# Properties

 Label 756.2.bo.b Level 756 Weight 2 Character orbit 756.bo Analytic conductor 6.037 Analytic rank 0 Dimension 54 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.bo (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$54$$ Relative dimension: $$9$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$54q + 3q^{3} - 3q^{5} - 9q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$54q + 3q^{3} - 3q^{5} - 9q^{9} + 9q^{13} + 3q^{15} + 3q^{21} + 12q^{23} + 27q^{25} + 39q^{27} + 30q^{29} - 9q^{31} - 6q^{33} + 6q^{35} - 18q^{39} - 27q^{41} + 9q^{43} - 81q^{45} + 9q^{47} + 12q^{53} - 21q^{57} - 24q^{59} - 3q^{63} - 78q^{65} - 9q^{67} + 6q^{69} - 30q^{71} + 57q^{75} + 9q^{77} + 99q^{81} + 78q^{83} + 18q^{85} + 21q^{87} + 12q^{89} - 51q^{93} - 36q^{95} + 36q^{97} + 15q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
85.1 0 −1.72800 + 0.118316i 0 0.626647 3.55389i 0 −0.766044 + 0.642788i 0 2.97200 0.408902i 0
85.2 0 −1.46947 + 0.916867i 0 −0.137580 + 0.780255i 0 −0.766044 + 0.642788i 0 1.31871 2.69462i 0
85.3 0 −1.24763 1.20143i 0 0.123908 0.702715i 0 −0.766044 + 0.642788i 0 0.113154 + 2.99787i 0
85.4 0 −0.377044 + 1.69051i 0 0.491885 2.78962i 0 −0.766044 + 0.642788i 0 −2.71568 1.27480i 0
85.5 0 −0.368051 1.69249i 0 −0.269336 + 1.52748i 0 −0.766044 + 0.642788i 0 −2.72908 + 1.24585i 0
85.6 0 0.425197 + 1.67905i 0 −0.440759 + 2.49967i 0 −0.766044 + 0.642788i 0 −2.63841 + 1.42785i 0
85.7 0 0.794215 1.53923i 0 0.457658 2.59551i 0 −0.766044 + 0.642788i 0 −1.73845 2.44496i 0
85.8 0 1.64827 + 0.532173i 0 0.150583 0.853998i 0 −0.766044 + 0.642788i 0 2.43358 + 1.75433i 0
85.9 0 1.70918 0.280534i 0 −0.389665 + 2.20990i 0 −0.766044 + 0.642788i 0 2.84260 0.958966i 0
169.1 0 −1.72800 0.118316i 0 0.626647 + 3.55389i 0 −0.766044 0.642788i 0 2.97200 + 0.408902i 0
169.2 0 −1.46947 0.916867i 0 −0.137580 0.780255i 0 −0.766044 0.642788i 0 1.31871 + 2.69462i 0
169.3 0 −1.24763 + 1.20143i 0 0.123908 + 0.702715i 0 −0.766044 0.642788i 0 0.113154 2.99787i 0
169.4 0 −0.377044 1.69051i 0 0.491885 + 2.78962i 0 −0.766044 0.642788i 0 −2.71568 + 1.27480i 0
169.5 0 −0.368051 + 1.69249i 0 −0.269336 1.52748i 0 −0.766044 0.642788i 0 −2.72908 1.24585i 0
169.6 0 0.425197 1.67905i 0 −0.440759 2.49967i 0 −0.766044 0.642788i 0 −2.63841 1.42785i 0
169.7 0 0.794215 + 1.53923i 0 0.457658 + 2.59551i 0 −0.766044 0.642788i 0 −1.73845 + 2.44496i 0
169.8 0 1.64827 0.532173i 0 0.150583 + 0.853998i 0 −0.766044 0.642788i 0 2.43358 1.75433i 0
169.9 0 1.70918 + 0.280534i 0 −0.389665 2.20990i 0 −0.766044 0.642788i 0 2.84260 + 0.958966i 0
337.1 0 −1.70391 0.310978i 0 −2.38170 1.99849i 0 0.939693 + 0.342020i 0 2.80659 + 1.05975i 0
337.2 0 −1.09514 + 1.34189i 0 0.684464 + 0.574334i 0 0.939693 + 0.342020i 0 −0.601340 2.93911i 0
See all 54 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 673.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e 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.bo.b 54
27.e even 9 1 inner 756.2.bo.b 54

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.bo.b 54 1.a even 1 1 trivial
756.2.bo.b 54 27.e even 9 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{54} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(756, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database