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Properties

Label 567.2.ba.a.143.8
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 143.8
Character \(\chi\) \(=\) 567.143
Dual form 567.2.ba.a.341.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.594351 + 0.708320i) q^{2} +(0.198832 + 1.12763i) q^{4} +(0.386349 - 0.324185i) q^{5} +(0.529787 - 2.59217i) q^{7} +(-2.51844 - 1.45402i) q^{8} +0.466338i q^{10} +(-3.12271 + 3.72150i) q^{11} +(-1.60911 + 4.42100i) q^{13} +(1.52120 + 1.91591i) q^{14} +(0.374792 - 0.136413i) q^{16} +4.77392 q^{17} +6.37115i q^{19} +(0.442381 + 0.371202i) q^{20} +(-0.780028 - 4.42376i) q^{22} +(-1.22699 + 3.37113i) q^{23} +(-0.824071 + 4.67354i) q^{25} +(-2.17510 - 3.76739i) q^{26} +(3.02836 + 0.0819996i) q^{28} +(-0.997075 - 2.73944i) q^{29} +(-0.773023 + 0.136305i) q^{31} +(1.86308 - 5.11878i) q^{32} +(-2.83739 + 3.38147i) q^{34} +(-0.635660 - 1.17323i) q^{35} +(2.73726 - 4.74107i) q^{37} +(-4.51281 - 3.78670i) q^{38} +(-1.44437 + 0.254681i) q^{40} +(-5.44849 - 1.98309i) q^{41} +(-1.36729 + 7.75429i) q^{43} +(-4.81739 - 2.78132i) q^{44} +(-1.65857 - 2.87274i) q^{46} +(-1.33467 + 7.56931i) q^{47} +(-6.43865 - 2.74659i) q^{49} +(-2.82057 - 3.36143i) q^{50} +(-5.30521 - 0.935452i) q^{52} +(4.26125 + 2.46023i) q^{53} +2.45014i q^{55} +(-5.10329 + 5.75788i) q^{56} +(2.53301 + 0.921941i) q^{58} +(5.01169 + 1.82411i) q^{59} +(-7.07246 - 1.24706i) q^{61} +(0.362900 - 0.628561i) q^{62} +(2.91725 + 5.05283i) q^{64} +(0.811544 + 2.22970i) q^{65} +(6.70384 - 5.62519i) q^{67} +(0.949211 + 5.38324i) q^{68} +(1.20883 + 0.247060i) q^{70} +(11.3110 - 6.53043i) q^{71} +(8.42907 - 4.86653i) q^{73} +(1.73130 + 4.75671i) q^{74} +(-7.18433 + 1.26679i) q^{76} +(7.99238 + 10.0662i) q^{77} +(-3.63977 - 3.05413i) q^{79} +(0.100577 - 0.174205i) q^{80} +(4.64298 - 2.68063i) q^{82} +(-2.75845 + 1.00400i) q^{83} +(1.84440 - 1.54764i) q^{85} +(-4.67987 - 5.57725i) q^{86} +(13.2755 - 4.83188i) q^{88} -11.2094 q^{89} +(10.6075 + 6.51327i) q^{91} +(-4.04537 - 0.713308i) q^{92} +(-4.56823 - 5.44420i) q^{94} +(2.06543 + 2.46149i) q^{95} +(-2.41733 - 0.426241i) q^{97} +(5.77228 - 2.92818i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} + 9 q^{11} - 3 q^{14} + 3 q^{16} + 18 q^{17} - 18 q^{20} - 12 q^{22} + 6 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 18 q^{34}+ \cdots - 27 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.594351 + 0.708320i −0.420270 + 0.500858i −0.934089 − 0.357041i \(-0.883786\pi\)
0.513819 + 0.857898i \(0.328230\pi\)
\(3\) 0 0
\(4\) 0.198832 + 1.12763i 0.0994162 + 0.563817i
\(5\) 0.386349 − 0.324185i 0.172781 − 0.144980i −0.552297 − 0.833648i \(-0.686248\pi\)
0.725077 + 0.688667i \(0.241804\pi\)
\(6\) 0 0
\(7\) 0.529787 − 2.59217i 0.200241 − 0.979747i
