 2