Properties

Label 1152.2.f.d
Level $1152$
Weight $2$
Character orbit 1152.f
Analytic conductor $9.199$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,2,Mod(575,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.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: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} - 2 \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{5} - 2 \beta_{2} q^{7} + 2 \beta_1 q^{11} + \beta_1 q^{13} + \beta_{2} q^{17} + 4 \beta_{3} q^{19} + 4 q^{23} - 3 q^{25} + 5 \beta_{3} q^{29} - 6 \beta_{2} q^{31} + 2 \beta_1 q^{35} - 4 \beta_1 q^{37} - 3 \beta_{2} q^{41} + 8 \beta_{3} q^{43} + 12 q^{47} - q^{49} - 9 \beta_{3} q^{53} - 4 \beta_{2} q^{55} + 4 \beta_1 q^{61} - 2 \beta_{2} q^{65} + 4 \beta_{3} q^{67} + 4 q^{71} + 8 q^{73} + 8 \beta_{3} q^{77} - 2 \beta_{2} q^{79} + 6 \beta_1 q^{83} - \beta_1 q^{85} + 11 \beta_{2} q^{89} + 4 \beta_{3} q^{91} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{23} - 12 q^{25} + 48 q^{47} - 4 q^{49} + 16 q^{71} + 32 q^{73} - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0 0 0 −1.41421 0 2.82843i 0 0 0
575.2 0 0 0 −1.41421 0 2.82843i 0 0 0
575.3 0 0 0 1.41421 0 2.82843i 0 0 0
575.4 0 0 0 1.41421 0 2.82843i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.f.d yes 4
3.b odd 2 1 1152.2.f.a 4
4.b odd 2 1 1152.2.f.a 4
8.b even 2 1 inner 1152.2.f.d yes 4
8.d odd 2 1 1152.2.f.a 4
12.b even 2 1 inner 1152.2.f.d yes 4
16.e even 4 1 2304.2.c.b 2
16.e even 4 1 2304.2.c.g 2
16.f odd 4 1 2304.2.c.a 2
16.f odd 4 1 2304.2.c.h 2
24.f even 2 1 inner 1152.2.f.d yes 4
24.h odd 2 1 1152.2.f.a 4
48.i odd 4 1 2304.2.c.a 2
48.i odd 4 1 2304.2.c.h 2
48.k even 4 1 2304.2.c.b 2
48.k even 4 1 2304.2.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.f.a 4 3.b odd 2 1
1152.2.f.a 4 4.b odd 2 1
1152.2.f.a 4 8.d odd 2 1
1152.2.f.a 4 24.h odd 2 1
1152.2.f.d yes 4 1.a even 1 1 trivial
1152.2.f.d yes 4 8.b even 2 1 inner
1152.2.f.d yes 4 12.b even 2 1 inner
1152.2.f.d yes 4 24.f even 2 1 inner
2304.2.c.a 2 16.f odd 4 1
2304.2.c.a 2 48.i odd 4 1
2304.2.c.b 2 16.e even 4 1
2304.2.c.b 2 48.k even 4 1
2304.2.c.g 2 16.e even 4 1
2304.2.c.g 2 48.k even 4 1
2304.2.c.h 2 16.f odd 4 1
2304.2.c.h 2 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 8 \) Copy content Toggle raw display
\( T_{23} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$47$ \( (T - 12)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$71$ \( (T - 4)^{4} \) Copy content Toggle raw display
$73$ \( (T - 8)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 242)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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