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Properties

Label 567.2.ba.a.143.17
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 143.17
Character \(\chi\) \(=\) 567.143
Dual form 567.2.ba.a.341.17

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.07972 - 1.28676i) q^{2} +(-0.142658 - 0.809053i) q^{4} +(-2.87107 + 2.40911i) q^{5} +(-1.86335 - 1.87828i) q^{7} +(1.71431 + 0.989760i) q^{8} +6.29553i q^{10} +(-3.07307 + 3.66234i) q^{11} +(-1.10865 + 3.04598i) q^{13} +(-4.42878 + 0.369659i) q^{14} +(4.66853 - 1.69921i) q^{16} -3.86831 q^{17} +1.52130i q^{19} +(2.35868 + 1.97917i) q^{20} +(1.39450 + 7.90859i) q^{22} +(-0.124554 + 0.342208i) q^{23} +(1.57097 - 8.90940i) q^{25} +(2.72242 + 4.71536i) q^{26} +(-1.25381 + 1.77550i) q^{28} +(2.29492 + 6.30525i) q^{29} +(-0.692139 + 0.122043i) q^{31} +(1.50016 - 4.12165i) q^{32} +(-4.17668 + 4.97757i) q^{34} +(9.87478 + 0.903668i) q^{35} +(2.75197 - 4.76655i) q^{37} +(1.95755 + 1.64258i) q^{38} +(-7.30635 + 1.28831i) q^{40} +(-0.793900 - 0.288956i) q^{41} +(0.0136704 - 0.0775288i) q^{43} +(3.40143 + 1.96381i) q^{44} +(0.305856 + 0.529758i) q^{46} +(-0.617436 + 3.50165i) q^{47} +(-0.0558886 + 6.99978i) q^{49} +(-9.76803 - 11.6411i) q^{50} +(2.62252 + 0.462421i) q^{52} +(-7.25668 - 4.18964i) q^{53} -17.9182i q^{55} +(-1.33531 - 5.06423i) q^{56} +(10.5912 + 3.85488i) q^{58} +(-1.36168 - 0.495612i) q^{59} +(2.82040 + 0.497313i) q^{61} +(-0.590275 + 1.02239i) q^{62} +(1.28433 + 2.22452i) q^{64} +(-4.15511 - 11.4161i) q^{65} +(3.09944 - 2.60074i) q^{67} +(0.551844 + 3.12967i) q^{68} +(11.8248 - 11.7307i) q^{70} +(14.1956 - 8.19580i) q^{71} +(-10.4351 + 6.02468i) q^{73} +(-3.16204 - 8.68764i) q^{74} +(1.23082 - 0.217026i) q^{76} +(12.6051 - 1.05212i) q^{77} +(1.00589 + 0.844043i) q^{79} +(-9.31009 + 16.1256i) q^{80} +(-1.22900 + 0.709566i) q^{82} +(12.7610 - 4.64462i) q^{83} +(11.1062 - 9.31918i) q^{85} +(-0.0850005 - 0.101300i) q^{86} +(-8.89304 + 3.23680i) q^{88} -8.72878 q^{89} +(7.78701 - 3.59337i) q^{91} +(0.294633 + 0.0519517i) q^{92} +(3.83912 + 4.57528i) q^{94} +(-3.66499 - 4.36777i) q^{95} +(-4.18960 - 0.738740i) q^{97} +(8.94667 + 7.62970i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} + 9 q^{11} - 3 q^{14} + 3 q^{16} + 18 q^{17} - 18 q^{20} - 12 q^{22} + 6 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 18 q^{34}+ \cdots - 27 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.07972 − 1.28676i 0.763476 − 0.909875i −0.234587 − 0.972095i \(-0.575374\pi\)
0.998062 + 0.0622204i \(0.0198182\pi\)
\(3\) 0 0
\(4\) −0.142658 − 0.809053i −0.0713289 − 0.404527i
\(5\) −2.87107 + 2.40911i −1.28398 + 1.07739i −0.291298 + 0.956632i \(0.594087\pi\)
−0.992682 + 0.120755i \(0.961469\pi\)
\(6\) 0 0
\(7\) −1.86335 − 1.87828i −0.704278 − 0.709924i
