Properties

Label 2304.2.d.e
Level $2304$
Weight $2$
Character orbit 2304.d
Analytic conductor $18.398$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,2,Mod(1153,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.1153");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.d (of order \(2\), degree \(1\), not 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: \(18.3975326257\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1152)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} - 2 q^{7} + 2 \beta q^{11} + \beta q^{13} - 4 q^{17} - 2 \beta q^{19} - 8 q^{23} + q^{25} - 3 \beta q^{29} - 6 q^{31} - 2 \beta q^{35} + \beta q^{37} + 12 q^{41} - 6 \beta q^{43} + 8 q^{47} - 3 q^{49} + 3 \beta q^{53} - 8 q^{55} + 4 \beta q^{59} - 5 \beta q^{61} - 4 q^{65} - 4 \beta q^{67} - 2 q^{73} - 4 \beta q^{77} - 14 q^{79} - 6 \beta q^{83} - 4 \beta q^{85} - 8 q^{89} - 2 \beta q^{91} + 8 q^{95} - 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{7} - 8 q^{17} - 16 q^{23} + 2 q^{25} - 12 q^{31} + 24 q^{41} + 16 q^{47} - 6 q^{49} - 16 q^{55} - 8 q^{65} - 4 q^{73} - 28 q^{79} - 16 q^{89} + 16 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1153.1
1.00000i
1.00000i
0 0 0 2.00000i 0 −2.00000 0 0 0
1153.2 0 0 0 2.00000i 0 −2.00000 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

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{17} + 4 \) Copy content Toggle raw display
\( T_{23} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T + 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( (T + 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 144 \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 64 \) Copy content Toggle raw display
$61$ \( T^{2} + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T + 14)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T + 8)^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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