Properties

Label 2300.1.f.a
Level $2300$
Weight $1$
Character orbit 2300.f
Analytic conductor $1.148$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -115
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2300,1,Mod(1701,2300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2300.1701");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2300.f (of order \(2\), degree \(1\), 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: \(1.14784952906\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 460)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.460.1
Artin image: $C_4\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - i q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{7} - q^{9} - i q^{17} - i q^{23} + q^{29} - q^{31} - i q^{37} - q^{41} - 2 i q^{43} + i q^{53} + q^{59} + i q^{63} - i q^{67} - q^{71} + q^{81} + i q^{83} + 2 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 2 q^{29} - 2 q^{31} - 2 q^{41} + 2 q^{59} - 2 q^{71} + 2 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1701.1
1.00000i
1.00000i
0 0 0 0 0 1.00000i 0 −1.00000 0
1701.2 0 0 0 0 0 1.00000i 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
115.c odd 2 1 CM by \(\Q(\sqrt{-115}) \)
5.b even 2 1 inner
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2300.1.f.a 2
5.b even 2 1 inner 2300.1.f.a 2
5.c odd 4 1 460.1.d.a 1
5.c odd 4 1 460.1.d.b yes 1
20.e even 4 1 1840.1.g.a 1
20.e even 4 1 1840.1.g.b 1
23.b odd 2 1 inner 2300.1.f.a 2
115.c odd 2 1 CM 2300.1.f.a 2
115.e even 4 1 460.1.d.a 1
115.e even 4 1 460.1.d.b yes 1
460.k odd 4 1 1840.1.g.a 1
460.k odd 4 1 1840.1.g.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.1.d.a 1 5.c odd 4 1
460.1.d.a 1 115.e even 4 1
460.1.d.b yes 1 5.c odd 4 1
460.1.d.b yes 1 115.e even 4 1
1840.1.g.a 1 20.e even 4 1
1840.1.g.a 1 460.k odd 4 1
1840.1.g.b 1 20.e even 4 1
1840.1.g.b 1 460.k odd 4 1
2300.1.f.a 2 1.a even 1 1 trivial
2300.1.f.a 2 5.b even 2 1 inner
2300.1.f.a 2 23.b odd 2 1 inner
2300.1.f.a 2 115.c odd 2 1 CM

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2300, [\chi])\).

Hecke characteristic polynomials

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