# Properties

 Label 2240.2.bd.a Level $2240$ Weight $2$ Character orbit 2240.bd Analytic conductor $17.886$ Analytic rank $0$ Dimension $44$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.bd (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 560) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$44q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q - 12q^{11} - 8q^{15} - 8q^{19} + 24q^{27} + 12q^{29} + 28q^{37} + 44q^{43} - 44q^{49} + 8q^{51} - 12q^{53} - 24q^{59} - 16q^{61} - 28q^{63} - 40q^{65} + 28q^{67} + 40q^{69} - 12q^{77} + 16q^{79} + 20q^{81} + 16q^{85} + 88q^{93} + 32q^{95} - 28q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
561.1 0 −2.22016 + 2.22016i 0 0.707107 + 0.707107i 0 1.00000i 0 6.85821i 0
561.2 0 −2.18833 + 2.18833i 0 −0.707107 0.707107i 0 1.00000i 0 6.57756i 0
561.3 0 −1.92081 + 1.92081i 0 0.707107 + 0.707107i 0 1.00000i 0 4.37899i 0
561.4 0 −1.63220 + 1.63220i 0 −0.707107 0.707107i 0 1.00000i 0 2.32815i 0
561.5 0 −1.28029 + 1.28029i 0 0.707107 + 0.707107i 0 1.00000i 0 0.278310i 0
561.6 0 −0.989765 + 0.989765i 0 0.707107 + 0.707107i 0 1.00000i 0 1.04073i 0
561.7 0 −0.925827 + 0.925827i 0 −0.707107 0.707107i 0 1.00000i 0 1.28569i 0
561.8 0 −0.693100 + 0.693100i 0 −0.707107 0.707107i 0 1.00000i 0 2.03922i 0
561.9 0 −0.659301 + 0.659301i 0 0.707107 + 0.707107i 0 1.00000i 0 2.13064i 0
561.10 0 −0.448521 + 0.448521i 0 0.707107 + 0.707107i 0 1.00000i 0 2.59766i 0
561.11 0 −0.257753 + 0.257753i 0 −0.707107 0.707107i 0 1.00000i 0 2.86713i 0
561.12 0 0.296675 0.296675i 0 −0.707107 0.707107i 0 1.00000i 0 2.82397i 0
561.13 0 0.576404 0.576404i 0 0.707107 + 0.707107i 0 1.00000i 0 2.33552i 0
561.14 0 0.605289 0.605289i 0 −0.707107 0.707107i 0 1.00000i 0 2.26725i 0
561.15 0 0.839605 0.839605i 0 −0.707107 0.707107i 0 1.00000i 0 1.59013i 0
561.16 0 1.16279 1.16279i 0 0.707107 + 0.707107i 0 1.00000i 0 0.295857i 0
561.17 0 1.25769 1.25769i 0 0.707107 + 0.707107i 0 1.00000i 0 0.163545i 0
561.18 0 1.35880 1.35880i 0 −0.707107 0.707107i 0 1.00000i 0 0.692696i 0
561.19 0 1.42818 1.42818i 0 0.707107 + 0.707107i 0 1.00000i 0 1.07940i 0
561.20 0 1.67958 1.67958i 0 0.707107 + 0.707107i 0 1.00000i 0 2.64195i 0
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1681.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.bd.a 44
4.b odd 2 1 560.2.bd.a 44
16.e even 4 1 inner 2240.2.bd.a 44
16.f odd 4 1 560.2.bd.a 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bd.a 44 4.b odd 2 1
560.2.bd.a 44 16.f odd 4 1
2240.2.bd.a 44 1.a even 1 1 trivial
2240.2.bd.a 44 16.e even 4 1 inner