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Properties

Label 567.2.be.a.503.4
Level $567$
Weight $2$
Character 567.503
Analytic conductor $4.528$
Analytic rank $0$
Dimension $132$
Inner twists $4$

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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(62,567)] 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("567.62"); 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, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.be (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 503.4
Character \(\chi\) \(=\) 567.503
Dual form 567.2.be.a.62.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.735125 + 2.01974i) q^{2} +(-2.00685 - 1.68395i) q^{4} +(0.0677816 - 0.384409i) q^{5} +(2.64076 - 0.162359i) q^{7} +(1.15362 - 0.666045i) q^{8} +(0.726578 + 0.419490i) q^{10} +(4.77807 - 0.842503i) q^{11} +(-1.96127 - 5.38855i) q^{13} +(-1.61337 + 5.45301i) q^{14} +(-0.412653 - 2.34027i) q^{16} +(3.57147 - 6.18597i) q^{17} +(-0.398672 + 0.230173i) q^{19} +(-0.783352 + 0.657311i) q^{20} +(-1.81084 + 10.2698i) q^{22} +(-0.358944 + 0.427773i) q^{23} +(4.55529 + 1.65799i) q^{25} +12.3252 q^{26} +(-5.57303 - 4.12109i) q^{28} +(-1.73522 + 4.76747i) q^{29} +(-2.14159 + 2.55225i) q^{31} +(7.65380 + 1.34957i) q^{32} +(9.86858 + 11.7609i) q^{34} +(0.116583 - 1.02614i) q^{35} +(-1.07348 + 1.85933i) q^{37} +(-0.171817 - 0.974420i) q^{38} +(-0.177839 - 0.488609i) q^{40} +(0.917771 - 0.334041i) q^{41} +(1.16047 + 6.58138i) q^{43} +(-11.0076 - 6.35525i) q^{44} +(-0.600121 - 1.03944i) q^{46} +(-2.83834 + 2.38165i) q^{47} +(6.94728 - 0.857502i) q^{49} +(-6.69741 + 7.98167i) q^{50} +(-5.13806 + 14.1167i) q^{52} -3.13310i q^{53} -1.89384i q^{55} +(2.93831 - 1.94617i) q^{56} +(-8.35346 - 7.00938i) q^{58} +(-0.715255 + 4.05641i) q^{59} +(-7.89549 - 9.40948i) q^{61} +(-3.58054 - 6.20168i) q^{62} +(-5.97591 + 10.3506i) q^{64} +(-2.20434 + 0.388685i) q^{65} +(6.30401 - 2.29447i) q^{67} +(-17.5843 + 6.40016i) q^{68} +(1.98683 + 0.989808i) q^{70} +(-6.08593 - 3.51371i) q^{71} +(6.55906 - 3.78687i) q^{73} +(-2.96621 - 3.53499i) q^{74} +(1.18768 + 0.209419i) q^{76} +(12.4810 - 3.00061i) q^{77} +(-0.818714 - 0.297987i) q^{79} -0.927591 q^{80} +2.09922i q^{82} +(6.78293 + 2.46878i) q^{83} +(-2.13586 - 1.79220i) q^{85} +(-14.1458 - 2.49428i) q^{86} +(4.95095 - 4.15434i) q^{88} +(2.65800 + 4.60379i) q^{89} +(-6.05413 - 13.9115i) q^{91} +(1.44070 - 0.254033i) q^{92} +(-2.72378 - 7.48351i) q^{94} +(0.0614580 + 0.168854i) q^{95} +(-10.7873 + 1.90209i) q^{97} +(-3.37519 + 14.6621i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 12 q^{2} - 12 q^{4} - 6 q^{7} + 18 q^{8} + 18 q^{11} - 3 q^{14} - 24 q^{16} - 12 q^{22} - 12 q^{23} - 12 q^{25} - 12 q^{28} + 48 q^{29} + 6 q^{32} + 36 q^{35} - 6 q^{37} - 12 q^{43} + 18 q^{44}+ \cdots - 126 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.735125 + 2.01974i −0.519812 + 1.42817i 0.350916 + 0.936407i \(0.385870\pi\)
