Properties

Label 2320.1.bm.a
Level $2320$
Weight $1$
Character orbit 2320.bm
Analytic conductor $1.158$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
RM discriminant 29
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2320,1,Mod(753,2320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("2320.753");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2320.bm (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: \(1.15783082931\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.3625.1
Artin image: $C_4\wr C_2$
Artin field: Galois closure of 8.4.22632992000.1

$q$-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.

\(f(q)\) \(=\) \( q + i q^{5} + (i + 1) q^{7} - i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{5} + (i + 1) q^{7} - i q^{9} + ( - i + 1) q^{13} + ( - i + 1) q^{23} - q^{25} + i q^{29} + (i - 1) q^{35} + q^{45} + i q^{49} + (i - 1) q^{53} + 2 i q^{59} + ( - i + 1) q^{63} + (i + 1) q^{65} + ( - i - 1) q^{67} - q^{81} + (i - 1) q^{83} + 2 q^{91} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{7} + 2 q^{13} + 2 q^{23} - 2 q^{25} - 2 q^{35} + 2 q^{45} - 2 q^{53} + 2 q^{63} + 2 q^{65} - 2 q^{67} - 2 q^{81} - 2 q^{83} + 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(1857\) \(2031\)
\(\chi(n)\) \(-1\) \(1\) \(i\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
753.1
1.00000i
1.00000i
0 0 0 1.00000i 0 1.00000 1.00000i 0 1.00000i 0
1217.1 0 0 0 1.00000i 0 1.00000 + 1.00000i 0 1.00000i 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
29.b even 2 1 RM by \(\Q(\sqrt{29}) \)
5.c odd 4 1 inner
145.h odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2320.1.bm.a 2
4.b odd 2 1 145.1.h.a 2
5.c odd 4 1 inner 2320.1.bm.a 2
12.b even 2 1 1305.1.o.a 2
20.d odd 2 1 725.1.h.a 2
20.e even 4 1 145.1.h.a 2
20.e even 4 1 725.1.h.a 2
29.b even 2 1 RM 2320.1.bm.a 2
60.l odd 4 1 1305.1.o.a 2
116.d odd 2 1 145.1.h.a 2
145.h odd 4 1 inner 2320.1.bm.a 2
348.b even 2 1 1305.1.o.a 2
580.e odd 2 1 725.1.h.a 2
580.o even 4 1 145.1.h.a 2
580.o even 4 1 725.1.h.a 2
1740.v odd 4 1 1305.1.o.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.1.h.a 2 4.b odd 2 1
145.1.h.a 2 20.e even 4 1
145.1.h.a 2 116.d odd 2 1
145.1.h.a 2 580.o even 4 1
725.1.h.a 2 20.d odd 2 1
725.1.h.a 2 20.e even 4 1
725.1.h.a 2 580.e odd 2 1
725.1.h.a 2 580.o even 4 1
1305.1.o.a 2 12.b even 2 1
1305.1.o.a 2 60.l odd 4 1
1305.1.o.a 2 348.b even 2 1
1305.1.o.a 2 1740.v odd 4 1
2320.1.bm.a 2 1.a even 1 1 trivial
2320.1.bm.a 2 5.c odd 4 1 inner
2320.1.bm.a 2 29.b even 2 1 RM
2320.1.bm.a 2 145.h odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2320, [\chi])\).

Hecke characteristic polynomials

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