Properties

Label 2880.2.k.e
Level $2880$
Weight $2$
Character orbit 2880.k
Analytic conductor $22.997$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{12}^{3} q^{5} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} +O(q^{10})\) \( q -\zeta_{12}^{3} q^{5} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + ( -2 + 4 \zeta_{12}^{2} ) q^{11} + ( 2 - 4 \zeta_{12}^{2} ) q^{13} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + 2 \zeta_{12}^{3} q^{19} + ( 3 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{23} - q^{25} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{35} -6 \zeta_{12}^{3} q^{37} + ( -6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{41} + ( -3 + 6 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{43} + ( -3 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{47} + ( 5 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{49} + ( -6 + 12 \zeta_{12}^{2} ) q^{53} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{55} + 6 \zeta_{12}^{3} q^{59} + ( -4 + 8 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{61} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{65} + ( -3 + 6 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{67} + ( 6 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{71} + ( -4 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{73} + ( 6 - 12 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{77} + 12 q^{79} + ( 1 - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{83} + ( 2 - 4 \zeta_{12}^{2} ) q^{85} + ( 6 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{89} + ( -6 + 12 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{91} + 2 q^{95} + ( 4 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{7} + O(q^{10}) \) \( 4q - 12q^{7} + 12q^{23} - 4q^{25} + 24q^{31} - 24q^{41} - 12q^{47} + 20q^{49} + 24q^{71} - 16q^{73} + 48q^{79} + 24q^{89} + 8q^{95} + 16q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1441.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0 0 0 1.00000i 0 −4.73205 0 0 0
1441.2 0 0 0 1.00000i 0 −1.26795 0 0 0
1441.3 0 0 0 1.00000i 0 −4.73205 0 0 0
1441.4 0 0 0 1.00000i 0 −1.26795 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.k.e 4
3.b odd 2 1 320.2.d.a 4
4.b odd 2 1 2880.2.k.l 4
8.b even 2 1 inner 2880.2.k.e 4
8.d odd 2 1 2880.2.k.l 4
12.b even 2 1 320.2.d.b yes 4
15.d odd 2 1 1600.2.d.h 4
15.e even 4 1 1600.2.f.e 4
15.e even 4 1 1600.2.f.i 4
24.f even 2 1 320.2.d.b yes 4
24.h odd 2 1 320.2.d.a 4
48.i odd 4 1 1280.2.a.b 2
48.i odd 4 1 1280.2.a.p 2
48.k even 4 1 1280.2.a.c 2
48.k even 4 1 1280.2.a.m 2
60.h even 2 1 1600.2.d.b 4
60.l odd 4 1 1600.2.f.d 4
60.l odd 4 1 1600.2.f.h 4
120.i odd 2 1 1600.2.d.h 4
120.m even 2 1 1600.2.d.b 4
120.q odd 4 1 1600.2.f.d 4
120.q odd 4 1 1600.2.f.h 4
120.w even 4 1 1600.2.f.e 4
120.w even 4 1 1600.2.f.i 4
240.t even 4 1 6400.2.a.bf 2
240.t even 4 1 6400.2.a.ck 2
240.bm odd 4 1 6400.2.a.y 2
240.bm odd 4 1 6400.2.a.cd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.d.a 4 3.b odd 2 1
320.2.d.a 4 24.h odd 2 1
320.2.d.b yes 4 12.b even 2 1
320.2.d.b yes 4 24.f even 2 1
1280.2.a.b 2 48.i odd 4 1
1280.2.a.c 2 48.k even 4 1
1280.2.a.m 2 48.k even 4 1
1280.2.a.p 2 48.i odd 4 1
1600.2.d.b 4 60.h even 2 1
1600.2.d.b 4 120.m even 2 1
1600.2.d.h 4 15.d odd 2 1
1600.2.d.h 4 120.i odd 2 1
1600.2.f.d 4 60.l odd 4 1
1600.2.f.d 4 120.q odd 4 1
1600.2.f.e 4 15.e even 4 1
1600.2.f.e 4 120.w even 4 1
1600.2.f.h 4 60.l odd 4 1
1600.2.f.h 4 120.q odd 4 1
1600.2.f.i 4 15.e even 4 1
1600.2.f.i 4 120.w even 4 1
2880.2.k.e 4 1.a even 1 1 trivial
2880.2.k.e 4 8.b even 2 1 inner
2880.2.k.l 4 4.b odd 2 1
2880.2.k.l 4 8.d odd 2 1
6400.2.a.y 2 240.bm odd 4 1
6400.2.a.bf 2 240.t even 4 1
6400.2.a.cd 2 240.bm odd 4 1
6400.2.a.ck 2 240.t even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2880, [\chi])\):

\( T_{7}^{2} + 6 T_{7} + 6 \)
\( T_{11}^{2} + 12 \)
\( T_{23}^{2} - 6 T_{23} - 18 \)
\( T_{47}^{2} + 6 T_{47} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( 6 + 6 T + T^{2} )^{2} \)
$11$ \( ( 12 + T^{2} )^{2} \)
$13$ \( ( 12 + T^{2} )^{2} \)
$17$ \( ( -12 + T^{2} )^{2} \)
$19$ \( ( 4 + T^{2} )^{2} \)
$23$ \( ( -18 - 6 T + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( 24 - 12 T + T^{2} )^{2} \)
$37$ \( ( 36 + T^{2} )^{2} \)
$41$ \( ( 24 + 12 T + T^{2} )^{2} \)
$43$ \( 4 + 104 T^{2} + T^{4} \)
$47$ \( ( -18 + 6 T + T^{2} )^{2} \)
$53$ \( ( 108 + T^{2} )^{2} \)
$59$ \( ( 36 + T^{2} )^{2} \)
$61$ \( 144 + 168 T^{2} + T^{4} \)
$67$ \( 4 + 104 T^{2} + T^{4} \)
$71$ \( ( -72 - 12 T + T^{2} )^{2} \)
$73$ \( ( -92 + 8 T + T^{2} )^{2} \)
$79$ \( ( -12 + T )^{4} \)
$83$ \( 36 + 24 T^{2} + T^{4} \)
$89$ \( ( -12 - 12 T + T^{2} )^{2} \)
$97$ \( ( -92 - 8 T + T^{2} )^{2} \)
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