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Properties

Label 546.4.l.f
Level $546$
Weight $4$
Character orbit 546.l
Analytic conductor $32.215$
Analytic rank $0$
Dimension $10$
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 = [10,-10,-15,-20,-22] 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: \( x^{10} - 2 x^{9} + 245 x^{8} - 866 x^{7} + 46349 x^{6} - 164186 x^{5} + 3498412 x^{4} + \cdots + 2923781184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_1 q^{2} + 3 \beta_1 q^{3} + ( - 4 \beta_1 - 4) q^{4} + ( - \beta_{5} - 2) q^{5} + ( - 6 \beta_1 - 6) q^{6} + ( - 7 \beta_1 - 7) q^{7} + 8 q^{8} + ( - 9 \beta_1 - 9) q^{9} + ( - 2 \beta_{8} - 4 \beta_1) q^{10}+ \cdots + ( - 9 \beta_{5} - 9 \beta_{4} + \cdots - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} - 15 q^{3} - 20 q^{4} - 22 q^{5} - 30 q^{6} - 35 q^{7} + 80 q^{8} - 45 q^{9} + 22 q^{10} + 25 q^{11} + 120 q^{12} - 42 q^{13} + 140 q^{14} + 33 q^{15} - 80 q^{16} + 122 q^{17} + 180 q^{18}+ \cdots - 450 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 245 x^{8} - 866 x^{7} + 46349 x^{6} - 164186 x^{5} + 3498412 x^{4} + \cdots + 2923781184 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 13\!\cdots\!61 \nu^{9} + \cdots - 79\!\cdots\!76 ) / 91\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35\!\cdots\!31 \nu^{9} + \cdots - 92\!\cdots\!80 ) / 22\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 37\!\cdots\!45 \nu^{9} + \cdots + 83\!\cdots\!12 ) / 22\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 40\!\cdots\!59 \nu^{9} + \cdots - 73\!\cdots\!44 ) / 22\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53\!\cdots\!11 \nu^{9} + \cdots - 68\!\cdots\!32 ) / 22\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 42\!\cdots\!71 \nu^{9} + \cdots + 12\!\cdots\!44 ) / 11\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 75\!\cdots\!71 \nu^{9} + \cdots - 12\!\cdots\!88 ) / 13\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!75 \nu^{9} + \cdots + 34\!\cdots\!96 ) / 16\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 32\!\cdots\!59 \nu^{9} + \cdots - 75\!\cdots\!84 ) / 27\!\cdots\!24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} - 2\beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{8} + 12\beta_{7} - 2\beta_{4} - \beta_{3} - 11\beta_{2} + 294\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{6} + 20\beta_{5} - 41\beta_{4} + 41\beta_{3} + 80\beta_{2} + 160 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 132 \beta_{9} + 1278 \beta_{8} - 2364 \beta_{7} + 1278 \beta_{5} + 245 \beta_{4} + 490 \beta_{3} + \cdots - 37854 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5520 \beta_{9} + 12972 \beta_{8} + 1914 \beta_{7} - 5520 \beta_{6} + 32758 \beta_{4} + \cdots + 75912 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12316\beta_{6} - 71674\beta_{5} + 16631\beta_{4} - 16631\beta_{3} + 101702\beta_{2} + 1816330 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1031472 \beta_{9} - 2234556 \beta_{8} - 674442 \beta_{7} - 2234556 \beta_{5} - 2327227 \beta_{4} + \cdots - 10734840 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 7537716 \beta_{9} - 34074366 \beta_{8} + 66923436 \beta_{7} - 7537716 \beta_{6} + \cdots + 821816718 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 59691056 \beta_{6} + 120394052 \beta_{5} - 114805033 \beta_{4} + 114805033 \beta_{3} + 279186576 \beta_{2} + 525437272 \) 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
3.47993 − 6.02742i
−6.34551 + 10.9907i
−4.88554 + 8.46200i
2.50939 − 4.34640i
6.24172 − 10.8110i
3.47993 + 6.02742i
−6.34551 − 10.9907i
−4.88554 − 8.46200i
2.50939 + 4.34640i
6.24172 + 10.8110i
−1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −21.4943 −3.00000 + 5.19615i −3.50000 + 6.06218i 8.00000 −4.50000 + 7.79423i 21.4943 + 37.2292i
211.2 −1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −5.48832 −3.00000 + 5.19615i −3.50000 + 6.06218i 8.00000 −4.50000 + 7.79423i 5.48832 + 9.50604i
211.3 −1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −3.44123 −3.00000 + 5.19615i −3.50000 + 6.06218i 8.00000 −4.50000 + 7.79423i 3.44123 + 5.96039i
211.4 −1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i 1.76241 −3.00000 + 5.19615i −3.50000 + 6.06218i 8.00000 −4.50000 + 7.79423i −1.76241 − 3.05259i
211.5 −1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i 17.6614 −3.00000 + 5.19615i −3.50000 + 6.06218i 8.00000 −4.50000 + 7.79423i −17.6614 − 30.5905i
295.1 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −21.4943 −3.00000 − 5.19615i −3.50000 − 6.06218i 8.00000 −4.50000 − 7.79423i 21.4943 − 37.2292i
295.2 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −5.48832 −3.00000 − 5.19615i −3.50000 − 6.06218i 8.00000 −4.50000 − 7.79423i 5.48832 − 9.50604i
295.3 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −3.44123 −3.00000 − 5.19615i −3.50000 − 6.06218i 8.00000 −4.50000 − 7.79423i 3.44123 − 5.96039i
295.4 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i 1.76241 −3.00000 − 5.19615i −3.50000 − 6.06218i 8.00000 −4.50000 − 7.79423i −1.76241 + 3.05259i
295.5 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i 17.6614 −3.00000 − 5.19615i −3.50000 − 6.06218i 8.00000 −4.50000 − 7.79423i −17.6614 + 30.5905i
\(n\): 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.f ✓ 10
13.c even 3 1 inner 546.4.l.f ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.l.f ✓ 10 1.a even 1 1 trivial
546.4.l.f ✓ 10 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} + 11T_{5}^{4} - 349T_{5}^{3} - 2742T_{5}^{2} - 1323T_{5} + 12636 \) 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
$5$ \( (T^{5} + 11 T^{4} + \cdots + 12636)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7 T + 49)^{5} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 24047019058176 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 51\!\cdots\!57 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 29\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 338053821337600 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 33\!\cdots\!89 \) Copy content Toggle raw display
$31$ \( (T^{5} + 391 T^{4} + \cdots + 331785888480)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{5} + 510 T^{4} + \cdots - 616916249352)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 1084457086533)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 87\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 76\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 71\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 49762574785664)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 234693452531828)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 2074688373672)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
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