Properties

Label 3234.2.e.c.2155.8
Level $3234$
Weight $2$
Character 3234.2155
Analytic conductor $25.824$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3234,2,Mod(2155,3234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3234, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3234.2155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.e (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: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2155.8
Character \(\chi\) \(=\) 3234.2155
Dual form 3234.2.e.c.2155.17

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +0.357203i q^{5} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +0.357203i q^{5} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +0.357203 q^{10} +(-3.31411 - 0.129255i) q^{11} +1.00000i q^{12} +4.33393 q^{13} +0.357203 q^{15} +1.00000 q^{16} +0.347231 q^{17} +1.00000i q^{18} -7.81849 q^{19} -0.357203i q^{20} +(-0.129255 + 3.31411i) q^{22} +6.36865 q^{23} +1.00000 q^{24} +4.87241 q^{25} -4.33393i q^{26} +1.00000i q^{27} +7.67778i q^{29} -0.357203i q^{30} +6.20254i q^{31} -1.00000i q^{32} +(-0.129255 + 3.31411i) q^{33} -0.347231i q^{34} +1.00000 q^{36} -8.95377 q^{37} +7.81849i q^{38} -4.33393i q^{39} -0.357203 q^{40} +5.43717 q^{41} -2.91740i q^{43} +(3.31411 + 0.129255i) q^{44} -0.357203i q^{45} -6.36865i q^{46} +3.84135i q^{47} -1.00000i q^{48} -4.87241i q^{50} -0.347231i q^{51} -4.33393 q^{52} -2.73623 q^{53} +1.00000 q^{54} +(0.0461702 - 1.18381i) q^{55} +7.81849i q^{57} +7.67778 q^{58} +0.911386i q^{59} -0.357203 q^{60} +12.4897 q^{61} +6.20254 q^{62} -1.00000 q^{64} +1.54809i q^{65} +(3.31411 + 0.129255i) q^{66} -13.6448 q^{67} -0.347231 q^{68} -6.36865i q^{69} +6.18162 q^{71} -1.00000i q^{72} +11.4503 q^{73} +8.95377i q^{74} -4.87241i q^{75} +7.81849 q^{76} -4.33393 q^{78} -6.65938i q^{79} +0.357203i q^{80} +1.00000 q^{81} -5.43717i q^{82} +16.7247 q^{83} +0.124032i q^{85} -2.91740 q^{86} +7.67778 q^{87} +(0.129255 - 3.31411i) q^{88} +14.5026i q^{89} -0.357203 q^{90} -6.36865 q^{92} +6.20254 q^{93} +3.84135 q^{94} -2.79279i q^{95} -1.00000 q^{96} +18.1486i q^{97} +(3.31411 + 0.129255i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 24 q^{4} - 24 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 24 q^{4} - 24 q^{6} - 24 q^{9} + 24 q^{16} + 16 q^{17} - 32 q^{19} - 8 q^{22} + 24 q^{24} - 8 q^{25} - 8 q^{33} + 24 q^{36} + 16 q^{37} - 16 q^{41} + 24 q^{54} + 16 q^{55} - 16 q^{62} - 24 q^{64} - 64 q^{67} - 16 q^{68} + 64 q^{71} + 32 q^{76} + 24 q^{81} - 16 q^{83} + 8 q^{88} - 16 q^{93} + 64 q^{94} - 24 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3234\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(1079\) \(2059\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(5\) 0.357203i 0.159746i 0.996805 + 0.0798729i \(0.0254515\pi\)
−0.996805 + 0.0798729i \(0.974549\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0.357203 0.112957
\(11\) −3.31411 0.129255i −0.999240 0.0389718i
\(12\) 1.00000i 0.288675i
\(13\) 4.33393 1.20202 0.601008 0.799243i \(-0.294766\pi\)
0.601008 + 0.799243i \(0.294766\pi\)
\(14\) 0 0
\(15\) 0.357203 0.0922293
\(16\) 1.00000 0.250000
\(17\) 0.347231 0.0842160 0.0421080 0.999113i \(-0.486593\pi\)
0.0421080 + 0.999113i \(0.486593\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −7.81849 −1.79369 −0.896843 0.442349i \(-0.854145\pi\)
−0.896843 + 0.442349i \(0.854145\pi\)
\(20\) 0.357203i 0.0798729i
\(21\) 0 0
\(22\) −0.129255 + 3.31411i −0.0275572 + 0.706570i
\(23\) 6.36865 1.32795 0.663977 0.747753i \(-0.268867\pi\)
0.663977 + 0.747753i \(0.268867\pi\)
\(24\) 1.00000 0.204124
\(25\) 4.87241 0.974481
\(26\) 4.33393i 0.849954i
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 7.67778i 1.42573i 0.701302 + 0.712864i \(0.252602\pi\)
−0.701302 + 0.712864i \(0.747398\pi\)
\(30\) 0.357203i 0.0652160i
\(31\) 6.20254i 1.11401i 0.830510 + 0.557004i \(0.188049\pi\)
−0.830510 + 0.557004i \(0.811951\pi\)
\(32\) 1.00000i 0.176777i
\(33\) −0.129255 + 3.31411i −0.0225004 + 0.576912i
\(34\) 0.347231i 0.0595497i
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.95377 −1.47199 −0.735995 0.676987i \(-0.763285\pi\)
−0.735995 + 0.676987i \(0.763285\pi\)
\(38\) 7.81849i 1.26833i
\(39\) 4.33393i 0.693985i
\(40\) −0.357203 −0.0564787
\(41\) 5.43717 0.849143 0.424572 0.905394i \(-0.360425\pi\)
0.424572 + 0.905394i \(0.360425\pi\)
\(42\) 0 0
\(43\) 2.91740i 0.444899i −0.974944 0.222450i \(-0.928595\pi\)
0.974944 0.222450i \(-0.0714053\pi\)
\(44\) 3.31411 + 0.129255i 0.499620 + 0.0194859i
\(45\) 0.357203i 0.0532486i
\(46\) 6.36865i 0.939006i
\(47\) 3.84135i 0.560318i 0.959954 + 0.280159i \(0.0903872\pi\)
−0.959954 + 0.280159i \(0.909613\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 0 0
\(50\) 4.87241i 0.689062i
\(51\) 0.347231i 0.0486221i
\(52\) −4.33393 −0.601008
\(53\) −2.73623 −0.375850 −0.187925 0.982183i \(-0.560176\pi\)
−0.187925 + 0.982183i \(0.560176\pi\)
\(54\) 1.00000 0.136083
\(55\) 0.0461702 1.18381i 0.00622558 0.159624i
\(56\) 0 0
\(57\) 7.81849i 1.03558i
\(58\) 7.67778 1.00814
\(59\) 0.911386i 0.118652i 0.998239 + 0.0593261i \(0.0188952\pi\)
−0.998239 + 0.0593261i \(0.981105\pi\)
\(60\) −0.357203 −0.0461147
\(61\) 12.4897 1.59914 0.799570 0.600572i \(-0.205061\pi\)
