Properties

Label 2675.1.d.a
Level $2675$
Weight $1$
Character orbit 2675.d
Analytic conductor $1.335$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -107
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2675,1,Mod(2674,2675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2675, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2675.2674");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2675 = 5^{2} \cdot 107 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2675.d (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: \(1.33499890879\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 107)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.107.1
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.

\(f(q)\) \(=\) \( q - i q^{3} - q^{4} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{3} - q^{4} - q^{11} + i q^{12} - i q^{13} + q^{16} + q^{19} - i q^{23} - i q^{27} - q^{29} + i q^{33} + i q^{37} - q^{39} - q^{41} + q^{44} - i q^{47} - i q^{48} - q^{49} + i q^{52} - i q^{53} - i q^{57} - q^{61} - q^{64} - q^{69} - q^{76} + q^{79} - q^{81} + i q^{83} + 2 i q^{87} + q^{89} + i q^{92} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{11} + 2 q^{16} + 2 q^{19} - 4 q^{29} - 2 q^{39} - 2 q^{41} + 2 q^{44} - 2 q^{49} - 2 q^{61} - 2 q^{64} - 2 q^{69} - 2 q^{76} + 2 q^{79} - 2 q^{81} + 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2675\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(1927\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2674.1
1.00000i
1.00000i
0 1.00000i −1.00000 0 0 0 0 0 0
2674.2 0 1.00000i −1.00000 0 0 0 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
107.b odd 2 1 CM by \(\Q(\sqrt{-107}) \)
5.b even 2 1 inner
535.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2675.1.d.a 2
5.b even 2 1 inner 2675.1.d.a 2
5.c odd 4 1 107.1.b.a 1
5.c odd 4 1 2675.1.c.a 1
15.e even 4 1 963.1.b.a 1
20.e even 4 1 1712.1.g.a 1
107.b odd 2 1 CM 2675.1.d.a 2
535.d odd 2 1 inner 2675.1.d.a 2
535.e even 4 1 107.1.b.a 1
535.e even 4 1 2675.1.c.a 1
1605.j odd 4 1 963.1.b.a 1
2140.i odd 4 1 1712.1.g.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
107.1.b.a 1 5.c odd 4 1
107.1.b.a 1 535.e even 4 1
963.1.b.a 1 15.e even 4 1
963.1.b.a 1 1605.j odd 4 1
1712.1.g.a 1 20.e even 4 1
1712.1.g.a 1 2140.i odd 4 1
2675.1.c.a 1 5.c odd 4 1
2675.1.c.a 1 535.e even 4 1
2675.1.d.a 2 1.a even 1 1 trivial
2675.1.d.a 2 5.b even 2 1 inner
2675.1.d.a 2 107.b odd 2 1 CM
2675.1.d.a 2 535.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2675, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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