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Properties

Label 567.2.bd.a.17.11
Level $567$
Weight $2$
Character 567.17
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(17,567)] 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("567.17"); 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([11, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.bd (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 17.11
Character \(\chi\) \(=\) 567.17
Dual form 567.2.bd.a.467.11

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.0159182 + 0.00280680i) q^{2} +(-1.87914 + 0.683951i) q^{4} +(-3.75147 + 1.36542i) q^{5} +(0.157938 + 2.64103i) q^{7} +(0.0559891 - 0.0323253i) q^{8} +(0.0558840 - 0.0322646i) q^{10} +(1.76436 - 4.84753i) q^{11} +(-0.220791 - 0.606618i) q^{13} +(-0.00992693 - 0.0415971i) q^{14} +(3.06298 - 2.57014i) q^{16} +(-1.69383 - 2.93380i) q^{17} +(0.328436 + 0.189623i) q^{19} +(6.11565 - 5.13164i) q^{20} +(-0.0144792 + 0.0821158i) q^{22} +(0.993035 + 0.175099i) q^{23} +(8.37891 - 7.03074i) q^{25} +(0.00521724 + 0.00903653i) q^{26} +(-2.10312 - 4.85485i) q^{28} +(-1.97120 + 5.41582i) q^{29} +(-2.20741 - 6.06480i) q^{31} +(-0.124656 + 0.148560i) q^{32} +(0.0351973 + 0.0419465i) q^{34} +(-4.19863 - 9.69210i) q^{35} +6.16814 q^{37} +(-0.00576033 - 0.00209659i) q^{38} +(-0.165904 + 0.197716i) q^{40} +(-0.266252 + 0.0969077i) q^{41} +(-1.45052 - 8.22633i) q^{43} +10.3159i q^{44} -0.0162988 q^{46} +(-6.91258 - 2.51597i) q^{47} +(-6.95011 + 0.834239i) q^{49} +(-0.113643 + 0.135434i) q^{50} +(0.829794 + 0.988910i) q^{52} +(-1.71989 - 0.992981i) q^{53} +20.5944i q^{55} +(0.0942150 + 0.142764i) q^{56} +(0.0161767 - 0.0917427i) q^{58} +(2.62371 + 2.20155i) q^{59} +(-2.75545 + 7.57053i) q^{61} +(0.0521605 + 0.0903447i) q^{62} +(-3.99687 + 6.92277i) q^{64} +(1.65658 + 1.97424i) q^{65} +(1.63911 - 9.29584i) q^{67} +(5.18952 + 4.35453i) q^{68} +(0.0940381 + 0.142496i) q^{70} +(-8.57363 - 4.94999i) q^{71} -7.08887i q^{73} +(-0.0981854 + 0.0173127i) q^{74} +(-0.746871 - 0.131693i) q^{76} +(13.0811 + 3.89411i) q^{77} +(-0.222353 - 1.26102i) q^{79} +(-7.98133 + 13.8241i) q^{80} +(0.00396624 - 0.00228991i) q^{82} +(8.07527 + 2.93916i) q^{83} +(10.3602 + 8.69327i) q^{85} +(0.0461793 + 0.126877i) q^{86} +(-0.0579132 - 0.328442i) q^{88} +(2.67849 - 4.63928i) q^{89} +(1.56723 - 0.678924i) q^{91} +(-1.98581 + 0.350152i) q^{92} +(0.117097 + 0.0206474i) q^{94} +(-1.49103 - 0.262910i) q^{95} +(-7.39427 + 1.30381i) q^{97} +(0.108291 - 0.0327871i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} - 9 q^{10} - 9 q^{11} + 42 q^{14} - 15 q^{16} + 9 q^{17} - 9 q^{19} + 18 q^{20} - 12 q^{22} - 30 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31}+ \cdots + 180 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0159182 + 0.00280680i −0.0112558 + 0.00198471i −0.179273 − 0.983799i \(-0.557375\pi\)
0.168017 + 0.985784i \(0.446264\pi\)
\(3\) 0 0
