# Properties

 Label 1840.3.k.e Level $1840$ Weight $3$ Character orbit 1840.k Analytic conductor $50.136$ Analytic rank $0$ Dimension $48$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1840.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$50.1363686423$$ Analytic rank: $$0$$ Dimension: $$48$$ Twist minimal: no (minimal twist has level 920) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 128q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 128q^{9} - 8q^{23} - 240q^{25} + 72q^{29} - 32q^{31} + 40q^{35} + 96q^{39} - 104q^{41} - 128q^{47} - 344q^{49} - 80q^{55} - 248q^{59} + 292q^{69} - 208q^{71} + 224q^{73} - 288q^{77} + 184q^{81} + 48q^{87} - 672q^{93} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
321.1 0 −5.39985 0 2.23607i 0 8.04288i 0 20.1584 0
321.2 0 −5.39985 0 2.23607i 0 8.04288i 0 20.1584 0
321.3 0 −5.24002 0 2.23607i 0 3.70696i 0 18.4579 0
321.4 0 −5.24002 0 2.23607i 0 3.70696i 0 18.4579 0
321.5 0 −4.84326 0 2.23607i 0 4.25633i 0 14.4572 0
321.6 0 −4.84326 0 2.23607i 0 4.25633i 0 14.4572 0
321.7 0 −4.51681 0 2.23607i 0 1.75483i 0 11.4015 0
321.8 0 −4.51681 0 2.23607i 0 1.75483i 0 11.4015 0
321.9 0 −3.03006 0 2.23607i 0 6.21370i 0 0.181242 0
321.10 0 −3.03006 0 2.23607i 0 6.21370i 0 0.181242 0
321.11 0 −2.78603 0 2.23607i 0 9.63233i 0 −1.23801 0
321.12 0 −2.78603 0 2.23607i 0 9.63233i 0 −1.23801 0
321.13 0 −2.67503 0 2.23607i 0 10.9996i 0 −1.84421 0
321.14 0 −2.67503 0 2.23607i 0 10.9996i 0 −1.84421 0
321.15 0 −2.49795 0 2.23607i 0 1.88865i 0 −2.76024 0
321.16 0 −2.49795 0 2.23607i 0 1.88865i 0 −2.76024 0
321.17 0 −2.35475 0 2.23607i 0 5.28829i 0 −3.45517 0
321.18 0 −2.35475 0 2.23607i 0 5.28829i 0 −3.45517 0
321.19 0 −1.59835 0 2.23607i 0 7.14235i 0 −6.44527 0
321.20 0 −1.59835 0 2.23607i 0 7.14235i 0 −6.44527 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 321.48 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 1840.3.k.e 48
4.b odd 2 1 920.3.k.a 48
23.b odd 2 1 inner 1840.3.k.e 48
92.b even 2 1 920.3.k.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.3.k.a 48 4.b odd 2 1
920.3.k.a 48 92.b even 2 1
1840.3.k.e 48 1.a even 1 1 trivial
1840.3.k.e 48 23.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(1840, [\chi])$$.