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Properties

Label 546.4.i.f
Level $546$
Weight $4$
Character orbit 546.i
Analytic conductor $32.215$
Analytic rank $0$
Dimension $12$
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(79,546)] 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("546.79"); 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, 2, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,12,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.2150428631\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 387 x^{10} - 16 x^{9} + 118610 x^{8} - 17859 x^{7} + 10864988 x^{6} + \cdots + 35073049284 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} + (3 \beta_{2} - 3) q^{3} + (4 \beta_{2} - 4) q^{4} + (\beta_{6} - 2 \beta_{2}) q^{5} - 6 q^{6} + (\beta_{7} - \beta_{2} + 3) q^{7} - 8 q^{8} - 9 \beta_{2} q^{9} + ( - 4 \beta_{2} + 2 \beta_1 + 4) q^{10}+ \cdots + (9 \beta_{11} + 9 \beta_{9} + 9 \beta_{8} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} - 18 q^{3} - 24 q^{4} - 14 q^{5} - 72 q^{6} + 29 q^{7} - 96 q^{8} - 54 q^{9} + 28 q^{10} - 10 q^{11} - 72 q^{12} + 156 q^{13} + 44 q^{14} + 84 q^{15} - 96 q^{16} - 119 q^{17} + 108 q^{18}+ \cdots + 180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 387 x^{10} - 16 x^{9} + 118610 x^{8} - 17859 x^{7} + 10864988 x^{6} + \cdots + 35073049284 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 97\!\cdots\!57 \nu^{11} + \cdots - 14\!\cdots\!48 ) / 53\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 28\!\cdots\!29 \nu^{11} + \cdots - 12\!\cdots\!70 ) / 69\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 41\!\cdots\!63 \nu^{11} + \cdots + 31\!\cdots\!40 ) / 69\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 86\!\cdots\!06 \nu^{11} + \cdots - 19\!\cdots\!59 ) / 11\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 33\!\cdots\!87 \nu^{11} + \cdots - 10\!\cdots\!76 ) / 17\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21\!\cdots\!13 \nu^{11} + \cdots + 96\!\cdots\!82 ) / 69\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13\!\cdots\!86 \nu^{11} + \cdots - 51\!\cdots\!01 ) / 34\!\cdots\!03 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 37\!\cdots\!30 \nu^{11} + \cdots + 22\!\cdots\!76 ) / 69\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!24 \nu^{11} + \cdots - 89\!\cdots\!98 ) / 69\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22\!\cdots\!97 \nu^{11} + \cdots - 66\!\cdots\!84 ) / 69\!\cdots\!06 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + 2\beta_{10} + 6\beta_{9} + \beta_{8} - \beta_{7} + 3\beta_{6} - 2\beta_{5} - 7\beta_{4} - 126\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{11} + 39\beta_{9} + 22\beta_{8} + 17\beta_{7} + 224\beta_{6} + 39\beta_{4} - 2\beta_{3} - 224\beta _1 - 125 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 1447 \beta_{9} - 1004 \beta_{8} + 1447 \beta_{7} + 915 \beta_{5} + 1919 \beta_{4} - 395 \beta_{3} + \cdots - 29335 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 158 \beta_{11} - 1293 \beta_{10} - 8220 \beta_{9} - 15007 \beta_{8} - 13714 \beta_{7} - 59385 \beta_{6} + \cdots + 1293 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 121326 \beta_{11} - 295309 \beta_{10} - 166680 \beta_{9} + 109615 \beta_{8} - 276295 \beta_{7} + \cdots + 8117073 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1802658 \beta_{9} + 2256630 \beta_{8} + 1802658 \beta_{7} - 695172 \beta_{5} - 2951802 \beta_{4} + \cdots + 14683011 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 35190961 \beta_{11} + 87954869 \beta_{10} + 167073675 \beta_{9} + 53402959 \beta_{8} + \cdots - 87954869 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3578843 \beta_{11} + 274325772 \beta_{10} + 1429014633 \beta_{9} + 727771297 \beta_{8} + \cdots - 5527428962 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 32029958308 \beta_{9} - 22922575007 \beta_{8} + 32029958308 \beta_{7} + 25590730590 \beta_{5} + \cdots - 620091357583 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7083965437 \beta_{11} - 97178836389 \beta_{10} - 312288354471 \beta_{9} - 422248866934 \beta_{8} + \cdots + 97178836389 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
8.60453 + 14.9035i
5.31437 + 9.20476i
1.15987 + 2.00895i
−1.57985 − 2.73638i
−4.21637 − 7.30298i
−8.28254 − 14.3458i
8.60453 − 14.9035i
5.31437 − 9.20476i
1.15987 − 2.00895i
−1.57985 + 2.73638i
−4.21637 + 7.30298i
−8.28254 + 14.3458i
1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i −9.60453 + 16.6355i −6.00000 −5.33734 + 17.7345i −8.00000 −4.50000 + 7.79423i 19.2091 + 33.2711i
79.2 1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i −6.31437 + 10.9368i −6.00000 −0.858993 − 18.5003i −8.00000 −4.50000 + 7.79423i 12.6287 + 21.8736i
79.3 1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i −2.15987 + 3.74100i −6.00000 16.7072 − 7.99187i −8.00000 −4.50000 + 7.79423i 4.31974 + 7.48201i
79.4 1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 0.579850 − 1.00433i −6.00000 17.1146 + 7.07752i −8.00000 −4.50000 + 7.79423i −1.15970 − 2.00866i
79.5 1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 3.21637 − 5.57092i −6.00000 0.538262 + 18.5124i −8.00000 −4.50000 + 7.79423i −6.43275 − 11.1418i
79.6 1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 7.28254 − 12.6137i −6.00000 −13.6637 − 12.5021i −8.00000 −4.50000 + 7.79423i −14.5651 − 25.2275i
235.1 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −9.60453 − 16.6355i −6.00000 −5.33734 − 17.7345i −8.00000 −4.50000 − 7.79423i 19.2091 − 33.2711i
235.2 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −6.31437 − 10.9368i −6.00000 −0.858993 + 18.5003i −8.00000 −4.50000 − 7.79423i 12.6287 − 21.8736i
235.3 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −2.15987 − 3.74100i −6.00000 16.7072 + 7.99187i −8.00000 −4.50000 − 7.79423i 4.31974 − 7.48201i
235.4 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 0.579850 + 1.00433i −6.00000 17.1146 − 7.07752i −8.00000 −4.50000 − 7.79423i −1.15970 + 2.00866i
235.5 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 3.21637 + 5.57092i −6.00000 0.538262 − 18.5124i −8.00000 −4.50000 − 7.79423i −6.43275 + 11.1418i
235.6 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 7.28254 + 12.6137i −6.00000 −13.6637 + 12.5021i −8.00000 −4.50000 − 7.79423i −14.5651 + 25.2275i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.i.f ✓ 12
7.c even 3 1 inner 546.4.i.f ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.i.f ✓ 12 1.a even 1 1 trivial
546.4.i.f ✓ 12 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 14 T_{5}^{11} + 499 T_{5}^{10} + 2188 T_{5}^{9} + 119096 T_{5}^{8} + 560299 T_{5}^{7} + \cdots + 12964555044 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 12964555044 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T - 13)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{6} - 383 T^{5} + \cdots - 3818302011)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 57\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 82\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 9981093300360)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 16605035309968)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 22\!\cdots\!89 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 52\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 65\!\cdots\!70)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 20\!\cdots\!92)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 63\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 11\!\cdots\!06)^{2} \) Copy content Toggle raw display
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