# Properties

 Label 4140.2.s.b Level $4140$ Weight $2$ Character orbit 4140.s Analytic conductor $33.058$ Analytic rank $0$ Dimension $44$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4140.s (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.0580664368$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ over $$\Q(i)$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 4 q^{7}+O(q^{10})$$ 44 * q - 4 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 4 q^{7} - 4 q^{13} - 24 q^{25} + 32 q^{31} + 40 q^{37} - 8 q^{43} - 24 q^{55} + 64 q^{61} + 12 q^{67} - 84 q^{73} - 104 q^{85} - 48 q^{91} + 44 q^{97}+O(q^{100})$$ 44 * q - 4 * q^7 - 4 * q^13 - 24 * q^25 + 32 * q^31 + 40 * q^37 - 8 * q^43 - 24 * q^55 + 64 * q^61 + 12 * q^67 - 84 * q^73 - 104 * q^85 - 48 * q^91 + 44 * q^97

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
737.1 0 0 0 −2.01776 0.963656i 0 2.04752 + 2.04752i 0 0 0
737.2 0 0 0 −1.94065 1.11080i 0 −0.576954 0.576954i 0 0 0
737.3 0 0 0 −1.83311 + 1.28050i 0 −0.0692454 0.0692454i 0 0 0
737.4 0 0 0 −1.79713 + 1.33054i 0 −0.390251 0.390251i 0 0 0
737.5 0 0 0 −1.79146 + 1.33815i 0 −2.07691 2.07691i 0 0 0
737.6 0 0 0 −1.52812 1.63243i 0 −2.30479 2.30479i 0 0 0
737.7 0 0 0 −1.10909 1.94163i 0 2.66231 + 2.66231i 0 0 0
737.8 0 0 0 −1.07726 1.95947i 0 2.36049 + 2.36049i 0 0 0
737.9 0 0 0 −0.994073 + 2.00295i 0 −2.91813 2.91813i 0 0 0
737.10 0 0 0 −0.818756 + 2.08078i 0 0.623610 + 0.623610i 0 0 0
737.11 0 0 0 −0.692041 2.12628i 0 −0.357642 0.357642i 0 0 0
737.12 0 0 0 0.692041 + 2.12628i 0 −0.357642 0.357642i 0 0 0
737.13 0 0 0 0.818756 2.08078i 0 0.623610 + 0.623610i 0 0 0
737.14 0 0 0 0.994073 2.00295i 0 −2.91813 2.91813i 0 0 0
737.15 0 0 0 1.07726 + 1.95947i 0 2.36049 + 2.36049i 0 0 0
737.16 0 0 0 1.10909 + 1.94163i 0 2.66231 + 2.66231i 0 0 0
737.17 0 0 0 1.52812 + 1.63243i 0 −2.30479 2.30479i 0 0 0
737.18 0 0 0 1.79146 1.33815i 0 −2.07691 2.07691i 0 0 0
737.19 0 0 0 1.79713 1.33054i 0 −0.390251 0.390251i 0 0 0
737.20 0 0 0 1.83311 1.28050i 0 −0.0692454 0.0692454i 0 0 0
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2393.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4140.2.s.b 44
3.b odd 2 1 inner 4140.2.s.b 44
5.c odd 4 1 inner 4140.2.s.b 44
15.e even 4 1 inner 4140.2.s.b 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4140.2.s.b 44 1.a even 1 1 trivial
4140.2.s.b 44 3.b odd 2 1 inner
4140.2.s.b 44 5.c odd 4 1 inner
4140.2.s.b 44 15.e even 4 1 inner