\(4\) −1.87914 + 0.683951i −0.939570 + 0.341975i
\(5\) −3.75147 + 1.36542i −1.67771 + 0.610636i −0.992993 − 0.118175i \(-0.962296\pi\)
−0.684715 + 0.728811i \(0.740073\pi\)
\(6\) 0 0
\(7\) 0.157938 + 2.64103i 0.0596949 + 0.998217i
\(8\) 0.0559891 − 0.0323253i 0.0197951 − 0.0114287i
\(9\) 0 0
\(10\) 0.0558840 − 0.0322646i 0.0176721 − 0.0102030i
\(11\) 1.76436 − 4.84753i 0.531973 − 1.46158i −0.324746 − 0.945801i \(-0.605279\pi\)
0.856719 − 0.515783i \(-0.172499\pi\)
\(12\) 0 0
\(13\) −0.220791 − 0.606618i −0.0612364 − 0.168246i 0.905301 − 0.424770i \(-0.139645\pi\)
−0.966538 + 0.256524i \(0.917423\pi\)
\(14\) −0.00992693 − 0.0415971i −0.00265308 − 0.0111173i
\(15\) 0 0
\(16\) 3.06298 − 2.57014i 0.765744 − 0.642536i
\(17\) −1.69383 − 2.93380i −0.410815 − 0.711552i 0.584165 − 0.811635i \(-0.301422\pi\)
−0.994979 + 0.100084i \(0.968089\pi\)
\(18\) 0 0
\(19\) 0.328436 + 0.189623i 0.0753485 + 0.0435025i 0.537201 − 0.843454i \(-0.319482\pi\)
−0.461852 + 0.886957i \(0.652815\pi\)
\(20\) 6.11565 − 5.13164i 1.36750 − 1.14747i
\(21\) 0 0
\(22\) −0.0144792 + 0.0821158i −0.00308698 + 0.0175072i
\(23\) 0.993035 + 0.175099i 0.207062 + 0.0365106i 0.276217 − 0.961095i \(-0.410919\pi\)
−0.0691549 + 0.997606i \(0.522030\pi\)
\(24\) 0 0
\(25\) 8.37891 − 7.03074i 1.67578 − 1.40615i
\(26\) 0.00521724 + 0.00903653i 0.00102318 + 0.00177221i
\(27\) 0 0
\(28\) −2.10312 − 4.85485i −0.397453 − 0.917480i
\(29\) −1.97120 + 5.41582i −0.366042 + 1.00569i 0.610810 + 0.791777i \(0.290844\pi\)
−0.976852 + 0.213916i \(0.931378\pi\)
\(30\) 0 0
\(31\) −2.20741 − 6.06480i −0.396462 − 1.08927i −0.963995 − 0.265920i \(-0.914324\pi\)
0.567533 − 0.823351i \(-0.307898\pi\)
\(32\) −0.124656 + 0.148560i −0.0220363 + 0.0262619i
\(33\) 0 0
\(34\) 0.0351973 + 0.0419465i 0.00603628 + 0.00719376i
\(35\) −4.19863 − 9.69210i −0.709697 − 1.63826i
\(36\) 0 0
\(37\) 6.16814 1.01404 0.507018 − 0.861936i \(-0.330748\pi\)
0.507018 + 0.861936i \(0.330748\pi\)
\(38\) −0.00576033 − 0.00209659i −0.000934449 − 0.000340112i
\(39\) 0 0
\(40\) −0.165904 + 0.197716i −0.0262317 + 0.0312617i
\(41\) −0.266252 + 0.0969077i −0.0415816 + 0.0151344i −0.362727 − 0.931895i \(-0.618154\pi\)
0.321146 + 0.947030i \(0.395932\pi\)
\(42\) 0 0
\(43\) −1.45052 − 8.22633i −0.221203 − 1.25450i −0.869812 − 0.493383i \(-0.835760\pi\)
0.648610 − 0.761121i \(-0.275351\pi\)
\(44\) 10.3159i 1.55518i
\(45\) 0 0
\(46\) −0.0162988 −0.00240312
\(47\) −6.91258 − 2.51597i −1.00830 − 0.366992i −0.215522 − 0.976499i \(-0.569145\pi\)
−0.792781 + 0.609507i \(0.791368\pi\)
\(48\) 0 0
\(49\) −6.95011 + 0.834239i −0.992873 + 0.119177i
\(50\) −0.113643 + 0.135434i −0.0160715 + 0.0191533i
\(51\) 0 0
\(52\) 0.829794 + 0.988910i 0.115072 + 0.137137i
\(53\) −1.71989 − 0.992981i −0.236246 − 0.136396i 0.377204 − 0.926130i \(-0.376885\pi\)
−0.613450 + 0.789734i \(0.710219\pi\)
\(54\) 0 0
\(55\) 20.5944i 2.77695i
\(56\) 0.0942150 + 0.142764i 0.0125900 + 0.0190776i