0.799570 + 0.600572i \(0.205061\pi\)
\(62\) 6.20254 0.787723
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 1.54809i 0.192017i
\(66\) 3.31411 + 0.129255i 0.407938 + 0.0159102i
\(67\) −13.6448 −1.66697 −0.833487 0.552538i \(-0.813659\pi\)
−0.833487 + 0.552538i \(0.813659\pi\)
\(68\) −0.347231 −0.0421080
\(69\) 6.36865i 0.766695i
\(70\) 0 0
\(71\) 6.18162 0.733623 0.366812 0.930295i \(-0.380449\pi\)
0.366812 + 0.930295i \(0.380449\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 11.4503 1.34015 0.670076 0.742292i \(-0.266261\pi\)
0.670076 + 0.742292i \(0.266261\pi\)
\(74\) 8.95377i 1.04085i
\(75\) 4.87241i 0.562617i
\(76\) 7.81849 0.896843
\(77\) 0 0
\(78\) −4.33393 −0.490721
\(79\) 6.65938i 0.749238i −0.927179 0.374619i \(-0.877773\pi\)
0.927179 0.374619i \(-0.122227\pi\)
\(80\) 0.357203i 0.0399365i
\(81\) 1.00000 0.111111
\(82\) 5.43717i 0.600435i
\(83\) 16.7247 1.83577 0.917886 0.396845i \(-0.129895\pi\)
0.917886 + 0.396845i \(0.129895\pi\)
\(84\) 0 0
\(85\) 0.124032i 0.0134531i
\(86\) −2.91740 −0.314591
\(87\) 7.67778 0.823145
\(88\) 0.129255 3.31411i 0.0137786 0.353285i
\(89\) 14.5026i 1.53728i 0.639684 + 0.768638i \(0.279065\pi\)
−0.639684 + 0.768638i \(0.720935\pi\)
\(90\) −0.357203 −0.0376525
\(91\) 0 0
\(92\) −6.36865 −0.663977
\(93\) 6.20254 0.643173
\(94\) 3.84135 0.396205
\(95\) 2.79279i 0.286534i
\(96\) −1.00000 −0.102062
\(97\) 18.1486i 1.84271i 0.388725 + 0.921354i \(0.372915\pi\)
−0.388725 + 0.921354i \(0.627085\pi\)
\(98\) 0 0
\(99\) 3.31411 + 0.129255i 0.333080 + 0.0129906i
\(100\) −4.87241 −0.487241
\(101\) −7.40672 −0.736996 −0.368498 0.929629i \(-0.620128\pi\)
−0.368498 + 0.929629i \(0.620128\pi\)
\(102\) −0.347231 −0.0343810
\(103\) 0.972851i 0.0958579i 0.998851 + 0.0479289i \(0.0152621\pi\)
−0.998851 + 0.0479289i \(0.984738\pi\)
\(104\) 4.33393i 0.424977i
\(105\) 0 0
\(106\) 2.73623i 0.265766i
\(107\) 4.11580i 0.397889i −0.980011 0.198945i \(-0.936249\pi\)
0.980011 0.198945i \(-0.0637514\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 9.36206i 0.896723i −0.893852 0.448361i \(-0.852008\pi\)
0.893852 0.448361i \(-0.147992\pi\)
\(110\) −1.18381 0.0461702i −0.112872 0.00440215i
\(111\) 8.95377i 0.849854i
\(112\) 0 0
\(113\) 13.9856 1.31566 0.657828 0.753168i \(-0.271475\pi\)
0.657828 + 0.753168i \(0.271475\pi\)
\(114\) 7.81849 0.732269
\(115\) 2.27490i 0.212135i
\(116\) 7.67778i 0.712864i
\(117\) −4.33393 −0.400672
\(118\) 0.911386 0.0838998
\(119\) 0 0
\(120\) 0.357203i 0.0326080i
\(121\) 10.9666 + 0.856728i 0.996962 + 0.0778844i
\(122\) 12.4897i 1.13076i
\(123\) 5.43717i 0.490253i
\(124\) 6.20254i 0.557004i
\(125\) 3.52645i 0.315415i
\(126\) 0 0
\(127\) 15.1853i 1.34748i 0.738970 + 0.673739i \(0.235313\pi\)
−0.738970 + 0.673739i \(0.764687\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −2.91740 −0.256863
\(130\) 1.54809 0.135777
\(131\) 15.1562 1.32420 0.662100 0.749416i \(-0.269665\pi\)
0.662100 + 0.749416i \(0.269665\pi\)
\(132\) 0.129255 3.31411i 0.0112502 0.288456i
\(133\) 0 0
\(134\) 13.6448i 1.17873i
\(135\) −0.357203 −0.0307431
\(136\) 0.347231i 0.0297748i
\(137\) 13.2508 1.13210 0.566048 0.824373i \(-0.308472\pi\)
0.566048 + 0.824373i \(0.308472\pi\)
\(138\) −6.36865 −0.542135
\(139\) 2.10063 0.178173 0.0890866 0.996024i \(-0.471605\pi\)
0.0890866 + 0.996024i \(0.471605\pi\)
\(140\) 0 0
\(141\) 3.84135 0.323500
\(142\) 6.18162i 0.518750i
\(143\) −14.3631 0.560182i −1.20110 0.0468448i
\(144\) −1.00000 −0.0833333
\(145\) −2.74252 −0.227754
\(146\) 11.4503i 0.947631i
\(147\) 0 0
\(148\) 8.95377 0.735995
\(149\) 21.4377i 1.75624i −0.478439 0.878121i \(-0.658797\pi\)
0.478439 0.878121i \(-0.341203\pi\)
\(150\) −4.87241 −0.397830
\(151\) 19.3147i 1.57181i −0.618347 0.785905i \(-0.712197\pi\)
0.618347 0.785905i \(-0.287803\pi\)
\(152\) 7.81849i 0.634164i
\(153\) −0.347231 −0.0280720
\(154\) 0 0
\(155\) −2.21556 −0.177958
\(156\) 4.33393i 0.346992i
\(157\) 17.5668i 1.40198i 0.713170 + 0.700991i \(0.247259\pi\)
−0.713170 + 0.700991i \(0.752741\pi\)
\(158\) −6.65938 −0.529792
\(159\) 2.73623i 0.216997i
\(160\) 0.357203 0.0282393
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) −13.7943 −1.08046 −0.540228 0.841519i \(-0.681662\pi\)
−0.540228 + 0.841519i \(0.681662\pi\)
\(164\) −5.43717 −0.424572
\(165\) −1.18381 0.0461702i −0.0921592 0.00359434i
\(166\) 16.7247i 1.29809i
\(167\) 8.30859 0.642938 0.321469 0.946920i \(-0.395823\pi\)
0.321469 + 0.946920i \(0.395823\pi\)
\(168\) 0 0
\(169\) 5.78297 0.444844
\(170\) 0.124032 0.00951281
\(171\) 7.81849 0.597895
\(172\) 2.91740i 0.222450i
\(173\) 9.78367 0.743839 0.371920 0.928265i \(-0.378700\pi\)
0.371920 + 0.928265i \(0.378700\pi\)
\(174\) 7.67778i 0.582051i
\(175\) 0 0
\(176\) −3.31411 0.129255i −0.249810 0.00974295i
\(177\) 0.911386 0.0685039
\(178\) 14.5026 1.08702
\(179\) 5.65887 0.422964 0.211482 0.977382i \(-0.432171\pi\)
0.211482 + 0.977382i \(0.432171\pi\)
\(180\) 0.357203i 0.0266243i
\(181\) 13.6419i 1.01399i −0.861949 0.506995i \(-0.830756\pi\)
0.861949 0.506995i \(-0.169244\pi\)
\(182\) 0 0
\(183\) 12.4897i 0.923264i
\(184\) 6.36865i 0.469503i
\(185\) 3.19831i 0.235144i
\(186\) 6.20254i 0.454792i
\(187\) −1.15076 0.0448813i −0.0841520 0.00328205i
\(188\) 3.84135i 0.280159i
\(189\) 0 0
