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Properties

Label 546.4.l.d
Level $546$
Weight $4$
Character orbit 546.l
Analytic conductor $32.215$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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 = [8,8,12,-16,0] 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 198x^{6} + 10569x^{4} + 86475x^{2} + 147456 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} - 3 \beta_1 q^{3} + ( - 4 \beta_1 - 4) q^{4} + ( - \beta_{7} - \beta_{5} + \beta_{3}) q^{5} + ( - 6 \beta_1 - 6) q^{6} + ( - 7 \beta_1 - 7) q^{7} - 8 q^{8} + ( - 9 \beta_1 - 9) q^{9}+ \cdots + (9 \beta_{7} + 27 \beta_{6} + \cdots + 81) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 12 q^{3} - 16 q^{4} - 24 q^{6} - 28 q^{7} - 64 q^{8} - 36 q^{9} - 35 q^{11} - 96 q^{12} - 95 q^{13} - 112 q^{14} - 64 q^{16} - 37 q^{17} - 144 q^{18} - 20 q^{19} - 168 q^{21} + 70 q^{22}+ \cdots + 630 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 198x^{6} + 10569x^{4} + 86475x^{2} + 147456 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 198\nu^{5} + 10185\nu^{3} + 48459\nu - 22656 ) / 45312 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 27 \nu^{7} + 64 \nu^{6} - 5394 \nu^{5} + 8736 \nu^{4} - 296739 \nu^{3} + 197040 \nu^{2} + \cdots - 1706496 ) / 379488 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27\nu^{7} - 64\nu^{6} + 5394\nu^{5} - 11952\nu^{4} + 296739\nu^{3} - 515424\nu^{2} + 3102489\nu + 471552 ) / 379488 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27\nu^{7} + 64\nu^{6} + 5394\nu^{5} + 11952\nu^{4} + 296739\nu^{3} + 515424\nu^{2} + 3102489\nu - 471552 ) / 379488 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 89 \nu^{7} + 224 \nu^{6} - 16470 \nu^{5} + 46656 \nu^{4} - 764097 \nu^{3} + 2566176 \nu^{2} + \cdots + 13863552 ) / 758976 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 89 \nu^{7} - 224 \nu^{6} - 16470 \nu^{5} - 46656 \nu^{4} - 764097 \nu^{3} - 2566176 \nu^{2} + \cdots - 13863552 ) / 758976 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 143 \nu^{7} - 352 \nu^{6} + 27258 \nu^{5} - 57696 \nu^{4} + 1357575 \nu^{3} - 2323488 \nu^{2} + \cdots - 7980672 ) / 758976 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} - 4\beta_{6} + \beta_{5} - 7\beta_{4} + 10\beta_{3} - 144 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -35\beta_{7} - \beta_{6} - 36\beta_{5} + 12\beta_{4} - 23\beta_{3} - 70\beta_{2} - 88\beta _1 - 44 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 158\beta_{7} + 132\beta_{6} + 26\beta_{5} + 231\beta_{4} - 389\beta_{3} + 4368 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3396\beta_{7} + 453\beta_{6} + 3849\beta_{5} - 1483\beta_{4} + 1913\beta_{3} + 6792\beta_{2} + 14376\beta _1 + 7188 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -21453\beta_{7} - 13913\beta_{6} - 7540\beta_{5} - 21383\beta_{4} + 42836\beta_{3} - 421788 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 332086 \beta_{7} - 79509 \beta_{6} - 411595 \beta_{5} + 171414 \beta_{4} - 160672 \beta_{3} + \cdots - 952428 \) Copy content Toggle raw display

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

Character values

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

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

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
− 10.2622i
− 1.53352i
2.67545i
9.12023i
10.2622i
1.53352i
− 2.67545i
− 9.12023i
1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −17.7746 −3.00000 + 5.19615i −3.50000 + 6.06218i −8.00000 −4.50000 + 7.79423i −17.7746 − 30.7865i
211.2 1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −2.65614 −3.00000 + 5.19615i −3.50000 + 6.06218i −8.00000 −4.50000 + 7.79423i −2.65614 − 4.60057i
211.3 1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 4.63401 −3.00000 + 5.19615i −3.50000 + 6.06218i −8.00000 −4.50000 + 7.79423i 4.63401 + 8.02634i
211.4 1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 15.7967 −3.00000 + 5.19615i −3.50000 + 6.06218i −8.00000 −4.50000 + 7.79423i 15.7967 + 27.3607i
295.1 1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i −17.7746 −3.00000 − 5.19615i −3.50000 − 6.06218i −8.00000 −4.50000 − 7.79423i −17.7746 + 30.7865i
295.2 1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i −2.65614 −3.00000 − 5.19615i −3.50000 − 6.06218i −8.00000 −4.50000 − 7.79423i −2.65614 + 4.60057i
295.3 1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i 4.63401 −3.00000 − 5.19615i −3.50000 − 6.06218i −8.00000 −4.50000 − 7.79423i 4.63401 − 8.02634i
295.4 1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i 15.7967 −3.00000 − 5.19615i −3.50000 − 6.06218i −8.00000 −4.50000 − 7.79423i 15.7967 − 27.3607i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.4
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.d ✓ 8
13.c even 3 1 inner 546.4.l.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.l.d ✓ 8 1.a even 1 1 trivial
546.4.l.d ✓ 8 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 297T_{5}^{2} + 531T_{5} + 3456 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 297 T^{2} + \cdots + 3456)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7 T + 49)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 35 T^{7} + \cdots + 499388409 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 70692462910161 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 64139547986089 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 335289637764 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 16\!\cdots\!09 \) Copy content Toggle raw display
$31$ \( (T^{4} + 167 T^{3} + \cdots - 10246588)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 81\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 48\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( (T^{4} - 214 T^{3} + \cdots + 1883235753)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 244 T^{3} + \cdots + 5885775513)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 49\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{4} + 1544 T^{3} + \cdots - 15699377248)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 704 T^{3} + \cdots - 72380594431)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 892 T^{3} + \cdots - 120850291278)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 69\!\cdots\!69 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
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