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Properties

Label 567.2.ba.a.143.14
Level $567$
Weight $2$
Character 567.143
Analytic conductor $4.528$
Analytic rank $0$
Dimension $132$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(143,567)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("567.143"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([7, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.ba (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 143.14
Character \(\chi\) \(=\) 567.143
Dual form 567.2.ba.a.341.14

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.575801 - 0.686212i) q^{2} +(0.207955 + 1.17937i) q^{4} +(-0.100696 + 0.0844942i) q^{5} +(-1.70219 + 2.02548i) q^{7} +(2.48059 + 1.43217i) q^{8} +0.117751i q^{10} +(-2.57587 + 3.06980i) q^{11} +(1.02604 - 2.81901i) q^{13} +(0.409789 + 2.33433i) q^{14} +(0.160408 - 0.0583836i) q^{16} +0.344621 q^{17} +4.89841i q^{19} +(-0.120590 - 0.101187i) q^{20} +(0.623349 + 3.53519i) q^{22} +(-2.18009 + 5.98976i) q^{23} +(-0.865240 + 4.90702i) q^{25} +(-1.34365 - 2.32726i) q^{26} +(-2.74277 - 1.58630i) q^{28} +(-2.29851 - 6.31510i) q^{29} +(8.59607 - 1.51572i) q^{31} +(-1.90702 + 5.23950i) q^{32} +(0.198433 - 0.236483i) q^{34} +(0.000262488 - 0.347783i) q^{35} +(3.64300 - 6.30985i) q^{37} +(3.36135 + 2.82051i) q^{38} +(-0.370796 + 0.0653813i) q^{40} +(9.04416 + 3.29180i) q^{41} +(0.350643 - 1.98860i) q^{43} +(-4.15611 - 2.39953i) q^{44} +(2.85495 + 4.94491i) q^{46} +(-0.771769 + 4.37692i) q^{47} +(-1.20513 - 6.89548i) q^{49} +(2.86905 + 3.41920i) q^{50} +(3.53803 + 0.623850i) q^{52} +(-5.49931 - 3.17503i) q^{53} -0.526764i q^{55} +(-7.12325 + 2.58656i) q^{56} +(-5.65698 - 2.05897i) q^{58} +(-0.167214 - 0.0608610i) q^{59} +(-4.60007 - 0.811116i) q^{61} +(3.90952 - 6.77148i) q^{62} +(2.66805 + 4.62120i) q^{64} +(0.134872 + 0.370558i) q^{65} +(8.83064 - 7.40979i) q^{67} +(0.0716657 + 0.406437i) q^{68} +(-0.238502 - 0.200434i) q^{70} +(-0.373587 + 0.215690i) q^{71} +(1.31951 - 0.761822i) q^{73} +(-2.23226 - 6.13309i) q^{74} +(-5.77706 + 1.01865i) q^{76} +(-1.83321 - 10.4427i) q^{77} +(1.81204 + 1.52049i) q^{79} +(-0.0112194 + 0.0194325i) q^{80} +(7.46651 - 4.31079i) q^{82} +(6.81615 - 2.48087i) q^{83} +(-0.0347021 + 0.0291185i) q^{85} +(-1.16270 - 1.38565i) q^{86} +(-10.7861 + 3.92584i) q^{88} +15.4181 q^{89} +(3.96334 + 6.87669i) q^{91} +(-7.51752 - 1.32554i) q^{92} +(2.55911 + 3.04983i) q^{94} +(-0.413888 - 0.493252i) q^{95} +(-12.2061 - 2.15226i) q^{97} +(-5.42568 - 3.14345i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} + 9 q^{11} - 3 q^{14} + 3 q^{16} + 18 q^{17} - 18 q^{20} - 12 q^{22} + 6 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 18 q^{34}+ \cdots - 27 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{18}\right)\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.575801 − 0.686212i 0.407152 − 0.485225i −0.523034 − 0.852312i \(-0.675200\pi\)