−0.870728 + 0.491765i \(0.836352\pi\)
\(3\) 0 0
\(4\) −2.00685 − 1.68395i −1.00343 − 0.841975i
\(5\) 0.0677816 − 0.384409i 0.0303129 − 0.171913i −0.965893 − 0.258942i \(-0.916626\pi\)
0.996206 + 0.0870291i \(0.0277373\pi\)
\(6\) 0 0
\(7\) 2.64076 − 0.162359i 0.998115 − 0.0613658i
\(8\) 1.15362 − 0.666045i 0.407868 − 0.235483i
\(9\) 0 0
\(10\) 0.726578 + 0.419490i 0.229764 + 0.132654i
\(11\) 4.77807 − 0.842503i 1.44064 − 0.254024i 0.601909 − 0.798565i \(-0.294407\pi\)
0.838734 + 0.544541i \(0.183296\pi\)
\(12\) 0 0
\(13\) −1.96127 − 5.38855i −0.543959 − 1.49451i −0.841742 − 0.539881i \(-0.818469\pi\)
0.297783 − 0.954634i \(-0.403753\pi\)
\(14\) −1.61337 + 5.45301i −0.431192 + 1.45738i
\(15\) 0 0
\(16\) −0.412653 − 2.34027i −0.103163 − 0.585068i
\(17\) 3.57147 − 6.18597i 0.866209 − 1.50032i 0.000368279 − 1.00000i \(-0.499883\pi\)
0.865841 − 0.500319i \(-0.166784\pi\)
\(18\) 0 0
\(19\) −0.398672 + 0.230173i −0.0914616 + 0.0528054i −0.545033 − 0.838414i \(-0.683483\pi\)
0.453572 + 0.891220i \(0.350150\pi\)
\(20\) −0.783352 + 0.657311i −0.175163 + 0.146979i
\(21\) 0 0
\(22\) −1.81084 + 10.2698i −0.386073 + 2.18953i
\(23\) −0.358944 + 0.427773i −0.0748450 + 0.0891968i −0.802168 − 0.597099i \(-0.796320\pi\)
0.727323 + 0.686296i \(0.240764\pi\)
\(24\) 0 0
\(25\) 4.55529 + 1.65799i 0.911057 + 0.331598i
\(26\) 12.3252 2.41718
\(27\) 0 0
\(28\) −5.57303 − 4.12109i −1.05320 − 0.778812i
\(29\) −1.73522 + 4.76747i −0.322222 + 0.885298i 0.667794 + 0.744346i \(0.267239\pi\)
−0.990016 + 0.140952i \(0.954984\pi\)
\(30\) 0 0
\(31\) −2.14159 + 2.55225i −0.384641 + 0.458398i −0.923273 − 0.384143i \(-0.874497\pi\)
0.538632 + 0.842541i \(0.318941\pi\)
\(32\) 7.65380 + 1.34957i 1.35301 + 0.238573i
\(33\) 0 0
\(34\) 9.86858 + 11.7609i 1.69245 + 2.01698i
\(35\) 0.116583 − 1.02614i 0.0197062 − 0.173449i
\(36\) 0 0
\(37\) −1.07348 + 1.85933i −0.176479 + 0.305671i −0.940672 − 0.339317i \(-0.889804\pi\)
0.764193 + 0.644988i \(0.223138\pi\)
\(38\) −0.171817 − 0.974420i −0.0278723 − 0.158072i
\(39\) 0 0
\(40\) −0.177839 − 0.488609i −0.0281188 − 0.0772558i
\(41\) 0.917771 − 0.334041i 0.143332 − 0.0521685i −0.269358 − 0.963040i \(-0.586812\pi\)
0.412690 + 0.910872i \(0.364589\pi\)
\(42\) 0 0
\(43\) 1.16047 + 6.58138i 0.176971 + 1.00365i 0.935845 + 0.352413i \(0.114639\pi\)
−0.758874 + 0.651238i \(0.774250\pi\)
\(44\) −11.0076 − 6.35525i −1.65946 − 0.958090i
\(45\) 0 0
\(46\) −0.600121 − 1.03944i −0.0884830 − 0.153257i
\(47\) −2.83834 + 2.38165i −0.414014 + 0.347399i −0.825880 − 0.563845i \(-0.809321\pi\)
0.411867 + 0.911244i \(0.364877\pi\)
\(48\) 0 0
\(49\) 6.94728 − 0.857502i 0.992468 − 0.122500i
\(50\) −6.69741 + 7.98167i −0.947157 + 1.12878i
\(51\) 0 0
\(52\) −5.13806 + 14.1167i −0.712521 + 1.95763i
\(53\) − 3.13310i − 0.430364i −0.976574 − 0.215182i \(-0.930966\pi\)
0.976574 − 0.215182i \(-0.0690345\pi\)
\(54\) 0 0
\(55\) − 1.89384i − 0.255365i