\(190\) −2.79279 −0.202610
\(191\) −4.61622 −0.334018 −0.167009 0.985955i \(-0.553411\pi\)
−0.167009 + 0.985955i \(0.553411\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 5.24602i 0.377617i 0.982014 + 0.188809i \(0.0604626\pi\)
−0.982014 + 0.188809i \(0.939537\pi\)
\(194\) 18.1486 1.30299
\(195\) 1.54809 0.110861
\(196\) 0 0
\(197\) 2.35695i 0.167926i −0.996469 0.0839629i \(-0.973242\pi\)
0.996469 0.0839629i \(-0.0267577\pi\)
\(198\) 0.129255 3.31411i 0.00918574 0.235523i
\(199\) 4.46393i 0.316440i −0.987404 0.158220i \(-0.949424\pi\)
0.987404 0.158220i \(-0.0505755\pi\)
\(200\) 4.87241i 0.344531i
\(201\) 13.6448i 0.962428i
\(202\) 7.40672i 0.521135i
\(203\) 0 0
\(204\) 0.347231i 0.0243111i
\(205\) 1.94217i 0.135647i
\(206\) 0.972851 0.0677817
\(207\) −6.36865 −0.442652
\(208\) 4.33393 0.300504
\(209\) 25.9113 + 1.01058i 1.79232 + 0.0699032i
\(210\) 0 0
\(211\) 12.7624i 0.878602i 0.898340 + 0.439301i \(0.144774\pi\)
−0.898340 + 0.439301i \(0.855226\pi\)
\(212\) 2.73623 0.187925
\(213\) 6.18162i 0.423558i
\(214\) −4.11580 −0.281350
\(215\) 1.04210 0.0710708
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −9.36206 −0.634079
\(219\) 11.4503i 0.773737i
\(220\) −0.0461702 + 1.18381i −0.00311279 + 0.0798122i
\(221\) 1.50488 0.101229
\(222\) 8.95377 0.600937
\(223\) 8.18197i 0.547905i 0.961743 + 0.273953i \(0.0883311\pi\)
−0.961743 + 0.273953i \(0.911669\pi\)
\(224\) 0 0
\(225\) −4.87241 −0.324827
\(226\) 13.9856i 0.930310i
\(227\) −3.26368 −0.216618 −0.108309 0.994117i \(-0.534544\pi\)
−0.108309 + 0.994117i \(0.534544\pi\)
\(228\) 7.81849i 0.517792i
\(229\) 21.9790i 1.45241i 0.687476 + 0.726207i \(0.258719\pi\)
−0.687476 + 0.726207i \(0.741281\pi\)
\(230\) 2.27490 0.150002
\(231\) 0 0
\(232\) −7.67778 −0.504071
\(233\) 17.7061i 1.15996i 0.814630 + 0.579981i \(0.196940\pi\)
−0.814630 + 0.579981i \(0.803060\pi\)
\(234\) 4.33393i 0.283318i
\(235\) −1.37214 −0.0895085
\(236\) 0.911386i 0.0593261i
\(237\) −6.65938 −0.432573
\(238\) 0 0
\(239\) 2.53011i 0.163659i 0.996646 + 0.0818296i \(0.0260763\pi\)
−0.996646 + 0.0818296i \(0.973924\pi\)
\(240\) 0.357203 0.0230573
\(241\) 1.93061 0.124362 0.0621809 0.998065i \(-0.480194\pi\)
0.0621809 + 0.998065i \(0.480194\pi\)
\(242\) 0.856728 10.9666i 0.0550726 0.704959i
\(243\) 1.00000i 0.0641500i
\(244\) −12.4897 −0.799570
\(245\) 0 0
\(246\) −5.43717 −0.346661
\(247\) −33.8848 −2.15604
\(248\) −6.20254 −0.393862
\(249\) 16.7247i 1.05988i
\(250\) 3.52645 0.223032
\(251\) 24.7683i 1.56336i −0.623678 0.781681i \(-0.714362\pi\)
0.623678 0.781681i \(-0.285638\pi\)
\(252\) 0 0
\(253\) −21.1064 0.823179i −1.32695 0.0517528i
\(254\) 15.1853 0.952810
\(255\) 0.124032 0.00776718
\(256\) 1.00000 0.0625000
\(257\) 4.66839i 0.291206i −0.989343 0.145603i \(-0.953488\pi\)
0.989343 0.145603i \(-0.0465123\pi\)
\(258\) 2.91740i 0.181629i
\(259\) 0 0
\(260\) 1.54809i 0.0960086i
\(261\) 7.67778i 0.475243i
\(262\) 15.1562i 0.936351i
\(263\) 17.4163i 1.07394i 0.843602 + 0.536968i \(0.180430\pi\)
−0.843602 + 0.536968i \(0.819570\pi\)
\(264\) −3.31411 0.129255i −0.203969 0.00795509i
\(265\) 0.977388i 0.0600405i
\(266\) 0 0
\(267\) 14.5026 0.887547
\(268\) 13.6448 0.833487
\(269\) 9.65915i 0.588929i −0.955662 0.294464i \(-0.904859\pi\)
0.955662 0.294464i \(-0.0951413\pi\)
\(270\) 0.357203i 0.0217387i
\(271\) 2.73625 0.166215 0.0831076 0.996541i \(-0.473515\pi\)
0.0831076 + 0.996541i \(0.473515\pi\)
\(272\) 0.347231 0.0210540
\(273\) 0 0
\(274\) 13.2508i 0.800512i
\(275\) −16.1477 0.629782i −0.973741 0.0379773i
\(276\) 6.36865i 0.383348i
\(277\) 22.9739i 1.38037i −0.723634 0.690184i \(-0.757530\pi\)
0.723634 0.690184i \(-0.242470\pi\)
\(278\) 2.10063i 0.125987i
\(279\) 6.20254i 0.371336i
\(280\) 0 0
\(281\) 11.8340i 0.705954i 0.935632 + 0.352977i \(0.114831\pi\)
−0.935632 + 0.352977i \(0.885169\pi\)
\(282\) 3.84135i 0.228749i
\(283\) −2.38857 −0.141985 −0.0709927 0.997477i \(-0.522617\pi\)
−0.0709927 + 0.997477i \(0.522617\pi\)
\(284\) −6.18162 −0.366812
\(285\) −2.79279 −0.165430
\(286\) −0.560182 + 14.3631i −0.0331243 + 0.849308i
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) −16.8794 −0.992908
\(290\) 2.74252i 0.161047i
\(291\) 18.1486 1.06389
\(292\) −11.4503 −0.670076
\(293\) −21.8292 −1.27528 −0.637639 0.770335i \(-0.720089\pi\)
−0.637639 + 0.770335i \(0.720089\pi\)
\(294\) 0 0
\(295\) −0.325549 −0.0189542
\(296\) 8.95377i 0.520427i
\(297\) 0.129255 3.31411i 0.00750013 0.192304i
\(298\) −21.4377 −1.24185
\(299\) 27.6013 1.59622
\(300\) 4.87241i 0.281309i
\(301\) 0 0
\(302\) −19.3147 −1.11144
\(303\) 7.40672i 0.425505i
\(304\) −7.81849 −0.448421
\(305\) 4.46135i 0.255456i
\(306\) 0.347231i 0.0198499i
\(307\) 5.28799 0.301801 0.150901 0.988549i \(-0.451783\pi\)
0.150901 + 0.988549i \(0.451783\pi\)
\(308\) 0 0
\(309\) 0.972851 0.0553436
\(310\) 2.21556i 0.125836i
\(311\) 7.26261i 0.411825i 0.978570 + 0.205912i \(0.0660162\pi\)
−0.978570 + 0.205912i \(0.933984\pi\)
\(312\) 4.33393 0.245361
\(313\) 4.59656i 0.259813i 0.991526 + 0.129906i \(0.0414677\pi\)
−0.991526 + 0.129906i \(0.958532\pi\)
\(314\) 17.5668 0.991351
\(315\) 0 0
\(316\) 6.65938i 0.374619i
\(317\) 13.4968 0.758059 0.379029 0.925385i \(-0.376258\pi\)
0.379029 + 0.925385i \(0.376258\pi\)
\(318\) 2.73623 0.153440
