# Properties

 Label 2304.2.f.f Level $2304$ Weight $2$ Character orbit 2304.f Analytic conductor $18.398$ Analytic rank $0$ Dimension $4$ CM discriminant -4 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$18.3975326257$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{5} +O(q^{10})$$ $$q + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{5} -4 \zeta_{8}^{2} q^{13} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{17} + 13 q^{25} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{29} + 2 \zeta_{8}^{2} q^{37} + ( 9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{41} + 7 q^{49} + ( -9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{53} + 10 \zeta_{8}^{2} q^{61} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{65} -16 q^{73} + 18 \zeta_{8}^{2} q^{85} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{89} -8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 52q^{25} + 28q^{49} - 64q^{73} - 32q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i
0 0 0 −4.24264 0 0 0 0 0
1151.2 0 0 0 −4.24264 0 0 0 0 0
1151.3 0 0 0 4.24264 0 0 0 0 0
1151.4 0 0 0 4.24264 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
3.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.f.f 4
3.b odd 2 1 inner 2304.2.f.f 4
4.b odd 2 1 CM 2304.2.f.f 4
8.b even 2 1 inner 2304.2.f.f 4
8.d odd 2 1 inner 2304.2.f.f 4
12.b even 2 1 inner 2304.2.f.f 4
16.e even 4 1 144.2.c.a 2
16.e even 4 1 576.2.c.a 2
16.f odd 4 1 144.2.c.a 2
16.f odd 4 1 576.2.c.a 2
24.f even 2 1 inner 2304.2.f.f 4
24.h odd 2 1 inner 2304.2.f.f 4
48.i odd 4 1 144.2.c.a 2
48.i odd 4 1 576.2.c.a 2
48.k even 4 1 144.2.c.a 2
48.k even 4 1 576.2.c.a 2
80.i odd 4 1 3600.2.o.a 4
80.j even 4 1 3600.2.o.a 4
80.k odd 4 1 3600.2.h.b 2
80.q even 4 1 3600.2.h.b 2
80.s even 4 1 3600.2.o.a 4
80.t odd 4 1 3600.2.o.a 4
112.j even 4 1 7056.2.h.b 2
112.l odd 4 1 7056.2.h.b 2
144.u even 12 2 1296.2.s.h 4
144.v odd 12 2 1296.2.s.h 4
144.w odd 12 2 1296.2.s.h 4
144.x even 12 2 1296.2.s.h 4
240.t even 4 1 3600.2.h.b 2
240.z odd 4 1 3600.2.o.a 4
240.bb even 4 1 3600.2.o.a 4
240.bd odd 4 1 3600.2.o.a 4
240.bf even 4 1 3600.2.o.a 4
240.bm odd 4 1 3600.2.h.b 2
336.v odd 4 1 7056.2.h.b 2
336.y even 4 1 7056.2.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.c.a 2 16.e even 4 1
144.2.c.a 2 16.f odd 4 1
144.2.c.a 2 48.i odd 4 1
144.2.c.a 2 48.k even 4 1
576.2.c.a 2 16.e even 4 1
576.2.c.a 2 16.f odd 4 1
576.2.c.a 2 48.i odd 4 1
576.2.c.a 2 48.k even 4 1
1296.2.s.h 4 144.u even 12 2
1296.2.s.h 4 144.v odd 12 2
1296.2.s.h 4 144.w odd 12 2
1296.2.s.h 4 144.x even 12 2
2304.2.f.f 4 1.a even 1 1 trivial
2304.2.f.f 4 3.b odd 2 1 inner
2304.2.f.f 4 4.b odd 2 1 CM
2304.2.f.f 4 8.b even 2 1 inner
2304.2.f.f 4 8.d odd 2 1 inner
2304.2.f.f 4 12.b even 2 1 inner
2304.2.f.f 4 24.f even 2 1 inner
2304.2.f.f 4 24.h odd 2 1 inner
3600.2.h.b 2 80.k odd 4 1
3600.2.h.b 2 80.q even 4 1
3600.2.h.b 2 240.t even 4 1
3600.2.h.b 2 240.bm odd 4 1
3600.2.o.a 4 80.i odd 4 1
3600.2.o.a 4 80.j even 4 1
3600.2.o.a 4 80.s even 4 1
3600.2.o.a 4 80.t odd 4 1
3600.2.o.a 4 240.z odd 4 1
3600.2.o.a 4 240.bb even 4 1
3600.2.o.a 4 240.bd odd 4 1
3600.2.o.a 4 240.bf even 4 1
7056.2.h.b 2 112.j even 4 1
7056.2.h.b 2 112.l odd 4 1
7056.2.h.b 2 336.v odd 4 1
7056.2.h.b 2 336.y even 4 1

## Hecke kernels

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

 $$T_{5}^{2} - 18$$ $$T_{7}$$ $$T_{23}$$ $$T_{43}$$