# Properties

 Label 1728.3.q.l Level $1728$ Weight $3$ Character orbit 1728.q Analytic conductor $47.085$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 288) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24 q + 60 q^{25} + 72 q^{29} + 36 q^{41} - 132 q^{49} - 96 q^{61} - 576 q^{65} + 24 q^{73} + 432 q^{77} + 96 q^{85} + 252 q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1 0 0 0 −7.38813 4.26554i 0 −1.36942 2.37191i 0 0 0
449.2 0 0 0 −7.38813 4.26554i 0 1.36942 + 2.37191i 0 0 0
449.3 0 0 0 −4.45884 2.57431i 0 −1.35076 2.33958i 0 0 0
449.4 0 0 0 −4.45884 2.57431i 0 1.35076 + 2.33958i 0 0 0
449.5 0 0 0 0.439631 + 0.253821i 0 6.44757 + 11.1675i 0 0 0
449.6 0 0 0 0.439631 + 0.253821i 0 −6.44757 11.1675i 0 0 0
449.7 0 0 0 1.15965 + 0.669525i 0 −0.328661 0.569258i 0 0 0
449.8 0 0 0 1.15965 + 0.669525i 0 0.328661 + 0.569258i 0 0 0
449.9 0 0 0 3.32266 + 1.91834i 0 −2.70775 4.68995i 0 0 0
449.10 0 0 0 3.32266 + 1.91834i 0 2.70775 + 4.68995i 0 0 0
449.11 0 0 0 6.92504 + 3.99817i 0 −6.10647 10.5767i 0 0 0
449.12 0 0 0 6.92504 + 3.99817i 0 6.10647 + 10.5767i 0 0 0
1601.1 0 0 0 −7.38813 + 4.26554i 0 −1.36942 + 2.37191i 0 0 0
1601.2 0 0 0 −7.38813 + 4.26554i 0 1.36942 2.37191i 0 0 0
1601.3 0 0 0 −4.45884 + 2.57431i 0 −1.35076 + 2.33958i 0 0 0
1601.4 0 0 0 −4.45884 + 2.57431i 0 1.35076 2.33958i 0 0 0
1601.5 0 0 0 0.439631 0.253821i 0 6.44757 11.1675i 0 0 0
1601.6 0 0 0 0.439631 0.253821i 0 −6.44757 + 11.1675i 0 0 0
1601.7 0 0 0 1.15965 0.669525i 0 −0.328661 + 0.569258i 0 0 0
1601.8 0 0 0 1.15965 0.669525i 0 0.328661 0.569258i 0 0 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1601.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.q.l 24
3.b odd 2 1 576.3.q.k 24
4.b odd 2 1 inner 1728.3.q.l 24
8.b even 2 1 864.3.q.b 24
8.d odd 2 1 864.3.q.b 24
9.c even 3 1 576.3.q.k 24
9.d odd 6 1 inner 1728.3.q.l 24
12.b even 2 1 576.3.q.k 24
24.f even 2 1 288.3.q.a 24
24.h odd 2 1 288.3.q.a 24
36.f odd 6 1 576.3.q.k 24
36.h even 6 1 inner 1728.3.q.l 24
72.j odd 6 1 864.3.q.b 24
72.j odd 6 1 2592.3.e.j 24
72.l even 6 1 864.3.q.b 24
72.l even 6 1 2592.3.e.j 24
72.n even 6 1 288.3.q.a 24
72.n even 6 1 2592.3.e.j 24
72.p odd 6 1 288.3.q.a 24
72.p odd 6 1 2592.3.e.j 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.q.a 24 24.f even 2 1
288.3.q.a 24 24.h odd 2 1
288.3.q.a 24 72.n even 6 1
288.3.q.a 24 72.p odd 6 1
576.3.q.k 24 3.b odd 2 1
576.3.q.k 24 9.c even 3 1
576.3.q.k 24 12.b even 2 1
576.3.q.k 24 36.f odd 6 1
864.3.q.b 24 8.b even 2 1
864.3.q.b 24 8.d odd 2 1
864.3.q.b 24 72.j odd 6 1
864.3.q.b 24 72.l even 6 1
1728.3.q.l 24 1.a even 1 1 trivial
1728.3.q.l 24 4.b odd 2 1 inner
1728.3.q.l 24 9.d odd 6 1 inner
1728.3.q.l 24 36.h even 6 1 inner
2592.3.e.j 24 72.j odd 6 1
2592.3.e.j 24 72.l even 6 1
2592.3.e.j 24 72.n even 6 1
2592.3.e.j 24 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{12} - \cdots$$ $$10\!\cdots\!70$$$$T_{7}^{12} +$$$$13\!\cdots\!00$$$$T_{7}^{10} +$$$$11\!\cdots\!65$$$$T_{7}^{8} +$$$$55\!\cdots\!00$$$$T_{7}^{6} +$$$$18\!\cdots\!08$$$$T_{7}^{4} +$$$$77\!\cdots\!72$$$$T_{7}^{2} +$$$$29\!\cdots\!16$$">$$T_{7}^{24} + \cdots$$