Properties

Label 570.2.d.b.229.1
Level $570$
Weight $2$
Character 570.229
Analytic conductor $4.551$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [570,2,Mod(229,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.229");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 229.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 570.229
Dual form 570.2.d.b.229.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +(-1.00000 - 2.00000i) q^{10} -4.00000 q^{11} +1.00000i q^{12} -4.00000i q^{13} +(-1.00000 - 2.00000i) q^{15} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} +(-2.00000 + 1.00000i) q^{20} +4.00000i q^{22} +1.00000 q^{24} +(3.00000 - 4.00000i) q^{25} -4.00000 q^{26} +1.00000i q^{27} +2.00000 q^{29} +(-2.00000 + 1.00000i) q^{30} -1.00000i q^{32} +4.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} -1.00000i q^{38} -4.00000 q^{39} +(1.00000 + 2.00000i) q^{40} -12.0000 q^{41} +6.00000i q^{43} +4.00000 q^{44} +(-2.00000 + 1.00000i) q^{45} -1.00000i q^{48} +7.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} -6.00000 q^{51} +4.00000i q^{52} -14.0000i q^{53} +1.00000 q^{54} +(-8.00000 + 4.00000i) q^{55} -1.00000i q^{57} -2.00000i q^{58} +10.0000 q^{59} +(1.00000 + 2.00000i) q^{60} -6.00000 q^{61} -1.00000 q^{64} +(-4.00000 - 8.00000i) q^{65} +4.00000 q^{66} +4.00000i q^{67} +6.00000i q^{68} +12.0000 q^{71} -1.00000i q^{72} +8.00000i q^{73} +4.00000 q^{74} +(-4.00000 - 3.00000i) q^{75} -1.00000 q^{76} +4.00000i q^{78} +8.00000 q^{79} +(2.00000 - 1.00000i) q^{80} +1.00000 q^{81} +12.0000i q^{82} +12.0000i q^{83} +(-6.00000 - 12.0000i) q^{85} +6.00000 q^{86} -2.00000i q^{87} -4.00000i q^{88} +8.00000 q^{89} +(1.00000 + 2.00000i) q^{90} +(2.00000 - 1.00000i) q^{95} -1.00000 q^{96} +10.0000i q^{97} -7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{9} - 2 q^{10} - 8 q^{11} - 2 q^{15} + 2 q^{16} + 2 q^{19} - 4 q^{20} + 2 q^{24} + 6 q^{25} - 8 q^{26} + 4 q^{29} - 4 q^{30} - 12 q^{34} + 2 q^{36} - 8 q^{39} + 2 q^{40} - 24 q^{41} + 8 q^{44} - 4 q^{45} + 14 q^{49} - 8 q^{50} - 12 q^{51} + 2 q^{54} - 16 q^{55} + 20 q^{59} + 2 q^{60} - 12 q^{61} - 2 q^{64} - 8 q^{65} + 8 q^{66} + 24 q^{71} + 8 q^{74} - 8 q^{75} - 2 q^{76} + 16 q^{79} + 4 q^{80} + 2 q^{81} - 12 q^{85} + 12 q^{86} + 16 q^{89} + 2 q^{90} + 4 q^{95} - 2 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 2.00000 1.00000i 0.894427 0.447214i
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) −1.00000 2.00000i −0.316228 0.632456i
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 4.00000i 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) −1.00000 2.00000i −0.258199 0.516398i
\(16\) 1.00000 0.250000
\(17\) 6.00000i 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 1.00000 0.229416
\(20\) −2.00000 + 1.00000i −0.447214 + 0.223607i
\(21\) 0 0
\(22\) 4.00000i 0.852803i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 1.00000 0.204124
\(25\) 3.00000 4.00000i 0.600000 0.800000i
\(26\) −4.00000 −0.784465
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −2.00000 + 1.00000i −0.365148 + 0.182574i
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 4.00000i 0.696311i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 1.00000i 0.162221i
\(39\) −4.00000 −0.640513
\(40\) 1.00000 + 2.00000i 0.158114 + 0.316228i
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 4.00000 0.603023
\(45\) −2.00000 + 1.00000i −0.298142 + 0.149071i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 7.00000 1.00000
\(50\) −4.00000 3.00000i −0.565685 0.424264i
\(51\) −6.00000 −0.840168
\(52\) 4.00000i 0.554700i
\(53\) 14.0000i 1.92305i −0.274721 0.961524i \(-0.588586\pi\)
0.274721 0.961524i \(-0.411414\pi\)
\(54\) 1.00000 0.136083
\(55\) −8.00000 + 4.00000i −1.07872 + 0.539360i
\(56\) 0 0
\(57\) 1.00000i 0.132453i
\(58\) 2.00000i 0.262613i
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 1.00000 + 2.00000i 0.129099 + 0.258199i
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −4.00000 8.00000i −0.496139 0.992278i
\(66\) 4.00000 0.492366
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 6.00000i 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 8.00000i 0.936329i 0.883641 + 0.468165i \(0.155085\pi\)
−0.883641 + 0.468165i \(0.844915\pi\)
\(74\) 4.00000 0.464991
\(75\) −4.00000 3.00000i −0.461880 0.346410i
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 4.00000i 0.452911i
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 2.00000 1.00000i 0.223607 0.111803i
