Properties

Label 165.1.l.a
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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Newspace parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 165.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0823457270844\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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 8.0.16471125.2

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

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.

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 −1.00000 1.00000i 1.00000i 0 0 0 1.00000 0
98.1 0 −1.00000 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.a 2
3.b odd 2 1 165.1.l.b yes 2
4.b odd 2 1 2640.1.ch.b 2
5.b even 2 1 825.1.l.b 2
5.c odd 4 1 165.1.l.b yes 2
5.c odd 4 1 825.1.l.a 2
11.b odd 2 1 CM 165.1.l.a 2
11.c even 5 4 1815.1.v.b 8
11.d odd 10 4 1815.1.v.b 8
12.b even 2 1 2640.1.ch.a 2
15.d odd 2 1 825.1.l.a 2
15.e even 4 1 inner 165.1.l.a 2
15.e even 4 1 825.1.l.b 2
20.e even 4 1 2640.1.ch.a 2
33.d even 2 1 165.1.l.b yes 2
33.f even 10 4 1815.1.v.a 8
33.h odd 10 4 1815.1.v.a 8
44.c even 2 1 2640.1.ch.b 2
55.d odd 2 1 825.1.l.b 2
55.e even 4 1 165.1.l.b yes 2
55.e even 4 1 825.1.l.a 2
55.k odd 20 4 1815.1.v.a 8
55.l even 20 4 1815.1.v.a 8
60.l odd 4 1 2640.1.ch.b 2
132.d odd 2 1 2640.1.ch.a 2
165.d even 2 1 825.1.l.a 2
165.l odd 4 1 inner 165.1.l.a 2
165.l odd 4 1 825.1.l.b 2
165.u odd 20 4 1815.1.v.b 8
165.v even 20 4 1815.1.v.b 8
220.i odd 4 1 2640.1.ch.a 2
660.q even 4 1 2640.1.ch.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.1.l.a 2 1.a even 1 1 trivial
165.1.l.a 2 11.b odd 2 1 CM
165.1.l.a 2 15.e even 4 1 inner
165.1.l.a 2 165.l odd 4 1 inner
165.1.l.b yes 2 3.b odd 2 1
165.1.l.b yes 2 5.c odd 4 1
165.1.l.b yes 2 33.d even 2 1
165.1.l.b yes 2 55.e even 4 1
825.1.l.a 2 5.c odd 4 1
825.1.l.a 2 15.d odd 2 1
825.1.l.a 2 55.e even 4 1
825.1.l.a 2 165.d even 2 1
825.1.l.b 2 5.b even 2 1
825.1.l.b 2 15.e even 4 1
825.1.l.b 2 55.d odd 2 1
825.1.l.b 2 165.l odd 4 1
1815.1.v.a 8 33.f even 10 4
1815.1.v.a 8 33.h odd 10 4
1815.1.v.a 8 55.k odd 20 4
1815.1.v.a 8 55.l even 20 4
1815.1.v.b 8 11.c even 5 4
1815.1.v.b 8 11.d odd 10 4
1815.1.v.b 8 165.u odd 20 4
1815.1.v.b 8 165.v even 20 4
2640.1.ch.a 2 12.b even 2 1
2640.1.ch.a 2 20.e even 4 1
2640.1.ch.a 2 132.d odd 2 1
2640.1.ch.a 2 220.i odd 4 1
2640.1.ch.b 2 4.b odd 2 1
2640.1.ch.b 2 44.c even 2 1
2640.1.ch.b 2 60.l odd 4 1
2640.1.ch.b 2 660.q even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{2} - 2 T_{23} + 2 \) acting on \(S_{1}^{\mathrm{new}}(165, [\chi])\).

Hecke characteristic polynomials

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