# Properties

 Label 756.2.ck.a Level 756 Weight 2 Character orbit 756.ck 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.ck (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$144$$ Relative dimension: $$24$$ over $$\Q(\zeta_{18})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$144q + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$144q + 6q^{9} - 6q^{11} + 12q^{15} + 33q^{21} + 21q^{23} - 6q^{29} + 27q^{35} + 39q^{39} - 54q^{47} + 18q^{49} - 9q^{51} - 45q^{53} + 3q^{57} + 45q^{59} + 39q^{63} + 24q^{65} - 36q^{69} + 36q^{71} + 45q^{75} + 21q^{77} - 18q^{79} + 18q^{81} + 36q^{85} - 45q^{87} + 9q^{91} - 48q^{93} - 66q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 0 −1.70107 0.326131i 0 −0.752649 + 4.26848i 0 −2.02371 1.70429i 0 2.78728 + 1.10954i 0
5.2 0 −1.69135 + 0.373296i 0 −0.203680 + 1.15512i 0 2.02312 + 1.70498i 0 2.72130 1.26275i 0
5.3 0 −1.63320 + 0.576754i 0 0.109190 0.619249i 0 −2.49443 + 0.881937i 0 2.33471 1.88391i 0
5.4 0 −1.63242 + 0.578955i 0 0.649280 3.68225i 0 0.647047 2.56541i 0 2.32962 1.89020i 0
5.5 0 −1.61581 0.623825i 0 −0.221654 + 1.25706i 0 2.55927 0.670941i 0 2.22168 + 2.01597i 0
5.6 0 −1.47292 0.911314i 0 0.688949 3.90722i 0 0.608733 + 2.57477i 0 1.33902 + 2.68459i 0
5.7 0 −1.33105 1.10829i 0 0.180656 1.02455i 0 −1.77393 1.96295i 0 0.543390 + 2.95038i 0
5.8 0 −0.792956 + 1.53988i 0 −0.192493 + 1.09168i 0 −0.137506 2.64218i 0 −1.74244 2.44211i 0
5.9 0 −0.573178 1.63446i 0 −0.0962103 + 0.545636i 0 2.05240 1.66962i 0 −2.34293 + 1.87368i 0
5.10 0 −0.538039 + 1.64636i 0 −0.430811 + 2.44325i 0 −0.571962 + 2.58319i 0 −2.42103 1.77162i 0
5.11 0 −0.230632 1.71663i 0 0.315692 1.79038i 0 0.645961 + 2.56568i 0 −2.89362 + 0.791819i 0
5.12 0 −0.0439893 1.73149i 0 −0.310888 + 1.76313i 0 −1.91547 + 1.82510i 0 −2.99613 + 0.152334i 0
5.13 0 0.212510 + 1.71896i 0 0.455707 2.58444i 0 2.64406 0.0947211i 0 −2.90968 + 0.730593i 0
5.14 0 0.277790 + 1.70963i 0 0.630118 3.57358i 0 −2.41159 + 1.08823i 0 −2.84567 + 0.949836i 0
5.15 0 0.671662 + 1.59652i 0 −0.158784 + 0.900507i 0 −0.241705 2.63469i 0 −2.09774 + 2.14464i 0
5.16 0 0.727904 1.57167i 0 −0.591212 + 3.35293i 0 −1.25160 2.33098i 0 −1.94031 2.28805i 0
5.17 0 1.06196 1.36830i 0 0.449866 2.55131i 0 −2.53584 0.754675i 0 −0.744465 2.90616i 0
5.18 0 1.11789 + 1.32299i 0 −0.682386 + 3.87000i 0 2.62129 + 0.358920i 0 −0.500623 + 2.95793i 0
5.19 0 1.26963 + 1.17815i 0 0.263400 1.49382i 0 0.0912000 + 2.64418i 0 0.223916 + 2.99163i 0
5.20 0 1.37168 1.05759i 0 −0.403931 + 2.29081i 0 2.60966 + 0.435529i 0 0.762996 2.90135i 0
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 605.24 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.ba even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.ck.a yes 144
7.d odd 6 1 756.2.ca.a 144
27.f odd 18 1 756.2.ca.a 144
189.ba even 18 1 inner 756.2.ck.a yes 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.ca.a 144 7.d odd 6 1
756.2.ca.a 144 27.f odd 18 1
756.2.ck.a yes 144 1.a even 1 1 trivial
756.2.ck.a yes 144 189.ba even 18 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