0.930187 + 0.367086i \(0.119645\pi\)
\(3\) 0 0
\(4\) 0.207955 + 1.17937i 0.103978 + 0.589686i
\(5\) −0.100696 + 0.0844942i −0.0450327 + 0.0377870i −0.665026 − 0.746820i \(-0.731580\pi\)
0.619994 + 0.784607i \(0.287135\pi\)
\(6\) 0 0
\(7\) −1.70219 + 2.02548i −0.643366 + 0.765559i
\(8\) 2.48059 + 1.43217i 0.877021 + 0.506348i
\(9\) 0 0
\(10\) 0.117751i 0.0372361i
\(11\) −2.57587 + 3.06980i −0.776654 + 0.925580i −0.998777 − 0.0494386i \(-0.984257\pi\)
0.222123 + 0.975019i \(0.428701\pi\)
\(12\) 0 0
\(13\) 1.02604 − 2.81901i 0.284571 − 0.781852i −0.712231 − 0.701945i \(-0.752315\pi\)
0.996802 − 0.0799075i \(-0.0254625\pi\)
\(14\) 0.409789 + 2.33433i 0.109521 + 0.623877i
\(15\) 0 0
\(16\) 0.160408 − 0.0583836i 0.0401019 − 0.0145959i
\(17\) 0.344621 0.0835829 0.0417914 − 0.999126i \(-0.486693\pi\)
0.0417914 + 0.999126i \(0.486693\pi\)
\(18\) 0 0
\(19\) 4.89841i 1.12377i 0.827214 + 0.561887i \(0.189924\pi\)
−0.827214 + 0.561887i \(0.810076\pi\)
\(20\) −0.120590 − 0.101187i −0.0269649 − 0.0226262i
\(21\) 0 0
\(22\) 0.623349 + 3.53519i 0.132898 + 0.753704i
\(23\) −2.18009 + 5.98976i −0.454581 + 1.24895i 0.474887 + 0.880047i \(0.342489\pi\)
−0.929468 + 0.368904i \(0.879733\pi\)
\(24\) 0 0
\(25\) −0.865240 + 4.90702i −0.173048 + 0.981404i
\(26\) −1.34365 − 2.32726i −0.263511 − 0.456414i
\(27\) 0 0
\(28\) −2.74277 − 1.58630i −0.518335 − 0.299783i
\(29\) −2.29851 − 6.31510i −0.426822 − 1.17268i −0.947731 − 0.319072i \(-0.896629\pi\)
0.520908 − 0.853613i \(-0.325593\pi\)
\(30\) 0 0
\(31\) 8.59607 − 1.51572i 1.54390 − 0.272231i 0.664125 − 0.747622i \(-0.268804\pi\)
0.879775 + 0.475391i \(0.157693\pi\)
\(32\) −1.90702 + 5.23950i −0.337117 + 0.926222i
\(33\) 0 0
\(34\) 0.198433 − 0.236483i 0.0340310 − 0.0405565i
\(35\) 0.000262488 − 0.347783i 4.43686e−5 − 0.0587861i
\(36\) 0 0
\(37\) 3.64300 − 6.30985i 0.598905 − 1.03733i −0.394078 − 0.919077i \(-0.628936\pi\)
0.992983 − 0.118257i \(-0.0377305\pi\)
\(38\) 3.36135 + 2.82051i 0.545283 + 0.457547i
\(39\) 0 0
\(40\) −0.370796 + 0.0653813i −0.0586280 + 0.0103377i
\(41\) 9.04416 + 3.29180i 1.41246 + 0.514094i 0.931851 − 0.362841i \(-0.118193\pi\)
0.480609 + 0.876935i \(0.340415\pi\)
\(42\) 0 0
\(43\) 0.350643 − 1.98860i 0.0534726 − 0.303258i −0.946328 − 0.323207i \(-0.895239\pi\)
0.999801 + 0.0199486i \(0.00635026\pi\)
\(44\) −4.15611 − 2.39953i −0.626556 − 0.361743i
\(45\) 0 0
\(46\) 2.85495 + 4.94491i 0.420939 + 0.729088i
\(47\) −0.771769 + 4.37692i −0.112574 + 0.638439i 0.875349 + 0.483492i \(0.160632\pi\)
−0.987923 + 0.154947i \(0.950479\pi\)
\(48\) 0 0
\(49\) −1.20513 − 6.89548i −0.172161 − 0.985069i
\(50\) 2.86905 + 3.41920i 0.405745 + 0.483549i
\(51\) 0 0
\(52\) 3.53803 + 0.623850i 0.490637 + 0.0865125i
\(53\) −5.49931 − 3.17503i −0.755388 − 0.436123i 0.0722494 − 0.997387i \(-0.476982\pi\)
−0.827637 + 0.561263i \(0.810316\pi\)
\(54\) 0 0
\(55\) − 0.526764i − 0.0710288i
