# Properties

 Label 4140.2.s.a 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 12 q^{7}+O(q^{10})$$ 44 * q - 12 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100})$$ 44 * q - 12 * q^7 - 4 * q^13 + 24 * q^25 - 48 * q^37 + 8 * q^43 + 40 * q^55 - 96 * q^61 - 44 * q^67 + 76 * q^73 + 72 * q^85 - 48 * q^91 - 20 * 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 −2.23603 + 0.0138422i 0 2.36435 + 2.36435i 0 0 0
737.2 0 0 0 −2.21141 + 0.331134i 0 −1.17089 1.17089i 0 0 0
737.3 0 0 0 −2.20494 + 0.371802i 0 −3.41792 3.41792i 0 0 0
737.4 0 0 0 −2.19878 0.406643i 0 1.26371 + 1.26371i 0 0 0
737.5 0 0 0 −1.99005 1.01966i 0 −1.87601 1.87601i 0 0 0
737.6 0 0 0 −1.77166 + 1.36427i 0 3.33197 + 3.33197i 0 0 0
737.7 0 0 0 −1.54074 1.62053i 0 −0.122636 0.122636i 0 0 0
737.8 0 0 0 −1.13155 + 1.92862i 0 2.43117 + 2.43117i 0 0 0
737.9 0 0 0 −0.340177 2.21004i 0 −3.49741 3.49741i 0 0 0
737.10 0 0 0 −0.210429 + 2.22614i 0 −0.752416 0.752416i 0 0 0
737.11 0 0 0 −0.00942680 + 2.23605i 0 −1.55392 1.55392i 0 0 0
737.12 0 0 0 0.00942680 2.23605i 0 −1.55392 1.55392i 0 0 0
737.13 0 0 0 0.210429 2.22614i 0 −0.752416 0.752416i 0 0 0
737.14 0 0 0 0.340177 + 2.21004i 0 −3.49741 3.49741i 0 0 0
737.15 0 0 0 1.13155 1.92862i 0 2.43117 + 2.43117i 0 0 0
737.16 0 0 0 1.54074 + 1.62053i 0 −0.122636 0.122636i 0 0 0
737.17 0 0 0 1.77166 1.36427i 0 3.33197 + 3.33197i 0 0 0
737.18 0 0 0 1.99005 + 1.01966i 0 −1.87601 1.87601i 0 0 0
737.19 0 0 0 2.19878 + 0.406643i 0 1.26371 + 1.26371i 0 0 0
737.20 0 0 0 2.20494 0.371802i 0 −3.41792 3.41792i 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.a 44
3.b odd 2 1 inner 4140.2.s.a 44
5.c odd 4 1 inner 4140.2.s.a 44
15.e even 4 1 inner 4140.2.s.a 44

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