# Properties

 Label 4140.3.d.b Level $4140$ Weight $3$ Character orbit 4140.d Analytic conductor $112.807$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 4140.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$112.806829445$$ Analytic rank: $$0$$ Dimension: $$32$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q+O(q^{10})$$ 32 * q $$\operatorname{Tr}(f)(q) =$$ $$32 q - 24 q^{13} - 160 q^{25} - 28 q^{31} - 260 q^{49} + 120 q^{55} - 296 q^{73} - 60 q^{85}+O(q^{100})$$ 32 * q - 24 * q^13 - 160 * q^25 - 28 * q^31 - 260 * q^49 + 120 * q^55 - 296 * q^73 - 60 * q^85

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2161.1 0 0 0 2.23607i 0 13.2033i 0 0 0
2161.2 0 0 0 2.23607i 0 9.55915i 0 0 0
2161.3 0 0 0 2.23607i 0 8.25712i 0 0 0
2161.4 0 0 0 2.23607i 0 7.68094i 0 0 0
2161.5 0 0 0 2.23607i 0 6.51414i 0 0 0
2161.6 0 0 0 2.23607i 0 4.50512i 0 0 0
2161.7 0 0 0 2.23607i 0 1.15266i 0 0 0
2161.8 0 0 0 2.23607i 0 0.246464i 0 0 0
2161.9 0 0 0 2.23607i 0 0.246464i 0 0 0
2161.10 0 0 0 2.23607i 0 1.15266i 0 0 0
2161.11 0 0 0 2.23607i 0 4.50512i 0 0 0
2161.12 0 0 0 2.23607i 0 6.51414i 0 0 0
2161.13 0 0 0 2.23607i 0 7.68094i 0 0 0
2161.14 0 0 0 2.23607i 0 8.25712i 0 0 0
2161.15 0 0 0 2.23607i 0 9.55915i 0 0 0
2161.16 0 0 0 2.23607i 0 13.2033i 0 0 0
2161.17 0 0 0 2.23607i 0 13.2033i 0 0 0
2161.18 0 0 0 2.23607i 0 9.55915i 0 0 0
2161.19 0 0 0 2.23607i 0 8.25712i 0 0 0
2161.20 0 0 0 2.23607i 0 7.68094i 0 0 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2161.32 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
23.b odd 2 1 inner
69.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4140.3.d.b 32
3.b odd 2 1 inner 4140.3.d.b 32
23.b odd 2 1 inner 4140.3.d.b 32
69.c even 2 1 inner 4140.3.d.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4140.3.d.b 32 1.a even 1 1 trivial
4140.3.d.b 32 3.b odd 2 1 inner
4140.3.d.b 32 23.b odd 2 1 inner
4140.3.d.b 32 69.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{16} + 457 T_{7}^{14} + 79883 T_{7}^{12} + 6914455 T_{7}^{10} + 313947096 T_{7}^{8} + 7108236140 T_{7}^{6} + 64495964848 T_{7}^{4} + 77210147648 T_{7}^{2} + 4453693696$$ acting on $$S_{3}^{\mathrm{new}}(4140, [\chi])$$.