# Properties

 Label 756.2.bq.a Level 756 Weight 2 Character orbit 756.bq 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.bq (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.

 $$\operatorname{Tr}(f)(q) =$$ $$144q + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$144q + 6q^{9} + 6q^{11} - 12q^{15} + 48q^{17} + 33q^{21} - 21q^{23} + 6q^{29} + 18q^{33} - 9q^{35} + 9q^{39} - 12q^{41} - 12q^{45} + 18q^{47} - 18q^{49} - 9q^{51} + 15q^{53} + 3q^{57} - 15q^{59} - 36q^{61} + 3q^{63} + 36q^{65} + 30q^{69} - 12q^{71} + 18q^{73} - 51q^{75} - 3q^{77} + 18q^{79} - 6q^{81} - 36q^{85} + 33q^{87} + 144q^{89} + 9q^{91} + 48q^{93} - 30q^{95} - 72q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 0 −1.73181 0.0288799i 0 2.86672 + 1.04340i 0 2.28241 + 1.33814i 0 2.99833 + 0.100029i 0
25.2 0 −1.67230 0.451029i 0 −0.947523 0.344870i 0 −1.17643 + 2.36981i 0 2.59315 + 1.50851i 0
25.3 0 −1.63641 0.567600i 0 −2.66918 0.971502i 0 −0.492567 2.59950i 0 2.35566 + 1.85765i 0
25.4 0 −1.47274 + 0.911614i 0 −3.21812 1.17130i 0 −2.64556 + 0.0321766i 0 1.33792 2.68514i 0
25.5 0 −1.38976 + 1.03372i 0 3.13641 + 1.14156i 0 −2.54733 0.714936i 0 0.862856 2.87324i 0
25.6 0 −1.36056 1.07186i 0 2.22913 + 0.811338i 0 −2.26967 1.35962i 0 0.702252 + 2.91665i 0
25.7 0 −1.26745 + 1.18050i 0 −0.0700823 0.0255079i 0 1.65572 2.06363i 0 0.212841 2.99244i 0
25.8 0 −1.22028 + 1.22920i 0 −0.946810 0.344611i 0 0.0913265 + 2.64417i 0 −0.0218551 2.99992i 0
25.9 0 −1.05320 1.37506i 0 −1.13419 0.412813i 0 2.58962 0.542074i 0 −0.781555 + 2.89641i 0
25.10 0 −0.277537 1.70967i 0 −0.774544 0.281911i 0 2.22920 + 1.42502i 0 −2.84595 + 0.948993i 0
25.11 0 −0.247313 1.71430i 0 −0.839073 0.305398i 0 −1.83758 + 1.90350i 0 −2.87767 + 0.847940i 0
25.12 0 −0.0456125 + 1.73145i 0 2.25335 + 0.820154i 0 −0.197469 2.63837i 0 −2.99584 0.157952i 0
25.13 0 0.0685939 + 1.73069i 0 −3.85868 1.40444i 0 2.56653 0.642577i 0 −2.99059 + 0.237430i 0
25.14 0 0.280333 + 1.70921i 0 −0.659448 0.240019i 0 −2.50760 + 0.843777i 0 −2.84283 + 0.958297i 0
25.15 0 0.568135 1.63622i 0 0.924390 + 0.336451i 0 −1.51517 2.16893i 0 −2.35445 1.85919i 0
25.16 0 0.898399 1.48084i 0 −3.88836 1.41525i 0 0.934219 2.47533i 0 −1.38576 2.66077i 0
25.17 0 0.910101 + 1.47367i 0 2.21872 + 0.807546i 0 2.43789 + 1.02795i 0 −1.34343 + 2.68238i 0
25.18 0 0.939844 1.45489i 0 2.93786 + 1.06929i 0 0.0512030 + 2.64526i 0 −1.23339 2.73473i 0
25.19 0 1.52533 + 0.820597i 0 −1.98623 0.722928i 0 0.905880 + 2.48584i 0 1.65324 + 2.50336i 0
25.20 0 1.52539 0.820472i 0 −3.26720 1.18916i 0 −0.0267571 + 2.64562i 0 1.65365 2.50309i 0
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 625.24 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.w 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.bq.a yes 144
7.c even 3 1 756.2.bp.a 144
27.e even 9 1 756.2.bp.a 144
189.w even 9 1 inner 756.2.bq.a yes 144

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

## 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