Properties

Label 2304.1.t.a
Level $2304$
Weight $1$
Character orbit 2304.t
Analytic conductor $1.150$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2304.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.5184.1

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{3} + \zeta_{12}^{5} q^{5} -\zeta_{12} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \zeta_{12}^{5} q^{5} -\zeta_{12} q^{7} + q^{9} -\zeta_{12}^{4} q^{11} + \zeta_{12}^{5} q^{13} -\zeta_{12}^{5} q^{15} + \zeta_{12} q^{21} -\zeta_{12}^{5} q^{23} - q^{27} + \zeta_{12} q^{29} + \zeta_{12}^{5} q^{31} + \zeta_{12}^{4} q^{33} + q^{35} -\zeta_{12}^{5} q^{39} + \zeta_{12}^{2} q^{41} + \zeta_{12}^{4} q^{43} + \zeta_{12}^{5} q^{45} + \zeta_{12} q^{47} + \zeta_{12}^{3} q^{55} + \zeta_{12}^{2} q^{59} + \zeta_{12} q^{61} -\zeta_{12} q^{63} -\zeta_{12}^{4} q^{65} + \zeta_{12}^{2} q^{67} + \zeta_{12}^{5} q^{69} + 2 \zeta_{12}^{3} q^{71} + \zeta_{12}^{5} q^{77} -\zeta_{12} q^{79} + q^{81} -\zeta_{12}^{4} q^{83} -\zeta_{12} q^{87} + q^{91} -\zeta_{12}^{5} q^{93} + \zeta_{12}^{4} q^{97} -\zeta_{12}^{4} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{3} + 4 q^{9} + 2 q^{11} - 4 q^{27} - 2 q^{33} + 4 q^{35} + 2 q^{41} - 2 q^{43} + 2 q^{59} + 2 q^{65} + 2 q^{67} + 4 q^{81} + 2 q^{83} + 4 q^{91} - 2 q^{97} + 2 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
895.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −1.00000 0 −0.866025 0.500000i 0 −0.866025 + 0.500000i 0 1.00000 0
895.2 0 −1.00000 0 0.866025 + 0.500000i 0 0.866025 0.500000i 0 1.00000 0
1663.1 0 −1.00000 0 −0.866025 + 0.500000i 0 −0.866025 0.500000i 0 1.00000 0
1663.2 0 −1.00000 0 0.866025 0.500000i 0 0.866025 + 0.500000i 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
8.d odd 2 1 inner
9.c even 3 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.1.t.a 4
4.b odd 2 1 2304.1.t.b 4
8.b even 2 1 2304.1.t.b 4
8.d odd 2 1 inner 2304.1.t.a 4
9.c even 3 1 inner 2304.1.t.a 4
16.e even 4 1 288.1.o.a 4
16.e even 4 1 576.1.o.a 4
16.f odd 4 1 288.1.o.a 4
16.f odd 4 1 576.1.o.a 4
36.f odd 6 1 2304.1.t.b 4
48.i odd 4 1 864.1.o.a 4
48.i odd 4 1 1728.1.o.a 4
48.k even 4 1 864.1.o.a 4
48.k even 4 1 1728.1.o.a 4
72.n even 6 1 2304.1.t.b 4
72.p odd 6 1 inner 2304.1.t.a 4
144.u even 12 1 864.1.o.a 4
144.u even 12 1 1728.1.o.a 4
144.u even 12 1 2592.1.g.b 2
144.v odd 12 1 288.1.o.a 4
144.v odd 12 1 576.1.o.a 4
144.v odd 12 1 2592.1.g.a 2
144.w odd 12 1 864.1.o.a 4
144.w odd 12 1 1728.1.o.a 4
144.w odd 12 1 2592.1.g.b 2
144.x even 12 1 288.1.o.a 4
144.x even 12 1 576.1.o.a 4
144.x even 12 1 2592.1.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.1.o.a 4 16.e even 4 1
288.1.o.a 4 16.f odd 4 1
288.1.o.a 4 144.v odd 12 1
288.1.o.a 4 144.x even 12 1
576.1.o.a 4 16.e even 4 1
576.1.o.a 4 16.f odd 4 1
576.1.o.a 4 144.v odd 12 1
576.1.o.a 4 144.x even 12 1
864.1.o.a 4 48.i odd 4 1
864.1.o.a 4 48.k even 4 1
864.1.o.a 4 144.u even 12 1
864.1.o.a 4 144.w odd 12 1
1728.1.o.a 4 48.i odd 4 1
1728.1.o.a 4 48.k even 4 1
1728.1.o.a 4 144.u even 12 1
1728.1.o.a 4 144.w odd 12 1
2304.1.t.a 4 1.a even 1 1 trivial
2304.1.t.a 4 8.d odd 2 1 inner
2304.1.t.a 4 9.c even 3 1 inner
2304.1.t.a 4 72.p odd 6 1 inner
2304.1.t.b 4 4.b odd 2 1
2304.1.t.b 4 8.b even 2 1
2304.1.t.b 4 36.f odd 6 1
2304.1.t.b 4 72.n even 6 1
2592.1.g.a 2 144.v odd 12 1
2592.1.g.a 2 144.x even 12 1
2592.1.g.b 2 144.u even 12 1
2592.1.g.b 2 144.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} - T_{11} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2304, [\chi])\).

Hecke characteristic polynomials

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