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Properties

Label 165.1.l.b
Level $165$
Weight $1$
Character orbit 165.l
Analytic conductor $0.082$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [165,1,Mod(32,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.32");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 165.l (of order \(4\), degree \(2\), 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: \(0.0823457270844\)
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: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.12375.1
Artin image: $C_4\wr C_2$
Artin field: Galois closure of 8.0.16471125.1

$q$-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.

\(f(q)\) \(=\) \( q - i q^{3} + i q^{4} - i q^{5} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{3} + i q^{4} - i q^{5} - q^{9} + i q^{11} + q^{12} - q^{15} - q^{16} + q^{20} + (i - 1) q^{23} - q^{25} + i q^{27} + q^{33} - i q^{36} + ( - i + 1) q^{37} - q^{44} + i q^{45} + ( - i - 1) q^{47} + i q^{48} - i q^{49} + ( - i + 1) q^{53} + q^{55} + q^{59} - i q^{60} - i q^{64} + (i - 1) q^{67} + (i + 1) q^{69} + i q^{75} + i q^{80} + q^{81} + ( - i - 1) q^{92} + (i - 1) q^{97} - i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 2 q^{12} - 2 q^{15} - 2 q^{16} + 2 q^{20} - 2 q^{23} - 2 q^{25} + 2 q^{33} + 2 q^{37} - 2 q^{44} - 2 q^{47} + 2 q^{53} + 2 q^{55} + 4 q^{59} - 2 q^{67} + 2 q^{69} + 2 q^{81} - 2 q^{92} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(67\)
\(\chi(n)\) \(-1\) \(-1\) \(-i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
− 1.00000i
1.00000i
0 1.00000i − 1.00000i 1.00000i 0 0 0 −1.00000 0
98.1 0 − 1.00000i 1.00000i − 1.00000i 0 0 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
15.e even 4 1 inner
165.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.1.l.b yes 2
3.b odd 2 1 165.1.l.a ✓ 2
4.b odd 2 1 2640.1.ch.a 2
5.b even 2 1 825.1.l.a 2
5.c odd 4 1 165.1.l.a ✓ 2
5.c odd 4 1 825.1.l.b 2
11.b odd 2 1 CM 165.1.l.b yes 2
11.c even 5 4 1815.1.v.a 8
11.d odd 10 4 1815.1.v.a 8
12.b even 2 1 2640.1.ch.b 2
15.d odd 2 1 825.1.l.b 2
15.e even 4 1 inner 165.1.l.b yes 2
15.e even 4 1 825.1.l.a 2
20.e even 4 1 2640.1.ch.b 2
33.d even 2 1 165.1.l.a ✓ 2
33.f even 10 4 1815.1.v.b 8
33.h odd 10 4 1815.1.v.b 8
44.c even 2 1 2640.1.ch.a 2
55.d odd 2 1 825.1.l.a 2
55.e even 4 1 165.1.l.a ✓ 2
55.e even 4 1 825.1.l.b 2
55.k odd 20 4 1815.1.v.b 8
55.l even 20 4 1815.1.v.b 8
60.l odd 4 1 2640.1.ch.a 2
132.d odd 2 1 2640.1.ch.b 2
165.d even 2 1 825.1.l.b 2
165.l odd 4 1 inner 165.1.l.b yes 2
165.l odd 4 1 825.1.l.a 2
165.u odd 20 4 1815.1.v.a 8
165.v even 20 4 1815.1.v.a 8
220.i odd 4 1 2640.1.ch.b 2
660.q even 4 1 2640.1.ch.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.1.l.a ✓ 2 3.b odd 2 1
165.1.l.a ✓ 2 5.c odd 4 1
165.1.l.a ✓ 2 33.d even 2 1
165.1.l.a ✓ 2 55.e even 4 1
165.1.l.b yes 2 1.a even 1 1 trivial
165.1.l.b yes 2 11.b odd 2 1 CM
165.1.l.b yes 2 15.e even 4 1 inner
165.1.l.b yes 2 165.l odd 4 1 inner
825.1.l.a 2 5.b even 2 1
825.1.l.a 2 15.e even 4 1
825.1.l.a 2 55.d odd 2 1
825.1.l.a 2 165.l odd 4 1
825.1.l.b 2 5.c odd 4 1
825.1.l.b 2 15.d odd 2 1
825.1.l.b 2 55.e even 4 1
825.1.l.b 2 165.d even 2 1
1815.1.v.a 8 11.c even 5 4
1815.1.v.a 8 11.d odd 10 4
1815.1.v.a 8 165.u odd 20 4
1815.1.v.a 8 165.v even 20 4
1815.1.v.b 8 33.f even 10 4
1815.1.v.b 8 33.h odd 10 4
1815.1.v.b 8 55.k odd 20 4
1815.1.v.b 8 55.l even 20 4
2640.1.ch.a 2 4.b odd 2 1
2640.1.ch.a 2 44.c even 2 1
2640.1.ch.a 2 60.l odd 4 1
2640.1.ch.a 2 660.q even 4 1
2640.1.ch.b 2 12.b even 2 1
2640.1.ch.b 2 20.e even 4 1
2640.1.ch.b 2 132.d odd 2 1
2640.1.ch.b 2 220.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{2} + 2T_{23} + 2 \) acting on \(S_{1}^{\mathrm{new}}(165, [\chi])\). 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
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$53$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$59$ \( (T - 2)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
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