\(319\) 0.992391 25.4450i 0.0555632 1.42465i
\(320\) 0.357203i 0.0199682i
\(321\) −4.11580 −0.229721
\(322\) 0 0
\(323\) −2.71483 −0.151057
\(324\) −1.00000 −0.0555556
\(325\) 21.1167 1.17134
\(326\) 13.7943i 0.763998i
\(327\) −9.36206 −0.517723
\(328\) 5.43717i 0.300217i
\(329\) 0 0
\(330\) −0.0461702 + 1.18381i −0.00254158 + 0.0651664i
\(331\) −6.39942 −0.351744 −0.175872 0.984413i \(-0.556275\pi\)
−0.175872 + 0.984413i \(0.556275\pi\)
\(332\) −16.7247 −0.917886
\(333\) 8.95377 0.490663
\(334\) 8.30859i 0.454626i
\(335\) 4.87395i 0.266292i
\(336\) 0 0
\(337\) 13.2274i 0.720544i −0.932847 0.360272i \(-0.882684\pi\)
0.932847 0.360272i \(-0.117316\pi\)
\(338\) 5.78297i 0.314552i
\(339\) 13.9856i 0.759595i
\(340\) 0.124032i 0.00672657i
\(341\) 0.801708 20.5559i 0.0434149 1.11316i
\(342\) 7.81849i 0.422776i
\(343\) 0 0
\(344\) 2.91740 0.157296
\(345\) 2.27490 0.122476
\(346\) 9.78367i 0.525974i
\(347\) 30.5044i 1.63756i 0.574105 + 0.818782i \(0.305350\pi\)
−0.574105 + 0.818782i \(0.694650\pi\)
\(348\) −7.67778 −0.411572
\(349\) −0.480174 −0.0257031 −0.0128516 0.999917i \(-0.504091\pi\)
−0.0128516 + 0.999917i \(0.504091\pi\)
\(350\) 0 0
\(351\) 4.33393i 0.231328i
\(352\) −0.129255 + 3.31411i −0.00688931 + 0.176642i
\(353\) 4.03853i 0.214949i −0.994208 0.107475i \(-0.965724\pi\)
0.994208 0.107475i \(-0.0342764\pi\)
\(354\) 0.911386i 0.0484396i
\(355\) 2.20809i 0.117193i
\(356\) 14.5026i 0.768638i
\(357\) 0 0
\(358\) 5.65887i 0.299081i
\(359\) 6.57967i 0.347262i 0.984811 + 0.173631i \(0.0555500\pi\)
−0.984811 + 0.173631i \(0.944450\pi\)
\(360\) 0.357203 0.0188262
\(361\) 42.1289 2.21731
\(362\) −13.6419 −0.717000
\(363\) 0.856728 10.9666i 0.0449666 0.575597i
\(364\) 0 0
\(365\) 4.09006i 0.214084i
\(366\) −12.4897 −0.652846
\(367\) 11.0084i 0.574632i 0.957836 + 0.287316i \(0.0927630\pi\)
−0.957836 + 0.287316i \(0.907237\pi\)
\(368\) 6.36865 0.331989
\(369\) −5.43717 −0.283048
\(370\) −3.19831 −0.166272
\(371\) 0 0
\(372\) −6.20254 −0.321587
\(373\) 25.8257i 1.33720i −0.743621 0.668601i \(-0.766893\pi\)
0.743621 0.668601i \(-0.233107\pi\)
\(374\) −0.0448813 + 1.15076i −0.00232076 + 0.0595044i
\(375\) 3.52645 0.182105
\(376\) −3.84135 −0.198102
\(377\) 33.2750i 1.71375i
\(378\) 0 0
\(379\) 23.0073 1.18180 0.590902 0.806743i \(-0.298772\pi\)
0.590902 + 0.806743i \(0.298772\pi\)
\(380\) 2.79279i 0.143267i
\(381\) 15.1853 0.777966
\(382\) 4.61622i 0.236186i
\(383\) 17.2499i 0.881429i 0.897647 + 0.440715i \(0.145275\pi\)
−0.897647 + 0.440715i \(0.854725\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 5.24602 0.267016
\(387\) 2.91740i 0.148300i
\(388\) 18.1486i 0.921354i
\(389\) 18.5596 0.941011 0.470506 0.882397i \(-0.344072\pi\)
0.470506 + 0.882397i \(0.344072\pi\)
\(390\) 1.54809i 0.0783907i
\(391\) 2.21139 0.111835
\(392\) 0 0
\(393\) 15.1562i 0.764527i
\(394\) −2.35695 −0.118741
\(395\) 2.37875 0.119688
\(396\) −3.31411 0.129255i −0.166540 0.00649530i
\(397\) 6.86138i 0.344363i −0.985065 0.172181i \(-0.944918\pi\)
0.985065 0.172181i \(-0.0550815\pi\)
\(398\) −4.46393 −0.223757
\(399\) 0 0
\(400\) 4.87241 0.243620
\(401\) 9.50963 0.474888 0.237444 0.971401i \(-0.423690\pi\)
0.237444 + 0.971401i \(0.423690\pi\)
\(402\) 13.6448 0.680540
\(403\) 26.8814i 1.33906i
\(404\) 7.40672 0.368498
\(405\) 0.357203i 0.0177495i
\(406\) 0 0
\(407\) 29.6737 + 1.15732i 1.47087 + 0.0573661i
\(408\) 0.347231 0.0171905
\(409\) −4.61520 −0.228207 −0.114103 0.993469i \(-0.536400\pi\)
−0.114103 + 0.993469i \(0.536400\pi\)
\(410\) 1.94217 0.0959170
\(411\) 13.2508i 0.653615i
\(412\) 0.972851i 0.0479289i
\(413\) 0 0
\(414\) 6.36865i 0.313002i
\(415\) 5.97410i 0.293257i
\(416\) 4.33393i 0.212489i
\(417\) 2.10063i 0.102868i
\(418\) 1.01058 25.9113i 0.0494290 1.26736i
\(419\) 15.9343i 0.778442i 0.921144 + 0.389221i \(0.127256\pi\)
−0.921144 + 0.389221i \(0.872744\pi\)
\(420\) 0 0
\(421\) −19.4154 −0.946247 −0.473123 0.880996i \(-0.656873\pi\)
−0.473123 + 0.880996i \(0.656873\pi\)
\(422\) 12.7624 0.621266
\(423\) 3.84135i 0.186773i
\(424\) 2.73623i 0.132883i
\(425\) 1.69185 0.0820669
\(426\) −6.18162 −0.299501
\(427\) 0 0
\(428\) 4.11580i 0.198945i
\(429\) −0.560182 + 14.3631i −0.0270458 + 0.693457i
\(430\) 1.04210i 0.0502547i
\(431\) 7.41070i 0.356961i 0.983943 + 0.178480i \(0.0571181\pi\)
−0.983943 + 0.178480i \(0.942882\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 19.3089i 0.927928i −0.885854 0.463964i \(-0.846427\pi\)
0.885854 0.463964i \(-0.153573\pi\)
\(434\) 0 0
\(435\) 2.74252i 0.131494i
\(436\) 9.36206i 0.448361i
\(437\) −49.7932 −2.38193
\(438\) −11.4503 −0.547115
\(439\) −32.5688 −1.55442 −0.777212 0.629238i \(-0.783367\pi\)
−0.777212 + 0.629238i \(0.783367\pi\)
\(440\) 1.18381 + 0.0461702i 0.0564358 + 0.00220108i
\(441\) 0 0
\(442\) 1.50488i 0.0715797i
\(443\) −29.8320 −1.41736 −0.708681 0.705529i \(-0.750710\pi\)
−0.708681 + 0.705529i \(0.750710\pi\)
\(444\) 8.95377i 0.424927i
\(445\) −5.18038 −0.245573
\(446\) 8.18197 0.387427
\(447\) −21.4377 −1.01397
\(448\) 0 0
\(449\) −4.45126 −0.210068 −0.105034 0.994469i \(-0.533495\pi\)
−0.105034 + 0.994469i \(0.533495\pi\)
\(450\) 4.87241i 0.229687i
\(451\) −18.0194 0.702781i −0.848498 0.0330926i
\(452\) −13.9856 −0.657828
\(453\) −19.3147 −0.907485
\(454\) 3.26368i 0.153172i