\(8\) 1.71431 + 0.989760i 0.606101 + 0.349933i
\(9\) 0 0
\(10\) 6.29553i 1.99082i
\(11\) −3.07307 + 3.66234i −0.926565 + 1.10424i 0.0677435 + 0.997703i \(0.478420\pi\)
−0.994309 + 0.106535i \(0.966024\pi\)
\(12\) 0 0
\(13\) −1.10865 + 3.04598i −0.307483 + 0.844804i 0.685662 + 0.727920i \(0.259513\pi\)
−0.993146 + 0.116884i \(0.962709\pi\)
\(14\) −4.42878 + 0.369659i −1.18364 + 0.0987955i
\(15\) 0 0
\(16\) 4.66853 − 1.69921i 1.16713 − 0.424802i
\(17\) −3.86831 −0.938202 −0.469101 − 0.883144i \(-0.655422\pi\)
−0.469101 + 0.883144i \(0.655422\pi\)
\(18\) 0 0
\(19\) 1.52130i 0.349011i 0.984656 + 0.174506i \(0.0558327\pi\)
−0.984656 + 0.174506i \(0.944167\pi\)
\(20\) 2.35868 + 1.97917i 0.527417 + 0.442555i
\(21\) 0 0
\(22\) 1.39450 + 7.90859i 0.297308 + 1.68612i
\(23\) −0.124554 + 0.342208i −0.0259712 + 0.0713553i −0.952001 − 0.306096i \(-0.900977\pi\)
0.926029 + 0.377451i \(0.123199\pi\)
\(24\) 0 0
\(25\) 1.57097 − 8.90940i 0.314194 − 1.78188i
\(26\) 2.72242 + 4.71536i 0.533910 + 0.924758i
\(27\) 0 0
\(28\) −1.25381 + 1.77550i −0.236948 + 0.335537i
\(29\) 2.29492 + 6.30525i 0.426157 + 1.17086i 0.948127 + 0.317892i \(0.102975\pi\)
−0.521970 + 0.852964i \(0.674803\pi\)
\(30\) 0 0
\(31\) −0.692139 + 0.122043i −0.124312 + 0.0219195i −0.235458 − 0.971885i \(-0.575659\pi\)
0.111146 + 0.993804i \(0.464548\pi\)
\(32\) 1.50016 − 4.12165i 0.265193 − 0.728612i
\(33\) 0 0
\(34\) −4.17668 + 4.97757i −0.716294 + 0.853647i
\(35\) 9.87478 + 0.903668i 1.66914 + 0.152748i
\(36\) 0 0
\(37\) 2.75197 − 4.76655i 0.452421 − 0.783616i −0.546115 − 0.837710i \(-0.683894\pi\)
0.998536 + 0.0540945i \(0.0172272\pi\)
\(38\) 1.95755 + 1.64258i 0.317556 + 0.266461i
\(39\) 0 0
\(40\) −7.30635 + 1.28831i −1.15524 + 0.203699i
\(41\) −0.793900 − 0.288956i −0.123986 − 0.0451273i 0.279282 − 0.960209i \(-0.409904\pi\)
−0.403268 + 0.915082i \(0.632126\pi\)
\(42\) 0 0
\(43\) 0.0136704 − 0.0775288i 0.00208472 − 0.0118230i −0.983748 − 0.179557i \(-0.942534\pi\)
0.985832 + 0.167734i \(0.0536448\pi\)
\(44\) 3.40143 + 1.96381i 0.512784 + 0.296056i
\(45\) 0 0
\(46\) 0.305856 + 0.529758i 0.0450960 + 0.0781086i
\(47\) −0.617436 + 3.50165i −0.0900622 + 0.510768i 0.906087 + 0.423092i \(0.139055\pi\)
−0.996149 + 0.0876764i \(0.972056\pi\)
\(48\) 0 0
\(49\) −0.0558886 + 6.99978i −0.00798409 + 0.999968i
\(50\) −9.76803 − 11.6411i −1.38141 − 1.64630i
\(51\) 0 0
\(52\) 2.62252 + 0.462421i 0.363678 + 0.0641262i
\(53\) −7.25668 − 4.18964i −0.996781 − 0.575492i −0.0894868 − 0.995988i \(-0.528523\pi\)
−0.907294 + 0.420496i \(0.861856\pi\)
\(54\) 0 0
\(55\) − 17.9182i − 2.41609i
\(56\) −1.33531 − 5.06423i −0.178438 − 0.676736i
\(57\) 0 0
\(58\) 10.5912 + 3.85488i 1.39069 + 0.506170i
