↑

Properties

Label 324.2.h.d
Level $324$
Weight $2$
Character orbit 324.h
Analytic conductor $2.587$
Analytic rank $0$
Dimension $8$
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [324,2,Mod(107,324)] 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("324.107"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,2,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.58715302549\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12960000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{3}) q^{2} + ( - \beta_{2} - \beta_1) q^{4} + (\beta_{5} + \beta_{4} + \beta_{3}) q^{5} + (2 \beta_{6} - \beta_{2} - 1) q^{7} + (\beta_{7} - 2 \beta_{4}) q^{8} + (\beta_{6} + \beta_1 - 3) q^{10}+ \cdots + 8 \beta_{7} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} - 20 q^{10} - 8 q^{13} + 14 q^{16} - 6 q^{22} + 60 q^{28} - 20 q^{34} - 32 q^{37} + 10 q^{40} - 48 q^{46} + 32 q^{49} + 4 q^{52} - 20 q^{58} + 16 q^{61} + 44 q^{64} - 30 q^{70} + 40 q^{73}+ \cdots - 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 8\nu^{4} - 16\nu^{2} + 19 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} - 12\nu^{3} + 11\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 2\nu^{5} - 6\nu^{3} - 3\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 16\nu^{5} - 40\nu^{3} + 23\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{6} + 16\nu^{4} - 40\nu^{2} - 3 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{7} + 3\nu^{5} - 8\nu^{3} + 2\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} + 2\beta_{5} + \beta_{4} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - 5\beta_{2} + \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 4\beta_{5} - \beta_{4} - 4\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - 3\beta_{2} + 2\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{7} - 2\beta_{5} - 13\beta_{4} - 13\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -8\beta_{6} + 8\beta_{2} + 8\beta _1 - 23 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 34\beta_{7} - 34\beta_{5} - 29\beta_{4} - 5\beta_{3} ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
1.40126 + 0.809017i
0.535233 + 0.309017i
−0.535233 − 0.309017i
−1.40126 − 0.809017i
1.40126 − 0.809017i
0.535233 − 0.309017i
−0.535233 + 0.309017i
−1.40126 + 0.809017i
−1.40126 + 0.190983i 0 1.92705 − 0.535233i 1.93649 − 1.11803i 0 3.35410 + 1.93649i −2.59808 + 1.11803i 0 −2.50000 + 1.93649i
107.2 −0.535233 − 1.30902i 0 −1.42705 + 1.40126i 1.93649 − 1.11803i 0 −3.35410 − 1.93649i 2.59808 + 1.11803i 0 −2.50000 − 1.93649i
107.3 0.535233 + 1.30902i 0 −1.42705 + 1.40126i −1.93649 + 1.11803i 0 −3.35410 − 1.93649i −2.59808 − 1.11803i 0 −2.50000 − 1.93649i
107.4 1.40126 − 0.190983i 0 1.92705 − 0.535233i −1.93649 + 1.11803i 0 3.35410 + 1.93649i 2.59808 − 1.11803i 0 −2.50000 + 1.93649i
215.1 −1.40126 − 0.190983i 0 1.92705 + 0.535233i 1.93649 + 1.11803i 0 3.35410 − 1.93649i −2.59808 − 1.11803i 0 −2.50000 − 1.93649i
215.2 −0.535233 + 1.30902i 0 −1.42705 − 1.40126i 1.93649 + 1.11803i 0 −3.35410 + 1.93649i 2.59808 − 1.11803i 0 −2.50000 + 1.93649i
215.3 0.535233 − 1.30902i 0 −1.42705 − 1.40126i −1.93649 − 1.11803i 0 −3.35410 + 1.93649i −2.59808 + 1.11803i 0 −2.50000 + 1.93649i
215.4 1.40126 + 0.190983i 0 1.92705 + 0.535233i −1.93649 − 1.11803i 0 3.35410 − 1.93649i 2.59808 + 1.11803i 0 −2.50000 − 1.93649i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
12.b even 2 1 inner
36.f odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.2.h.d 8
3.b odd 2 1 inner 324.2.h.d 8
4.b odd 2 1 inner 324.2.h.d 8
9.c even 3 1 108.2.b.a ✓ 4
9.c even 3 1 inner 324.2.h.d 8
9.d odd 6 1 108.2.b.a ✓ 4
9.d odd 6 1 inner 324.2.h.d 8
12.b even 2 1 inner 324.2.h.d 8
36.f odd 6 1 108.2.b.a ✓ 4
36.f odd 6 1 inner 324.2.h.d 8
36.h even 6 1 108.2.b.a ✓ 4
36.h even 6 1 inner 324.2.h.d 8
72.j odd 6 1 1728.2.c.c 4
72.l even 6 1 1728.2.c.c 4
72.n even 6 1 1728.2.c.c 4
72.p odd 6 1 1728.2.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.b.a ✓ 4 9.c even 3 1
108.2.b.a ✓ 4 9.d odd 6 1
108.2.b.a ✓ 4 36.f odd 6 1
108.2.b.a ✓ 4 36.h even 6 1
324.2.h.d 8 1.a even 1 1 trivial
324.2.h.d 8 3.b odd 2 1 inner
324.2.h.d 8 4.b odd 2 1 inner
324.2.h.d 8 9.c even 3 1 inner
324.2.h.d 8 9.d odd 6 1 inner
324.2.h.d 8 12.b even 2 1 inner
324.2.h.d 8 36.f odd 6 1 inner
324.2.h.d 8 36.h even 6 1 inner
1728.2.c.c 4 72.j odd 6 1
1728.2.c.c 4 72.l even 6 1
1728.2.c.c 4 72.n even 6 1
1728.2.c.c 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{4} - 5T_{5}^{2} + 25 \) Copy content Toggle raw display
\( T_{7}^{4} - 15T_{7}^{2} + 225 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{6} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 15 T^{2} + 225)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 48 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 20 T^{2} + 400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 15 T^{2} + 225)^{2} \) Copy content Toggle raw display
$37$ \( (T + 4)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 80 T^{2} + 6400)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 60 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 12 T^{2} + 144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 5)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 12 T^{2} + 144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 60 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$73$ \( (T - 5)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - 60 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 147 T^{2} + 21609)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 11 T + 121)^{4} \) Copy content Toggle raw display
show more
show less