Properties

Label 2640.1.ch.a
Level $2640$
Weight $1$
Character orbit 2640.ch
Analytic conductor $1.318$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2640.ch (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.31753163335\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.12375.1
Artin image: $C_2\times C_4\wr C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2640\mathbb{Z}\right)^\times\).

\(n\) \(661\) \(881\) \(991\) \(1057\) \(1201\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(i\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
1.00000i
1.00000i
0 1.00000i 0 1.00000i 0 0 0 −1.00000 0
2177.1 0 1.00000i 0 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 2640.1.ch.a 2
3.b odd 2 1 2640.1.ch.b 2
4.b odd 2 1 165.1.l.b yes 2
5.c odd 4 1 2640.1.ch.b 2
11.b odd 2 1 CM 2640.1.ch.a 2
12.b even 2 1 165.1.l.a 2
15.e even 4 1 inner 2640.1.ch.a 2
20.d odd 2 1 825.1.l.a 2
20.e even 4 1 165.1.l.a 2
20.e even 4 1 825.1.l.b 2
33.d even 2 1 2640.1.ch.b 2
44.c even 2 1 165.1.l.b yes 2
44.g even 10 4 1815.1.v.a 8
44.h odd 10 4 1815.1.v.a 8
55.e even 4 1 2640.1.ch.b 2
60.h even 2 1 825.1.l.b 2
60.l odd 4 1 165.1.l.b yes 2
60.l odd 4 1 825.1.l.a 2
132.d odd 2 1 165.1.l.a 2
132.n odd 10 4 1815.1.v.b 8
132.o even 10 4 1815.1.v.b 8
165.l odd 4 1 inner 2640.1.ch.a 2
220.g even 2 1 825.1.l.a 2
220.i odd 4 1 165.1.l.a 2
220.i odd 4 1 825.1.l.b 2
220.v even 20 4 1815.1.v.b 8
220.w odd 20 4 1815.1.v.b 8
660.g odd 2 1 825.1.l.b 2
660.q even 4 1 165.1.l.b yes 2
660.q even 4 1 825.1.l.a 2
660.bp odd 20 4 1815.1.v.a 8
660.bv even 20 4 1815.1.v.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.1.l.a 2 12.b even 2 1
165.1.l.a 2 20.e even 4 1
165.1.l.a 2 132.d odd 2 1
165.1.l.a 2 220.i odd 4 1
165.1.l.b yes 2 4.b odd 2 1
165.1.l.b yes 2 44.c even 2 1
165.1.l.b yes 2 60.l odd 4 1
165.1.l.b yes 2 660.q even 4 1
825.1.l.a 2 20.d odd 2 1
825.1.l.a 2 60.l odd 4 1
825.1.l.a 2 220.g even 2 1
825.1.l.a 2 660.q even 4 1
825.1.l.b 2 20.e even 4 1
825.1.l.b 2 60.h even 2 1
825.1.l.b 2 220.i odd 4 1
825.1.l.b 2 660.g odd 2 1
1815.1.v.a 8 44.g even 10 4
1815.1.v.a 8 44.h odd 10 4
1815.1.v.a 8 660.bp odd 20 4
1815.1.v.a 8 660.bv even 20 4
1815.1.v.b 8 132.n odd 10 4
1815.1.v.b 8 132.o even 10 4
1815.1.v.b 8 220.v even 20 4
1815.1.v.b 8 220.w odd 20 4
2640.1.ch.a 2 1.a even 1 1 trivial
2640.1.ch.a 2 11.b odd 2 1 CM
2640.1.ch.a 2 15.e even 4 1 inner
2640.1.ch.a 2 165.l odd 4 1 inner
2640.1.ch.b 2 3.b odd 2 1
2640.1.ch.b 2 5.c odd 4 1
2640.1.ch.b 2 33.d even 2 1
2640.1.ch.b 2 55.e 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 \(S_{1}^{\mathrm{new}}(2640, [\chi])\).

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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