# Properties

 Label 756.2.bo.a Level 756 Weight 2 Character orbit 756.bo Analytic conductor 6.037 Analytic rank 0 Dimension 54 CM no Inner twists 2

# Learn more about

## 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.

 $$\operatorname{Tr}(f)(q) =$$ $$54q - 3q^{3} - 3q^{5} + 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$54q - 3q^{3} - 3q^{5} + 3q^{9} - 9q^{13} + 3q^{15} - 3q^{21} - 45q^{25} + 15q^{27} + 6q^{29} - 9q^{31} + 6q^{33} + 6q^{35} + 30q^{39} + 9q^{41} + 9q^{43} - 21q^{45} - 27q^{47} - 108q^{51} - 84q^{53} - 57q^{57} - 66q^{59} + 3q^{63} + 30q^{65} + 63q^{67} + 42q^{69} + 42q^{71} + 3q^{75} - 9q^{77} + 36q^{79} + 3q^{81} + 12q^{83} + 18q^{85} + 39q^{87} + 12q^{89} + 21q^{93} + 120q^{95} + 18q^{97} + 39q^{99} + O(q^{100})$$

## 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 −1.72198 + 0.186471i 0 −0.509284 + 2.88830i 0 0.766044 0.642788i 0 2.93046 0.642199i 0
85.2 0 −1.57816 0.713728i 0 0.103874 0.589101i 0 0.766044 0.642788i 0 1.98118 + 2.25276i 0
85.3 0 −1.42882 + 0.979012i 0 −0.0286316 + 0.162378i 0 0.766044 0.642788i 0 1.08307 2.79767i 0
85.4 0 −0.576212 1.63340i 0 0.746163 4.23170i 0 0.766044 0.642788i 0 −2.33596 + 1.88236i 0
85.5 0 −0.550546 1.64222i 0 −0.678318 + 3.84693i 0 0.766044 0.642788i 0 −2.39380 + 1.80824i 0
85.6 0 −0.195914 + 1.72094i 0 0.348691 1.97752i 0 0.766044 0.642788i 0 −2.92324 0.674312i 0
85.7 0 1.33395 + 1.10480i 0 −0.457942 + 2.59712i 0 0.766044 0.642788i 0 0.558836 + 2.94749i 0
85.8 0 1.48997 0.883169i 0 −0.192063 + 1.08924i 0 0.766044 0.642788i 0 1.44003 2.63179i 0
85.9 0 1.61438 0.627515i 0 0.586259 3.32484i 0 0.766044 0.642788i 0 2.21245 2.02610i 0
169.1 0 −1.72198 0.186471i 0 −0.509284 2.88830i 0 0.766044 + 0.642788i 0 2.93046 + 0.642199i 0
169.2 0 −1.57816 + 0.713728i 0 0.103874 + 0.589101i 0 0.766044 + 0.642788i 0 1.98118 2.25276i 0
169.3 0 −1.42882 0.979012i 0 −0.0286316 0.162378i 0 0.766044 + 0.642788i 0 1.08307 + 2.79767i 0
169.4 0 −0.576212 + 1.63340i 0 0.746163 + 4.23170i 0 0.766044 + 0.642788i 0 −2.33596 1.88236i 0
169.5 0 −0.550546 + 1.64222i 0 −0.678318 3.84693i 0 0.766044 + 0.642788i 0 −2.39380 1.80824i 0
169.6 0 −0.195914 1.72094i 0 0.348691 + 1.97752i 0 0.766044 + 0.642788i 0 −2.92324 + 0.674312i 0
169.7 0 1.33395 1.10480i 0 −0.457942 2.59712i 0 0.766044 + 0.642788i 0 0.558836 2.94749i 0
169.8 0 1.48997 + 0.883169i 0 −0.192063 1.08924i 0 0.766044 + 0.642788i 0 1.44003 + 2.63179i 0
169.9 0 1.61438 + 0.627515i 0 0.586259 + 3.32484i 0 0.766044 + 0.642788i 0 2.21245 + 2.02610i 0
337.1 0 −1.69816 + 0.340939i 0 −2.75833 2.31451i 0 −0.939693 0.342020i 0 2.76752 1.15794i 0
337.2 0 −1.55705 0.758672i 0 0.267700 + 0.224627i 0 −0.939693 0.342020i 0 1.84883 + 2.36259i 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.a 54
27.e even 9 1 inner 756.2.bo.a 54

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