\(455\) 0 0
\(456\) −7.81849 −0.366135
\(457\) 27.1160i 1.26843i −0.773156 0.634216i \(-0.781323\pi\)
0.773156 0.634216i \(-0.218677\pi\)
\(458\) 21.9790 1.02701
\(459\) 0.347231i 0.0162074i
\(460\) 2.27490i 0.106068i
\(461\) 12.7571 0.594158 0.297079 0.954853i \(-0.403987\pi\)
0.297079 + 0.954853i \(0.403987\pi\)
\(462\) 0 0
\(463\) −26.8679 −1.24866 −0.624329 0.781162i \(-0.714627\pi\)
−0.624329 + 0.781162i \(0.714627\pi\)
\(464\) 7.67778i 0.356432i
\(465\) 2.21556i 0.102744i
\(466\) 17.7061 0.820217
\(467\) 35.2058i 1.62913i 0.580073 + 0.814565i \(0.303024\pi\)
−0.580073 + 0.814565i \(0.696976\pi\)
\(468\) 4.33393 0.200336
\(469\) 0 0
\(470\) 1.37214i 0.0632920i
\(471\) 17.5668 0.809434
\(472\) −0.911386 −0.0419499
\(473\) −0.377088 + 9.66857i −0.0173385 + 0.444561i
\(474\) 6.65938i 0.305875i
\(475\) −38.0949 −1.74791
\(476\) 0 0
\(477\) 2.73623 0.125283
\(478\) 2.53011 0.115724
\(479\) 31.8131 1.45358 0.726789 0.686861i \(-0.241012\pi\)
0.726789 + 0.686861i \(0.241012\pi\)
\(480\) 0.357203i 0.0163040i
\(481\) −38.8050 −1.76936
\(482\) 1.93061i 0.0879370i
\(483\) 0 0
\(484\) −10.9666 0.856728i −0.498481 0.0389422i
\(485\) −6.48271 −0.294365
\(486\) −1.00000 −0.0453609
\(487\) −28.4612 −1.28970 −0.644849 0.764310i \(-0.723080\pi\)
−0.644849 + 0.764310i \(0.723080\pi\)
\(488\) 12.4897i 0.565382i
\(489\) 13.7943i 0.623802i
\(490\) 0 0
\(491\) 34.2454i 1.54547i 0.634727 + 0.772737i \(0.281113\pi\)
−0.634727 + 0.772737i \(0.718887\pi\)
\(492\) 5.43717i 0.245127i
\(493\) 2.66597i 0.120069i
\(494\) 33.8848i 1.52455i
\(495\) −0.0461702 + 1.18381i −0.00207519 + 0.0532082i
\(496\) 6.20254i 0.278502i
\(497\) 0 0
\(498\) −16.7247 −0.749451
\(499\) −7.93504 −0.355221 −0.177611 0.984101i \(-0.556837\pi\)
−0.177611 + 0.984101i \(0.556837\pi\)
\(500\) 3.52645i 0.157708i
\(501\) 8.30859i 0.371200i
\(502\) −24.7683 −1.10546
\(503\) −7.71704 −0.344086 −0.172043 0.985089i \(-0.555037\pi\)
−0.172043 + 0.985089i \(0.555037\pi\)
\(504\) 0 0
\(505\) 2.64570i 0.117732i
\(506\) −0.823179 + 21.1064i −0.0365948 + 0.938293i
\(507\) 5.78297i 0.256831i
\(508\) 15.1853i 0.673739i
\(509\) 33.2386i 1.47328i 0.676288 + 0.736638i \(0.263588\pi\)
−0.676288 + 0.736638i \(0.736412\pi\)
\(510\) 0.124032i 0.00549222i
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 7.81849i 0.345195i
\(514\) −4.66839 −0.205914
\(515\) −0.347505 −0.0153129
\(516\) 2.91740 0.128431
\(517\) 0.496513 12.7306i 0.0218366 0.559892i
\(518\) 0 0
\(519\) 9.78367i 0.429456i
\(520\) −1.54809 −0.0678883
\(521\) 9.99903i 0.438066i −0.975717 0.219033i \(-0.929710\pi\)
0.975717 0.219033i \(-0.0702902\pi\)
\(522\) −7.67778 −0.336047
\(523\) 4.45341 0.194734 0.0973669 0.995249i \(-0.468958\pi\)
0.0973669 + 0.995249i \(0.468958\pi\)
\(524\) −15.1562 −0.662100
\(525\) 0 0
\(526\) 17.4163 0.759388
\(527\) 2.15372i 0.0938173i
\(528\) −0.129255 + 3.31411i −0.00562510 + 0.144228i
\(529\) 17.5597 0.763464
\(530\) −0.977388 −0.0424550
\(531\) 0.911386i 0.0395508i
\(532\) 0 0
\(533\) 23.5643 1.02068
\(534\) 14.5026i 0.627590i
\(535\) 1.47017 0.0635611
\(536\) 13.6448i 0.589365i
\(537\) 5.65887i 0.244198i
\(538\) −9.65915 −0.416436
\(539\) 0 0
\(540\) 0.357203 0.0153716
\(541\) 36.1740i 1.55524i 0.628732 + 0.777622i \(0.283574\pi\)
−0.628732 + 0.777622i \(0.716426\pi\)
\(542\) 2.73625i 0.117532i
\(543\) −13.6419 −0.585428
\(544\) 0.347231i 0.0148874i
\(545\) 3.34415 0.143248
\(546\) 0 0
\(547\) 37.3673i 1.59771i −0.601523 0.798855i \(-0.705439\pi\)
0.601523 0.798855i \(-0.294561\pi\)
\(548\) −13.2508 −0.566048
\(549\) −12.4897 −0.533047
\(550\) −0.629782 + 16.1477i −0.0268540 + 0.688539i
\(551\) 60.0287i 2.55731i
\(552\) 6.36865 0.271068
\(553\) 0 0
\(554\) −22.9739 −0.976068
\(555\) −3.19831 −0.135761
\(556\) −2.10063 −0.0890866
\(557\) 41.1608i 1.74404i 0.489470 + 0.872020i \(0.337190\pi\)
−0.489470 + 0.872020i \(0.662810\pi\)
\(558\) −6.20254 −0.262574
\(559\) 12.6438i 0.534776i
\(560\) 0 0
\(561\) −0.0448813 + 1.15076i −0.00189489 + 0.0485852i
\(562\) 11.8340 0.499185
\(563\) −19.1043 −0.805151 −0.402575 0.915387i \(-0.631885\pi\)
−0.402575 + 0.915387i \(0.631885\pi\)
\(564\) −3.84135 −0.161750
\(565\) 4.99570i 0.210171i
\(566\) 2.38857i 0.100399i
\(567\) 0 0
\(568\) 6.18162i 0.259375i
\(569\) 42.6242i 1.78690i 0.449164 + 0.893449i \(0.351722\pi\)
−0.449164 + 0.893449i \(0.648278\pi\)
\(570\) 2.79279i 0.116977i
\(571\) 8.53860i 0.357329i 0.983910 + 0.178665i \(0.0571777\pi\)
−0.983910 + 0.178665i \(0.942822\pi\)
\(572\) 14.3631 + 0.560182i 0.600552 + 0.0234224i
\(573\) 4.61622i 0.192845i
\(574\) 0 0
\(575\) 31.0306 1.29407
\(576\) 1.00000 0.0416667
\(577\) 31.5935i 1.31525i 0.753344 + 0.657627i \(0.228440\pi\)
−0.753344 + 0.657627i \(0.771560\pi\)
\(578\) 16.8794i 0.702092i
\(579\) 5.24602 0.218017
\(580\) 2.74252 0.113877
\(581\) 0 0
\(582\) 18.1486i 0.752282i
\(583\) 9.06815 + 0.353671i 0.375565 + 0.0146476i
\(584\) 11.4503i 0.473815i
\(585\) 1.54809i 0.0640057i
\(586\) 21.8292i 0.901758i
\(587\) 19.9085i 0.821710i −0.911701 0.410855i \(-0.865230\pi\)
0.911701 0.410855i \(-0.134770\pi\)
\(588\) 0 0
\(589\) 48.4945i 1.99818i
\(590\) 0.325549i 0.0134027i
\(591\) −2.35695 −0.0969520
\(592\) −8.95377 −0.367998
\(593\) 34.5395 1.41837 0.709184 0.705023i \(-0.249063\pi\)
