Properties

Label 2304.2.d.h
Level $2304$
Weight $2$
Character orbit 2304.d
Analytic conductor $18.398$
Analytic rank $1$
Dimension $2$
CM discriminant -4
Inner twists $4$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 4 i q^{5} +O(q^{10})\) \( q + 4 i q^{5} -6 i q^{13} -8 q^{17} -11 q^{25} + 4 i q^{29} + 2 i q^{37} -8 q^{41} -7 q^{49} -4 i q^{53} -10 i q^{61} + 24 q^{65} -6 q^{73} -32 i q^{85} + 16 q^{89} -18 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 16q^{17} - 22q^{25} - 16q^{41} - 14q^{49} + 48q^{65} - 12q^{73} + 32q^{89} - 36q^{97} + O(q^{100}) \)

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 4.00000i 0 0 0 0 0
1153.2 0 0 0 4.00000i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
8.b even 2 1 inner
8.d odd 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.h 2
3.b odd 2 1 2304.2.d.l 2
4.b odd 2 1 CM 2304.2.d.h 2
8.b even 2 1 inner 2304.2.d.h 2
8.d odd 2 1 inner 2304.2.d.h 2
12.b even 2 1 2304.2.d.l 2
16.e even 4 1 288.2.a.a 1
16.e even 4 1 576.2.a.i 1
16.f odd 4 1 288.2.a.a 1
16.f odd 4 1 576.2.a.i 1
24.f even 2 1 2304.2.d.l 2
24.h odd 2 1 2304.2.d.l 2
48.i odd 4 1 288.2.a.e yes 1
48.i odd 4 1 576.2.a.a 1
48.k even 4 1 288.2.a.e yes 1
48.k even 4 1 576.2.a.a 1
80.i odd 4 1 7200.2.f.n 2
80.j even 4 1 7200.2.f.n 2
80.k odd 4 1 7200.2.a.bf 1
80.q even 4 1 7200.2.a.bf 1
80.s even 4 1 7200.2.f.n 2
80.t odd 4 1 7200.2.f.n 2
144.u even 12 2 2592.2.i.a 2
144.v odd 12 2 2592.2.i.x 2
144.w odd 12 2 2592.2.i.a 2
144.x even 12 2 2592.2.i.x 2
240.t even 4 1 7200.2.a.be 1
240.z odd 4 1 7200.2.f.q 2
240.bb even 4 1 7200.2.f.q 2
240.bd odd 4 1 7200.2.f.q 2
240.bf even 4 1 7200.2.f.q 2
240.bm odd 4 1 7200.2.a.be 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.a.a 1 16.e even 4 1
288.2.a.a 1 16.f odd 4 1
288.2.a.e yes 1 48.i odd 4 1
288.2.a.e yes 1 48.k even 4 1
576.2.a.a 1 48.i odd 4 1
576.2.a.a 1 48.k even 4 1
576.2.a.i 1 16.e even 4 1
576.2.a.i 1 16.f odd 4 1
2304.2.d.h 2 1.a even 1 1 trivial
2304.2.d.h 2 4.b odd 2 1 CM
2304.2.d.h 2 8.b even 2 1 inner
2304.2.d.h 2 8.d odd 2 1 inner
2304.2.d.l 2 3.b odd 2 1
2304.2.d.l 2 12.b even 2 1
2304.2.d.l 2 24.f even 2 1
2304.2.d.l 2 24.h odd 2 1
2592.2.i.a 2 144.u even 12 2
2592.2.i.a 2 144.w odd 12 2
2592.2.i.x 2 144.v odd 12 2
2592.2.i.x 2 144.x even 12 2
7200.2.a.be 1 240.t even 4 1
7200.2.a.be 1 240.bm odd 4 1
7200.2.a.bf 1 80.k odd 4 1
7200.2.a.bf 1 80.q even 4 1
7200.2.f.n 2 80.i odd 4 1
7200.2.f.n 2 80.j even 4 1
7200.2.f.n 2 80.s even 4 1
7200.2.f.n 2 80.t odd 4 1
7200.2.f.q 2 240.z odd 4 1
7200.2.f.q 2 240.bb even 4 1
7200.2.f.q 2 240.bd odd 4 1
7200.2.f.q 2 240.bf even 4 1

Hecke kernels

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

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