\(56\) −7.12325 + 2.58656i −0.951884 + 0.345644i
\(57\) 0 0
\(58\) −5.65698 − 2.05897i −0.742798 − 0.270356i
\(59\) −0.167214 − 0.0608610i −0.0217694 − 0.00792343i 0.331112 − 0.943591i \(-0.392576\pi\)
−0.352882 + 0.935668i \(0.614798\pi\)
\(60\) 0 0
\(61\) −4.60007 − 0.811116i −0.588978 − 0.103853i −0.128787 − 0.991672i \(-0.541108\pi\)
−0.460192 + 0.887820i \(0.652219\pi\)
\(62\) 3.90952 − 6.77148i 0.496509 − 0.859979i
\(63\) 0 0
\(64\) 2.66805 + 4.62120i 0.333506 + 0.577649i
\(65\) 0.134872 + 0.370558i 0.0167288 + 0.0459620i
\(66\) 0 0
\(67\) 8.83064 − 7.40979i 1.07883 − 0.905249i 0.0830108 − 0.996549i \(-0.473546\pi\)
0.995824 + 0.0912992i \(0.0291020\pi\)
\(68\) 0.0716657 + 0.406437i 0.00869075 + 0.0492877i
\(69\) 0 0
\(70\) −0.238502 − 0.200434i −0.0285064 − 0.0239564i
\(71\) −0.373587 + 0.215690i −0.0443366 + 0.0255977i −0.522004 − 0.852943i \(-0.674816\pi\)
0.477668 + 0.878540i \(0.341482\pi\)
\(72\) 0 0
\(73\) 1.31951 − 0.761822i 0.154437 − 0.0891645i −0.420790 − 0.907158i \(-0.638247\pi\)
0.575227 + 0.817994i \(0.304914\pi\)
\(74\) −2.23226 − 6.13309i −0.259495 − 0.712957i
\(75\) 0 0
\(76\) −5.77706 + 1.01865i −0.662674 + 0.116847i
\(77\) −1.83321 − 10.4427i −0.208914 − 1.19006i
\(78\) 0 0
\(79\) 1.81204 + 1.52049i 0.203871 + 0.171068i 0.739007 − 0.673698i \(-0.235295\pi\)
−0.535136 + 0.844766i \(0.679740\pi\)
\(80\) −0.0112194 + 0.0194325i −0.00125436 + 0.00217262i
\(81\) 0 0
\(82\) 7.46651 − 4.31079i 0.824538 − 0.476047i
\(83\) 6.81615 − 2.48087i 0.748169 − 0.272311i 0.0603341 − 0.998178i \(-0.480783\pi\)
0.687835 + 0.725867i \(0.258561\pi\)
\(84\) 0 0
\(85\) −0.0347021 + 0.0291185i −0.00376397 + 0.00315834i
\(86\) −1.16270 − 1.38565i −0.125377 − 0.149418i
\(87\) 0 0
\(88\) −10.7861 + 3.92584i −1.14981 + 0.418496i
\(89\) 15.4181 1.63431 0.817156 − 0.576417i \(-0.195550\pi\)
0.817156 + 0.576417i \(0.195550\pi\)
\(90\) 0 0
\(91\) 3.96334 + 6.87669i 0.415471 + 0.720873i
\(92\) −7.51752 − 1.32554i −0.783756 − 0.138197i
\(93\) 0 0
\(94\) 2.55911 + 3.04983i 0.263952 + 0.314566i
\(95\) −0.413888 − 0.493252i −0.0424640 − 0.0506066i
\(96\) 0 0
\(97\) −12.2061 − 2.15226i −1.23934 − 0.218529i −0.484708 − 0.874676i \(-0.661074\pi\)
−0.754634 + 0.656147i \(0.772185\pi\)
\(98\) −5.42568 − 3.14345i −0.548076 − 0.317536i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 567.2.ba.a.143.14 132
3.2 odd 2 189.2.ba.a.101.9 ✓ 132
7.5 odd 6 567.2.bd.a.467.9 132
21.5 even 6 189.2.bd.a.47.14 yes 132
27.4 even 9 189.2.bd.a.185.14 yes 132
27.23 odd 18 567.2.bd.a.17.9 132
189.131 even 18 inner 567.2.ba.a.341.14 132
189.166 odd 18 189.2.ba.a.131.9 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.9 ✓ 132 3.2 odd 2
189.2.ba.a.131.9 yes 132 189.166 odd 18
189.2.bd.a.47.14 yes 132 21.5 even 6
189.2.bd.a.185.14 yes 132 27.4 even 9
567.2.ba.a.143.14 132 1.1 even 1 trivial
567.2.ba.a.341.14 132 189.131 even 18 inner
567.2.bd.a.17.9 132 27.23 odd 18
567.2.bd.a.467.9 132 7.5 odd 6