0.709184 + 0.705023i \(0.249063\pi\)
\(594\) −3.31411 0.129255i −0.135979 0.00530339i
\(595\) 0 0
\(596\) 21.4377i 0.878121i
\(597\) −4.46393 −0.182697
\(598\) 27.6013i 1.12870i
\(599\) −12.8576 −0.525346 −0.262673 0.964885i \(-0.584604\pi\)
−0.262673 + 0.964885i \(0.584604\pi\)
\(600\) 4.87241 0.198915
\(601\) 11.2038 0.457011 0.228505 0.973543i \(-0.426616\pi\)
0.228505 + 0.973543i \(0.426616\pi\)
\(602\) 0 0
\(603\) 13.6448 0.555658
\(604\) 19.3147i 0.785905i
\(605\) −0.306026 + 3.91729i −0.0124417 + 0.159261i
\(606\) 7.40672 0.300877
\(607\) 3.49287 0.141771 0.0708857 0.997484i \(-0.477417\pi\)
0.0708857 + 0.997484i \(0.477417\pi\)
\(608\) 7.81849i 0.317082i
\(609\) 0 0
\(610\) 4.46135 0.180635
\(611\) 16.6481i 0.673511i
\(612\) 0.347231 0.0140360
\(613\) 31.2095i 1.26054i 0.776377 + 0.630269i \(0.217056\pi\)
−0.776377 + 0.630269i \(0.782944\pi\)
\(614\) 5.28799i 0.213406i
\(615\) 1.94217 0.0783159
\(616\) 0 0
\(617\) −35.4070 −1.42543 −0.712717 0.701452i \(-0.752536\pi\)
−0.712717 + 0.701452i \(0.752536\pi\)
\(618\) 0.972851i 0.0391338i
\(619\) 0.132999i 0.00534570i −0.999996 0.00267285i \(-0.999149\pi\)
0.999996 0.00267285i \(-0.000850795\pi\)
\(620\) 2.21556 0.0889791
\(621\) 6.36865i 0.255565i
\(622\) 7.26261 0.291204
\(623\) 0 0
\(624\) 4.33393i 0.173496i
\(625\) 23.1024 0.924095
\(626\) 4.59656 0.183715
\(627\) 1.01058 25.9113i 0.0403586 1.03480i
\(628\) 17.5668i 0.700991i
\(629\) −3.10903 −0.123965
\(630\) 0 0
\(631\) 5.18583 0.206445 0.103222 0.994658i \(-0.467085\pi\)
0.103222 + 0.994658i \(0.467085\pi\)
\(632\) 6.65938 0.264896
\(633\) 12.7624 0.507261
\(634\) 13.4968i 0.536028i
\(635\) −5.42423 −0.215254
\(636\) 2.73623i 0.108499i
\(637\) 0 0
\(638\) −25.4450 0.992391i −1.00738 0.0392891i
\(639\) −6.18162 −0.244541
\(640\) −0.357203 −0.0141197
\(641\) 13.0446 0.515232 0.257616 0.966247i \(-0.417063\pi\)
0.257616 + 0.966247i \(0.417063\pi\)
\(642\) 4.11580i 0.162438i
\(643\) 7.54506i 0.297548i 0.988871 + 0.148774i \(0.0475327\pi\)
−0.988871 + 0.148774i \(0.952467\pi\)
\(644\) 0 0
\(645\) 1.04210i 0.0410328i
\(646\) 2.71483i 0.106813i
\(647\) 18.9501i 0.745006i 0.928031 + 0.372503i \(0.121500\pi\)
−0.928031 + 0.372503i \(0.878500\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0.117801 3.02043i 0.00462409 0.118562i
\(650\) 21.1167i 0.828264i
\(651\) 0 0
\(652\) 13.7943 0.540228
\(653\) 16.3241 0.638810 0.319405 0.947618i \(-0.396517\pi\)
0.319405 + 0.947618i \(0.396517\pi\)
\(654\) 9.36206i 0.366085i
\(655\) 5.41382i 0.211535i
\(656\) 5.43717 0.212286
\(657\) −11.4503 −0.446717
\(658\) 0 0
\(659\) 43.7066i 1.70257i −0.524706 0.851283i \(-0.675825\pi\)
0.524706 0.851283i \(-0.324175\pi\)
\(660\) 1.18381 + 0.0461702i 0.0460796 + 0.00179717i
\(661\) 10.3498i 0.402560i 0.979534 + 0.201280i \(0.0645101\pi\)
−0.979534 + 0.201280i \(0.935490\pi\)
\(662\) 6.39942i 0.248721i
\(663\) 1.50488i 0.0584446i
\(664\) 16.7247i 0.649043i
\(665\) 0 0
\(666\) 8.95377i 0.346951i
\(667\) 48.8971i 1.89330i
\(668\) −8.30859 −0.321469
\(669\) 8.18197 0.316333
\(670\) −4.87395 −0.188297
\(671\) −41.3921 1.61435i −1.59793 0.0623214i
\(672\) 0 0
\(673\) 34.3241i 1.32310i 0.749902 + 0.661549i \(0.230101\pi\)
−0.749902 + 0.661549i \(0.769899\pi\)
\(674\) −13.2274 −0.509502
\(675\) 4.87241i 0.187539i
\(676\) −5.78297 −0.222422
\(677\) 18.1171 0.696298 0.348149 0.937439i \(-0.386810\pi\)
0.348149 + 0.937439i \(0.386810\pi\)
\(678\) −13.9856 −0.537115
\(679\) 0 0
\(680\) −0.124032 −0.00475641
\(681\) 3.26368i 0.125064i
\(682\) −20.5559 0.801708i −0.787125 0.0306990i
\(683\) 27.7424 1.06153 0.530767 0.847518i \(-0.321904\pi\)
0.530767 + 0.847518i \(0.321904\pi\)
\(684\) −7.81849 −0.298948
\(685\) 4.73323i 0.180848i
\(686\) 0 0
\(687\) 21.9790 0.838552
\(688\) 2.91740i 0.111225i
\(689\) −11.8586 −0.451778
\(690\) 2.27490i 0.0866039i
\(691\) 15.2646i 0.580693i 0.956922 + 0.290347i \(0.0937706\pi\)
−0.956922 + 0.290347i \(0.906229\pi\)
\(692\) −9.78367 −0.371920
\(693\) 0 0
\(694\) 30.5044 1.15793
\(695\) 0.750351i 0.0284624i
\(696\) 7.67778i 0.291026i
\(697\) 1.88796 0.0715114
\(698\) 0.480174i 0.0181749i
\(699\) 17.7061 0.669704
\(700\) 0 0
\(701\) 7.78140i 0.293900i −0.989144 0.146950i \(-0.953054\pi\)
0.989144 0.146950i \(-0.0469456\pi\)
\(702\) 4.33393 0.163574
\(703\) 70.0050 2.64029
\(704\) 3.31411 + 0.129255i 0.124905 + 0.00487148i
\(705\) 1.37214i 0.0516777i
\(706\) −4.03853 −0.151992
\(707\) 0 0
\(708\) −0.911386 −0.0342520
\(709\) −1.43232 −0.0537919 −0.0268960 0.999638i \(-0.508562\pi\)
−0.0268960 + 0.999638i \(0.508562\pi\)
\(710\) 2.20809 0.0828682
\(711\) 6.65938i 0.249746i
\(712\) −14.5026 −0.543509
\(713\) 39.5018i 1.47935i
\(714\) 0 0
\(715\) 0.200098 5.13054i 0.00748326 0.191871i
\(716\) −5.65887 −0.211482
\(717\) 2.53011 0.0944886
\(718\) 6.57967 0.245551
\(719\) 49.3021i 1.83866i −0.393491 0.919328i \(-0.628733\pi\)
0.393491 0.919328i \(-0.371267\pi\)
\(720\) 0.357203i 0.0133122i
\(721\) 0 0
\(722\) 42.1289i 1.56787i
\(723\) 1.93061i 0.0718003i
\(724\) 13.6419i 0.506995i
\(725\) 37.4093i 1.38935i
\(726\) −10.9666 0.856728i −0.407008 0.0317962i
\(727\) 26.7798i 0.993209i −0.867977 0.496604i \(-0.834580\pi\)
0.867977 0.496604i \(-0.165420\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 4.09006 0.151380
