# Properties

 Label 4140.2.n.b Level $4140$ Weight $2$ Character orbit 4140.n Analytic conductor $33.058$ Analytic rank $0$ Dimension $32$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$33.0580664368$$ 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 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 56 q^{25} + 16 q^{31} - 96 q^{49} - 16 q^{55} - 40 q^{85}+O(q^{100})$$ 32 * q + 56 * q^25 + 16 * q^31 - 96 * q^49 - 16 * q^55 - 40 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2069.1 0 0 0 −2.23315 0.114288i 0 −3.07318 0 0 0
2069.2 0 0 0 −2.23315 0.114288i 0 3.07318 0 0 0
2069.3 0 0 0 −2.23315 + 0.114288i 0 −3.07318 0 0 0
2069.4 0 0 0 −2.23315 + 0.114288i 0 3.07318 0 0 0
2069.5 0 0 0 −1.83744 1.27428i 0 −0.920125 0 0 0
2069.6 0 0 0 −1.83744 1.27428i 0 0.920125 0 0 0
2069.7 0 0 0 −1.83744 + 1.27428i 0 −0.920125 0 0 0
2069.8 0 0 0 −1.83744 + 1.27428i 0 0.920125 0 0 0
2069.9 0 0 0 −1.64510 1.51448i 0 −2.05693 0 0 0
2069.10 0 0 0 −1.64510 1.51448i 0 2.05693 0 0 0
2069.11 0 0 0 −1.64510 + 1.51448i 0 −2.05693 0 0 0
2069.12 0 0 0 −1.64510 + 1.51448i 0 2.05693 0 0 0
2069.13 0 0 0 −1.55901 1.60296i 0 −1.21571 0 0 0
2069.14 0 0 0 −1.55901 1.60296i 0 1.21571 0 0 0
2069.15 0 0 0 −1.55901 + 1.60296i 0 −1.21571 0 0 0
2069.16 0 0 0 −1.55901 + 1.60296i 0 1.21571 0 0 0
2069.17 0 0 0 1.55901 1.60296i 0 −1.21571 0 0 0
2069.18 0 0 0 1.55901 1.60296i 0 1.21571 0 0 0
2069.19 0 0 0 1.55901 + 1.60296i 0 −1.21571 0 0 0
2069.20 0 0 0 1.55901 + 1.60296i 0 1.21571 0 0 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2069.32 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
5.b even 2 1 inner
15.d odd 2 1 inner
23.b odd 2 1 inner
69.c even 2 1 inner
115.c odd 2 1 inner
345.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4140.2.n.b 32
3.b odd 2 1 inner 4140.2.n.b 32
5.b even 2 1 inner 4140.2.n.b 32
15.d odd 2 1 inner 4140.2.n.b 32
23.b odd 2 1 inner 4140.2.n.b 32
69.c even 2 1 inner 4140.2.n.b 32
115.c odd 2 1 inner 4140.2.n.b 32
345.h even 2 1 inner 4140.2.n.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4140.2.n.b 32 1.a even 1 1 trivial
4140.2.n.b 32 3.b odd 2 1 inner
4140.2.n.b 32 5.b even 2 1 inner
4140.2.n.b 32 15.d odd 2 1 inner
4140.2.n.b 32 23.b odd 2 1 inner
4140.2.n.b 32 69.c even 2 1 inner
4140.2.n.b 32 115.c odd 2 1 inner
4140.2.n.b 32 345.h even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 16T_{7}^{6} + 73T_{7}^{4} - 110T_{7}^{2} + 50$$ acting on $$S_{2}^{\mathrm{new}}(4140, [\chi])$$.