# Properties

 Label 2268.4.f.a Level $2268$ Weight $4$ Character orbit 2268.f Analytic conductor $133.816$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2268 = 2^{2} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2268.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$133.816331893$$ Analytic rank: $$0$$ Dimension: $$48$$ Twist minimal: no (minimal twist has level 252) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 12q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 12q^{7} + 1200q^{25} + 336q^{37} - 168q^{43} - 636q^{49} + 1176q^{67} - 408q^{79} + 720q^{85} - 1080q^{91} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1133.1 0 0 0 −21.1141 0 −13.5800 + 12.5930i 0 0 0
1133.2 0 0 0 −21.1141 0 −13.5800 12.5930i 0 0 0
1133.3 0 0 0 −18.2402 0 18.5185 0.256813i 0 0 0
1133.4 0 0 0 −18.2402 0 18.5185 + 0.256813i 0 0 0
1133.5 0 0 0 −16.5975 0 −11.9446 14.1537i 0 0 0
1133.6 0 0 0 −16.5975 0 −11.9446 + 14.1537i 0 0 0
1133.7 0 0 0 −15.6490 0 −3.87814 18.1097i 0 0 0
1133.8 0 0 0 −15.6490 0 −3.87814 + 18.1097i 0 0 0
1133.9 0 0 0 −12.0714 0 14.8834 + 11.0221i 0 0 0
1133.10 0 0 0 −12.0714 0 14.8834 11.0221i 0 0 0
1133.11 0 0 0 −10.9938 0 11.4708 14.5403i 0 0 0
1133.12 0 0 0 −10.9938 0 11.4708 + 14.5403i 0 0 0
1133.13 0 0 0 −10.3297 0 2.73026 + 18.3179i 0 0 0
1133.14 0 0 0 −10.3297 0 2.73026 18.3179i 0 0 0
1133.15 0 0 0 −7.06893 0 −18.4258 1.86777i 0 0 0
1133.16 0 0 0 −7.06893 0 −18.4258 + 1.86777i 0 0 0
1133.17 0 0 0 −5.99994 0 −15.8480 9.58340i 0 0 0
1133.18 0 0 0 −5.99994 0 −15.8480 + 9.58340i 0 0 0
1133.19 0 0 0 −4.68539 0 −7.47522 16.9446i 0 0 0
1133.20 0 0 0 −4.68539 0 −7.47522 + 16.9446i 0 0 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1133.48 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
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.4.f.a 48
3.b odd 2 1 inner 2268.4.f.a 48
7.b odd 2 1 inner 2268.4.f.a 48
9.c even 3 1 252.4.x.a 48
9.c even 3 1 756.4.x.a 48
9.d odd 6 1 252.4.x.a 48
9.d odd 6 1 756.4.x.a 48
21.c even 2 1 inner 2268.4.f.a 48
63.l odd 6 1 252.4.x.a 48
63.l odd 6 1 756.4.x.a 48
63.o even 6 1 252.4.x.a 48
63.o even 6 1 756.4.x.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.x.a 48 9.c even 3 1
252.4.x.a 48 9.d odd 6 1
252.4.x.a 48 63.l odd 6 1
252.4.x.a 48 63.o even 6 1
756.4.x.a 48 9.c even 3 1
756.4.x.a 48 9.d odd 6 1
756.4.x.a 48 63.l odd 6 1
756.4.x.a 48 63.o even 6 1
2268.4.f.a 48 1.a even 1 1 trivial
2268.4.f.a 48 3.b odd 2 1 inner
2268.4.f.a 48 7.b odd 2 1 inner
2268.4.f.a 48 21.c even 2 1 inner