Properties

Label 2304.2.d.g
Level $2304$
Weight $2$
Character orbit 2304.d
Analytic conductor $18.398$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{5} -2 q^{7} +O(q^{10})\) \( q + 2 i q^{5} -2 q^{7} + 4 i q^{11} -2 i q^{13} + 4 q^{17} + 4 i q^{19} + 8 q^{23} + q^{25} -6 i q^{29} -6 q^{31} -4 i q^{35} -2 i q^{37} -12 q^{41} + 12 i q^{43} -8 q^{47} -3 q^{49} + 6 i q^{53} -8 q^{55} + 8 i q^{59} + 10 i q^{61} + 4 q^{65} + 8 i q^{67} -2 q^{73} -8 i q^{77} -14 q^{79} -12 i q^{83} + 8 i q^{85} + 8 q^{89} + 4 i q^{91} -8 q^{95} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{7} + O(q^{10}) \) \( 2q - 4q^{7} + 8q^{17} + 16q^{23} + 2q^{25} - 12q^{31} - 24q^{41} - 16q^{47} - 6q^{49} - 16q^{55} + 8q^{65} - 4q^{73} - 28q^{79} + 16q^{89} - 16q^{95} - 4q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1153.1
1.00000i
1.00000i
0 0 0 2.00000i 0 −2.00000 0 0 0
1153.2 0 0 0 2.00000i 0 −2.00000 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 2304.2.d.g 2
3.b odd 2 1 2304.2.d.e 2
4.b odd 2 1 2304.2.d.p 2
8.b even 2 1 inner 2304.2.d.g 2
8.d odd 2 1 2304.2.d.p 2
12.b even 2 1 2304.2.d.n 2
16.e even 4 1 1152.2.a.f yes 1
16.e even 4 1 1152.2.a.q yes 1
16.f odd 4 1 1152.2.a.d 1
16.f odd 4 1 1152.2.a.o yes 1
24.f even 2 1 2304.2.d.n 2
24.h odd 2 1 2304.2.d.e 2
48.i odd 4 1 1152.2.a.g yes 1
48.i odd 4 1 1152.2.a.p yes 1
48.k even 4 1 1152.2.a.e yes 1
48.k even 4 1 1152.2.a.n yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.a.d 1 16.f odd 4 1
1152.2.a.e yes 1 48.k even 4 1
1152.2.a.f yes 1 16.e even 4 1
1152.2.a.g yes 1 48.i odd 4 1
1152.2.a.n yes 1 48.k even 4 1
1152.2.a.o yes 1 16.f odd 4 1
1152.2.a.p yes 1 48.i odd 4 1
1152.2.a.q yes 1 16.e even 4 1
2304.2.d.e 2 3.b odd 2 1
2304.2.d.e 2 24.h odd 2 1
2304.2.d.g 2 1.a even 1 1 trivial
2304.2.d.g 2 8.b even 2 1 inner
2304.2.d.n 2 12.b even 2 1
2304.2.d.n 2 24.f even 2 1
2304.2.d.p 2 4.b odd 2 1
2304.2.d.p 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} + 4 \)
\( T_{7} + 2 \)
\( T_{11}^{2} + 16 \)
\( T_{17} - 4 \)
\( T_{23} - 8 \)