\(731\) 1.01301i 0.0374676i
\(732\) 12.4897i 0.461632i
\(733\) −3.04403 −0.112434 −0.0562170 0.998419i \(-0.517904\pi\)
−0.0562170 + 0.998419i \(0.517904\pi\)
\(734\) 11.0084 0.406326
\(735\) 0 0
\(736\) 6.36865i 0.234751i
\(737\) 45.2202 + 1.76365i 1.66571 + 0.0649650i
\(738\) 5.43717i 0.200145i
\(739\) 9.04683i 0.332793i −0.986059 0.166397i \(-0.946787\pi\)
0.986059 0.166397i \(-0.0532132\pi\)
\(740\) 3.19831i 0.117572i
\(741\) 33.8848i 1.24479i
\(742\) 0 0
\(743\) 38.9413i 1.42862i −0.699830 0.714309i \(-0.746741\pi\)
0.699830 0.714309i \(-0.253259\pi\)
\(744\) 6.20254i 0.227396i
\(745\) 7.65759 0.280552
\(746\) −25.8257 −0.945545
\(747\) −16.7247 −0.611924
\(748\) 1.15076 + 0.0448813i 0.0420760 + 0.00164102i
\(749\) 0 0
\(750\) 3.52645i 0.128768i
\(751\) −1.90337 −0.0694549 −0.0347274 0.999397i \(-0.511056\pi\)
−0.0347274 + 0.999397i \(0.511056\pi\)
\(752\) 3.84135i 0.140079i
\(753\) −24.7683 −0.902608
\(754\) 33.2750 1.21180
\(755\) 6.89927 0.251090
\(756\) 0 0
\(757\) 15.1242 0.549697 0.274848 0.961488i \(-0.411372\pi\)
0.274848 + 0.961488i \(0.411372\pi\)
\(758\) 23.0073i 0.835662i
\(759\) −0.823179 + 21.1064i −0.0298795 + 0.766113i
\(760\) 2.79279 0.101305
\(761\) −8.47032 −0.307049 −0.153524 0.988145i \(-0.549062\pi\)
−0.153524 + 0.988145i \(0.549062\pi\)
\(762\) 15.1853i 0.550105i
\(763\) 0 0
\(764\) 4.61622 0.167009
\(765\) 0.124032i 0.00448438i
\(766\) 17.2499 0.623265
\(767\) 3.94988i 0.142622i
\(768\) 1.00000i 0.0360844i
\(769\) −40.4068 −1.45711 −0.728553 0.684990i \(-0.759807\pi\)
−0.728553 + 0.684990i \(0.759807\pi\)
\(770\) 0 0
\(771\) −4.66839 −0.168128
\(772\) 5.24602i 0.188809i
\(773\) 36.4379i 1.31058i −0.755378 0.655290i \(-0.772547\pi\)
0.755378 0.655290i \(-0.227453\pi\)
\(774\) 2.91740 0.104864
\(775\) 30.2213i 1.08558i
\(776\) −18.1486 −0.651495
\(777\) 0 0
\(778\) 18.5596i 0.665395i
\(779\) −42.5105 −1.52310
\(780\) −1.54809 −0.0554306
\(781\) −20.4865 0.799005i −0.733066 0.0285906i
\(782\) 2.21139i 0.0790793i
\(783\) −7.67778 −0.274382
\(784\) 0 0
\(785\) −6.27490 −0.223961
\(786\) −15.1562 −0.540602
\(787\) 12.6092 0.449470 0.224735 0.974420i \(-0.427848\pi\)
0.224735 + 0.974420i \(0.427848\pi\)
\(788\) 2.35695i 0.0839629i
\(789\) 17.4163 0.620037
\(790\) 2.37875i 0.0846320i
\(791\) 0 0
\(792\) −0.129255 + 3.31411i −0.00459287 + 0.117762i
\(793\) 54.1295 1.92219
\(794\) −6.86138 −0.243501
\(795\) −0.977388 −0.0346644
\(796\) 4.46393i 0.158220i
\(797\) 17.2558i 0.611231i −0.952155 0.305616i \(-0.901138\pi\)
0.952155 0.305616i \(-0.0988622\pi\)
\(798\) 0 0
\(799\) 1.33384i 0.0471877i
\(800\) 4.87241i 0.172266i
\(801\) 14.5026i 0.512425i
\(802\) 9.50963i 0.335797i
\(803\) −37.9474 1.48000i −1.33913 0.0522282i
\(804\) 13.6448i 0.481214i
\(805\) 0 0
\(806\) 26.8814 0.946856
\(807\) −9.65915 −0.340018
\(808\) 7.40672i 0.260567i
\(809\) 25.3566i 0.891492i 0.895160 + 0.445746i \(0.147061\pi\)
−0.895160 + 0.445746i \(0.852939\pi\)
\(810\) 0.357203 0.0125508
\(811\) −29.5935 −1.03917 −0.519583 0.854420i \(-0.673913\pi\)
−0.519583 + 0.854420i \(0.673913\pi\)
\(812\) 0 0
\(813\) 2.73625i 0.0959643i
\(814\) 1.15732 29.6737i 0.0405640 1.04006i
\(815\) 4.92738i 0.172598i
\(816\) 0.347231i 0.0121555i
\(817\) 22.8097i 0.798009i
\(818\) 4.61520i 0.161367i
\(819\) 0 0
\(820\) 1.94217i 0.0678235i
\(821\) 3.66644i 0.127960i −0.997951 0.0639798i \(-0.979621\pi\)
0.997951 0.0639798i \(-0.0203793\pi\)
\(822\) −13.2508 −0.462176
\(823\) −40.4699 −1.41069 −0.705347 0.708862i \(-0.749209\pi\)
−0.705347 + 0.708862i \(0.749209\pi\)
\(824\) −0.972851 −0.0338909
\(825\) −0.629782 + 16.1477i −0.0219262 + 0.562190i
\(826\) 0 0
\(827\) 46.6107i 1.62081i 0.585869 + 0.810406i \(0.300753\pi\)
−0.585869 + 0.810406i \(0.699247\pi\)
\(828\) 6.36865 0.221326
\(829\) 33.1897i 1.15273i 0.817194 + 0.576363i \(0.195529\pi\)
−0.817194 + 0.576363i \(0.804471\pi\)
\(830\) 5.97410 0.207364
\(831\) −22.9739 −0.796956
\(832\) −4.33393 −0.150252
\(833\) 0 0
\(834\) −2.10063 −0.0727389
\(835\) 2.96785i 0.102707i
\(836\) −25.9113 1.01058i −0.896161 0.0349516i
\(837\) −6.20254 −0.214391
\(838\) 15.9343 0.550442
\(839\) 5.48280i 0.189287i −0.995511 0.0946436i \(-0.969829\pi\)
0.995511 0.0946436i \(-0.0301712\pi\)
\(840\) 0 0
\(841\) −29.9484 −1.03270
\(842\) 19.4154i 0.669097i
\(843\) 11.8340 0.407583
\(844\) 12.7624i 0.439301i
\(845\) 2.06569i 0.0710620i
\(846\) −3.84135 −0.132068
\(847\) 0 0
\(848\) −2.73623 −0.0939625
\(849\) 2.38857i 0.0819753i
\(850\) 1.69185i 0.0580300i
\(851\) −57.0234 −1.95474
\(852\) 6.18162i 0.211779i
\(853\) −26.9597 −0.923083 −0.461541 0.887119i \(-0.652703\pi\)
−0.461541 + 0.887119i \(0.652703\pi\)
\(854\) 0 0
\(855\) 2.79279i 0.0955113i
\(856\) 4.11580 0.140675
\(857\) −5.65832 −0.193285 −0.0966423 0.995319i \(-0.530810\pi\)
−0.0966423 + 0.995319i \(0.530810\pi\)
\(858\) 14.3631 + 0.560182i 0.490348 + 0.0191243i
\(859\) 19.7245i 0.672992i 0.941685 + 0.336496i \(0.109242\pi\)
−0.941685 + 0.336496i \(0.890758\pi\)
\(860\) −1.04210 −0.0355354
\(861\) 0 0
\(862\) 7.41070 0.252409
\(863\) −2.24915 −0.0765621 −0.0382810 0.999267i \(-0.512188\pi\)
−0.0382810 + 0.999267i \(0.512188\pi\)
\(864\) 1.00000 0.0340207
\(865\) 3.49475i 0.118825i
\(866\) −19.3089 −0.656144
\(867\) 16.8794i 0.573256i
\(868\) 0 0