\(8\) −2.51844 − 1.45402i −0.890401 − 0.514073i
\(9\) 0 0
\(10\) 0.466338i 0.147469i
\(11\) −3.12271 + 3.72150i −0.941533 + 1.12208i 0.0508281 + 0.998707i \(0.483814\pi\)
−0.992361 + 0.123368i \(0.960631\pi\)
\(12\) 0 0
\(13\) −1.60911 + 4.42100i −0.446287 + 1.22616i 0.489003 + 0.872282i \(0.337361\pi\)
−0.935290 + 0.353882i \(0.884862\pi\)
\(14\) 1.52120 + 1.91591i 0.406559 + 0.512050i
\(15\) 0 0
\(16\) 0.374792 − 0.136413i 0.0936980 − 0.0341033i
\(17\) 4.77392 1.15785 0.578923 − 0.815382i \(-0.303473\pi\)
0.578923 + 0.815382i \(0.303473\pi\)
\(18\) 0 0
\(19\) 6.37115i 1.46164i 0.682569 + 0.730821i \(0.260863\pi\)
−0.682569 + 0.730821i \(0.739137\pi\)
\(20\) 0.442381 + 0.371202i 0.0989195 + 0.0830033i
\(21\) 0 0
\(22\) −0.780028 − 4.42376i −0.166302 − 0.943148i
\(23\) −1.22699 + 3.37113i −0.255845 + 0.702929i 0.743567 + 0.668661i \(0.233132\pi\)
−0.999413 + 0.0342682i \(0.989090\pi\)
\(24\) 0 0
\(25\) −0.824071 + 4.67354i −0.164814 + 0.934708i
\(26\) −2.17510 − 3.76739i −0.426573 − 0.738846i
\(27\) 0 0
\(28\) 3.02836 + 0.0819996i 0.572305 + 0.0154965i
\(29\) −0.997075 − 2.73944i −0.185152 − 0.508701i 0.812039 − 0.583604i \(-0.198358\pi\)
−0.997191 + 0.0749022i \(0.976136\pi\)
\(30\) 0 0
\(31\) −0.773023 + 0.136305i −0.138839 + 0.0244811i −0.242636 − 0.970117i \(-0.578012\pi\)
0.103797 + 0.994599i \(0.466901\pi\)
\(32\) 1.86308 − 5.11878i 0.329349 − 0.904880i
\(33\) 0 0
\(34\) −2.83739 + 3.38147i −0.486608 + 0.579916i
\(35\) −0.635660 − 1.17323i −0.107446 − 0.198312i
\(36\) 0 0
\(37\) 2.73726 − 4.74107i 0.450002 − 0.779426i −0.548383 − 0.836227i \(-0.684757\pi\)
0.998386 + 0.0568005i \(0.0180899\pi\)
\(38\) −4.51281 − 3.78670i −0.732075 − 0.614284i
\(39\) 0 0
\(40\) −1.44437 + 0.254681i −0.228374 + 0.0402686i
\(41\) −5.44849 − 1.98309i −0.850912 − 0.309707i −0.120500 − 0.992713i \(-0.538450\pi\)
−0.730412 + 0.683007i \(0.760672\pi\)
\(42\) 0 0
\(43\) −1.36729 + 7.75429i −0.208510 + 1.18252i 0.683310 + 0.730128i \(0.260540\pi\)
−0.891820 + 0.452390i \(0.850571\pi\)
\(44\) −4.81739 − 2.78132i −0.726249 − 0.419300i
\(45\) 0 0
\(46\) −1.65857 − 2.87274i −0.244544 − 0.423562i
\(47\) −1.33467 + 7.56931i −0.194682 + 1.10410i 0.718188 + 0.695849i \(0.244972\pi\)
−0.912871 + 0.408249i \(0.866139\pi\)
\(48\) 0 0
\(49\) −6.43865 − 2.74659i −0.919807 − 0.392370i
\(50\) −2.82057 − 3.36143i −0.398889 − 0.475378i
\(51\) 0 0
\(52\) −5.30521 − 0.935452i −0.735700 − 0.129724i
\(53\) 4.26125 + 2.46023i 0.585328 + 0.337939i 0.763248 − 0.646106i \(-0.223604\pi\)
−0.177920 + 0.984045i \(0.556937\pi\)
\(54\) 0 0
\(55\) 2.45014i 0.330376i
\(56\) −5.10329 + 5.75788i −0.681956 + 0.769429i
\(57\) 0 0
\(58\) 2.53301 + 0.921941i 0.332601 + 0.121057i
