Properties

Label 2304.2.c.i
Level $2304$
Weight $2$
Character orbit 2304.c
Analytic conductor $18.398$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 72)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{5} + ( -2 + 4 \zeta_{24}^{4} ) q^{7} +O(q^{10})\) \( q + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{5} + ( -2 + 4 \zeta_{24}^{4} ) q^{7} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{11} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{13} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{17} + 4 \zeta_{24}^{6} q^{19} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{23} - q^{25} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{29} + ( -2 + 4 \zeta_{24}^{4} ) q^{31} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{35} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{41} + 8 \zeta_{24}^{6} q^{43} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{47} -5 q^{49} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{53} + ( 4 - 8 \zeta_{24}^{4} ) q^{55} + ( 8 \zeta_{24} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} ) q^{59} + ( -16 \zeta_{24}^{2} + 8 \zeta_{24}^{6} ) q^{61} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{65} + 4 \zeta_{24}^{6} q^{67} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{71} + 4 q^{73} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{77} + ( 2 - 4 \zeta_{24}^{4} ) q^{79} + ( 10 \zeta_{24} + 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} ) q^{83} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{85} + ( -5 \zeta_{24} + 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} ) q^{89} + 12 \zeta_{24}^{6} q^{91} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{95} + 8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{25} - 40q^{49} + 32q^{73} + 64q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2303.1
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
−0.965926 0.258819i
0 0 0 2.44949i 0 3.46410i 0 0 0
2303.2 0 0 0 2.44949i 0 3.46410i 0 0 0
2303.3 0 0 0 2.44949i 0 3.46410i 0 0 0
2303.4 0 0 0 2.44949i 0 3.46410i 0 0 0
2303.5 0 0 0 2.44949i 0 3.46410i 0 0 0
2303.6 0 0 0 2.44949i 0 3.46410i 0 0 0
2303.7 0 0 0 2.44949i 0 3.46410i 0 0 0
2303.8 0 0 0 2.44949i 0 3.46410i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2303.8
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
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h 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.c.i 8
3.b odd 2 1 inner 2304.2.c.i 8
4.b odd 2 1 inner 2304.2.c.i 8
8.b even 2 1 inner 2304.2.c.i 8
8.d odd 2 1 inner 2304.2.c.i 8
12.b even 2 1 inner 2304.2.c.i 8
16.e even 4 1 72.2.f.a 4
16.e even 4 1 288.2.f.a 4
16.f odd 4 1 72.2.f.a 4
16.f odd 4 1 288.2.f.a 4
24.f even 2 1 inner 2304.2.c.i 8
24.h odd 2 1 inner 2304.2.c.i 8
48.i odd 4 1 72.2.f.a 4
48.i odd 4 1 288.2.f.a 4
48.k even 4 1 72.2.f.a 4
48.k even 4 1 288.2.f.a 4
