Properties

Label 2304.1.t.b
Level $2304$
Weight $1$
Character orbit 2304.t
Analytic conductor $1.150$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,1,Mod(895,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.895");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2304.t (of order \(6\), degree \(2\), 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: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.5184.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + q^{3} + \zeta_{12}^{5} q^{5} + \zeta_{12} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \zeta_{12}^{5} q^{5} + \zeta_{12} q^{7} + q^{9} + \zeta_{12}^{4} q^{11} + \zeta_{12}^{5} q^{13} + \zeta_{12}^{5} q^{15} + \zeta_{12} q^{21} + \zeta_{12}^{5} q^{23} + q^{27} + \zeta_{12} q^{29} - \zeta_{12}^{5} q^{31} + \zeta_{12}^{4} q^{33} - q^{35} + \zeta_{12}^{5} q^{39} + \zeta_{12}^{2} q^{41} - \zeta_{12}^{4} q^{43} + \zeta_{12}^{5} q^{45} - \zeta_{12} q^{47} - \zeta_{12}^{3} q^{55} - \zeta_{12}^{2} q^{59} + \zeta_{12} q^{61} + \zeta_{12} q^{63} - \zeta_{12}^{4} q^{65} - \zeta_{12}^{2} q^{67} + \zeta_{12}^{5} q^{69} - \zeta_{12}^{3} q^{71} + \zeta_{12}^{5} q^{77} + \zeta_{12} q^{79} + q^{81} + \zeta_{12}^{4} q^{83} + \zeta_{12} q^{87} - q^{91} - \zeta_{12}^{5} q^{93} + \zeta_{12}^{4} q^{97} + \zeta_{12}^{4} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} - 2 q^{11} + 4 q^{27} - 2 q^{33} - 4 q^{35} + 2 q^{41} + 2 q^{43} - 2 q^{59} + 2 q^{65} - 2 q^{67} + 4 q^{81} - 2 q^{83} - 4 q^{91} - 2 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
895.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 1.00000 0 −0.866025 0.500000i 0 0.866025 0.500000i 0 1.00000 0
895.2 0 1.00000 0 0.866025 + 0.500000i 0 −0.866025 + 0.500000i 0 1.00000 0
1663.1 0 1.00000 0 −0.866025 + 0.500000i 0 0.866025 + 0.500000i 0 1.00000 0
1663.2 0 1.00000 0 0.866025 0.500000i 0 −0.866025 0.500000i 0 1.00000 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
9.c even 3 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.1.t.b 4
4.b odd 2 1 2304.1.t.a 4
8.b even 2 1 2304.1.t.a 4
8.d odd 2 1 inner 2304.1.t.b 4
9.c even 3 1 inner 2304.1.t.b 4
16.e even 4 1 288.1.o.a 4
16.e even 4 1 576.1.o.a 4
16.f odd 4 1 288.1.o.a 4
16.f odd 4 1 576.1.o.a 4
36.f odd 6 1 2304.1.t.a 4
48.i odd 4 1 864.1.o.a 4
48.i odd 4 1 1728.1.o.a 4
48.k even 4 1 864.1.o.a 4
48.k even 4 1 1728.1.o.a 4
72.n even 6 1 2304.1.t.a 4
72.p odd 6 1 inner 2304.1.t.b 4
144.u even 12 1 864.1.o.a 4
144.u even 12 1 1728.1.o.a 4
144.u even 12 1 2592.1.g.b 2
144.v odd 12 1 288.1.o.a 4
144.v odd 12 1 576.1.o.a 4
144.v odd 12 1 2592.1.g.a 2
144.w odd 12 1 864.1.o.a 4
144.w odd 12 1 1728.1.o.a 4
144.w odd 12 1 2592.1.g.b 2
144.x even 12 1 288.1.o.a 4
144.x even 12 1 576.1.o.a 4
144.x even 12 1 2592.1.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.1.o.a 4 16.e even 4 1
288.1.o.a 4 16.f odd 4 1
288.1.o.a 4 144.v odd 12 1
288.1.o.a 4 144.x even 12 1
576.1.o.a 4 16.e even 4 1
576.1.o.a 4 16.f odd 4 1
576.1.o.a 4 144.v odd 12 1
576.1.o.a 4 144.x even 12 1
864.1.o.a 4 48.i odd 4 1
864.1.o.a 4 48.k even 4 1
864.1.o.a 4 144.u even 12 1
864.1.o.a 4 144.w odd 12 1
1728.1.o.a 4 48.i odd 4 1
1728.1.o.a 4 48.k even 4 1
1728.1.o.a 4 144.u even 12 1
1728.1.o.a 4 144.w odd 12 1
2304.1.t.a 4 4.b odd 2 1
2304.1.t.a 4 8.b even 2 1
2304.1.t.a 4 36.f odd 6 1
2304.1.t.a 4 72.n even 6 1
2304.1.t.b 4 1.a even 1 1 trivial
2304.1.t.b 4 8.d odd 2 1 inner
2304.1.t.b 4 9.c even 3 1 inner
2304.1.t.b 4 72.p odd 6 1 inner
2592.1.g.a 2 144.v odd 12 1
2592.1.g.a 2 144.x even 12 1
2592.1.g.b 2 144.u even 12 1
2592.1.g.b 2 144.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} + T_{11} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$29$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$31$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$67$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$83$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
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