Properties

Label 1728.4.f.e
Level $1728$
Weight $4$
Character orbit 1728.f
Analytic conductor $101.955$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,4,Mod(863,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.863");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.f (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{5} - 13 \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{5} - 13 \beta_{3} q^{7} + \beta_{6} q^{11} - 13 \beta_1 q^{13} - \beta_{7} q^{17} + 47 \beta_{2} q^{19} - \beta_{4} q^{23} + 115 q^{25} - 12 \beta_{5} q^{29} + 172 \beta_{3} q^{31} + 13 \beta_{6} q^{35} + 19 \beta_1 q^{37} - 10 \beta_{7} q^{41} - 114 \beta_{2} q^{43} - 11 \beta_{4} q^{47} + 174 q^{49} - 34 \beta_{5} q^{53} - 240 \beta_{3} q^{55} + 45 \beta_{6} q^{59} - 271 \beta_1 q^{61} - 13 \beta_{7} q^{65} - 11 \beta_{2} q^{67} - 22 \beta_{4} q^{71} + 391 q^{73} + 13 \beta_{5} q^{77} - 413 \beta_{3} q^{79} + 8 \beta_{6} q^{83} - 240 \beta_1 q^{85} + 49 \beta_{7} q^{89} - 169 \beta_{2} q^{91} + 47 \beta_{4} q^{95} - 427 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 920 q^{25} + 1392 q^{49} + 3128 q^{73} - 3416 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} - 10\nu^{3} + 7\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -3\nu^{6} - 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -5\nu^{7} + 16\nu^{5} - 44\nu^{3} + 31\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -8\nu^{6} + 24\nu^{4} - 56\nu^{2} + 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 18\nu^{7} - 48\nu^{5} + 132\nu^{3} - 6\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 3\beta_{5} + 12\beta_{3} - 12\beta_{2} ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{6} + \beta_{4} - 36\beta _1 + 36 ) / 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 24\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{6} - \beta_{4} - 28\beta _1 - 28 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} - 15\beta_{5} + 132\beta_{3} + 132\beta_{2} ) / 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{4} - 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} - 39\beta_{5} - 348\beta_{3} + 348\beta_{2} ) / 48 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
863.1
0.535233 0.309017i
1.40126 + 0.809017i
1.40126 0.809017i
0.535233 + 0.309017i
−1.40126 + 0.809017i
−0.535233 0.309017i
−0.535233 + 0.309017i
−1.40126 0.809017i
0 0 0 −15.4919 0 13.0000i 0 0 0
863.2 0 0 0 −15.4919 0 13.0000i 0 0 0
863.3 0 0 0 −15.4919 0 13.0000i 0 0 0
863.4 0 0 0 −15.4919 0 13.0000i 0 0 0
863.5 0 0 0 15.4919 0 13.0000i 0 0 0
863.6 0 0 0 15.4919 0 13.0000i 0 0 0
863.7 0 0 0 15.4919 0 13.0000i 0 0 0
863.8 0 0 0 15.4919 0 13.0000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 863.8
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
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.f.e 8
3.b odd 2 1 inner 1728.4.f.e 8
4.b odd 2 1 inner 1728.4.f.e 8
8.b even 2 1 inner 1728.4.f.e 8
8.d odd 2 1 inner 1728.4.f.e 8
12.b even 2 1 inner 1728.4.f.e 8
24.f even 2 1 inner 1728.4.f.e 8
24.h odd 2 1 inner 1728.4.f.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.4.f.e 8 1.a even 1 1 trivial
1728.4.f.e 8 3.b odd 2 1 inner
1728.4.f.e 8 4.b odd 2 1 inner
1728.4.f.e 8 8.b even 2 1 inner
1728.4.f.e 8 8.d odd 2 1 inner
1728.4.f.e 8 12.b even 2 1 inner
1728.4.f.e 8 24.f even 2 1 inner
1728.4.f.e 8 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} - 240 \) Copy content Toggle raw display
\( T_{7}^{2} + 169 \) Copy content Toggle raw display
\( T_{23}^{2} - 720 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - 240)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 169)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 240)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 507)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 720)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6627)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 720)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 34560)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 29584)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1083)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 72000)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 38988)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 87120)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 277440)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 486000)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 220323)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 363)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 348480)^{4} \) Copy content Toggle raw display
$73$ \( (T - 391)^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 170569)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 15360)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 1728720)^{4} \) Copy content Toggle raw display
$97$ \( (T + 427)^{8} \) Copy content Toggle raw display
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