Properties

Label 2880.1.r.a
Level 2880
Weight 1
Character orbit 2880.r
Analytic conductor 1.437
Analytic rank 0
Dimension 4
Projective image \(D_{4}\)
CM discriminant -15
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.r (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: \(\Q(\zeta_{8})\)
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 4.0.92160.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{8}^{3} q^{5} +O(q^{10})\) \( q -\zeta_{8}^{3} q^{5} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{17} + ( -1 - \zeta_{8}^{2} ) q^{19} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{23} -\zeta_{8}^{2} q^{25} + 2 \zeta_{8}^{2} q^{31} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{47} + q^{49} + ( -1 + \zeta_{8}^{2} ) q^{61} -2 \zeta_{8}^{3} q^{83} + ( -1 - \zeta_{8}^{2} ) q^{85} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{19} + 4q^{49} - 4q^{61} - 4q^{85} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
−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
559.2 0 0 0 0.707107 + 0.707107i 0 0 0 0 0
1999.1 0 0 0 −0.707107 + 0.707107i 0 0 0 0 0
1999.2 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner
80.k odd 4 1 inner
240.t even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.1.r.a 4
3.b odd 2 1 inner 2880.1.r.a 4
4.b odd 2 1 720.1.r.a 4
5.b even 2 1 inner 2880.1.r.a 4
12.b even 2 1 720.1.r.a 4
15.d odd 2 1 CM 2880.1.r.a 4
16.e even 4 1 720.1.r.a 4
16.f odd 4 1 inner 2880.1.r.a 4
20.d odd 2 1 720.1.r.a 4
20.e even 4 2 3600.1.bo.a 4
48.i odd 4 1 720.1.r.a 4
48.k even 4 1 inner 2880.1.r.a 4
60.h even 2 1 720.1.r.a 4
60.l odd 4 2 3600.1.bo.a 4
80.i odd 4 1 3600.1.bo.a 4
80.k odd 4 1 inner 2880.1.r.a 4
80.q even 4 1 720.1.r.a 4
80.t odd 4 1 3600.1.bo.a 4
240.t even 4 1 inner 2880.1.r.a 4
240.bb even 4 1 3600.1.bo.a 4
240.bf even 4 1 3600.1.bo.a 4
240.bm odd 4 1 720.1.r.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.r.a 4 4.b odd 2 1
720.1.r.a 4 12.b even 2 1
720.1.r.a 4 16.e even 4 1
720.1.r.a 4 20.d odd 2 1
720.1.r.a 4 48.i odd 4 1
720.1.r.a 4 60.h even 2 1
720.1.r.a 4 80.q even 4 1
720.1.r.a 4 240.bm odd 4 1
2880.1.r.a 4 1.a even 1 1 trivial
2880.1.r.a 4 3.b odd 2 1 inner
2880.1.r.a 4 5.b even 2 1 inner
2880.1.r.a 4 15.d odd 2 1 CM
2880.1.r.a 4 16.f odd 4 1 inner
2880.1.r.a 4 48.k even 4 1 inner
2880.1.r.a 4 80.k odd 4 1 inner
2880.1.r.a 4 240.t even 4 1 inner
3600.1.bo.a 4 20.e even 4 2
3600.1.bo.a 4 60.l odd 4 2
3600.1.bo.a 4 80.i odd 4 1
3600.1.bo.a 4 80.t odd 4 1
3600.1.bo.a 4 240.bb even 4 1
3600.1.bo.a 4 240.bf even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2880, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( 1 + T^{4} \)
$7$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$11$ \( ( 1 + T^{4} )^{2} \)
$13$ \( ( 1 + T^{4} )^{2} \)
$17$ \( ( 1 + T^{4} )^{2} \)
$19$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
$23$ \( ( 1 + T^{4} )^{2} \)
$29$ \( ( 1 + T^{4} )^{2} \)
$31$ \( ( 1 + T^{2} )^{4} \)
$37$ \( ( 1 + T^{4} )^{2} \)
$41$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$43$ \( ( 1 + T^{4} )^{2} \)
$47$ \( ( 1 + T^{4} )^{2} \)
$53$ \( ( 1 + T^{4} )^{2} \)
$59$ \( ( 1 + T^{4} )^{2} \)
$61$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
$67$ \( ( 1 + T^{4} )^{2} \)
$71$ \( ( 1 + T^{2} )^{4} \)
$73$ \( ( 1 + T^{2} )^{4} \)
$79$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$83$ \( ( 1 + T^{4} )^{2} \)
$89$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$97$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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