Properties

Label 2880.1.bk.a
Level $2880$
Weight $1$
Character orbit 2880.bk
Analytic conductor $1.437$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2880.bk (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 720)
Projective image \(D_{4}\)
Projective field Galois closure of 4.0.13500.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{8}^{3} q^{5} +O(q^{10})\) \( q + \zeta_{8}^{3} q^{5} + ( -1 - \zeta_{8}^{2} ) q^{13} -2 \zeta_{8} q^{17} -\zeta_{8}^{2} q^{25} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{29} + ( 1 - \zeta_{8}^{2} ) q^{37} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{41} -\zeta_{8}^{2} q^{49} -2 \zeta_{8}^{3} q^{53} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{65} + ( -1 - \zeta_{8}^{2} ) q^{73} + 2 q^{85} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{89} + ( -1 + \zeta_{8}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{13} + 4q^{37} - 4q^{73} + 8q^{85} - 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(-1\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1727.1
0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
0 0 0 −0.707107 0.707107i 0 0 0 0 0
1727.2 0 0 0 0.707107 + 0.707107i 0 0 0 0 0
2303.1 0 0 0 −0.707107 + 0.707107i 0 0 0 0 0
2303.2 0 0 0 0.707107 0.707107i 0 0 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
5.c odd 4 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
60.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.1.bk.a 4
3.b odd 2 1 inner 2880.1.bk.a 4
4.b odd 2 1 CM 2880.1.bk.a 4
5.c odd 4 1 inner 2880.1.bk.a 4
8.b even 2 1 720.1.bk.a 4
8.d odd 2 1 720.1.bk.a 4
12.b even 2 1 inner 2880.1.bk.a 4
15.e even 4 1 inner 2880.1.bk.a 4
20.e even 4 1 inner 2880.1.bk.a 4
24.f even 2 1 720.1.bk.a 4
24.h odd 2 1 720.1.bk.a 4
40.e odd 2 1 3600.1.bk.a 4
40.f even 2 1 3600.1.bk.a 4
40.i odd 4 1 720.1.bk.a 4
40.i odd 4 1 3600.1.bk.a 4
40.k even 4 1 720.1.bk.a 4
40.k even 4 1 3600.1.bk.a 4
60.l odd 4 1 inner 2880.1.bk.a 4
120.i odd 2 1 3600.1.bk.a 4
120.m even 2 1 3600.1.bk.a 4
120.q odd 4 1 720.1.bk.a 4
120.q odd 4 1 3600.1.bk.a 4
120.w even 4 1 720.1.bk.a 4
120.w even 4 1 3600.1.bk.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.bk.a 4 8.b even 2 1
720.1.bk.a 4 8.d odd 2 1
720.1.bk.a 4 24.f even 2 1
720.1.bk.a 4 24.h odd 2 1
720.1.bk.a 4 40.i odd 4 1
720.1.bk.a 4 40.k even 4 1
720.1.bk.a 4 120.q odd 4 1
720.1.bk.a 4 120.w even 4 1
2880.1.bk.a 4 1.a even 1 1 trivial
2880.1.bk.a 4 3.b odd 2 1 inner
2880.1.bk.a 4 4.b odd 2 1 CM
2880.1.bk.a 4 5.c odd 4 1 inner
2880.1.bk.a 4 12.b even 2 1 inner
2880.1.bk.a 4 15.e even 4 1 inner
2880.1.bk.a 4 20.e even 4 1 inner
2880.1.bk.a 4 60.l odd 4 1 inner
3600.1.bk.a 4 40.e odd 2 1
3600.1.bk.a 4 40.f even 2 1
3600.1.bk.a 4 40.i odd 4 1
3600.1.bk.a 4 40.k even 4 1
3600.1.bk.a 4 120.i odd 2 1
3600.1.bk.a 4 120.m even 2 1
3600.1.bk.a 4 120.q odd 4 1
3600.1.bk.a 4 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} + 2 T_{13} + 2 \) acting on \(S_{1}^{\mathrm{new}}(2880, [\chi])\).