\(57\) 0 0
\(58\) 0.0161767 − 0.0917427i 0.00212411 − 0.0120464i
\(59\) 2.62371 + 2.20155i 0.341578 + 0.286618i 0.797398 − 0.603454i \(-0.206209\pi\)
−0.455820 + 0.890072i \(0.650654\pi\)
\(60\) 0 0
\(61\) −2.75545 + 7.57053i −0.352799 + 0.969307i 0.628668 + 0.777674i \(0.283601\pi\)
−0.981467 + 0.191633i \(0.938622\pi\)
\(62\) 0.0521605 + 0.0903447i 0.00662439 + 0.0114738i
\(63\) 0 0
\(64\) −3.99687 + 6.92277i −0.499608 + 0.865347i
\(65\) 1.65658 + 1.97424i 0.205474 + 0.244874i
\(66\) 0 0
\(67\) 1.63911 − 9.29584i 0.200249 − 1.13567i −0.704494 − 0.709710i \(-0.748826\pi\)
0.904743 − 0.425958i \(-0.140063\pi\)
\(68\) 5.18952 + 4.35453i 0.629322 + 0.528064i
\(69\) 0 0
\(70\) 0.0940381 + 0.142496i 0.0112397 + 0.0170315i
\(71\) −8.57363 − 4.94999i −1.01750 − 0.587456i −0.104123 − 0.994564i \(-0.533204\pi\)
−0.913380 + 0.407109i \(0.866537\pi\)
\(72\) 0 0
\(73\) − 7.08887i − 0.829690i −0.909892 − 0.414845i \(-0.863836\pi\)
0.909892 − 0.414845i \(-0.136164\pi\)
\(74\) −0.0981854 + 0.0173127i −0.0114138 + 0.00201256i
\(75\) 0 0
\(76\) −0.746871 − 0.131693i −0.0856719 − 0.0151063i
\(77\) 13.0811 + 3.89411i 1.49073 + 0.443775i
\(78\) 0 0
\(79\) −0.222353 − 1.26102i −0.0250166 − 0.141876i 0.969741 − 0.244135i \(-0.0785041\pi\)
−0.994758 + 0.102259i \(0.967393\pi\)
\(80\) −7.98133 + 13.8241i −0.892340 + 1.54558i
\(81\) 0 0
\(82\) 0.00396624 − 0.00228991i 0.000437998 − 0.000252878i
\(83\) 8.07527 + 2.93916i 0.886376 + 0.322614i 0.744780 − 0.667310i \(-0.232554\pi\)
0.141596 + 0.989925i \(0.454777\pi\)
\(84\) 0 0
\(85\) 10.3602 + 8.69327i 1.12373 + 0.942918i
\(86\) 0.0461793 + 0.126877i 0.00497964 + 0.0136815i
\(87\) 0 0
\(88\) −0.0579132 − 0.328442i −0.00617356 − 0.0350120i
\(89\) 2.67849 − 4.63928i 0.283920 − 0.491763i −0.688427 − 0.725306i \(-0.741698\pi\)
0.972347 + 0.233543i \(0.0750318\pi\)
\(90\) 0 0
\(91\) 1.56723 − 0.678924i 0.164290 − 0.0711706i
\(92\) −1.98581 + 0.350152i −0.207035 + 0.0365059i
\(93\) 0 0
\(94\) 0.117097 + 0.0206474i 0.0120777 + 0.00212962i
\(95\) −1.49103 − 0.262910i −0.152977 − 0.0269739i
\(96\) 0 0
\(97\) −7.39427 + 1.30381i −0.750774 + 0.132382i −0.535925 − 0.844265i \(-0.680037\pi\)
−0.214849 + 0.976647i \(0.568926\pi\)
\(98\) 0.108291 − 0.0327871i 0.0109391 − 0.00331200i
\(99\) 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.bd.a.17.11 132
3.2 odd 2 189.2.bd.a.185.12 yes 132
7.5 odd 6 567.2.ba.a.341.12 132
21.5 even 6 189.2.ba.a.131.11 yes 132
27.7 even 9 189.2.ba.a.101.11 ✓ 132
27.20 odd 18 567.2.ba.a.143.12 132
189.47 even 18 inner 567.2.bd.a.467.11 132
189.61 odd 18 189.2.bd.a.47.12 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.11 ✓ 132 27.7 even 9
189.2.ba.a.131.11 yes 132 21.5 even 6
189.2.bd.a.47.12 yes 132 189.61 odd 18
189.2.bd.a.185.12 yes 132 3.2 odd 2
567.2.ba.a.143.12 132 27.20 odd 18
567.2.ba.a.341.12 132 7.5 odd 6
567.2.bd.a.17.11 132 1.1 even 1 trivial
567.2.bd.a.467.11 132 189.47 even 18 inner