\(81\) 1.00000 0.111111
\(82\) 12.0000i 1.32518i
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) −6.00000 12.0000i −0.650791 1.30158i
\(86\) 6.00000 0.646997
\(87\) 2.00000i 0.214423i
\(88\) 4.00000i 0.426401i
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 1.00000 + 2.00000i 0.105409 + 0.210819i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.00000 1.00000i 0.205196 0.102598i
\(96\) −1.00000 −0.102062
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 7.00000i 0.707107i
\(99\) 4.00000 0.402015
\(100\) −3.00000 + 4.00000i −0.300000 + 0.400000i
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 6.00000i 0.594089i
\(103\) 2.00000i 0.197066i 0.995134 + 0.0985329i \(0.0314150\pi\)
−0.995134 + 0.0985329i \(0.968585\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −14.0000 −1.35980
\(107\) 20.0000i 1.93347i −0.255774 0.966736i \(-0.582330\pi\)
0.255774 0.966736i \(-0.417670\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 4.00000 + 8.00000i 0.381385 + 0.762770i
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 10.0000i 0.940721i −0.882474 0.470360i \(-0.844124\pi\)
0.882474 0.470360i \(-0.155876\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 4.00000i 0.369800i
\(118\) 10.0000i 0.920575i
\(119\) 0 0
\(120\) 2.00000 1.00000i 0.182574 0.0912871i
\(121\) 5.00000 0.454545
\(122\) 6.00000i 0.543214i
\(123\) 12.0000i 1.08200i
\(124\) 0 0
\(125\) 2.00000 11.0000i 0.178885 0.983870i
\(126\) 0 0
\(127\) 10.0000i 0.887357i 0.896186 + 0.443678i \(0.146327\pi\)
−0.896186 + 0.443678i \(0.853673\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 6.00000 0.528271
\(130\) −8.00000 + 4.00000i −0.701646 + 0.350823i
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 1.00000 + 2.00000i 0.0860663 + 0.172133i
\(136\) 6.00000 0.514496
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.0000i 1.00702i
\(143\) 16.0000i 1.33799i
\(144\) −1.00000 −0.0833333
\(145\) 4.00000 2.00000i 0.332182 0.166091i
\(146\) 8.00000 0.662085
\(147\) 7.00000i 0.577350i
\(148\) 4.00000i 0.328798i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −3.00000 + 4.00000i −0.244949 + 0.326599i
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) 6.00000i 0.478852i −0.970915 0.239426i \(-0.923041\pi\)
0.970915 0.239426i \(-0.0769593\pi\)
\(158\) 8.00000i 0.636446i
\(159\) −14.0000 −1.11027
\(160\) −1.00000 2.00000i −0.0790569 0.158114i
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) 10.0000i 0.783260i −0.920123 0.391630i \(-0.871911\pi\)
0.920123 0.391630i \(-0.128089\pi\)
\(164\) 12.0000 0.937043
\(165\) 4.00000 + 8.00000i 0.311400 + 0.622799i
\(166\) 12.0000 0.931381
\(167\) 12.0000i 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) −12.0000 + 6.00000i −0.920358 + 0.460179i
\(171\) −1.00000 −0.0764719
\(172\) 6.00000i 0.457496i
\(173\) 2.00000i 0.152057i −0.997106 0.0760286i \(-0.975776\pi\)
0.997106 0.0760286i \(-0.0242240\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 10.0000i 0.751646i
\(178\) 8.00000i 0.599625i
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 2.00000 1.00000i 0.149071 0.0745356i
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 6.00000i 0.443533i
\(184\) 0 0
\(185\) 4.00000 + 8.00000i 0.294086 + 0.588172i
\(186\) 0 0
\(187\) 24.0000i 1.75505i
\(188\) 0 0
\(189\) 0 0
\(190\) −1.00000 2.00000i −0.0725476 0.145095i
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 10.0000i 0.719816i −0.932988 0.359908i \(-0.882808\pi\)
0.932988 0.359908i \(-0.117192\pi\)
\(194\) 10.0000 0.717958
\(195\) −8.00000 + 4.00000i −0.572892 + 0.286446i
\(196\) −7.00000 −0.500000
\(197\) 10.0000i 0.712470i 0.934396 + 0.356235i \(0.115940\pi\)
−0.934396 + 0.356235i \(0.884060\pi\)
\(198\) 4.00000i 0.284268i
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 4.00000 + 3.00000i 0.282843 + 0.212132i
\(201\) 4.00000 0.282138
\(202\) 12.0000i 0.844317i
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) −24.0000 + 12.0000i −1.67623 + 0.838116i
\(206\) 2.00000 0.139347
\(207\) 0 0
\(208\) 4.00000i 0.277350i
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 14.0000i 0.961524i
\(213\) 12.0000i 0.822226i
\(214\) −20.0000 −1.36717
\(215\) 6.00000 + 12.0000i 0.409197 + 0.818393i
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 14.0000i 0.948200i
\(219\) 8.00000 0.540590
\(220\) 8.00000 4.00000i 0.539360 0.269680i
\(221\) −24.0000 −1.61441
