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Properties

Label 567.2.ba.a.341.12
Level $567$
Weight $2$
Character 567.341
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 341.12
Character \(\chi\) \(=\) 567.341
Dual form 567.2.ba.a.143.12

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.0103898 + 0.0123821i) q^{2} +(0.347251 - 1.96936i) q^{4} +(-3.05823 - 2.56616i) q^{5} +(-2.57348 - 0.614149i) q^{7} +(0.0559891 - 0.0323253i) q^{8} -0.0645293i q^{10} +(3.31590 + 3.95174i) q^{11} +(0.220791 + 0.606618i) q^{13} +(-0.0191336 - 0.0382461i) q^{14} +(-3.75730 - 1.36754i) q^{16} -3.38766 q^{17} +0.379246i q^{19} +(-6.11565 + 5.13164i) q^{20} +(-0.0144792 + 0.0821158i) q^{22} +(-0.344877 - 0.947543i) q^{23} +(1.89935 + 10.7717i) q^{25} +(-0.00521724 + 0.00903653i) q^{26} +(-2.10312 + 4.85485i) q^{28} +(-1.97120 + 5.41582i) q^{29} +(-6.35598 - 1.12073i) q^{31} +(-0.0663282 - 0.182235i) q^{32} +(-0.0351973 - 0.0419465i) q^{34} +(6.29429 + 8.48217i) q^{35} +(-3.08407 - 5.34176i) q^{37} +(-0.00469587 + 0.00394030i) q^{38} +(-0.254179 - 0.0448186i) q^{40} +(0.266252 - 0.0969077i) q^{41} +(-1.45052 - 8.22633i) q^{43} +(8.93384 - 5.15796i) q^{44} +(0.00814938 - 0.0141151i) q^{46} +(-1.27739 - 7.24446i) q^{47} +(6.24564 + 3.16101i) q^{49} +(-0.113643 + 0.135434i) q^{50} +(1.27132 - 0.224168i) q^{52} +(1.71989 - 0.992981i) q^{53} -20.5944i q^{55} +(-0.163940 + 0.0488030i) q^{56} +(-0.0875398 + 0.0318619i) q^{58} +(3.21846 - 1.17142i) q^{59} +(-7.93400 + 1.39898i) q^{61} +(-0.0521605 - 0.0903447i) q^{62} +(-3.99687 + 6.92277i) q^{64} +(0.881448 - 2.42176i) q^{65} +(7.23088 + 6.06743i) q^{67} +(-1.17637 + 6.67152i) q^{68} +(-0.0396306 + 0.166065i) q^{70} +(-8.57363 - 4.94999i) q^{71} +(-6.13914 - 3.54444i) q^{73} +(0.0340994 - 0.0936874i) q^{74} +(0.746871 + 0.131693i) q^{76} +(-6.10647 - 12.2062i) q^{77} +(-0.980903 + 0.823075i) q^{79} +(7.98133 + 13.8241i) q^{80} +(0.00396624 + 0.00228991i) q^{82} +(-8.07527 - 2.93916i) q^{83} +(10.3602 + 8.69327i) q^{85} +(0.0867887 - 0.103431i) q^{86} +(0.313396 + 0.114067i) q^{88} +5.35698 q^{89} +(-0.195648 - 1.69672i) q^{91} +(-1.98581 + 0.350152i) q^{92} +(0.0764298 - 0.0910855i) q^{94} +(0.973203 - 1.15982i) q^{95} +(7.39427 - 1.30381i) q^{97} +(0.0257512 + 0.110177i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} + 9 q^{11} - 3 q^{14} + 3 q^{16} + 18 q^{17} - 18 q^{20} - 12 q^{22} + 6 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 18 q^{34}+ \cdots - 27 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0103898 + 0.0123821i 0.00734672 + 0.00875548i 0.769705 − 0.638399i \(-0.220403\pi\)
−0.762359 + 0.647155i \(0.775959\pi\)
\(3\) 0 0
\(4\) 0.347251 − 1.96936i 0.173625 − 0.984679i
