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Properties

Label 1152.2.p.c
Level $1152$
Weight $2$
Character orbit 1152.p
Analytic conductor $9.199$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,2,Mod(191,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.p (of order \(6\), degree \(2\), 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: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{3}) q^{3} + ( - \beta_{6} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{3}) q^{3} + ( - \beta_{6} - \beta_1) q^{9} + ( - \beta_{5} + 2 \beta_{4}) q^{11} + (\beta_{2} - 6 \beta_1 + 3) q^{17} + (\beta_{7} - \beta_{4} + 3 \beta_{3}) q^{19} + ( - 5 \beta_1 + 5) q^{25} + (\beta_{7} - \beta_{5}) q^{27} + (\beta_{6} + \beta_{2} - 5 \beta_1 + 4) q^{33} + ( - 2 \beta_{6} - 3 \beta_1 + 6) q^{41} + ( - 3 \beta_{7} - \beta_{5} + \cdots + 2 \beta_{3}) q^{43}+ \cdots + (3 \beta_{5} - 5 \beta_{4} - 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{9} + 20 q^{25} + 12 q^{33} + 36 q^{41} - 28 q^{49} + 36 q^{57} + 8 q^{73} + 28 q^{81} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{24}^{7} - 2\zeta_{24}^{6} + \zeta_{24}^{5} - \zeta_{24}^{3} - \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{24}^{7} + 3\zeta_{24}^{6} - \zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( 2\beta_{7} + 3\beta_{6} + \beta_{5} + 3\beta_{4} + \beta_{3} ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( -\beta_{7} - 2\beta_{5} + 3\beta_{4} - 2\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} - \beta_{5} + 5\beta_{3} - 3\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} + 3\beta_{6} + 2\beta_{5} + 3\beta_{4} - 4\beta_{3} - 3\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{7} - \beta_{5} - \beta_{3} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( 2\beta_{7} - 3\beta_{6} + \beta_{5} + 3\beta_{4} + \beta_{3} ) / 12 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(\beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−0.965926 − 0.258819i
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
0 −1.57313 + 0.724745i 0 0 0 0 0 1.94949 − 2.28024i 0
191.2 0 −0.158919 − 1.72474i 0 0 0 0 0 −2.94949 + 0.548188i 0
191.3 0 0.158919 + 1.72474i 0 0 0 0 0 −2.94949 + 0.548188i 0
191.4 0 1.57313 − 0.724745i 0 0 0 0 0 1.94949 − 2.28024i 0
959.1 0 −1.57313 − 0.724745i 0 0 0 0 0 1.94949 + 2.28024i 0
959.2 0 −0.158919 + 1.72474i 0 0 0 0 0 −2.94949 − 0.548188i 0
959.3 0 0.158919 − 1.72474i 0 0 0 0 0 −2.94949 − 0.548188i 0
959.4 0 1.57313 + 0.724745i 0 0 0 0 0 1.94949 + 2.28024i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner
72.j odd 6 1 inner
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.p.c ✓ 8
3.b odd 2 1 3456.2.p.c 8
4.b odd 2 1 inner 1152.2.p.c ✓ 8
8.b even 2 1 inner 1152.2.p.c ✓ 8
8.d odd 2 1 CM 1152.2.p.c ✓ 8
9.c even 3 1 3456.2.p.c 8
9.d odd 6 1 inner 1152.2.p.c ✓ 8
12.b even 2 1 3456.2.p.c 8
24.f even 2 1 3456.2.p.c 8
24.h odd 2 1 3456.2.p.c 8
36.f odd 6 1 3456.2.p.c 8
36.h even 6 1 inner 1152.2.p.c ✓ 8
72.j odd 6 1 inner 1152.2.p.c ✓ 8
72.l even 6 1 inner 1152.2.p.c ✓ 8
72.n even 6 1 3456.2.p.c 8
72.p odd 6 1 3456.2.p.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.p.c ✓ 8 1.a even 1 1 trivial
1152.2.p.c ✓ 8 4.b odd 2 1 inner
1152.2.p.c ✓ 8 8.b even 2 1 inner
1152.2.p.c ✓ 8 8.d odd 2 1 CM
1152.2.p.c ✓ 8 9.d odd 6 1 inner
1152.2.p.c ✓ 8 36.h even 6 1 inner
1152.2.p.c ✓ 8 72.j odd 6 1 inner
1152.2.p.c ✓ 8 72.l even 6 1 inner
3456.2.p.c 8 3.b odd 2 1
3456.2.p.c 8 9.c even 3 1
3456.2.p.c 8 12.b even 2 1
3456.2.p.c 8 24.f even 2 1
3456.2.p.c 8 24.h odd 2 1
3456.2.p.c 8 36.f odd 6 1
3456.2.p.c 8 72.n even 6 1
3456.2.p.c 8 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 \(S_{2}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{8} - 30T_{11}^{6} + 891T_{11}^{4} - 270T_{11}^{2} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 30 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 70 T^{2} + 361)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 42 T^{2} + 225)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 18 T^{3} + \cdots + 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 186 T^{6} + \cdots + 10556001 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} - 318 T^{6} + \cdots + 395254161 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} + 330 T^{6} + \cdots + 276922881 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T - 215)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 324 T^{2} + 104976)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 10 T^{3} + \cdots + 36481)^{2} \) Copy content Toggle raw display
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