\(56\) 2.93831 − 1.94617i 0.392648 − 0.260068i
\(57\) 0 0
\(58\) −8.35346 − 7.00938i −1.09686 − 0.920377i
\(59\) −0.715255 + 4.05641i −0.0931183 + 0.528100i 0.902189 + 0.431340i \(0.141959\pi\)
−0.995308 + 0.0967600i \(0.969152\pi\)
\(60\) 0 0
\(61\) −7.89549 − 9.40948i −1.01091 − 1.20476i −0.978704 − 0.205279i \(-0.934190\pi\)
−0.0322101 − 0.999481i \(-0.510255\pi\)
\(62\) −3.58054 − 6.20168i −0.454730 − 0.787615i
\(63\) 0 0
\(64\) −5.97591 + 10.3506i −0.746989 + 1.29382i
\(65\) −2.20434 + 0.388685i −0.273415 + 0.0482104i
\(66\) 0 0
\(67\) 6.30401 − 2.29447i 0.770158 − 0.280314i 0.0730953 − 0.997325i \(-0.476712\pi\)
0.697062 + 0.717011i \(0.254490\pi\)
\(68\) −17.5843 + 6.40016i −2.13241 + 0.776133i
\(69\) 0 0
\(70\) 1.98683 + 0.989808i 0.237471 + 0.118305i
\(71\) −6.08593 − 3.51371i −0.722267 − 0.417001i 0.0933194 − 0.995636i \(-0.470252\pi\)
−0.815587 + 0.578635i \(0.803586\pi\)
\(72\) 0 0
\(73\) 6.55906 − 3.78687i 0.767680 − 0.443220i −0.0643665 − 0.997926i \(-0.520503\pi\)
0.832046 + 0.554706i \(0.187169\pi\)
\(74\) −2.96621 − 3.53499i −0.344815 − 0.410935i
\(75\) 0 0
\(76\) 1.18768 + 0.209419i 0.136236 + 0.0240221i
\(77\) 12.4810 − 3.00061i 1.42234 − 0.341952i
\(78\) 0 0
\(79\) −0.818714 − 0.297987i −0.0921125 − 0.0335262i 0.295553 − 0.955326i \(-0.404496\pi\)
−0.387665 + 0.921800i \(0.626718\pi\)
\(80\) −0.927591 −0.103708
\(81\) 0 0
\(82\) 2.09922i 0.231820i
\(83\) 6.78293 + 2.46878i 0.744523 + 0.270984i 0.686299 − 0.727319i \(-0.259234\pi\)
0.0582238 + 0.998304i \(0.481456\pi\)
\(84\) 0 0
\(85\) −2.13586 − 1.79220i −0.231667 − 0.194391i
\(86\) −14.1458 − 2.49428i −1.52538 − 0.268965i
\(87\) 0 0
\(88\) 4.95095 − 4.15434i 0.527773 − 0.442854i
\(89\) 2.65800 + 4.60379i 0.281748 + 0.488001i 0.971815 − 0.235744i \(-0.0757526\pi\)
−0.690068 + 0.723745i \(0.742419\pi\)
\(90\) 0 0
\(91\) −6.05413 − 13.9115i −0.634646 − 1.45832i
\(92\) 1.44070 − 0.254033i 0.150203 − 0.0264848i
\(93\) 0 0
\(94\) −2.72378 − 7.48351i −0.280936 − 0.771865i
\(95\) 0.0614580 + 0.168854i 0.00630546 + 0.0173241i
\(96\) 0 0
\(97\) −10.7873 + 1.90209i −1.09528 + 0.193128i −0.691964 − 0.721932i \(-0.743254\pi\)
−0.403318 + 0.915060i \(0.632143\pi\)
\(98\) −3.37519 + 14.6621i −0.340946 + 1.48109i
\(99\) 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.be.a.503.4 132
3.2 odd 2 189.2.be.a.104.20 yes 132
7.6 odd 2 inner 567.2.be.a.503.3 132
21.20 even 2 189.2.be.a.104.19 yes 132
27.7 even 9 189.2.be.a.20.19 ✓ 132
27.20 odd 18 inner 567.2.be.a.62.3 132
189.20 even 18 inner 567.2.be.a.62.4 132
189.34 odd 18 189.2.be.a.20.20 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.be.a.20.19 ✓ 132 27.7 even 9
189.2.be.a.20.20 yes 132 189.34 odd 18
189.2.be.a.104.19 yes 132 21.20 even 2
189.2.be.a.104.20 yes 132 3.2 odd 2
567.2.be.a.62.3 132 27.20 odd 18 inner
567.2.be.a.62.4 132 189.20 even 18 inner
567.2.be.a.503.3 132 7.6 odd 2 inner
567.2.be.a.503.4 132 1.1 even 1 trivial