\(222\) 4.00000i 0.268462i
\(223\) 10.0000i 0.669650i 0.942280 + 0.334825i \(0.108677\pi\)
−0.942280 + 0.334825i \(0.891323\pi\)
\(224\) 0 0
\(225\) −3.00000 + 4.00000i −0.200000 + 0.266667i
\(226\) −10.0000 −0.665190
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 1.00000i 0.0662266i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.00000i 0.131306i
\(233\) 14.0000i 0.917170i 0.888650 + 0.458585i \(0.151644\pi\)
−0.888650 + 0.458585i \(0.848356\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) 8.00000i 0.519656i
\(238\) 0 0
\(239\) −10.0000 −0.646846 −0.323423 0.946254i \(-0.604834\pi\)
−0.323423 + 0.946254i \(0.604834\pi\)
\(240\) −1.00000 2.00000i −0.0645497 0.129099i
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 5.00000i 0.321412i
\(243\) 1.00000i 0.0641500i
\(244\) 6.00000 0.384111
\(245\) 14.0000 7.00000i 0.894427 0.447214i
\(246\) 12.0000 0.765092
\(247\) 4.00000i 0.254514i
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) −11.0000 2.00000i −0.695701 0.126491i
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 10.0000 0.627456
\(255\) −12.0000 + 6.00000i −0.751469 + 0.375735i
\(256\) 1.00000 0.0625000
\(257\) 2.00000i 0.124757i 0.998053 + 0.0623783i \(0.0198685\pi\)
−0.998053 + 0.0623783i \(0.980131\pi\)
\(258\) 6.00000i 0.373544i
\(259\) 0 0
\(260\) 4.00000 + 8.00000i 0.248069 + 0.496139i
\(261\) −2.00000 −0.123797
\(262\) 20.0000i 1.23560i
\(263\) 16.0000i 0.986602i 0.869859 + 0.493301i \(0.164210\pi\)
−0.869859 + 0.493301i \(0.835790\pi\)
\(264\) −4.00000 −0.246183
\(265\) −14.0000 28.0000i −0.860013 1.72003i
\(266\) 0 0
\(267\) 8.00000i 0.489592i
\(268\) 4.00000i 0.244339i
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 2.00000 1.00000i 0.121716 0.0608581i
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 6.00000i 0.363803i
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) −12.0000 + 16.0000i −0.723627 + 0.964836i
\(276\) 0 0
\(277\) 14.0000i 0.841178i 0.907251 + 0.420589i \(0.138177\pi\)
−0.907251 + 0.420589i \(0.861823\pi\)
\(278\) 12.0000i 0.719712i
\(279\) 0 0
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) 2.00000i 0.118888i −0.998232 0.0594438i \(-0.981067\pi\)
0.998232 0.0594438i \(-0.0189327\pi\)
\(284\) −12.0000 −0.712069
\(285\) −1.00000 2.00000i −0.0592349 0.118470i
\(286\) 16.0000 0.946100
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) −2.00000 4.00000i −0.117444 0.234888i
\(291\) 10.0000 0.586210
\(292\) 8.00000i 0.468165i
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) −7.00000 −0.408248
\(295\) 20.0000 10.0000i 1.16445 0.582223i
\(296\) −4.00000 −0.232495
\(297\) 4.00000i 0.232104i
\(298\) 0 0
\(299\) 0 0
\(300\) 4.00000 + 3.00000i 0.230940 + 0.173205i
\(301\) 0 0
\(302\) 16.0000i 0.920697i
\(303\) 12.0000i 0.689382i
\(304\) 1.00000 0.0573539
\(305\) −12.0000 + 6.00000i −0.687118 + 0.343559i
\(306\) 6.00000 0.342997
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 2.00000 0.113776
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 4.00000i 0.226455i
\(313\) 4.00000i 0.226093i 0.993590 + 0.113047i \(0.0360610\pi\)
−0.993590 + 0.113047i \(0.963939\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 18.0000i 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 14.0000i 0.785081i
\(319\) −8.00000 −0.447914
\(320\) −2.00000 + 1.00000i −0.111803 + 0.0559017i
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) 6.00000i 0.333849i
\(324\) −1.00000 −0.0555556
\(325\) −16.0000 12.0000i −0.887520 0.665640i
\(326\) −10.0000 −0.553849
\(327\) 14.0000i 0.774202i
\(328\) 12.0000i 0.662589i
\(329\) 0 0
\(330\) 8.00000 4.00000i 0.440386 0.220193i
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 12.0000i 0.658586i
\(333\) 4.00000i 0.219199i
\(334\) −12.0000 −0.656611
\(335\) 4.00000 + 8.00000i 0.218543 + 0.437087i
\(336\) 0 0
\(337\) 18.0000i 0.980522i −0.871576 0.490261i \(-0.836901\pi\)
0.871576 0.490261i \(-0.163099\pi\)
\(338\) 3.00000i 0.163178i
\(339\) −10.0000 −0.543125
\(340\) 6.00000 + 12.0000i 0.325396 + 0.650791i
\(341\) 0 0
\(342\) 1.00000i 0.0540738i
\(343\) 0 0
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) 32.0000i 1.71785i −0.512101 0.858925i \(-0.671133\pi\)
0.512101 0.858925i \(-0.328867\pi\)
\(348\) 2.00000i 0.107211i
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 4.00000i 0.213201i
\(353\) 22.0000i 1.17094i 0.810693 + 0.585471i \(0.199090\pi\)
−0.810693 + 0.585471i \(0.800910\pi\)
\(354\) −10.0000 −0.531494
