↑

Properties

Label 546.4.l.i
Level $546$
Weight $4$
Character orbit 546.l
Analytic conductor $32.215$
Analytic rank $0$
Dimension $10$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [546,4,Mod(211,546)] 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("546.211"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.l (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,10,-15,-20,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.2150428631\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 450 x^{8} - 47 x^{7} + 157097 x^{6} + 59961 x^{5} + 20236961 x^{4} + 84562978 x^{3} + \cdots + 6692712481 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{3} + 2) q^{2} + ( - 3 \beta_{3} - 3) q^{3} + 4 \beta_{3} q^{4} + (\beta_{2} + 1) q^{5} - 6 \beta_{3} q^{6} + 7 \beta_{3} q^{7} - 8 q^{8} + 9 \beta_{3} q^{9} + (2 \beta_{3} - 2 \beta_1 + 2) q^{10}+ \cdots + ( - 9 \beta_{6} + 54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} - 15 q^{3} - 20 q^{4} + 8 q^{5} + 30 q^{6} - 35 q^{7} - 80 q^{8} - 45 q^{9} + 8 q^{10} - 29 q^{11} + 120 q^{12} + 34 q^{13} - 140 q^{14} - 12 q^{15} - 80 q^{16} - 92 q^{17} - 180 q^{18}+ \cdots + 522 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 450 x^{8} - 47 x^{7} + 157097 x^{6} + 59961 x^{5} + 20236961 x^{4} + 84562978 x^{3} + \cdots + 6692712481 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11012026384650 \nu^{9} - 62905162500000 \nu^{8} + 332923475692500 \nu^{7} + \cdots + 15\!\cdots\!23 ) / 83\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14\!\cdots\!19 \nu^{9} + \cdots - 53\!\cdots\!92 ) / 52\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20\!\cdots\!99 \nu^{9} + \cdots - 36\!\cdots\!81 ) / 66\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21\!\cdots\!15 \nu^{9} + \cdots - 15\!\cdots\!20 ) / 24\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 88\!\cdots\!69 \nu^{9} + \cdots + 16\!\cdots\!12 ) / 73\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 99\!\cdots\!77 \nu^{9} + \cdots + 29\!\cdots\!22 ) / 66\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 14\!\cdots\!39 \nu^{9} + \cdots + 19\!\cdots\!63 ) / 66\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 77\!\cdots\!60 \nu^{9} + \cdots - 15\!\cdots\!05 ) / 22\!\cdots\!02 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{9} - 2\beta_{8} + \beta_{7} - \beta_{5} + 181\beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -12\beta_{9} + 12\beta_{8} - 24\beta_{7} + 15\beta_{6} + 13\beta_{5} + 255\beta_{2} - 12\beta _1 - 63 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 867 \beta_{9} + 389 \beta_{8} + 389 \beta_{7} - 478 \beta_{5} + 234 \beta_{4} - 45448 \beta_{3} + \cdots - 45448 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 467 \beta_{9} - 10494 \beta_{8} + 5247 \beta_{7} - 6501 \beta_{6} - 5247 \beta_{5} + \cdots - 69302 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 127638 \beta_{9} + 127638 \beta_{8} - 255276 \beta_{7} + 102285 \beta_{6} + 252537 \beta_{5} + \cdots + 12237454 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2191716 \beta_{9} + 1869522 \beta_{8} + 1869522 \beta_{7} - 322194 \beta_{5} + 2203416 \beta_{4} + \cdots - 40440393 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 34563626 \beta_{9} - 80628752 \beta_{8} + 40314376 \beta_{7} - 36089964 \beta_{6} + \cdots - 121688553 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 627765183 \beta_{9} + 627765183 \beta_{8} - 1255530366 \beta_{7} + 706821054 \beta_{6} + \cdots + 19112256891 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{3}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
7.75590 + 13.4336i
7.65882 + 13.2655i
−0.959200 − 1.66138i
−5.02328 − 8.70057i
−8.93225 − 15.4711i
7.75590 − 13.4336i
7.65882 − 13.2655i
−0.959200 + 1.66138i
−5.02328 + 8.70057i
−8.93225 + 15.4711i
1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −14.5118 3.00000 − 5.19615i −3.50000 + 6.06218i −8.00000 −4.50000 + 7.79423i −14.5118 − 25.1352i
211.2 1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −14.3176 3.00000 − 5.19615i −3.50000 + 6.06218i −8.00000 −4.50000 + 7.79423i −14.3176 − 24.7989i
211.3 1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i 2.91840 3.00000 − 5.19615i −3.50000 + 6.06218i −8.00000 −4.50000 + 7.79423i 2.91840 + 5.05482i
211.4 1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i 11.0466 3.00000 − 5.19615i −3.50000 + 6.06218i −8.00000 −4.50000 + 7.79423i 11.0466 + 19.1332i
211.5 1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i 18.8645 3.00000 − 5.19615i −3.50000 + 6.06218i −8.00000 −4.50000 + 7.79423i 18.8645 + 32.6743i
295.1 1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −14.5118 3.00000 + 5.19615i −3.50000 − 6.06218i −8.00000 −4.50000 − 7.79423i −14.5118 + 25.1352i
295.2 1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −14.3176 3.00000 + 5.19615i −3.50000 − 6.06218i −8.00000 −4.50000 − 7.79423i −14.3176 + 24.7989i
295.3 1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i 2.91840 3.00000 + 5.19615i −3.50000 − 6.06218i −8.00000 −4.50000 − 7.79423i 2.91840 − 5.05482i
295.4 1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i 11.0466 3.00000 + 5.19615i −3.50000 − 6.06218i −8.00000 −4.50000 − 7.79423i 11.0466 − 19.1332i
295.5 1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i 18.8645 3.00000 + 5.19615i −3.50000 − 6.06218i −8.00000 −4.50000 − 7.79423i 18.8645 − 32.6743i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.4.l.i ✓ 10
13.c even 3 1 inner 546.4.l.i ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.l.i ✓ 10 1.a even 1 1 trivial
546.4.l.i ✓ 10 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 4T_{5}^{4} - 443T_{5}^{3} + 1095T_{5}^{2} + 43902T_{5} - 126360 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{5} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{5} \) Copy content Toggle raw display
$5$ \( (T^{5} - 4 T^{4} + \cdots - 126360)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7 T + 49)^{5} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 9758751210000 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 51\!\cdots\!57 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 621494579022849 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 90\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( (T^{5} - 72 T^{4} + \cdots - 29950123216)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 91\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots - 1053080573130)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 3638197564839)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 72\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 68\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{5} - 800 T^{4} + \cdots + 250880950800)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 165683620847680)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots + 46938297773862)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 59\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
show more
show less