Properties

Label 2304.2.k.d
Level $2304$
Weight $2$
Character orbit 2304.k
Analytic conductor $18.398$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 768)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{8}^{2} ) q^{5} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{8}^{2} ) q^{5} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} + 4 \zeta_{8}^{3} q^{11} + ( -3 - 3 \zeta_{8}^{2} ) q^{13} + 6 q^{17} -2 \zeta_{8} q^{19} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{23} -3 \zeta_{8}^{2} q^{25} + ( 4 + 4 \zeta_{8}^{2} ) q^{29} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{31} + 12 \zeta_{8} q^{35} + ( -3 + 3 \zeta_{8}^{2} ) q^{37} + 10 \zeta_{8}^{2} q^{41} + 6 \zeta_{8}^{3} q^{43} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{47} -11 q^{49} + ( 4 - 4 \zeta_{8}^{2} ) q^{53} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{55} + ( -3 - 3 \zeta_{8}^{2} ) q^{61} -12 q^{65} + 4 \zeta_{8} q^{67} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{71} + 16 \zeta_{8}^{2} q^{73} + ( -12 - 12 \zeta_{8}^{2} ) q^{77} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{79} + 16 \zeta_{8} q^{83} + ( 12 - 12 \zeta_{8}^{2} ) q^{85} + 14 \zeta_{8}^{2} q^{89} -18 \zeta_{8}^{3} q^{91} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{95} -4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{5} + O(q^{10}) \) \( 4q + 8q^{5} - 12q^{13} + 24q^{17} + 16q^{29} - 12q^{37} - 44q^{49} + 16q^{53} - 12q^{61} - 48q^{65} - 48q^{77} + 48q^{85} - 16q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−0.707107 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 0 0 2.00000 2.00000i 0 4.24264i 0 0 0
577.2 0 0 0 2.00000 2.00000i 0 4.24264i 0 0 0
1729.1 0 0 0 2.00000 + 2.00000i 0 4.24264i 0 0 0
1729.2 0 0 0 2.00000 + 2.00000i 0 4.24264i 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 inner
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.k.d 4
3.b odd 2 1 768.2.j.a 4
4.b odd 2 1 inner 2304.2.k.d 4
8.b even 2 1 2304.2.k.a 4
8.d odd 2 1 2304.2.k.a 4
12.b even 2 1 768.2.j.a 4
16.e even 4 1 2304.2.k.a 4
16.e even 4 1 inner 2304.2.k.d 4
16.f odd 4 1 2304.2.k.a 4
16.f odd 4 1 inner 2304.2.k.d 4
24.f even 2 1 768.2.j.d yes 4
24.h odd 2 1 768.2.j.d yes 4
32.g even 8 1 9216.2.a.e 2
32.g even 8 1 9216.2.a.q 2
32.h odd 8 1 9216.2.a.e 2
32.h odd 8 1 9216.2.a.q 2
48.i odd 4 1 768.2.j.a 4
48.i odd 4 1 768.2.j.d yes 4
48.k even 4 1 768.2.j.a 4
48.k even 4 1 768.2.j.d yes 4
96.o even 8 1 3072.2.a.d 2
96.o even 8 1 3072.2.a.f 2
96.o even 8 2 3072.2.d.d 4
96.p odd 8 1 3072.2.a.d 2
96.p odd 8 1 3072.2.a.f 2
96.p odd 8 2 3072.2.d.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.a 4 3.b odd 2 1
768.2.j.a 4 12.b even 2 1
768.2.j.a 4 48.i odd 4 1
768.2.j.a 4 48.k even 4 1
768.2.j.d yes 4 24.f even 2 1
768.2.j.d yes 4 24.h odd 2 1
768.2.j.d yes 4 48.i odd 4 1
768.2.j.d yes 4 48.k even 4 1
2304.2.k.a 4 8.b even 2 1
2304.2.k.a 4 8.d odd 2 1
2304.2.k.a 4 16.e even 4 1
2304.2.k.a 4 16.f odd 4 1
2304.2.k.d 4 1.a even 1 1 trivial
2304.2.k.d 4 4.b odd 2 1 inner
2304.2.k.d 4 16.e even 4 1 inner
2304.2.k.d 4 16.f odd 4 1 inner
3072.2.a.d 2 96.o even 8 1
3072.2.a.d 2 96.p odd 8 1
3072.2.a.f 2 96.o even 8 1
3072.2.a.f 2 96.p odd 8 1
3072.2.d.d 4 96.o even 8 2
3072.2.d.d 4 96.p odd 8 2
9216.2.a.e 2 32.g even 8 1
9216.2.a.e 2 32.h odd 8 1
9216.2.a.q 2 32.g even 8 1
9216.2.a.q 2 32.h odd 8 1

Hecke kernels

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

\( T_{5}^{2} - 4 T_{5} + 8 \)
\( T_{7}^{2} + 18 \)
\( T_{13}^{2} + 6 T_{13} + 18 \)