# Properties

 Label 756.4.x.a Level $756$ Weight $4$ Character orbit 756.x Analytic conductor $44.605$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 756.x (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$44.6054439643$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 252) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 6q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 6q^{7} + 12q^{11} + 408q^{23} - 600q^{25} + 84q^{29} + 336q^{37} + 84q^{43} + 318q^{49} - 2964q^{65} - 588q^{67} - 2400q^{77} + 204q^{79} - 360q^{85} - 1080q^{91} - 300q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
125.1 0 0 0 −10.5571 + 18.2854i 0 17.6959 + 5.46406i 0 0 0
125.2 0 0 0 −9.12012 + 15.7965i 0 −9.48165 15.9091i 0 0 0
125.3 0 0 0 −8.29874 + 14.3738i 0 18.2297 + 3.26749i 0 0 0
125.4 0 0 0 −7.82452 + 13.5525i 0 −13.7444 + 12.4134i 0 0 0
125.5 0 0 0 −6.03570 + 10.4541i 0 2.10370 18.4004i 0 0 0
125.6 0 0 0 −5.49690 + 9.52092i 0 6.85688 17.2042i 0 0 0
125.7 0 0 0 −5.16485 + 8.94579i 0 −17.2289 + 6.79448i 0 0 0
125.8 0 0 0 −3.53447 + 6.12188i 0 10.8305 + 15.0234i 0 0 0
125.9 0 0 0 −2.99997 + 5.19610i 0 −0.375477 + 18.5165i 0 0 0
125.10 0 0 0 −2.34269 + 4.05766i 0 −10.9369 + 14.9461i 0 0 0
125.11 0 0 0 −2.20656 + 3.82187i 0 −17.7037 + 5.43851i 0 0 0
125.12 0 0 0 −0.330097 + 0.571745i 0 −15.6440 9.91296i 0 0 0
125.13 0 0 0 0.330097 0.571745i 0 −0.762896 18.5045i 0 0 0
125.14 0 0 0 2.20656 3.82187i 0 13.5618 12.6126i 0 0 0
125.15 0 0 0 2.34269 4.05766i 0 18.4121 1.99859i 0 0 0
125.16 0 0 0 2.99997 5.19610i 0 16.2235 + 8.93305i 0 0 0
125.17 0 0 0 3.53447 6.12188i 0 7.59538 + 16.8911i 0 0 0
125.18 0 0 0 5.16485 8.94579i 0 14.4986 11.5234i 0 0 0
125.19 0 0 0 5.49690 9.52092i 0 −18.3277 2.66385i 0 0 0
125.20 0 0 0 6.03570 10.4541i 0 −16.9871 7.37834i 0 0 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 629.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.4.x.a 48
3.b odd 2 1 252.4.x.a 48
7.b odd 2 1 inner 756.4.x.a 48
9.c even 3 1 252.4.x.a 48
9.c even 3 1 2268.4.f.a 48
9.d odd 6 1 inner 756.4.x.a 48
9.d odd 6 1 2268.4.f.a 48
21.c even 2 1 252.4.x.a 48
63.l odd 6 1 252.4.x.a 48
63.l odd 6 1 2268.4.f.a 48
63.o even 6 1 inner 756.4.x.a 48
63.o even 6 1 2268.4.f.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.x.a 48 3.b odd 2 1
252.4.x.a 48 9.c even 3 1
252.4.x.a 48 21.c even 2 1
252.4.x.a 48 63.l odd 6 1
756.4.x.a 48 1.a even 1 1 trivial
756.4.x.a 48 7.b odd 2 1 inner
756.4.x.a 48 9.d odd 6 1 inner
756.4.x.a 48 63.o even 6 1 inner
2268.4.f.a 48 9.c even 3 1
2268.4.f.a 48 9.d odd 6 1
2268.4.f.a 48 63.l odd 6 1
2268.4.f.a 48 63.o even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(756, [\chi])$$.