\(355\) 24.0000 12.0000i 1.27379 0.636894i
\(356\) −8.00000 −0.423999
\(357\) 0 0
\(358\) 6.00000i 0.317110i
\(359\) −34.0000 −1.79445 −0.897226 0.441572i \(-0.854421\pi\)
−0.897226 + 0.441572i \(0.854421\pi\)
\(360\) −1.00000 2.00000i −0.0527046 0.105409i
\(361\) 1.00000 0.0526316
\(362\) 10.0000i 0.525588i
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) 8.00000 + 16.0000i 0.418739 + 0.837478i
\(366\) 6.00000 0.313625
\(367\) 12.0000i 0.626395i 0.949688 + 0.313197i \(0.101400\pi\)
−0.949688 + 0.313197i \(0.898600\pi\)
\(368\) 0 0
\(369\) 12.0000 0.624695
\(370\) 8.00000 4.00000i 0.415900 0.207950i
\(371\) 0 0
\(372\) 0 0
\(373\) 12.0000i 0.621336i 0.950518 + 0.310668i \(0.100553\pi\)
−0.950518 + 0.310668i \(0.899447\pi\)
\(374\) 24.0000 1.24101
\(375\) −11.0000 2.00000i −0.568038 0.103280i
\(376\) 0 0
\(377\) 8.00000i 0.412021i
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) −2.00000 + 1.00000i −0.102598 + 0.0512989i
\(381\) 10.0000 0.512316
\(382\) 6.00000i 0.306987i
\(383\) 36.0000i 1.83951i 0.392488 + 0.919757i \(0.371614\pi\)
−0.392488 + 0.919757i \(0.628386\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 6.00000i 0.304997i
\(388\) 10.0000i 0.507673i
\(389\) 4.00000 0.202808 0.101404 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) 4.00000 + 8.00000i 0.202548 + 0.405096i
\(391\) 0 0
\(392\) 7.00000i 0.353553i
\(393\) 20.0000i 1.00887i
\(394\) 10.0000 0.503793
\(395\) 16.0000 8.00000i 0.805047 0.402524i
\(396\) −4.00000 −0.201008
\(397\) 18.0000i 0.903394i −0.892171 0.451697i \(-0.850819\pi\)
0.892171 0.451697i \(-0.149181\pi\)
\(398\) 4.00000i 0.200502i
\(399\) 0 0
\(400\) 3.00000 4.00000i 0.150000 0.200000i
\(401\) 28.0000 1.39825 0.699127 0.714998i \(-0.253572\pi\)
0.699127 + 0.714998i \(0.253572\pi\)
\(402\) 4.00000i 0.199502i
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) 2.00000 1.00000i 0.0993808 0.0496904i
\(406\) 0 0
\(407\) 16.0000i 0.793091i
\(408\) 6.00000i 0.297044i
\(409\) 34.0000 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 12.0000 + 24.0000i 0.592638 + 1.18528i
\(411\) 6.00000 0.295958
\(412\) 2.00000i 0.0985329i
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000 + 24.0000i 0.589057 + 1.17811i
\(416\) −4.00000 −0.196116
\(417\) 12.0000i 0.587643i
\(418\) 4.00000i 0.195646i
\(419\) −8.00000 −0.390826 −0.195413 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) 8.00000i 0.389434i
\(423\) 0 0
\(424\) 14.0000 0.679900
\(425\) −24.0000 18.0000i −1.16417 0.873128i
\(426\) −12.0000 −0.581402
\(427\) 0 0
\(428\) 20.0000i 0.966736i
\(429\) 16.0000 0.772487
\(430\) 12.0000 6.00000i 0.578691 0.289346i
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 18.0000i 0.865025i −0.901628 0.432512i \(-0.857627\pi\)
0.901628 0.432512i \(-0.142373\pi\)
\(434\) 0 0
\(435\) −2.00000 4.00000i −0.0958927 0.191785i
\(436\) −14.0000 −0.670478
\(437\) 0 0
\(438\) 8.00000i 0.382255i
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) −4.00000 8.00000i −0.190693 0.381385i
\(441\) −7.00000 −0.333333
\(442\) 24.0000i 1.14156i
\(443\) 12.0000i 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) −4.00000 −0.189832
\(445\) 16.0000 8.00000i 0.758473 0.379236i
\(446\) 10.0000 0.473514
\(447\) 0 0
\(448\) 0 0
\(449\) 32.0000 1.51017 0.755087 0.655625i \(-0.227595\pi\)
0.755087 + 0.655625i \(0.227595\pi\)
\(450\) 4.00000 + 3.00000i 0.188562 + 0.141421i
\(451\) 48.0000 2.26023
\(452\) 10.0000i 0.470360i
\(453\) 16.0000i 0.751746i
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 32.0000i 1.49690i 0.663193 + 0.748448i \(0.269201\pi\)
−0.663193 + 0.748448i \(0.730799\pi\)
\(458\) 10.0000i 0.467269i
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 24.0000 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) 24.0000i 1.11059i 0.831654 + 0.555294i \(0.187394\pi\)
−0.831654 + 0.555294i \(0.812606\pi\)
\(468\) 4.00000i 0.184900i
\(469\) 0 0
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 10.0000i 0.460287i
\(473\) 24.0000i 1.10352i
\(474\) −8.00000 −0.367452
\(475\) 3.00000 4.00000i 0.137649 0.183533i
\(476\) 0 0
\(477\) 14.0000i 0.641016i
\(478\) 10.0000i 0.457389i
\(479\) −38.0000 −1.73626 −0.868132 0.496333i \(-0.834679\pi\)
−0.868132 + 0.496333i \(0.834679\pi\)
\(480\) −2.00000 + 1.00000i −0.0912871 + 0.0456435i
\(481\) 16.0000 0.729537
\(482\) 26.0000i 1.18427i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 10.0000 + 20.0000i 0.454077 + 0.908153i
