Properties

Label 2304.3.b.a
Level $2304$
Weight $3$
Character orbit 2304.b
Analytic conductor $62.779$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,3,Mod(127,2304)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.b (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: \(62.7794529086\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 288)
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.

\(f(q)\) \(=\) \( q + 2 \beta q^{5} - 2 \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{5} - 2 \beta q^{7} - 16 q^{11} - \beta q^{13} - 24 q^{17} + 24 q^{19} + 16 \beta q^{23} + 9 q^{25} - 22 \beta q^{29} + 26 \beta q^{31} + 16 q^{35} - 9 \beta q^{37} + 8 q^{41} - 56 q^{43} - 16 \beta q^{47} + 33 q^{49} - 18 \beta q^{53} - 32 \beta q^{55} + 32 q^{59} - 31 \beta q^{61} + 8 q^{65} + 80 q^{67} - 64 \beta q^{71} + 66 q^{73} + 32 \beta q^{77} + 10 \beta q^{79} - 16 q^{83} - 48 \beta q^{85} - 144 q^{89} - 8 q^{91} + 48 \beta q^{95} + 94 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{11} - 48 q^{17} + 48 q^{19} + 18 q^{25} + 32 q^{35} + 16 q^{41} - 112 q^{43} + 66 q^{49} + 64 q^{59} + 16 q^{65} + 160 q^{67} + 132 q^{73} - 32 q^{83} - 288 q^{89} - 16 q^{91} + 188 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.00000i
1.00000i
0 0 0 4.00000i 0 4.00000i 0 0 0
127.2 0 0 0 4.00000i 0 4.00000i 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.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.b.a 2
3.b odd 2 1 2304.3.b.i 2
4.b odd 2 1 2304.3.b.h 2
8.b even 2 1 2304.3.b.h 2
8.d odd 2 1 inner 2304.3.b.a 2
12.b even 2 1 2304.3.b.b 2
16.e even 4 1 288.3.g.a 2
16.e even 4 1 576.3.g.h 2
16.f odd 4 1 288.3.g.a 2
16.f odd 4 1 576.3.g.h 2
24.f even 2 1 2304.3.b.i 2
24.h odd 2 1 2304.3.b.b 2
48.i odd 4 1 288.3.g.c yes 2
48.i odd 4 1 576.3.g.d 2
48.k even 4 1 288.3.g.c yes 2
48.k even 4 1 576.3.g.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.g.a 2 16.e even 4 1
288.3.g.a 2 16.f odd 4 1
288.3.g.c yes 2 48.i odd 4 1
288.3.g.c yes 2 48.k even 4 1
576.3.g.d 2 48.i odd 4 1
576.3.g.d 2 48.k even 4 1
576.3.g.h 2 16.e even 4 1
576.3.g.h 2 16.f odd 4 1
2304.3.b.a 2 1.a even 1 1 trivial
2304.3.b.a 2 8.d odd 2 1 inner
2304.3.b.b 2 12.b even 2 1
2304.3.b.b 2 24.h odd 2 1
2304.3.b.h 2 4.b odd 2 1
2304.3.b.h 2 8.b even 2 1
2304.3.b.i 2 3.b odd 2 1
2304.3.b.i 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} + 16 \) Copy content Toggle raw display
\( T_{17} + 24 \) Copy content Toggle raw display
\( T_{19} - 24 \) 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
$5$ \( T^{2} + 16 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T + 24)^{2} \) Copy content Toggle raw display
$19$ \( (T - 24)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1024 \) Copy content Toggle raw display
$29$ \( T^{2} + 1936 \) Copy content Toggle raw display
$31$ \( T^{2} + 2704 \) Copy content Toggle raw display
$37$ \( T^{2} + 324 \) Copy content Toggle raw display
$41$ \( (T - 8)^{2} \) Copy content Toggle raw display
$43$ \( (T + 56)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 1024 \) Copy content Toggle raw display
$53$ \( T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( (T - 32)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 3844 \) Copy content Toggle raw display
$67$ \( (T - 80)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 16384 \) Copy content Toggle raw display
$73$ \( (T - 66)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 400 \) Copy content Toggle raw display
$83$ \( (T + 16)^{2} \) Copy content Toggle raw display
$89$ \( (T + 144)^{2} \) Copy content Toggle raw display
$97$ \( (T - 94)^{2} \) Copy content Toggle raw display
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