Properties

Label 47.1.b.a
Level $47$
Weight $1$
Character orbit 47.b
Self dual yes
Analytic conductor $0.023$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -47
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [47,1,Mod(46,47)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(47, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("47.46");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 47.b (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: \(0.0234560555938\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.2209.1
Artin image: $D_5$
Artin field: Galois closure of 5.1.2209.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} - \beta q^{3} + ( - \beta + 1) q^{4} - q^{6} + (\beta - 1) q^{7} - q^{8} + \beta q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} - \beta q^{3} + ( - \beta + 1) q^{4} - q^{6} + (\beta - 1) q^{7} - q^{8} + \beta q^{9} + q^{12} + ( - \beta + 2) q^{14} - \beta q^{17} + q^{18} - q^{21} + \beta q^{24} + q^{25} - q^{27} + (\beta - 2) q^{28} + q^{32} - q^{34} - q^{36} - \beta q^{37} + ( - \beta + 1) q^{42} + q^{47} + ( - \beta + 1) q^{49} + (\beta - 1) q^{50} + (\beta + 1) q^{51} + (\beta - 1) q^{53} + ( - \beta + 1) q^{54} + ( - \beta + 1) q^{56} + (\beta - 1) q^{59} + (\beta - 1) q^{61} + q^{63} + (\beta - 1) q^{64} + q^{68} - \beta q^{71} - \beta q^{72} - q^{74} - \beta q^{75} - \beta q^{79} + 2 q^{83} + (\beta - 1) q^{84} + (\beta - 1) q^{89} + (\beta - 1) q^{94} - \beta q^{96} + (\beta - 1) q^{97} + (\beta - 2) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{3} + q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{3} + q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + q^{9} + 2 q^{12} + 3 q^{14} - q^{17} + 2 q^{18} - 2 q^{21} + q^{24} + 2 q^{25} - 2 q^{27} - 3 q^{28} + 2 q^{32} - 2 q^{34} - 2 q^{36} - q^{37} + q^{42} + 2 q^{47} + q^{49} - q^{50} + 3 q^{51} - q^{53} + q^{54} + q^{56} - q^{59} - q^{61} + 2 q^{63} - q^{64} + 2 q^{68} - q^{71} - q^{72} - 2 q^{74} - q^{75} - q^{79} + 4 q^{83} - q^{84} - q^{89} - q^{94} - q^{96} - q^{97} - 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/47\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
−0.618034
1.61803
−1.61803 0.618034 1.61803 0 −1.00000 −1.61803 −1.00000 −0.618034 0
46.2 0.618034 −1.61803 −0.618034 0 −1.00000 0.618034 −1.00000 1.61803 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
47.b odd 2 1 CM by \(\Q(\sqrt{-47}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 47.1.b.a 2
3.b odd 2 1 423.1.d.a 2
4.b odd 2 1 752.1.g.a 2
5.b even 2 1 1175.1.d.c 2
5.c odd 4 2 1175.1.b.b 4
7.b odd 2 1 2303.1.d.c 2
7.c even 3 2 2303.1.f.c 4
7.d odd 6 2 2303.1.f.b 4
8.b even 2 1 3008.1.g.b 2
8.d odd 2 1 3008.1.g.a 2
9.c even 3 2 3807.1.f.b 4
9.d odd 6 2 3807.1.f.a 4
47.b odd 2 1 CM 47.1.b.a 2
47.c even 23 22 2209.1.d.a 44
47.d odd 46 22 2209.1.d.a 44
141.c even 2 1 423.1.d.a 2
188.b even 2 1 752.1.g.a 2
235.b odd 2 1 1175.1.d.c 2
235.e even 4 2 1175.1.b.b 4
329.c even 2 1 2303.1.d.c 2
329.f odd 6 2 2303.1.f.c 4
329.g even 6 2 2303.1.f.b 4
376.e odd 2 1 3008.1.g.b 2
376.h even 2 1 3008.1.g.a 2
423.f odd 6 2 3807.1.f.b 4
423.g even 6 2 3807.1.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.1.b.a 2 1.a even 1 1 trivial
47.1.b.a 2 47.b odd 2 1 CM
423.1.d.a 2 3.b odd 2 1
423.1.d.a 2 141.c even 2 1
752.1.g.a 2 4.b odd 2 1
752.1.g.a 2 188.b even 2 1
1175.1.b.b 4 5.c odd 4 2
1175.1.b.b 4 235.e even 4 2
1175.1.d.c 2 5.b even 2 1
1175.1.d.c 2 235.b odd 2 1
2209.1.d.a 44 47.c even 23 22
2209.1.d.a 44 47.d odd 46 22
2303.1.d.c 2 7.b odd 2 1
2303.1.d.c 2 329.c even 2 1
2303.1.f.b 4 7.d odd 6 2
2303.1.f.b 4 329.g even 6 2
2303.1.f.c 4 7.c even 3 2
2303.1.f.c 4 329.f odd 6 2
3008.1.g.a 2 8.d odd 2 1
3008.1.g.a 2 376.h even 2 1
3008.1.g.b 2 8.b even 2 1
3008.1.g.b 2 376.e odd 2 1
3807.1.f.a 4 9.d odd 6 2
3807.1.f.a 4 423.g even 6 2
3807.1.f.b 4 9.c even 3 2
3807.1.f.b 4 423.f odd 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(47, [\chi])\).

Hecke characteristic polynomials

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