\(486\) −1.00000 −0.0453609
\(487\) 18.0000i 0.815658i 0.913058 + 0.407829i \(0.133714\pi\)
−0.913058 + 0.407829i \(0.866286\pi\)
\(488\) 6.00000i 0.271607i
\(489\) −10.0000 −0.452216
\(490\) −7.00000 14.0000i −0.316228 0.632456i
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 12.0000i 0.541002i
\(493\) 12.0000i 0.540453i
\(494\) −4.00000 −0.179969
\(495\) 8.00000 4.00000i 0.359573 0.179787i
\(496\) 0 0
\(497\) 0 0
\(498\) 12.0000i 0.537733i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) −2.00000 + 11.0000i −0.0894427 + 0.491935i
\(501\) −12.0000 −0.536120
\(502\) 24.0000i 1.07117i
\(503\) 16.0000i 0.713405i 0.934218 + 0.356702i \(0.116099\pi\)
−0.934218 + 0.356702i \(0.883901\pi\)
\(504\) 0 0
\(505\) 24.0000 12.0000i 1.06799 0.533993i
\(506\) 0 0
\(507\) 3.00000i 0.133235i
\(508\) 10.0000i 0.443678i
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 6.00000 + 12.0000i 0.265684 + 0.531369i
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 1.00000i 0.0441511i
\(514\) 2.00000 0.0882162
\(515\) 2.00000 + 4.00000i 0.0881305 + 0.176261i
\(516\) −6.00000 −0.264135
\(517\) 0 0
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 8.00000 4.00000i 0.350823 0.175412i
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 0 0
\(528\) 4.00000i 0.174078i
\(529\) 23.0000 1.00000
\(530\) −28.0000 + 14.0000i −1.21624 + 0.608121i
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) 48.0000i 2.07911i
\(534\) −8.00000 −0.346194
\(535\) −20.0000 40.0000i −0.864675 1.72935i
\(536\) −4.00000 −0.172774
\(537\) 6.00000i 0.258919i
\(538\) 2.00000i 0.0862261i
\(539\) −28.0000 −1.20605
\(540\) −1.00000 2.00000i −0.0430331 0.0860663i
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 28.0000i 1.20270i
\(543\) 10.0000i 0.429141i
\(544\) −6.00000 −0.257248
\(545\) 28.0000 14.0000i 1.19939 0.599694i
\(546\) 0 0
\(547\) 24.0000i 1.02617i −0.858339 0.513083i \(-0.828503\pi\)
0.858339 0.513083i \(-0.171497\pi\)
\(548\) 6.00000i 0.256307i
\(549\) 6.00000 0.256074
\(550\) 16.0000 + 12.0000i 0.682242 + 0.511682i
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) 0 0
\(554\) 14.0000 0.594803
\(555\) 8.00000 4.00000i 0.339581 0.169791i
\(556\) −12.0000 −0.508913
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 12.0000i 0.506189i
\(563\) 36.0000i 1.51722i −0.651546 0.758610i \(-0.725879\pi\)
0.651546 0.758610i \(-0.274121\pi\)
\(564\) 0 0
\(565\) −10.0000 20.0000i −0.420703 0.841406i
\(566\) −2.00000 −0.0840663
\(567\) 0 0
\(568\) 12.0000i 0.503509i
\(569\) −32.0000 −1.34151 −0.670755 0.741679i \(-0.734030\pi\)
−0.670755 + 0.741679i \(0.734030\pi\)
\(570\) −2.00000 + 1.00000i −0.0837708 + 0.0418854i
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 16.0000i 0.668994i
\(573\) 6.00000i 0.250654i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 8.00000i 0.333044i 0.986038 + 0.166522i \(0.0532537\pi\)
−0.986038 + 0.166522i \(0.946746\pi\)
\(578\) 19.0000i 0.790296i
\(579\) −10.0000 −0.415586
\(580\) −4.00000 + 2.00000i −0.166091 + 0.0830455i
\(581\) 0 0
\(582\) 10.0000i 0.414513i
\(583\) 56.0000i 2.31928i
\(584\) −8.00000 −0.331042
\(585\) 4.00000 + 8.00000i 0.165380 + 0.330759i
\(586\) 14.0000 0.578335
\(587\) 32.0000i 1.32078i −0.750922 0.660391i \(-0.770391\pi\)
0.750922 0.660391i \(-0.229609\pi\)
\(588\) 7.00000i 0.288675i
\(589\) 0 0
\(590\) −10.0000 20.0000i −0.411693 0.823387i
\(591\) 10.0000 0.411345
\(592\) 4.00000i 0.164399i
\(593\) 22.0000i 0.903432i −0.892162 0.451716i \(-0.850812\pi\)
0.892162 0.451716i \(-0.149188\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 0 0
\(597\) 4.00000i 0.163709i
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 3.00000 4.00000i 0.122474 0.163299i
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) −16.0000 −0.651031
\(605\) 10.0000 5.00000i 0.406558 0.203279i
\(606\) −12.0000 −0.487467
\(607\) 30.0000i 1.21766i −0.793300 0.608831i \(-0.791639\pi\)
0.793300 0.608831i \(-0.208361\pi\)
\(608\) 1.00000i 0.0405554i
\(609\) 0 0
\(610\) 6.00000 + 12.0000i 0.242933 + 0.485866i
\(611\) 0 0
\(612\) 6.00000i 0.242536i
\(613\) 46.0000i 1.85792i −0.370177 0.928961i \(-0.620703\pi\)
0.370177 0.928961i \(-0.379297\pi\)
\(614\) 0 0
\(615\) 12.0000 + 24.0000i 0.483887 + 0.967773i
\(616\) 0 0
\(617\) 6.00000i 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) 2.00000i 0.0804518i
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 18.0000i 0.721734i
