Properties

Label 2448.2.be.h
Level $2448$
Weight $2$
Character orbit 2448.be
Analytic conductor $19.547$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2448,2,Mod(1441,2448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2448, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2448.1441");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2448.be (of order \(4\), 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: \(19.5473784148\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1224)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 i - 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 i - 3) q^{7} + ( - 2 i + 2) q^{11} + 2 q^{13} + ( - 4 i + 1) q^{17} + (i - 1) q^{23} - 5 i q^{25} + (4 i + 4) q^{29} + (i + 1) q^{31} + ( - i + 1) q^{41} + 4 i q^{43} + 6 q^{47} - 11 i q^{49} + 10 i q^{53} + 12 i q^{59} + ( - 4 i + 4) q^{61} - 8 q^{67} + (11 i + 11) q^{71} + (7 i + 7) q^{73} + 12 i q^{77} + (9 i - 9) q^{79} - 4 i q^{83} + 12 q^{89} + (6 i - 6) q^{91} + (i + 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{7} + 4 q^{11} + 4 q^{13} + 2 q^{17} - 2 q^{23} + 8 q^{29} + 2 q^{31} + 2 q^{41} + 12 q^{47} + 8 q^{61} - 16 q^{67} + 22 q^{71} + 14 q^{73} - 18 q^{79} + 24 q^{89} - 12 q^{91} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2448\mathbb{Z}\right)^\times\).

\(n\) \(613\) \(1361\) \(1873\) \(2143\)
\(\chi(n)\) \(1\) \(1\) \(i\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1441.1
1.00000i
1.00000i
0 0 0 0 0 −3.00000 + 3.00000i 0 0 0
1585.1 0 0 0 0 0 −3.00000 3.00000i 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
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2448.2.be.h 2
3.b odd 2 1 2448.2.be.g 2
4.b odd 2 1 1224.2.w.c 2
12.b even 2 1 1224.2.w.d yes 2
17.c even 4 1 inner 2448.2.be.h 2
51.f odd 4 1 2448.2.be.g 2
68.f odd 4 1 1224.2.w.c 2
204.l even 4 1 1224.2.w.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.2.w.c 2 4.b odd 2 1
1224.2.w.c 2 68.f odd 4 1
1224.2.w.d yes 2 12.b even 2 1
1224.2.w.d yes 2 204.l even 4 1
2448.2.be.g 2 3.b odd 2 1
2448.2.be.g 2 51.f odd 4 1
2448.2.be.h 2 1.a even 1 1 trivial
2448.2.be.h 2 17.c even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2448, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} + 18 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} + 8 \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 17 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$29$ \( T^{2} - 8T + 32 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 100 \) Copy content Toggle raw display
$59$ \( T^{2} + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T + 32 \) Copy content Toggle raw display
$67$ \( (T + 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 22T + 242 \) Copy content Toggle raw display
$73$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$79$ \( T^{2} + 18T + 162 \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
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