Properties

Label 207.1.d.a
Level $207$
Weight $1$
Character orbit 207.d
Self dual yes
Analytic conductor $0.103$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -23
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [207,1,Mod(91,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("207.91");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 207.d (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: \(0.103306457615\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 23)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.23.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.14283.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{8} - q^{13} - q^{16} - q^{23} + q^{25} - q^{26} + q^{29} - q^{31} + q^{41} - q^{46} + q^{47} + q^{49} + q^{50} + q^{58} - 2 q^{59} - q^{62} + q^{64} + q^{71} - q^{73} + q^{82} + q^{94} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
0
1.00000 0 0 0 0 0 −1.00000 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
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.1.d.a 1
3.b odd 2 1 23.1.b.a 1
4.b odd 2 1 3312.1.c.a 1
9.c even 3 2 1863.1.f.a 2
9.d odd 6 2 1863.1.f.b 2
12.b even 2 1 368.1.f.a 1
15.d odd 2 1 575.1.d.a 1
15.e even 4 2 575.1.c.a 2
21.c even 2 1 1127.1.d.b 1
21.g even 6 2 1127.1.f.a 2
21.h odd 6 2 1127.1.f.b 2
23.b odd 2 1 CM 207.1.d.a 1
24.f even 2 1 1472.1.f.a 1
24.h odd 2 1 1472.1.f.b 1
33.d even 2 1 2783.1.d.b 1
33.f even 10 4 2783.1.f.a 4
33.h odd 10 4 2783.1.f.c 4
39.d odd 2 1 3887.1.d.b 1
39.f even 4 2 3887.1.c.a 2
39.h odd 6 2 3887.1.h.a 2
39.i odd 6 2 3887.1.h.c 2
39.k even 12 4 3887.1.j.e 4
69.c even 2 1 23.1.b.a 1
69.g even 22 10 529.1.d.a 10
69.h odd 22 10 529.1.d.a 10
92.b even 2 1 3312.1.c.a 1
207.f odd 6 2 1863.1.f.a 2
207.g even 6 2 1863.1.f.b 2
276.h odd 2 1 368.1.f.a 1
345.h even 2 1 575.1.d.a 1
345.l odd 4 2 575.1.c.a 2
483.c odd 2 1 1127.1.d.b 1
483.m even 6 2 1127.1.f.b 2
483.o odd 6 2 1127.1.f.a 2
552.b even 2 1 1472.1.f.b 1
552.h odd 2 1 1472.1.f.a 1
759.h odd 2 1 2783.1.d.b 1
759.j odd 10 4 2783.1.f.a 4
759.m even 10 4 2783.1.f.c 4
897.g even 2 1 3887.1.d.b 1
897.m odd 4 2 3887.1.c.a 2
897.n even 6 2 3887.1.h.a 2
897.t even 6 2 3887.1.h.c 2
897.w odd 12 4 3887.1.j.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.1.b.a 1 3.b odd 2 1
23.1.b.a 1 69.c even 2 1
207.1.d.a 1 1.a even 1 1 trivial
207.1.d.a 1 23.b odd 2 1 CM
368.1.f.a 1 12.b even 2 1
368.1.f.a 1 276.h odd 2 1
529.1.d.a 10 69.g even 22 10
529.1.d.a 10 69.h odd 22 10
575.1.c.a 2 15.e even 4 2
575.1.c.a 2 345.l odd 4 2
575.1.d.a 1 15.d odd 2 1
575.1.d.a 1 345.h even 2 1
1127.1.d.b 1 21.c even 2 1
1127.1.d.b 1 483.c odd 2 1
1127.1.f.a 2 21.g even 6 2
1127.1.f.a 2 483.o odd 6 2
1127.1.f.b 2 21.h odd 6 2
1127.1.f.b 2 483.m even 6 2
1472.1.f.a 1 24.f even 2 1
1472.1.f.a 1 552.h odd 2 1
1472.1.f.b 1 24.h odd 2 1
1472.1.f.b 1 552.b even 2 1
1863.1.f.a 2 9.c even 3 2
1863.1.f.a 2 207.f odd 6 2
1863.1.f.b 2 9.d odd 6 2
1863.1.f.b 2 207.g even 6 2
2783.1.d.b 1 33.d even 2 1
2783.1.d.b 1 759.h odd 2 1
2783.1.f.a 4 33.f even 10 4
2783.1.f.a 4 759.j odd 10 4
2783.1.f.c 4 33.h odd 10 4
2783.1.f.c 4 759.m even 10 4
3312.1.c.a 1 4.b odd 2 1
3312.1.c.a 1 92.b even 2 1
3887.1.c.a 2 39.f even 4 2
3887.1.c.a 2 897.m odd 4 2
3887.1.d.b 1 39.d odd 2 1
3887.1.d.b 1 897.g even 2 1
3887.1.h.a 2 39.h odd 6 2
3887.1.h.a 2 897.n even 6 2
3887.1.h.c 2 39.i odd 6 2
3887.1.h.c 2 897.t even 6 2
3887.1.j.e 4 39.k even 12 4
3887.1.j.e 4 897.w odd 12 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(207, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) 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
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T - 1 \) Copy content Toggle raw display
$31$ \( T + 1 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 1 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 1 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T + 2 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T - 1 \) Copy content Toggle raw display
$73$ \( T + 1 \) 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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