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 4.00000 0.159872
\(627\) 4.00000i 0.159745i
\(628\) 6.00000i 0.239426i
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 28.0000 1.11466 0.557331 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) 8.00000i 0.318223i
\(633\) 8.00000i 0.317971i
\(634\) −18.0000 −0.714871
\(635\) 10.0000 + 20.0000i 0.396838 + 0.793676i
\(636\) 14.0000 0.555136
\(637\) 28.0000i 1.10940i
\(638\) 8.00000i 0.316723i
\(639\) −12.0000 −0.474713
\(640\) 1.00000 + 2.00000i 0.0395285 + 0.0790569i
\(641\) 36.0000 1.42191 0.710957 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) 20.0000i 0.789337i
\(643\) 26.0000i 1.02534i −0.858586 0.512670i \(-0.828656\pi\)
0.858586 0.512670i \(-0.171344\pi\)
\(644\) 0 0
\(645\) 12.0000 6.00000i 0.472500 0.236250i
\(646\) −6.00000 −0.236067
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −40.0000 −1.57014
\(650\) −12.0000 + 16.0000i −0.470679 + 0.627572i
\(651\) 0 0
\(652\) 10.0000i 0.391630i
\(653\) 18.0000i 0.704394i 0.935926 + 0.352197i \(0.114565\pi\)
−0.935926 + 0.352197i \(0.885435\pi\)
\(654\) −14.0000 −0.547443
\(655\) −40.0000 + 20.0000i −1.56293 + 0.781465i
\(656\) −12.0000 −0.468521
\(657\) 8.00000i 0.312110i
\(658\) 0 0
\(659\) 26.0000 1.01282 0.506408 0.862294i \(-0.330973\pi\)
0.506408 + 0.862294i \(0.330973\pi\)
\(660\) −4.00000 8.00000i −0.155700 0.311400i
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 8.00000i 0.310929i
\(663\) 24.0000i 0.932083i
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 0 0
\(668\) 12.0000i 0.464294i
\(669\) 10.0000 0.386622
\(670\) 8.00000 4.00000i 0.309067 0.154533i
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) 26.0000i 1.00223i 0.865382 + 0.501113i \(0.167076\pi\)
−0.865382 + 0.501113i \(0.832924\pi\)
\(674\) −18.0000 −0.693334
\(675\) 4.00000 + 3.00000i 0.153960 + 0.115470i
\(676\) 3.00000 0.115385
\(677\) 30.0000i 1.15299i 0.817099 + 0.576497i \(0.195581\pi\)
−0.817099 + 0.576497i \(0.804419\pi\)
\(678\) 10.0000i 0.384048i
\(679\) 0 0
\(680\) 12.0000 6.00000i 0.460179 0.230089i
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 12.0000i 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 1.00000 0.0382360
\(685\) 6.00000 + 12.0000i 0.229248 + 0.458496i
\(686\) 0 0
\(687\) 10.0000i 0.381524i
\(688\) 6.00000i 0.228748i
\(689\) −56.0000 −2.13343
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 2.00000i 0.0760286i
\(693\) 0 0
\(694\) −32.0000 −1.21470
\(695\) 24.0000 12.0000i 0.910372 0.455186i
\(696\) 2.00000 0.0758098
\(697\) 72.0000i 2.72719i
\(698\) 34.0000i 1.28692i
\(699\) 14.0000 0.529529
\(700\) 0 0
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 4.00000i 0.150970i
\(703\) 4.00000i 0.150863i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 22.0000 0.827981
\(707\) 0 0
\(708\) 10.0000i 0.375823i
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) −12.0000 24.0000i −0.450352 0.900704i
\(711\) −8.00000 −0.300023
\(712\) 8.00000i 0.299813i
\(713\) 0 0
\(714\) 0 0
\(715\) 16.0000 + 32.0000i 0.598366 + 1.19673i
\(716\) 6.00000 0.224231
\(717\) 10.0000i 0.373457i
\(718\) 34.0000i 1.26887i
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) −2.00000 + 1.00000i −0.0745356 + 0.0372678i
\(721\) 0 0
\(722\) 1.00000i 0.0372161i
\(723\) 26.0000i 0.966950i
\(724\) 10.0000 0.371647
\(725\) 6.00000 8.00000i 0.222834 0.297113i
\(726\) −5.00000 −0.185567
\(727\) 40.0000i 1.48352i 0.670667 + 0.741759i \(0.266008\pi\)
−0.670667 + 0.741759i \(0.733992\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 16.0000 8.00000i 0.592187 0.296093i
\(731\) 36.0000 1.33151
\(732\) 6.00000i 0.221766i
\(733\) 26.0000i 0.960332i −0.877178 0.480166i \(-0.840576\pi\)
0.877178 0.480166i \(-0.159424\pi\)
\(734\) 12.0000 0.442928
\(735\) −7.00000 14.0000i −0.258199 0.516398i
\(736\) 0 0
\(737\) 16.0000i 0.589368i
\(738\) 12.0000i 0.441726i
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) −4.00000 8.00000i −0.147043 0.294086i
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) 44.0000i 1.61420i −0.590412 0.807102i \(-0.701035\pi\)
0.590412 0.807102i \(-0.298965\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 12.0000 0.439351
\(747\) 12.0000i 0.439057i
\(748\) 24.0000i 0.877527i
\(749\) 0 0
\(750\) −2.00000 + 11.0000i −0.0730297 + 0.401663i
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 24.0000i 0.874609i
\(754\) −8.00000 −0.291343
\(755\) 32.0000 16.0000i 1.16460 0.582300i
