Properties

Label 1872.1.i.b
Level $1872$
Weight $1$
Character orbit 1872.i
Self dual yes
Analytic conductor $0.934$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -4, -52, 13
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1872,1,Mod(415,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1872.415");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1872.i (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: yes
Analytic conductor: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 208)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{13})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.7488.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q - q^{13} + 2 q^{17} + q^{25} + 2 q^{29} - q^{49} - 2 q^{53} + 2 q^{61}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(1\) \(0\) \(0\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1
0
0 0 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
13.b even 2 1 RM by \(\Q(\sqrt{13}) \)
52.b odd 2 1 CM by \(\Q(\sqrt{-13}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.1.i.b 1
3.b odd 2 1 208.1.c.a 1
4.b odd 2 1 CM 1872.1.i.b 1
12.b even 2 1 208.1.c.a 1
13.b even 2 1 RM 1872.1.i.b 1
24.f even 2 1 832.1.c.a 1
24.h odd 2 1 832.1.c.a 1
39.d odd 2 1 208.1.c.a 1
39.f even 4 2 2704.1.d.a 1
39.h odd 6 2 2704.1.y.b 2
39.i odd 6 2 2704.1.y.b 2
39.k even 12 4 2704.1.bb.a 2
48.i odd 4 2 3328.1.h.a 2
48.k even 4 2 3328.1.h.a 2
52.b odd 2 1 CM 1872.1.i.b 1
156.h even 2 1 208.1.c.a 1
156.l odd 4 2 2704.1.d.a 1
156.p even 6 2 2704.1.y.b 2
156.r even 6 2 2704.1.y.b 2
156.v odd 12 4 2704.1.bb.a 2
312.b odd 2 1 832.1.c.a 1
312.h even 2 1 832.1.c.a 1
624.v even 4 2 3328.1.h.a 2
624.bi odd 4 2 3328.1.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
208.1.c.a 1 3.b odd 2 1
208.1.c.a 1 12.b even 2 1
208.1.c.a 1 39.d odd 2 1
208.1.c.a 1 156.h even 2 1
832.1.c.a 1 24.f even 2 1
832.1.c.a 1 24.h odd 2 1
832.1.c.a 1 312.b odd 2 1
832.1.c.a 1 312.h even 2 1
1872.1.i.b 1 1.a even 1 1 trivial
1872.1.i.b 1 4.b odd 2 1 CM
1872.1.i.b 1 13.b even 2 1 RM
1872.1.i.b 1 52.b odd 2 1 CM
2704.1.d.a 1 39.f even 4 2
2704.1.d.a 1 156.l odd 4 2
2704.1.y.b 2 39.h odd 6 2
2704.1.y.b 2 39.i odd 6 2
2704.1.y.b 2 156.p even 6 2
2704.1.y.b 2 156.r even 6 2
2704.1.bb.a 2 39.k even 12 4
2704.1.bb.a 2 156.v odd 12 4
3328.1.h.a 2 48.i odd 4 2
3328.1.h.a 2 48.k even 4 2
3328.1.h.a 2 624.v even 4 2
3328.1.h.a 2 624.bi odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{1}^{\mathrm{new}}(1872, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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