\(5\) −3.05823 − 2.56616i −1.36768 − 1.14762i −0.973526 − 0.228575i \(-0.926593\pi\)
−0.394154 − 0.919045i \(-0.628962\pi\)
\(6\) 0 0
\(7\) −2.57348 − 0.614149i −0.972686 − 0.232127i
\(8\) 0.0559891 − 0.0323253i 0.0197951 − 0.0114287i
\(9\) 0 0
\(10\) − 0.0645293i − 0.0204059i
\(11\) 3.31590 + 3.95174i 0.999782 + 1.19149i 0.981461 + 0.191662i \(0.0613878\pi\)
0.0183214 + 0.999832i \(0.494168\pi\)
\(12\) 0 0
\(13\) 0.220791 + 0.606618i 0.0612364 + 0.168246i 0.966538 − 0.256524i \(-0.0825773\pi\)
−0.905301 + 0.424770i \(0.860355\pi\)
\(14\) −0.0191336 − 0.0382461i −0.00511367 − 0.0102217i
\(15\) 0 0
\(16\) −3.75730 − 1.36754i −0.939324 − 0.341886i
\(17\) −3.38766 −0.821629 −0.410815 − 0.911719i \(-0.634756\pi\)
−0.410815 + 0.911719i \(0.634756\pi\)
\(18\) 0 0
\(19\) 0.379246i 0.0870049i 0.999053 + 0.0435025i \(0.0138516\pi\)
−0.999053 + 0.0435025i \(0.986148\pi\)
\(20\) −6.11565 + 5.13164i −1.36750 + 1.14747i
\(21\) 0 0
\(22\) −0.0144792 + 0.0821158i −0.00308698 + 0.0175072i
\(23\) −0.344877 − 0.947543i −0.0719119 − 0.197576i 0.898530 − 0.438913i \(-0.144636\pi\)
−0.970441 + 0.241337i \(0.922414\pi\)
\(24\) 0 0
\(25\) 1.89935 + 10.7717i 0.379869 + 2.15434i
\(26\) −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
\(31\) −6.35598 − 1.12073i −1.14157 − 0.201289i −0.429276 − 0.903173i \(-0.641231\pi\)
−0.712291 + 0.701884i \(0.752342\pi\)
\(32\) −0.0663282 − 0.182235i −0.0117253 − 0.0322150i
\(33\) 0 0
\(34\) −0.0351973 − 0.0419465i −0.00603628 − 0.00719376i
\(35\) 6.29429 + 8.48217i 1.06393 + 1.43375i
\(36\) 0 0
\(37\) −3.08407 − 5.34176i −0.507018 − 0.878181i −0.999967 − 0.00812261i \(-0.997414\pi\)
0.492949 − 0.870058i \(-0.335919\pi\)
\(38\) −0.00469587 + 0.00394030i −0.000761770 + 0.000639201i
\(39\) 0 0
\(40\) −0.254179 − 0.0448186i −0.0401892 − 0.00708645i
\(41\) 0.266252 − 0.0969077i 0.0415816 − 0.0151344i −0.321146 − 0.947030i \(-0.604068\pi\)
0.362727 + 0.931895i \(0.381846\pi\)
\(42\) 0 0
\(43\) −1.45052 − 8.22633i −0.221203 − 1.25450i −0.869812 − 0.493383i \(-0.835760\pi\)
0.648610 − 0.761121i \(-0.275351\pi\)
\(44\) 8.93384 − 5.15796i 1.34683 − 0.777591i
\(45\) 0 0
\(46\) 0.00814938 − 0.0141151i 0.00120156 − 0.00208116i
\(47\) −1.27739 − 7.24446i −0.186327 − 1.05671i −0.924239 − 0.381815i \(-0.875299\pi\)
0.737912 − 0.674897i \(-0.235812\pi\)
\(48\) 0 0
\(49\) 6.24564 + 3.16101i 0.892235 + 0.451572i
\(50\) −0.113643 + 0.135434i −0.0160715 + 0.0191533i
\(51\) 0 0
\(52\) 1.27132 − 0.224168i 0.176300 − 0.0310865i
\(53\) 1.71989 − 0.992981i 0.236246 − 0.136396i −0.377204 − 0.926130i \(-0.623115\pi\)
0.613450 + 0.789734i \(0.289781\pi\)
\(54\) 0 0
\(55\) − 20.5944i − 2.77695i
\(56\) −0.163940 + 0.0488030i −0.0219074 + 0.00652158i
\(57\) 0 0