\(756\) 0 0
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 12.0000i 0.435860i
\(759\) 0 0
\(760\) 1.00000 + 2.00000i 0.0362738 + 0.0725476i
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 10.0000i 0.362262i
\(763\) 0 0
\(764\) −6.00000 −0.217072
\(765\) 6.00000 + 12.0000i 0.216930 + 0.433861i
\(766\) 36.0000 1.30073
\(767\) 40.0000i 1.44432i
\(768\) 1.00000i 0.0360844i
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) 10.0000i 0.359908i
\(773\) 38.0000i 1.36677i −0.730061 0.683383i \(-0.760508\pi\)
0.730061 0.683383i \(-0.239492\pi\)
\(774\) −6.00000 −0.215666
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 4.00000i 0.143407i
\(779\) −12.0000 −0.429945
\(780\) 8.00000 4.00000i 0.286446 0.143223i
\(781\) −48.0000 −1.71758
\(782\) 0 0
\(783\) 2.00000i 0.0714742i
\(784\) 7.00000 0.250000
\(785\) −6.00000 12.0000i −0.214149 0.428298i
\(786\) 20.0000 0.713376
\(787\) 48.0000i 1.71102i −0.517790 0.855508i \(-0.673245\pi\)
0.517790 0.855508i \(-0.326755\pi\)
\(788\) 10.0000i 0.356235i
\(789\) 16.0000 0.569615
\(790\) −8.00000 16.0000i −0.284627 0.569254i
\(791\) 0 0
\(792\) 4.00000i 0.142134i
\(793\) 24.0000i 0.852265i
\(794\) −18.0000 −0.638796
\(795\) −28.0000 + 14.0000i −0.993058 + 0.496529i
\(796\) −4.00000 −0.141776
\(797\) 42.0000i 1.48772i 0.668338 + 0.743858i \(0.267006\pi\)
−0.668338 + 0.743858i \(0.732994\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −4.00000 3.00000i −0.141421 0.106066i
\(801\) −8.00000 −0.282666
\(802\) 28.0000i 0.988714i
\(803\) 32.0000i 1.12926i
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) 2.00000i 0.0704033i
\(808\) 12.0000i 0.422159i
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) −1.00000 2.00000i −0.0351364 0.0702728i
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 0 0
\(813\) 28.0000i 0.982003i
\(814\) −16.0000 −0.560800
\(815\) −10.0000 20.0000i −0.350285 0.700569i
\(816\) −6.00000 −0.210042
\(817\) 6.00000i 0.209913i
\(818\) 34.0000i 1.18878i
\(819\) 0 0
\(820\) 24.0000 12.0000i 0.838116 0.419058i
\(821\) −4.00000 −0.139601 −0.0698005 0.997561i \(-0.522236\pi\)
−0.0698005 + 0.997561i \(0.522236\pi\)
\(822\) 6.00000i 0.209274i
\(823\) 24.0000i 0.836587i −0.908312 0.418294i \(-0.862628\pi\)
0.908312 0.418294i \(-0.137372\pi\)
\(824\) −2.00000 −0.0696733
\(825\) 16.0000 + 12.0000i 0.557048 + 0.417786i
\(826\) 0 0
\(827\) 12.0000i 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) 0 0
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 24.0000 12.0000i 0.833052 0.416526i
\(831\) 14.0000 0.485655
\(832\) 4.00000i 0.138675i
\(833\) 42.0000i 1.45521i
\(834\) −12.0000 −0.415526
\(835\) −12.0000 24.0000i −0.415277 0.830554i
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) 8.00000i 0.276355i
\(839\) −8.00000 −0.276191 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 14.0000i 0.482472i
\(843\) 12.0000i 0.413302i
\(844\) 8.00000 0.275371
\(845\) −6.00000 + 3.00000i −0.206406 + 0.103203i
\(846\) 0 0
\(847\) 0 0
\(848\) 14.0000i 0.480762i
\(849\) −2.00000 −0.0686398
\(850\) −18.0000 + 24.0000i −0.617395 + 0.823193i
\(851\) 0 0
\(852\) 12.0000i 0.411113i
\(853\) 50.0000i 1.71197i −0.517003 0.855984i \(-0.672952\pi\)
0.517003 0.855984i \(-0.327048\pi\)
\(854\) 0 0
\(855\) −2.00000 + 1.00000i −0.0683986 + 0.0341993i
\(856\) 20.0000 0.683586
\(857\) 42.0000i 1.43469i 0.696717 + 0.717346i \(0.254643\pi\)
−0.696717 + 0.717346i \(0.745357\pi\)
\(858\) 16.0000i 0.546231i
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) −6.00000 12.0000i −0.204598 0.409197i
\(861\) 0 0
\(862\) 12.0000i 0.408722i
\(863\) 24.0000i 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 1.00000 0.0340207
\(865\) −2.00000 4.00000i −0.0680020 0.136004i
\(866\) −18.0000 −0.611665
\(867\) 19.0000i 0.645274i
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) −4.00000 + 2.00000i −0.135613 + 0.0678064i
\(871\) 16.0000 0.542139
\(872\) 14.0000i 0.474100i
\(873\) 10.0000i 0.338449i
\(874\) 0 0
\(875\) 0 0
\(876\) −8.00000 −0.270295
\(877\) 28.0000i 0.945493i 0.881199 + 0.472746i \(0.156737\pi\)
−0.881199 + 0.472746i \(0.843263\pi\)
\(878\) 16.0000i 0.539974i
\(879\) 14.0000 0.472208
\(880\) −8.00000 + 4.00000i −0.269680 + 0.134840i
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 7.00000i 0.235702i
\(883\) 2.00000i 0.0673054i −0.999434 0.0336527i \(-0.989286\pi\)
0.999434 0.0336527i \(-0.0107140\pi\)
\(884\) 24.0000 0.807207
\(885\) −10.0000 20.0000i −0.336146 0.672293i
