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Properties

Label 342.6.g.a
Level $342$
Weight $6$
Character orbit 342.g
Analytic conductor $54.851$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [342,6,Mod(163,342)] 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("342.163"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 342.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-12,0,-48,-14] 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: \(54.8512663760\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 133x^{4} - 60x^{3} + 17689x^{2} - 3990x + 900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 38)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \beta_{3} q^{2} + (16 \beta_{3} - 16) q^{4} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{5} + (4 \beta_{2} - 104) q^{7} + 64 q^{8} + (4 \beta_{5} + 20 \beta_{3} + \cdots - 20) q^{10} + (\beta_{4} - \beta_{2} + 156) q^{11}+ \cdots + ( - 256 \beta_{5} + 256 \beta_{4} + \cdots - 3328 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{2} - 48 q^{4} - 14 q^{5} - 624 q^{7} + 384 q^{8} - 56 q^{10} + 938 q^{11} - 736 q^{13} + 1248 q^{14} - 768 q^{16} + 1000 q^{17} + 5681 q^{19} + 448 q^{20} - 1876 q^{22} + 2168 q^{23} - 3187 q^{25}+ \cdots + 3796 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 133x^{4} - 60x^{3} + 17689x^{2} - 3990x + 900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} - 60 ) / 133 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 133\nu^{3} - 30\nu^{2} + 17689\nu ) / 3990 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 133\nu^{2} - 30\nu + 11837 ) / 133 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -89\nu^{5} - 11837\nu^{3} + 6660\nu^{2} - 1574321\nu + 355110 ) / 3990 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 89\beta_{3} - 89 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 133\beta_{2} + 60 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -133\beta_{5} + 133\beta_{4} - 11837\beta_{3} + 15\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 60\beta_{5} + 13320\beta_{3} - 17689\beta_{2} - 17689\beta _1 - 13320 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(-1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
5.70904 − 9.88835i
−5.82187 + 10.0838i
0.112825 − 0.195419i
5.70904 + 9.88835i
−5.82187 − 10.0838i
0.112825 + 0.195419i
−2.00000 + 3.46410i 0 −8.00000 − 13.8564i −34.6044 + 59.9365i 0 −195.345 64.0000 0 −138.418 − 239.746i
163.2 −2.00000 + 3.46410i 0 −8.00000 − 13.8564i −14.1445 + 24.4990i 0 −10.8501 64.0000 0 −56.5781 − 97.9961i
163.3 −2.00000 + 3.46410i 0 −8.00000 − 13.8564i 41.7489 − 72.3112i 0 −105.805 64.0000 0 166.996 + 289.245i
235.1 −2.00000 − 3.46410i 0 −8.00000 + 13.8564i −34.6044 − 59.9365i 0 −195.345 64.0000 0 −138.418 + 239.746i
235.2 −2.00000 − 3.46410i 0 −8.00000 + 13.8564i −14.1445 − 24.4990i 0 −10.8501 64.0000 0 −56.5781 + 97.9961i
235.3 −2.00000 − 3.46410i 0 −8.00000 + 13.8564i 41.7489 + 72.3112i 0 −105.805 64.0000 0 166.996 − 289.245i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.6.g.a 6
3.b odd 2 1 38.6.c.a ✓ 6
12.b even 2 1 304.6.i.a 6
19.c even 3 1 inner 342.6.g.a 6
57.f even 6 1 722.6.a.f 3
57.h odd 6 1 38.6.c.a ✓ 6
57.h odd 6 1 722.6.a.e 3
228.m even 6 1 304.6.i.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.c.a ✓ 6 3.b odd 2 1
38.6.c.a ✓ 6 57.h odd 6 1
304.6.i.a 6 12.b even 2 1
304.6.i.a 6 228.m even 6 1
342.6.g.a 6 1.a even 1 1 trivial
342.6.g.a 6 19.c even 3 1 inner
722.6.a.e 3 57.h odd 6 1
722.6.a.f 3 57.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 14T_{5}^{5} + 6379T_{5}^{4} + 240390T_{5}^{3} + 40518153T_{5}^{2} + 1010772108T_{5} + 26724402576 \) acting on \(S_{6}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4 T + 16)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 26724402576 \) Copy content Toggle raw display
$7$ \( (T^{3} + 312 T^{2} + \cdots + 224256)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 469 T^{2} + \cdots - 2905680)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 394380754796484 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 15\!\cdots\!99 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{3} - 198 T^{2} + \cdots + 281768267232)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 10030 T^{2} + \cdots + 34115602072)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 16\!\cdots\!29 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 26\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 29\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 44\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 56\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{3} + \cdots + 518524918263648)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 33\!\cdots\!21 \) Copy content Toggle raw display
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