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Properties

Label 342.4.g.f
Level $342$
Weight $4$
Character orbit 342.g
Analytic conductor $20.179$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [342,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 342.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-6,0,-12,1] 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: \(20.1786532220\)
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: \( x^{6} - x^{5} + 64x^{4} + 33x^{3} + 3984x^{2} - 945x + 225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q - 2 \beta_{4} q^{2} + (4 \beta_{4} - 4) q^{4} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{5} + (\beta_{2} + 9) q^{7} + 8 q^{8} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + \cdots + 2) q^{10}+ \cdots + (4 \beta_{5} + 438 \beta_{4} + \cdots + 36 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - 12 q^{4} + q^{5} + 52 q^{7} + 48 q^{8} + 2 q^{10} - 8 q^{11} + 129 q^{13} - 52 q^{14} - 48 q^{16} + 51 q^{17} + 40 q^{19} - 8 q^{20} + 8 q^{22} - 47 q^{23} - 338 q^{25} - 516 q^{26} - 104 q^{28}+ \cdots + 1354 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 64x^{4} + 33x^{3} + 3984x^{2} - 945x + 225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 64\nu^{4} + 4096\nu^{3} - 3984\nu^{2} + 945\nu - 60480 ) / 254031 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\nu^{5} - 1344\nu^{4} + 1339\nu^{3} - 83664\nu^{2} + 19845\nu - 3641036 ) / 169354 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1344\nu^{5} - 1339\nu^{4} + 85696\nu^{3} + 64832\nu^{2} + 5334576\nu + 4800 ) / 1270155 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 57792\nu^{5} - 57577\nu^{4} + 3684928\nu^{3} + 1517621\nu^{2} + 229386768\nu - 54410265 ) / 2540310 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{5} + 43\beta_{4} - 43 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 63\beta_{2} - 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 128\beta_{5} - 2737\beta_{4} - 128\beta_{3} - 48\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 224\beta_{5} - 3856\beta_{4} - 4017\beta_{2} - 4017\beta _1 + 3856 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\)

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
0.118706 − 0.205606i
4.16954 − 7.22186i
−3.78825 + 6.56144i
0.118706 + 0.205606i
4.16954 + 7.22186i
−3.78825 − 6.56144i
−1.00000 + 1.73205i 0 −2.00000 − 3.46410i −10.3546 + 17.9347i 0 8.76259 8.00000 0 −20.7092 − 35.8694i
163.2 −1.00000 + 1.73205i 0 −2.00000 − 3.46410i 2.96554 − 5.13646i 0 0.660916 8.00000 0 5.93108 + 10.2729i
163.3 −1.00000 + 1.73205i 0 −2.00000 − 3.46410i 7.88908 − 13.6643i 0 16.5765 8.00000 0 15.7782 + 27.3286i
235.1 −1.00000 − 1.73205i 0 −2.00000 + 3.46410i −10.3546 − 17.9347i 0 8.76259 8.00000 0 −20.7092 + 35.8694i
235.2 −1.00000 − 1.73205i 0 −2.00000 + 3.46410i 2.96554 + 5.13646i 0 0.660916 8.00000 0 5.93108 − 10.2729i
235.3 −1.00000 − 1.73205i 0 −2.00000 + 3.46410i 7.88908 + 13.6643i 0 16.5765 8.00000 0 15.7782 − 27.3286i
\(n\): 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 342.4.g.f 6
3.b odd 2 1 38.4.c.c ✓ 6
12.b even 2 1 304.4.i.e 6
19.c even 3 1 inner 342.4.g.f 6
57.f even 6 1 722.4.a.k 3
57.h odd 6 1 38.4.c.c ✓ 6
57.h odd 6 1 722.4.a.j 3
228.m even 6 1 304.4.i.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.c.c ✓ 6 3.b odd 2 1
38.4.c.c ✓ 6 57.h odd 6 1
304.4.i.e 6 12.b even 2 1
304.4.i.e 6 228.m even 6 1
342.4.g.f 6 1.a even 1 1 trivial
342.4.g.f 6 19.c even 3 1 inner
722.4.a.j 3 57.h odd 6 1
722.4.a.k 3 57.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - T_{5}^{5} + 357T_{5}^{4} - 3520T_{5}^{3} + 128674T_{5}^{2} - 689928T_{5} + 3755844 \) acting on \(S_{4}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} + \cdots + 3755844 \) Copy content Toggle raw display
$7$ \( (T^{3} - 26 T^{2} + \cdots - 96)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + 4 T^{2} + \cdots - 49980)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 129 T^{5} + \cdots + 129322384 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 11293737984 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 4555440036 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 5937750562500 \) Copy content Toggle raw display
$31$ \( (T^{3} + 50 T^{2} + \cdots + 3809848)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 188 T^{2} + \cdots - 88004)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 81183541856481 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 259289089231936 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 25671752892900 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 10857156800400 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 12\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 356206057990084 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 964239549849 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 68\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 713681971209 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{3} + 588 T^{2} + \cdots + 162474984)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 68\!\cdots\!25 \) Copy content Toggle raw display
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