\(58\) −0.0875398 + 0.0318619i −0.0114945 + 0.00418367i
\(59\) 3.21846 − 1.17142i 0.419007 − 0.152506i −0.123908 − 0.992294i \(-0.539543\pi\)
0.542915 + 0.839788i \(0.317321\pi\)
\(60\) 0 0
\(61\) −7.93400 + 1.39898i −1.01584 + 0.179121i −0.656692 − 0.754158i \(-0.728045\pi\)
−0.359152 + 0.933279i \(0.616934\pi\)
\(62\) −0.0521605 − 0.0903447i −0.00662439 − 0.0114738i
\(63\) 0 0
\(64\) −3.99687 + 6.92277i −0.499608 + 0.865347i
\(65\) 0.881448 − 2.42176i 0.109330 − 0.300382i
\(66\) 0 0
\(67\) 7.23088 + 6.06743i 0.883393 + 0.741255i 0.966874 − 0.255255i \(-0.0821593\pi\)
−0.0834809 + 0.996509i \(0.526604\pi\)
\(68\) −1.17637 + 6.67152i −0.142656 + 0.809041i
\(69\) 0 0
\(70\) −0.0396306 + 0.166065i −0.00473676 + 0.0198486i
\(71\) −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
\(73\) −6.13914 − 3.54444i −0.718533 − 0.414845i 0.0956798 − 0.995412i \(-0.469498\pi\)
−0.814212 + 0.580567i \(0.802831\pi\)
\(74\) 0.0340994 − 0.0936874i 0.00396398 − 0.0108909i
\(75\) 0 0
\(76\) 0.746871 + 0.131693i 0.0856719 + 0.0151063i
\(77\) −6.10647 − 12.2062i −0.695896 − 1.39103i
\(78\) 0 0
\(79\) −0.980903 + 0.823075i −0.110360 + 0.0926032i −0.696298 − 0.717753i \(-0.745171\pi\)
0.585938 + 0.810356i \(0.300726\pi\)
\(80\) 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 −0.141596 − 0.989925i \(-0.545223\pi\)
−0.744780 + 0.667310i \(0.767446\pi\)
\(84\) 0 0
\(85\) 10.3602 + 8.69327i 1.12373 + 0.942918i
\(86\) 0.0867887 − 0.103431i 0.00935867 − 0.0111532i
\(87\) 0 0
\(88\) 0.313396 + 0.114067i 0.0334081 + 0.0121595i
\(89\) 5.35698 0.567839 0.283920 − 0.958848i \(-0.408365\pi\)
0.283920 + 0.958848i \(0.408365\pi\)
\(90\) 0 0
\(91\) −0.195648 − 1.69672i −0.0205095 − 0.177865i
\(92\) −1.98581 + 0.350152i −0.207035 + 0.0365059i
\(93\) 0 0
\(94\) 0.0764298 − 0.0910855i 0.00788314 − 0.00939476i
\(95\) 0.973203 − 1.15982i 0.0998486 − 0.118995i
\(96\) 0 0
\(97\) 7.39427 − 1.30381i 0.750774 − 0.132382i 0.214849 − 0.976647i \(-0.431074\pi\)
0.535925 + 0.844265i \(0.319963\pi\)
\(98\) 0.0257512 + 0.110177i 0.00260127 + 0.0111295i
\(99\) 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.341.12 132
3.2 odd 2 189.2.ba.a.131.11 yes 132
7.3 odd 6 567.2.bd.a.17.11 132
21.17 even 6 189.2.bd.a.185.12 yes 132
27.7 even 9 189.2.bd.a.47.12 yes 132
27.20 odd 18 567.2.bd.a.467.11 132
189.101 even 18 inner 567.2.ba.a.143.12 132
189.115 odd 18 189.2.ba.a.101.11 ✓ 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.11 ✓ 132 189.115 odd 18
189.2.ba.a.131.11 yes 132 3.2 odd 2
189.2.bd.a.47.12 yes 132 27.7 even 9
189.2.bd.a.185.12 yes 132 21.17 even 6
567.2.ba.a.143.12 132 189.101 even 18 inner
567.2.ba.a.341.12 132 1.1 even 1 trivial
567.2.bd.a.17.11 132 7.3 odd 6
567.2.bd.a.467.11 132 27.20 odd 18