\(59\) −1.36168 − 0.495612i −0.177276 − 0.0645231i 0.251857 − 0.967764i \(-0.418959\pi\)
−0.429133 + 0.903241i \(0.641181\pi\)
\(60\) 0 0
\(61\) 2.82040 + 0.497313i 0.361116 + 0.0636745i 0.351263 − 0.936277i \(-0.385752\pi\)
0.00985324 + 0.999951i \(0.496864\pi\)
\(62\) −0.590275 + 1.02239i −0.0749650 + 0.129843i
\(63\) 0 0
\(64\) 1.28433 + 2.22452i 0.160541 + 0.278066i
\(65\) −4.15511 − 11.4161i −0.515378 − 1.41599i
\(66\) 0 0
\(67\) 3.09944 − 2.60074i 0.378657 − 0.317731i −0.433518 − 0.901145i \(-0.642728\pi\)
0.812175 + 0.583414i \(0.198284\pi\)
\(68\) 0.551844 + 3.12967i 0.0669210 + 0.379528i
\(69\) 0 0
\(70\) 11.8248 − 11.7307i 1.41333 − 1.40209i
\(71\) 14.1956 − 8.19580i 1.68470 − 0.972663i 0.726240 − 0.687441i \(-0.241266\pi\)
0.958461 − 0.285222i \(-0.0920674\pi\)
\(72\) 0 0
\(73\) −10.4351 + 6.02468i −1.22133 + 0.705136i −0.965202 − 0.261507i \(-0.915781\pi\)
−0.256129 + 0.966643i \(0.582447\pi\)
\(74\) −3.16204 − 8.68764i −0.367580 − 1.00992i
\(75\) 0 0
\(76\) 1.23082 − 0.217026i 0.141184 − 0.0248946i
\(77\) 12.6051 − 1.05212i 1.43648 − 0.119900i
\(78\) 0 0
\(79\) 1.00589 + 0.844043i 0.113172 + 0.0949623i 0.697617 − 0.716471i \(-0.254244\pi\)
−0.584446 + 0.811433i \(0.698688\pi\)
\(80\) −9.31009 + 16.1256i −1.04090 + 1.80289i
\(81\) 0 0
\(82\) −1.22900 + 0.709566i −0.135721 + 0.0783584i
\(83\) 12.7610 − 4.64462i 1.40070 − 0.509814i 0.472315 − 0.881430i \(-0.343419\pi\)
0.928386 + 0.371616i \(0.121196\pi\)
\(84\) 0 0
\(85\) 11.1062 − 9.31918i 1.20463 − 1.01081i
\(86\) −0.0850005 − 0.101300i −0.00916584 − 0.0109234i
\(87\) 0 0
\(88\) −8.89304 + 3.23680i −0.948002 + 0.345044i
\(89\) −8.72878 −0.925248 −0.462624 − 0.886555i \(-0.653092\pi\)
−0.462624 + 0.886555i \(0.653092\pi\)
\(90\) 0 0
\(91\) 7.78701 − 3.59337i 0.816300 − 0.376687i
\(92\) 0.294633 + 0.0519517i 0.0307176 + 0.00541634i
\(93\) 0 0
\(94\) 3.83912 + 4.57528i 0.395975 + 0.471904i
\(95\) −3.66499 − 4.36777i −0.376020 − 0.448123i
\(96\) 0 0
\(97\) −4.18960 − 0.738740i −0.425390 − 0.0750077i −0.0431451 − 0.999069i \(-0.513738\pi\)
−0.382245 + 0.924061i \(0.624849\pi\)
\(98\) 8.94667 + 7.62970i 0.903750 + 0.770716i
\(99\) 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.143.17 132
3.2 odd 2 189.2.ba.a.101.6 ✓ 132
7.5 odd 6 567.2.bd.a.467.6 132
21.5 even 6 189.2.bd.a.47.17 yes 132
27.4 even 9 189.2.bd.a.185.17 yes 132
27.23 odd 18 567.2.bd.a.17.6 132
189.131 even 18 inner 567.2.ba.a.341.17 132
189.166 odd 18 189.2.ba.a.131.6 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.6 ✓ 132 3.2 odd 2
189.2.ba.a.131.6 yes 132 189.166 odd 18
189.2.bd.a.47.17 yes 132 21.5 even 6
189.2.bd.a.185.17 yes 132 27.4 even 9
567.2.ba.a.143.17 132 1.1 even 1 trivial
567.2.ba.a.341.17 132 189.131 even 18 inner
567.2.bd.a.17.6 132 27.23 odd 18
567.2.bd.a.467.6 132 7.5 odd 6