# Properties

 Label 756.2.bp.a Level 756 Weight 2 Character orbit 756.bp Analytic conductor 6.037 Analytic rank 0 Dimension 144 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 756.bp (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$144$$ Relative dimension: $$24$$ over $$\Q(\zeta_{9})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$144q - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$144q - 12q^{9} - 12q^{11} - 12q^{15} - 24q^{17} - 3q^{21} + 15q^{23} + 6q^{29} + 18q^{33} + 18q^{35} + 18q^{39} - 12q^{41} + 6q^{45} + 18q^{47} + 36q^{49} + 18q^{51} + 15q^{53} + 3q^{57} + 30q^{59} + 18q^{61} + 3q^{63} + 9q^{65} + 30q^{69} - 12q^{71} - 36q^{73} + 102q^{75} + 69q^{77} + 18q^{79} + 12q^{81} - 36q^{85} + 78q^{87} - 72q^{89} - 18q^{91} - 60q^{93} + 42q^{95} - 72q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1 0 −1.72645 + 0.139144i 0 −0.604458 + 0.220005i 0 −2.43449 1.03598i 0 2.96128 0.480450i 0
193.2 0 −1.72611 0.143355i 0 1.14483 0.416685i 0 −1.12940 + 2.39259i 0 2.95890 + 0.494894i 0
193.3 0 −1.68394 + 0.405388i 0 1.57900 0.574710i 0 2.62692 + 0.315138i 0 2.67132 1.36530i 0
193.4 0 −1.67981 0.422185i 0 −3.86396 + 1.40636i 0 0.267801 2.63216i 0 2.64352 + 1.41838i 0
193.5 0 −1.35655 1.07693i 0 2.04264 0.743461i 0 0.979055 2.45794i 0 0.680440 + 2.92181i 0
193.6 0 −1.22275 + 1.22673i 0 −1.75075 + 0.637221i 0 2.59849 + 0.497853i 0 −0.00975629 2.99998i 0
193.7 0 −0.985812 + 1.42414i 0 −1.13435 + 0.412871i 0 −2.60567 0.458792i 0 −1.05635 2.80787i 0
193.8 0 −0.844572 1.51218i 0 −0.569103 + 0.207137i 0 2.00271 + 1.72892i 0 −1.57340 + 2.55430i 0
193.9 0 −0.699375 + 1.58457i 0 3.09474 1.12639i 0 −0.555998 + 2.58667i 0 −2.02175 2.21642i 0
193.10 0 −0.636453 1.61088i 0 −2.33777 + 0.850878i 0 −2.09399 + 1.61716i 0 −2.18985 + 2.05050i 0
193.11 0 −0.513768 + 1.65410i 0 −0.633570 + 0.230601i 0 1.95184 1.78614i 0 −2.47208 1.69965i 0
193.12 0 0.128228 1.72730i 0 3.80842 1.38615i 0 −2.63217 0.267690i 0 −2.96712 0.442975i 0
193.13 0 0.156536 1.72496i 0 −1.43093 + 0.520817i 0 −0.946337 2.47072i 0 −2.95099 0.540038i 0
193.14 0 0.378367 + 1.69022i 0 −3.32039 + 1.20852i 0 0.693513 + 2.55324i 0 −2.71368 + 1.27905i 0
193.15 0 0.557871 1.63975i 0 2.46711 0.897956i 0 1.73437 + 1.99799i 0 −2.37756 1.82954i 0
193.16 0 0.726734 + 1.57221i 0 1.04532 0.380466i 0 −2.56389 + 0.653064i 0 −1.94371 + 2.28516i 0
193.17 0 0.963882 + 1.43907i 0 −0.981872 + 0.357372i 0 0.459325 2.60557i 0 −1.14186 + 2.77419i 0
193.18 0 1.12192 1.31958i 0 −3.46735 + 1.26201i 0 2.63004 0.287949i 0 −0.482607 2.96093i 0
193.19 0 1.17989 + 1.26802i 0 3.75808 1.36783i 0 2.59104 0.535278i 0 −0.215726 + 2.99223i 0
193.20 0 1.44372 0.956902i 0 −2.40619 + 0.875781i 0 −1.62201 + 2.09024i 0 1.16868 2.76300i 0
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 709.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.u even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.bp.a 144
7.c even 3 1 756.2.bq.a yes 144
27.e even 9 1 756.2.bq.a yes 144
189.u even 9 1 inner 756.2.bp.a 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.bp.a 144 1.a even 1 1 trivial
756.2.bp.a 144 189.u even 9 1 inner
756.2.bq.a yes 144 7.c even 3 1
756.2.bq.a yes 144 27.e even 9 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(756, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database