\(869\) −0.860757 + 22.0699i −0.0291992 + 0.748669i
\(870\) 2.74252 0.0929803
\(871\) −59.1355 −2.00373
\(872\) 9.36206 0.317039
\(873\) 18.1486i 0.614236i
\(874\) 49.7932i 1.68428i
\(875\) 0 0
\(876\) 11.4503i 0.386869i
\(877\) 40.6162i 1.37151i 0.727831 + 0.685756i \(0.240528\pi\)
−0.727831 + 0.685756i \(0.759472\pi\)
\(878\) 32.5688i 1.09914i
\(879\) 21.8292i 0.736282i
\(880\) 0.0461702 1.18381i 0.00155640 0.0399061i
\(881\) 16.7791i 0.565302i 0.959223 + 0.282651i \(0.0912139\pi\)
−0.959223 + 0.282651i \(0.908786\pi\)
\(882\) 0 0
\(883\) 2.56752 0.0864041 0.0432020 0.999066i \(-0.486244\pi\)
0.0432020 + 0.999066i \(0.486244\pi\)
\(884\) −1.50488 −0.0506145
\(885\) 0.325549i 0.0109432i
\(886\) 29.8320i 1.00223i
\(887\) 11.5185 0.386755 0.193377 0.981124i \(-0.438056\pi\)
0.193377 + 0.981124i \(0.438056\pi\)
\(888\) −8.95377 −0.300469
\(889\) 0 0
\(890\) 5.18038i 0.173647i
\(891\) −3.31411 0.129255i −0.111027 0.00433020i
\(892\) 8.18197i 0.273953i
\(893\) 30.0335i 1.00503i
\(894\) 21.4377i 0.716983i
\(895\) 2.02136i 0.0675667i
\(896\) 0 0
\(897\) 27.6013i 0.921580i
\(898\) 4.45126i 0.148541i
\(899\) −47.6217 −1.58827
\(900\) 4.87241 0.162414
\(901\) −0.950105 −0.0316526
\(902\) −0.702781 + 18.0194i −0.0234000 + 0.599979i
\(903\) 0 0
\(904\) 13.9856i 0.465155i
\(905\) 4.87290 0.161981
\(906\) 19.3147i 0.641689i
\(907\) 26.8783 0.892478 0.446239 0.894914i \(-0.352763\pi\)
0.446239 + 0.894914i \(0.352763\pi\)
\(908\) 3.26368 0.108309
\(909\) 7.40672 0.245665
\(910\) 0 0
\(911\) −33.9723 −1.12555 −0.562776 0.826609i \(-0.690267\pi\)
−0.562776 + 0.826609i \(0.690267\pi\)
\(912\) 7.81849i 0.258896i
\(913\) −55.4273 2.16175i −1.83438 0.0715433i
\(914\) −27.1160 −0.896916
\(915\) 4.46135 0.147488
\(916\) 21.9790i 0.726207i
\(917\) 0 0
\(918\) 0.347231 0.0114603
\(919\) 49.8543i 1.64454i −0.569096 0.822271i \(-0.692707\pi\)
0.569096 0.822271i \(-0.307293\pi\)
\(920\) −2.27490 −0.0750012
\(921\) 5.28799i 0.174245i
\(922\) 12.7571i 0.420133i
\(923\) 26.7907 0.881828
\(924\) 0 0
\(925\) −43.6264 −1.43443
\(926\) 26.8679i 0.882934i
\(927\) 0.972851i 0.0319526i
\(928\) 7.67778 0.252036
\(929\) 7.49418i 0.245876i 0.992414 + 0.122938i \(0.0392316\pi\)
−0.992414 + 0.122938i \(0.960768\pi\)
\(930\) 2.21556 0.0726512
\(931\) 0 0
\(932\) 17.7061i 0.579981i
\(933\) 7.26261 0.237767
\(934\) 35.2058 1.15197
\(935\) 0.0160317 0.411055i 0.000524294 0.0134429i
\(936\) 4.33393i 0.141659i
\(937\) −51.6569 −1.68756 −0.843779 0.536690i \(-0.819674\pi\)
−0.843779 + 0.536690i \(0.819674\pi\)
\(938\) 0 0
\(939\) 4.59656 0.150003
\(940\) 1.37214 0.0447542
\(941\) 46.4654 1.51473 0.757364 0.652993i \(-0.226487\pi\)
0.757364 + 0.652993i \(0.226487\pi\)
\(942\) 17.5668i 0.572356i
\(943\) 34.6274 1.12762
\(944\) 0.911386i 0.0296631i
\(945\) 0 0
\(946\) 9.66857 + 0.377088i 0.314352 + 0.0122602i
\(947\) 39.3012 1.27712 0.638559 0.769573i \(-0.279531\pi\)
0.638559 + 0.769573i \(0.279531\pi\)
\(948\) 6.65938 0.216286
\(949\) 49.6247 1.61089
\(950\) 38.0949i 1.23596i
\(951\) 13.4968i 0.437665i
\(952\) 0 0
\(953\) 59.6687i 1.93286i −0.256931 0.966430i \(-0.582711\pi\)
0.256931 0.966430i \(-0.417289\pi\)
\(954\) 2.73623i 0.0885887i
\(955\) 1.64893i 0.0533580i
\(956\) 2.53011i 0.0818296i
\(957\) −25.4450 0.992391i −0.822519 0.0320794i
\(958\) 31.8131i 1.02783i
\(959\) 0 0
\(960\) −0.357203 −0.0115287
\(961\) −7.47149 −0.241016
\(962\) 38.8050i 1.25112i
\(963\) 4.11580i 0.132630i
\(964\) −1.93061 −0.0621809
\(965\) −1.87389 −0.0603228
\(966\) 0 0
\(967\) 40.0267i 1.28717i −0.765374 0.643586i \(-0.777446\pi\)
0.765374 0.643586i \(-0.222554\pi\)
\(968\) −0.856728 + 10.9666i −0.0275363 + 0.352479i
\(969\) 2.71483i 0.0872128i
\(970\) 6.48271i 0.208147i
\(971\) 8.33112i 0.267358i −0.991025 0.133679i \(-0.957321\pi\)
0.991025 0.133679i \(-0.0426791\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 0 0
\(974\) 28.4612i 0.911954i
\(975\) 21.1167i 0.676275i
\(976\) 12.4897 0.399785
\(977\) 8.84048 0.282832 0.141416 0.989950i \(-0.454834\pi\)
0.141416 + 0.989950i \(0.454834\pi\)
\(978\) 13.7943 0.441095
\(979\) 1.87454 48.0632i 0.0599104 1.53611i
\(980\) 0 0
\(981\) 9.36206i 0.298908i
\(982\) 34.2454 1.09281
\(983\) 40.8291i 1.30224i 0.758973 + 0.651122i \(0.225702\pi\)
−0.758973 + 0.651122i \(0.774298\pi\)
\(984\) 5.43717 0.173331
\(985\) 0.841909 0.0268255
\(986\) 2.66597 0.0849017
\(987\) 0 0
\(988\) 33.8848 1.07802
\(989\) 18.5799i 0.590806i
\(990\) 1.18381 + 0.0461702i 0.0376239 + 0.00146738i
\(991\) −17.4799 −0.555266 −0.277633 0.960687i \(-0.589550\pi\)
−0.277633 + 0.960687i \(0.589550\pi\)
\(992\) 6.20254 0.196931
\(993\) 6.39942i 0.203080i
\(994\) 0 0
\(995\) 1.59453 0.0505500
\(996\) 16.7247i 0.529942i
\(997\) 17.5927 0.557168 0.278584 0.960412i \(-0.410135\pi\)
0.278584 + 0.960412i \(0.410135\pi\)
\(998\) 7.93504i 0.251179i
\(999\) 8.95377i 0.283285i
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 3234.2.e.c.2155.8 24
7.6 odd 2 3234.2.e.d.2155.5 yes 24
11.10 odd 2 3234.2.e.d.2155.20 yes 24
77.76 even 2 inner 3234.2.e.c.2155.17 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.e.c.2155.8 24 1.1 even 1 trivial
3234.2.e.c.2155.17 yes 24 77.76 even 2 inner
3234.2.e.d.2155.5 yes 24 7.6 odd 2
3234.2.e.d.2155.20 yes 24 11.10 odd 2