\(886\) −12.0000 −0.403148
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 4.00000i 0.134231i
\(889\) 0 0
\(890\) −8.00000 16.0000i −0.268161 0.536321i
\(891\) −4.00000 −0.134005
\(892\) 10.0000i 0.334825i
\(893\) 0 0
\(894\) 0 0
\(895\) −12.0000 + 6.00000i −0.401116 + 0.200558i
\(896\) 0 0
\(897\) 0 0
\(898\) 32.0000i 1.06785i
\(899\) 0 0
\(900\) 3.00000 4.00000i 0.100000 0.133333i
\(901\) −84.0000 −2.79845
\(902\) 48.0000i 1.59823i
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) −20.0000 + 10.0000i −0.664822 + 0.332411i
\(906\) −16.0000 −0.531564
\(907\) 12.0000i 0.398453i −0.979953 0.199227i \(-0.936157\pi\)
0.979953 0.199227i \(-0.0638430\pi\)
\(908\) 12.0000i 0.398234i
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) 48.0000i 1.58857i
\(914\) 32.0000 1.05847
\(915\) 6.00000 + 12.0000i 0.198354 + 0.396708i
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 6.00000i 0.198030i
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 24.0000i 0.790398i
\(923\) 48.0000i 1.57994i
\(924\) 0 0
\(925\) 16.0000 + 12.0000i 0.526077 + 0.394558i
\(926\) 16.0000 0.525793
\(927\) 2.00000i 0.0656886i
\(928\) 2.00000i 0.0656532i
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) 14.0000i 0.458585i
\(933\) 18.0000i 0.589294i
\(934\) 24.0000 0.785304
\(935\) 24.0000 + 48.0000i 0.784884 + 1.56977i
\(936\) −4.00000 −0.130744
\(937\) 20.0000i 0.653372i 0.945133 + 0.326686i \(0.105932\pi\)
−0.945133 + 0.326686i \(0.894068\pi\)
\(938\) 0 0
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) 58.0000 1.89075 0.945373 0.325991i \(-0.105698\pi\)
0.945373 + 0.325991i \(0.105698\pi\)
\(942\) 6.00000i 0.195491i
\(943\) 0 0
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) 40.0000i 1.29983i −0.760009 0.649913i \(-0.774805\pi\)
0.760009 0.649913i \(-0.225195\pi\)
\(948\) 8.00000i 0.259828i
\(949\) 32.0000 1.03876
\(950\) −4.00000 3.00000i −0.129777 0.0973329i
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 42.0000i 1.36051i 0.732974 + 0.680257i \(0.238132\pi\)
−0.732974 + 0.680257i \(0.761868\pi\)
\(954\) 14.0000 0.453267
\(955\) 12.0000 6.00000i 0.388311 0.194155i
\(956\) 10.0000 0.323423
\(957\) 8.00000i 0.258603i
\(958\) 38.0000i 1.22772i
\(959\) 0 0
\(960\) 1.00000 + 2.00000i 0.0322749 + 0.0645497i
\(961\) −31.0000 −1.00000
\(962\) 16.0000i 0.515861i
\(963\) 20.0000i 0.644491i
\(964\) −26.0000 −0.837404
\(965\) −10.0000 20.0000i −0.321911 0.643823i
\(966\) 0 0
\(967\) 44.0000i 1.41494i 0.706741 + 0.707472i \(0.250165\pi\)
−0.706741 + 0.707472i \(0.749835\pi\)
\(968\) 5.00000i 0.160706i
\(969\) −6.00000 −0.192748
\(970\) 20.0000 10.0000i 0.642161 0.321081i
\(971\) 38.0000 1.21948 0.609739 0.792602i \(-0.291274\pi\)
0.609739 + 0.792602i \(0.291274\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 0 0
\(974\) 18.0000 0.576757
\(975\) −12.0000 + 16.0000i −0.384308 + 0.512410i
\(976\) −6.00000 −0.192055
\(977\) 34.0000i 1.08776i −0.839164 0.543878i \(-0.816955\pi\)
0.839164 0.543878i \(-0.183045\pi\)
\(978\) 10.0000i 0.319765i
\(979\) −32.0000 −1.02272
\(980\) −14.0000 + 7.00000i −0.447214 + 0.223607i
\(981\) −14.0000 −0.446986
\(982\) 36.0000i 1.14881i
\(983\) 56.0000i 1.78612i 0.449935 + 0.893061i \(0.351447\pi\)
−0.449935 + 0.893061i \(0.648553\pi\)
\(984\) −12.0000 −0.382546
\(985\) 10.0000 + 20.0000i 0.318626 + 0.637253i
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 4.00000i 0.127257i
\(989\) 0 0
\(990\) −4.00000 8.00000i −0.127128 0.254257i
\(991\) 48.0000 1.52477 0.762385 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) 0 0
\(993\) 8.00000i 0.253872i
\(994\) 0 0
\(995\) 8.00000 4.00000i 0.253617 0.126809i
\(996\) −12.0000 −0.380235
\(997\) 2.00000i 0.0633406i 0.999498 + 0.0316703i \(0.0100827\pi\)
−0.999498 + 0.0316703i \(0.989917\pi\)
\(998\) 4.00000i 0.126618i
\(999\) −4.00000 −0.126554
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 570.2.d.b.229.1 2
3.2 odd 2 1710.2.d.a.1369.2 2
5.2 odd 4 2850.2.a.s.1.1 1
5.3 odd 4 2850.2.a.k.1.1 1
5.4 even 2 inner 570.2.d.b.229.2 yes 2
15.2 even 4 8550.2.a.j.1.1 1
15.8 even 4 8550.2.a.bc.1.1 1
15.14 odd 2 1710.2.d.a.1369.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.d.b.229.1 2 1.1 even 1 trivial
570.2.d.b.229.2 yes 2 5.4 even 2 inner
1710.2.d.a.1369.1 2 15.14 odd 2
1710.2.d.a.1369.2 2 3.2 odd 2
2850.2.a.k.1.1 1 5.3 odd 4
2850.2.a.s.1.1 1 5.2 odd 4
8550.2.a.j.1.1 1 15.2 even 4
8550.2.a.bc.1.1 1 15.8 even 4