# Properties

 Label 4140.3.d.c Level $4140$ Weight $3$ Character orbit 4140.d Analytic conductor $112.807$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# 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: no (minimal twist has level 1380) 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 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 24 q^{13} - 64 q^{23} - 160 q^{25} + 60 q^{29} - 4 q^{31} + 60 q^{35} + 108 q^{41} - 136 q^{47} - 428 q^{49} + 120 q^{55} + 84 q^{59} - 188 q^{71} + 472 q^{73} + 120 q^{77} + 60 q^{85} - 80 q^{95}+O(q^{100})$$ 32 * q + 24 * q^13 - 64 * q^23 - 160 * q^25 + 60 * q^29 - 4 * q^31 + 60 * q^35 + 108 * q^41 - 136 * q^47 - 428 * q^49 + 120 * q^55 + 84 * q^59 - 188 * q^71 + 472 * q^73 + 120 * q^77 + 60 * q^85 - 80 * q^95

## 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 11.7218i 0 0 0
2161.2 0 0 0 2.23607i 0 10.9556i 0 0 0
2161.3 0 0 0 2.23607i 0 10.0406i 0 0 0
2161.4 0 0 0 2.23607i 0 4.23458i 0 0 0
2161.5 0 0 0 2.23607i 0 3.01539i 0 0 0
2161.6 0 0 0 2.23607i 0 2.98794i 0 0 0
2161.7 0 0 0 2.23607i 0 2.71701i 0 0 0
2161.8 0 0 0 2.23607i 0 1.24316i 0 0 0
2161.9 0 0 0 2.23607i 0 0.509961i 0 0 0
2161.10 0 0 0 2.23607i 0 2.07606i 0 0 0
2161.11 0 0 0 2.23607i 0 4.95389i 0 0 0
2161.12 0 0 0 2.23607i 0 8.75590i 0 0 0
2161.13 0 0 0 2.23607i 0 9.24833i 0 0 0
2161.14 0 0 0 2.23607i 0 11.3307i 0 0 0
2161.15 0 0 0 2.23607i 0 11.5587i 0 0 0
2161.16 0 0 0 2.23607i 0 11.8989i 0 0 0
2161.17 0 0 0 2.23607i 0 11.8989i 0 0 0
2161.18 0 0 0 2.23607i 0 11.5587i 0 0 0
2161.19 0 0 0 2.23607i 0 11.3307i 0 0 0
2161.20 0 0 0 2.23607i 0 9.24833i 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
23.b odd 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.c 32
3.b odd 2 1 1380.3.d.a 32
23.b odd 2 1 inner 4140.3.d.c 32
69.c even 2 1 1380.3.d.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.3.d.a 32 3.b odd 2 1
1380.3.d.a 32 69.c even 2 1
4140.3.d.c 32 1.a even 1 1 trivial
4140.3.d.c 32 23.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{32} + 998 T_{7}^{30} + 441911 T_{7}^{28} + 114285060 T_{7}^{26} + 19123041207 T_{7}^{24} + 2167506294182 T_{7}^{22} + 169468469656969 T_{7}^{20} + \cdots + 12\!\cdots\!16$$ acting on $$S_{3}^{\mathrm{new}}(4140, [\chi])$$.