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Properties

Label 546.4.l.c
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,-16,20] 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: \( x^{8} + 187x^{6} + 9250x^{4} + 108267x^{2} + 298116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} + 3 \beta_1 q^{3} + ( - 4 \beta_1 - 4) q^{4} + (\beta_{5} + \beta_{3} + 3) q^{5} + (6 \beta_1 + 6) q^{6} + (7 \beta_1 + 7) q^{7} - 8 q^{8} + ( - 9 \beta_1 - 9) q^{9} + (2 \beta_{3} - 6 \beta_1) q^{10}+ \cdots + (27 \beta_{7} - 27 \beta_{6} + \cdots + 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 12 q^{3} - 16 q^{4} + 20 q^{5} + 24 q^{6} + 28 q^{7} - 64 q^{8} - 36 q^{9} + 20 q^{10} - 7 q^{11} + 96 q^{12} - 69 q^{13} + 112 q^{14} - 30 q^{15} - 64 q^{16} + 99 q^{17} - 144 q^{18} - 30 q^{19}+ \cdots + 126 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 187x^{6} + 9250x^{4} + 108267x^{2} + 298116 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -11\nu^{7} - 2015\nu^{5} - 91754\nu^{3} - 621879\nu - 329238 ) / 658476 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 244 \nu^{7} + 190 \nu^{6} + 44203 \nu^{5} + 29542 \nu^{4} + 2035435 \nu^{3} + 1202485 \nu^{2} + \cdots + 13847106 ) / 2304666 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 411 \nu^{7} + 674 \nu^{6} + 74301 \nu^{5} + 129056 \nu^{4} + 3428592 \nu^{3} + 6388376 \nu^{2} + \cdots + 48344478 ) / 4609332 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 244 \nu^{7} + 190 \nu^{6} - 44203 \nu^{5} + 29542 \nu^{4} - 2035435 \nu^{3} + 1202485 \nu^{2} + \cdots + 13847106 ) / 2304666 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 411 \nu^{7} + 674 \nu^{6} - 74301 \nu^{5} + 129056 \nu^{4} - 3428592 \nu^{3} + 6388376 \nu^{2} + \cdots + 48344478 ) / 4609332 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 202 \nu^{7} + 622 \nu^{6} + 34207 \nu^{5} + 108841 \nu^{4} + 1301758 \nu^{3} + 4421749 \nu^{2} + \cdots + 19591572 ) / 2304666 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 481 \nu^{7} - 1244 \nu^{6} + 82519 \nu^{5} - 217682 \nu^{4} + 3245794 \nu^{3} - 8843498 \nu^{2} + \cdots - 36878478 ) / 4609332 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2\beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} + 3\beta_{4} + 2\beta_{3} + 3\beta_{2} + 2\beta _1 - 44 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{7} - 9\beta_{6} + 53\beta_{5} - 104\beta_{4} - 53\beta_{3} + 104\beta_{2} + 649\beta _1 + 329 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -175\beta_{7} + 175\beta_{6} - 80\beta_{5} - 431\beta_{4} - 80\beta_{3} - 431\beta_{2} - 175\beta _1 + 3882 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3 \beta_{7} - 3 \beta_{6} - 3784 \beta_{5} + 10774 \beta_{4} + 3784 \beta_{3} - 10774 \beta_{2} + \cdots - 48100 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14552 \beta_{7} - 14552 \beta_{6} - 219 \beta_{5} + 54092 \beta_{4} - 219 \beta_{3} + 54092 \beta_{2} + \cdots - 397999 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 75621 \beta_{7} + 75621 \beta_{6} + 307607 \beta_{5} - 1162643 \beta_{4} - 307607 \beta_{3} + \cdots + 6033509 ) / 3 \) 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
− 7.60004i
− 3.35160i
10.6759i
2.00779i
7.60004i
3.35160i
− 10.6759i
− 2.00779i
1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −5.66410 3.00000 − 5.19615i 3.50000 − 6.06218i −8.00000 −4.50000 + 7.79423i −5.66410 − 9.81051i
211.2 1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −0.509442 3.00000 − 5.19615i 3.50000 − 6.06218i −8.00000 −4.50000 + 7.79423i −0.509442 − 0.882379i
211.3 1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i 2.48023 3.00000 − 5.19615i 3.50000 − 6.06218i −8.00000 −4.50000 + 7.79423i 2.48023 + 4.29588i
211.4 1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i 13.6933 3.00000 − 5.19615i 3.50000 − 6.06218i −8.00000 −4.50000 + 7.79423i 13.6933 + 23.7175i
295.1 1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −5.66410 3.00000 + 5.19615i 3.50000 + 6.06218i −8.00000 −4.50000 − 7.79423i −5.66410 + 9.81051i
295.2 1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −0.509442 3.00000 + 5.19615i 3.50000 + 6.06218i −8.00000 −4.50000 − 7.79423i −0.509442 + 0.882379i
295.3 1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i 2.48023 3.00000 + 5.19615i 3.50000 + 6.06218i −8.00000 −4.50000 − 7.79423i 2.48023 − 4.29588i
295.4 1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i 13.6933 3.00000 + 5.19615i 3.50000 + 6.06218i −8.00000 −4.50000 − 7.79423i 13.6933 − 23.7175i
\(n\): 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.c ✓ 8
13.c even 3 1 inner 546.4.l.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.l.c ✓ 8 1.a even 1 1 trivial
546.4.l.c ✓ 8 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 10T_{5}^{3} - 63T_{5}^{2} + 163T_{5} + 98 \) 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
$5$ \( (T^{4} - 10 T^{3} + \cdots + 98)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7 T + 49)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 5900236969 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 81676026675441 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 217827224978841 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 261091240589124 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 31\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( (T^{4} - 199 T^{3} + \cdots + 305796474)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 35\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 37\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( (T^{4} + 450 T^{3} + \cdots + 1734838521)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 754 T^{3} + \cdots + 15380146431)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 59\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 46\!\cdots\!01 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{4} - 792 T^{3} + \cdots + 67855038576)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + \cdots - 8626513991)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 286 T^{3} + \cdots + 98762092344)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 24\!\cdots\!29 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
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