80.i odd 4 1 1800.2.m.c 8
80.i odd 4 1 7200.2.m.c 8
80.j even 4 1 1800.2.m.c 8
80.j even 4 1 7200.2.m.c 8
80.k odd 4 1 1800.2.b.c 4
80.k odd 4 1 7200.2.b.c 4
80.q even 4 1 1800.2.b.c 4
80.q even 4 1 7200.2.b.c 4
80.s even 4 1 1800.2.m.c 8
80.s even 4 1 7200.2.m.c 8
80.t odd 4 1 1800.2.m.c 8
80.t odd 4 1 7200.2.m.c 8
144.u even 12 1 648.2.l.a 4
144.u even 12 1 648.2.l.c 4
144.u even 12 1 2592.2.p.a 4
144.u even 12 1 2592.2.p.c 4
144.v odd 12 1 648.2.l.a 4
144.v odd 12 1 648.2.l.c 4
144.v odd 12 1 2592.2.p.a 4
144.v odd 12 1 2592.2.p.c 4
144.w odd 12 1 648.2.l.a 4
144.w odd 12 1 648.2.l.c 4
144.w odd 12 1 2592.2.p.a 4
144.w odd 12 1 2592.2.p.c 4
144.x even 12 1 648.2.l.a 4
144.x even 12 1 648.2.l.c 4
144.x even 12 1 2592.2.p.a 4
144.x even 12 1 2592.2.p.c 4
240.t even 4 1 1800.2.b.c 4
240.t even 4 1 7200.2.b.c 4
240.z odd 4 1 1800.2.m.c 8
240.z odd 4 1 7200.2.m.c 8
240.bb even 4 1 1800.2.m.c 8
240.bb even 4 1 7200.2.m.c 8
240.bd odd 4 1 1800.2.m.c 8
240.bd odd 4 1 7200.2.m.c 8
240.bf even 4 1 1800.2.m.c 8
240.bf even 4 1 7200.2.m.c 8
240.bm odd 4 1 1800.2.b.c 4
240.bm odd 4 1 7200.2.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.f.a 4 16.e even 4 1
72.2.f.a 4 16.f odd 4 1
72.2.f.a 4 48.i odd 4 1
72.2.f.a 4 48.k even 4 1
288.2.f.a 4 16.e even 4 1
288.2.f.a 4 16.f odd 4 1
288.2.f.a 4 48.i odd 4 1
288.2.f.a 4 48.k even 4 1
648.2.l.a 4 144.u even 12 1
648.2.l.a 4 144.v odd 12 1
648.2.l.a 4 144.w odd 12 1
648.2.l.a 4 144.x even 12 1
648.2.l.c 4 144.u even 12 1
648.2.l.c 4 144.v odd 12 1
648.2.l.c 4 144.w odd 12 1
648.2.l.c 4 144.x even 12 1
1800.2.b.c 4 80.k odd 4 1
1800.2.b.c 4 80.q even 4 1
1800.2.b.c 4 240.t even 4 1
1800.2.b.c 4 240.bm odd 4 1
1800.2.m.c 8 80.i odd 4 1
1800.2.m.c 8 80.j even 4 1
1800.2.m.c 8 80.s even 4 1
1800.2.m.c 8 80.t odd 4 1
1800.2.m.c 8 240.z odd 4 1
1800.2.m.c 8 240.bb even 4 1
1800.2.m.c 8 240.bd odd 4 1
1800.2.m.c 8 240.bf even 4 1
2304.2.c.i 8 1.a even 1 1 trivial
2304.2.c.i 8 3.b odd 2 1 inner
2304.2.c.i 8 4.b odd 2 1 inner
2304.2.c.i 8 8.b even 2 1 inner
2304.2.c.i 8 8.d odd 2 1 inner
2304.2.c.i 8 12.b even 2 1 inner
2304.2.c.i 8 24.f even 2 1 inner
2304.2.c.i 8 24.h odd 2 1 inner
2592.2.p.a 4 144.u even 12 1
2592.2.p.a 4 144.v odd 12 1
2592.2.p.a 4 144.w odd 12 1
2592.2.p.a 4 144.x even 12 1
2592.2.p.c 4 144.u even 12 1
2592.2.p.c 4 144.v odd 12 1
2592.2.p.c 4 144.w odd 12 1
2592.2.p.c 4 144.x even 12 1
7200.2.b.c 4 80.k odd 4 1
7200.2.b.c 4 80.q even 4 1
7200.2.b.c 4 240.t even 4 1
7200.2.b.c 4 240.bm odd 4 1
7200.2.m.c 8 80.i odd 4 1
7200.2.m.c 8 80.j even 4 1
7200.2.m.c 8 80.s even 4 1
7200.2.m.c 8 80.t odd 4 1
7200.2.m.c 8 240.z odd 4 1
7200.2.m.c 8 240.bb even 4 1
7200.2.m.c 8 240.bd odd 4 1
7200.2.m.c 8 240.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 \(S_{2}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} + 6 \)
\( T_{7}^{2} + 12 \)
\( T_{11}^{2} - 8 \)
\( T_{13}^{2} - 12 \)