\(59\) 5.01169 + 1.82411i 0.652467 + 0.237478i 0.646981 − 0.762507i \(-0.276031\pi\)
0.00548607 + 0.999985i \(0.498254\pi\)
\(60\) 0 0
\(61\) −7.07246 − 1.24706i −0.905535 − 0.159670i −0.298560 − 0.954391i \(-0.596506\pi\)
−0.606975 + 0.794721i \(0.707617\pi\)
\(62\) 0.362900 − 0.628561i 0.0460883 − 0.0798273i
\(63\) 0 0
\(64\) 2.91725 + 5.05283i 0.364656 + 0.631603i
\(65\) 0.811544 + 2.22970i 0.100660 + 0.276560i
\(66\) 0 0
\(67\) 6.70384 − 5.62519i 0.819005 − 0.687227i −0.133734 − 0.991017i \(-0.542697\pi\)
0.952739 + 0.303790i \(0.0982523\pi\)
\(68\) 0.949211 + 5.38324i 0.115109 + 0.652814i
\(69\) 0 0
\(70\) 1.20883 + 0.247060i 0.144482 + 0.0295293i
\(71\) 11.3110 − 6.53043i 1.34237 − 0.775019i 0.355217 − 0.934784i \(-0.384407\pi\)
0.987155 + 0.159764i \(0.0510735\pi\)
\(72\) 0 0
\(73\) 8.42907 − 4.86653i 0.986548 − 0.569584i 0.0823076 − 0.996607i \(-0.473771\pi\)
0.904241 + 0.427023i \(0.140438\pi\)
\(74\) 1.73130 + 4.75671i 0.201260 + 0.552956i
\(75\) 0 0
\(76\) −7.18433 + 1.26679i −0.824099 + 0.145311i
\(77\) 7.99238 + 10.0662i 0.910816 + 1.14715i
\(78\) 0 0
\(79\) −3.63977 − 3.05413i −0.409506 − 0.343617i 0.414648 − 0.909982i \(-0.363905\pi\)
−0.824154 + 0.566365i \(0.808349\pi\)
\(80\) 0.100577 − 0.174205i 0.0112449 − 0.0194767i
\(81\) 0 0
\(82\) 4.64298 − 2.68063i 0.512731 − 0.296026i
\(83\) −2.75845 + 1.00400i −0.302780 + 0.110203i −0.488942 − 0.872316i \(-0.662617\pi\)
0.186163 + 0.982519i \(0.440395\pi\)
\(84\) 0 0
\(85\) 1.84440 − 1.54764i 0.200053 − 0.167865i
\(86\) −4.67987 − 5.57725i −0.504643 − 0.601410i
\(87\) 0 0
\(88\) 13.2755 − 4.83188i 1.41517 − 0.515080i
\(89\) −11.2094 −1.18819 −0.594097 − 0.804394i \(-0.702490\pi\)
−0.594097 + 0.804394i \(0.702490\pi\)
\(90\) 0 0
\(91\) 10.6075 + 6.51327i 1.11197 + 0.682776i
\(92\) −4.04537 − 0.713308i −0.421759 − 0.0743675i
\(93\) 0 0
\(94\) −4.56823 − 5.44420i −0.471177 − 0.561527i
\(95\) 2.06543 + 2.46149i 0.211909 + 0.252543i
\(96\) 0 0
\(97\) −2.41733 − 0.426241i −0.245443 − 0.0432783i 0.0495730 − 0.998771i \(-0.484214\pi\)
−0.295016 + 0.955492i \(0.595325\pi\)
\(98\) 5.77228 − 2.92818i 0.583089 − 0.295791i
\(99\) 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.143.8 132
3.2 odd 2 189.2.ba.a.101.15 ✓ 132
7.5 odd 6 567.2.bd.a.467.15 132
21.5 even 6 189.2.bd.a.47.8 yes 132
27.4 even 9 189.2.bd.a.185.8 yes 132
27.23 odd 18 567.2.bd.a.17.15 132
189.131 even 18 inner 567.2.ba.a.341.8 132
189.166 odd 18 189.2.ba.a.131.15 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.15 ✓ 132 3.2 odd 2
189.2.ba.a.131.15 yes 132 189.166 odd 18
189.2.bd.a.47.8 yes 132 21.5 even 6
189.2.bd.a.185.8 yes 132 27.4 even 9
567.2.ba.a.143.8 132 1.1 even 1 trivial
567.2.ba.a.341.8 132 189.131 even 18 inner
567.2.bd.a.17.15 132 27.23 odd 18
567.2.bd.a.467.15 132 7.5 odd 6