Properties

Label 2880.2.h.f
Level $2880$
Weight $2$
Character orbit 2880.h
Analytic conductor $22.997$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 720)
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{24}^{6} q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} +O(q^{10})\) \( q -\zeta_{24}^{6} q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - 4 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{11} + ( 2 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{13} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{19} + ( 2 \zeta_{24} + 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{23} - q^{25} -6 \zeta_{24}^{6} q^{29} + ( 2 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{31} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{35} + ( -2 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{37} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{41} + ( 4 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 8 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{43} + ( -2 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{47} + q^{49} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{53} + ( -2 - \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{55} + ( \zeta_{24} - 4 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{59} + ( -2 - 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{61} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{65} + ( -4 + 8 \zeta_{24}^{4} ) q^{67} + ( -4 \zeta_{24} - 8 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{71} + ( -2 - 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{73} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{77} + ( 2 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{79} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{83} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 12 \zeta_{24}^{6} ) q^{89} + ( -6 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 12 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{91} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{95} + ( -2 - 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 16q^{13} - 8q^{25} - 16q^{37} + 8q^{49} - 16q^{61} - 16q^{73} - 16q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
0.965926 + 0.258819i
0.258819 + 0.965926i
−0.965926 0.258819i
−0.258819 0.965926i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.965926 0.258819i
0.258819 0.965926i
0 0 0 1.00000i 0 2.44949i 0 0 0
1151.2 0 0 0 1.00000i 0 2.44949i 0 0 0
1151.3 0 0 0 1.00000i 0 2.44949i 0 0 0
1151.4 0 0 0 1.00000i 0 2.44949i 0 0 0
1151.5 0 0 0 1.00000i 0 2.44949i 0 0 0
1151.6 0 0 0 1.00000i 0 2.44949i 0 0 0
1151.7 0 0 0 1.00000i 0 2.44949i 0 0 0
1151.8 0 0 0 1.00000i 0 2.44949i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.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.h.f 8
3.b odd 2 1 inner 2880.2.h.f 8
4.b odd 2 1 inner 2880.2.h.f 8
8.b even 2 1 720.2.h.a 8
8.d odd 2 1 720.2.h.a 8
12.b even 2 1 inner 2880.2.h.f 8
24.f even 2 1 720.2.h.a 8
24.h odd 2 1 720.2.h.a 8
40.e odd 2 1 3600.2.h.j 8
40.f even 2 1 3600.2.h.j 8
40.i odd 4 1 3600.2.o.c 8
40.i odd 4 1 3600.2.o.d 8
40.k even 4 1 3600.2.o.c 8
40.k even 4 1 3600.2.o.d 8
120.i odd 2 1 3600.2.h.j 8
120.m even 2 1 3600.2.h.j 8
120.q odd 4 1 3600.2.o.c 8
120.q odd 4 1 3600.2.o.d 8
120.w even 4 1 3600.2.o.c 8
120.w even 4 1 3600.2.o.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.h.a 8 8.b even 2 1
720.2.h.a 8 8.d odd 2 1
720.2.h.a 8 24.f even 2 1
720.2.h.a 8 24.h odd 2 1
2880.2.h.f 8 1.a even 1 1 trivial
2880.2.h.f 8 3.b odd 2 1 inner
2880.2.h.f 8 4.b odd 2 1 inner
2880.2.h.f 8 12.b even 2 1 inner
3600.2.h.j 8 40.e odd 2 1
3600.2.h.j 8 40.f even 2 1
3600.2.h.j 8 120.i odd 2 1
3600.2.h.j 8 120.m even 2 1
3600.2.o.c 8 40.i odd 4 1
3600.2.o.c 8 40.k even 4 1
3600.2.o.c 8 120.q odd 4 1
3600.2.o.c 8 120.w even 4 1
3600.2.o.d 8 40.i odd 4 1
3600.2.o.d 8 40.k even 4 1
3600.2.o.d 8 120.q odd 4 1
3600.2.o.d 8 120.w even 4 1

Hecke kernels

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

\( T_{7}^{2} + 6 \)
\( T_{11}^{4} - 36 T_{11}^{2} + 36 \)
\( T_{23}^{4} - 72 T_{23}^{2} + 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 1 + T^{2} )^{4} \)
$7$ \( ( 6 + T^{2} )^{4} \)
$11$ \( ( 36 - 36 T^{2} + T^{4} )^{2} \)
$13$ \( ( -14 - 4 T + T^{2} )^{4} \)
$17$ \( T^{8} \)
$19$ \( ( 24 + T^{2} )^{4} \)
$23$ \( ( 144 - 72 T^{2} + T^{4} )^{2} \)
$29$ \( ( 36 + T^{2} )^{4} \)
$31$ \( ( 144 + 72 T^{2} + T^{4} )^{2} \)
$37$ \( ( -14 + 4 T + T^{2} )^{4} \)
$41$ \( ( 18 + T^{2} )^{4} \)
$43$ \( ( 576 + 144 T^{2} + T^{4} )^{2} \)
$47$ \( ( 576 - 144 T^{2} + T^{4} )^{2} \)
$53$ \( ( 72 + T^{2} )^{4} \)
$59$ \( ( 36 - 36 T^{2} + T^{4} )^{2} \)
$61$ \( ( -68 + 4 T + T^{2} )^{4} \)
$67$ \( ( 48 + T^{2} )^{4} \)
$71$ \( ( 2304 - 288 T^{2} + T^{4} )^{2} \)
$73$ \( ( -68 + 4 T + T^{2} )^{4} \)
$79$ \( ( 144 + 72 T^{2} + T^{4} )^{2} \)
$83$ \( ( -216 + T^{2} )^{4} \)
$89$ \( ( 15876 + 324 T^{2} + T^{4} )^{2} \)
$97$ \( ( -68 + 4 T + T^{2} )^{4} \)
show more
show less