# 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 $8$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5}^{2} + 6$$ $$T_{7}^{2} + 12$$ $$T_{11}^{2} - 8$$ $$T_{13}^{2} - 12$$