# Properties

 Label 756.2.bf.d Level 756 Weight 2 Character orbit 756.bf Analytic conductor 6.037 Analytic rank 0 Dimension 32 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 2q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 2q^{7} + 4q^{10} + 20q^{16} - 6q^{19} + 20q^{22} + 20q^{25} - 24q^{28} + 8q^{34} - 2q^{37} + 52q^{40} + 24q^{46} - 10q^{49} + 16q^{52} + 16q^{55} - 80q^{58} + 48q^{64} + 42q^{67} + 32q^{70} - 18q^{73} - 40q^{76} - 6q^{79} + 8q^{82} - 8q^{85} - 80q^{88} + 8q^{91} - 8q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1 −1.41359 + 0.0418553i 0 1.99650 0.118333i 1.51137 + 0.872592i 0 1.00609 2.44699i −2.81728 + 0.250838i 0 −2.17299 1.17023i
271.2 −1.38947 0.263404i 0 1.86124 + 0.731982i −3.59250 2.07413i 0 −2.54686 + 0.716577i −2.39332 1.50732i 0 4.44533 + 3.82822i
271.3 −1.28874 0.582370i 0 1.32169 + 1.50104i 1.99770 + 1.15337i 0 0.0928632 + 2.64412i −0.829152 2.70416i 0 −1.90282 2.64979i
271.4 −1.15218 + 0.820056i 0 0.655017 1.88970i −2.47070 1.42646i 0 2.58673 0.555723i 0.794961 + 2.71441i 0 4.01646 0.382580i
271.5 −0.691316 1.23373i 0 −1.04416 + 1.70579i 0.766614 + 0.442605i 0 −2.19153 1.48229i 2.82633 + 0.108975i 0 0.0160812 1.25177i
271.6 −0.577244 + 1.29104i 0 −1.33358 1.49049i 3.11886 + 1.80067i 0 −0.838804 2.50926i 2.69409 0.861330i 0 −4.12508 + 2.98715i
271.7 −0.444821 + 1.34244i 0 −1.60427 1.19429i −1.00309 0.579135i 0 −0.250854 + 2.63383i 2.31687 1.62238i 0 1.22365 1.08897i
271.8 −0.271635 1.38788i 0 −1.85243 + 0.753993i −0.945181 0.545701i 0 2.64237 + 0.133712i 1.54964 + 2.36614i 0 −0.500624 + 1.46003i
271.9 0.271635 + 1.38788i 0 −1.85243 + 0.753993i 0.945181 + 0.545701i 0 2.64237 + 0.133712i −1.54964 2.36614i 0 −0.500624 + 1.46003i
271.10 0.444821 1.34244i 0 −1.60427 1.19429i 1.00309 + 0.579135i 0 −0.250854 + 2.63383i −2.31687 + 1.62238i 0 1.22365 1.08897i
271.11 0.577244 1.29104i 0 −1.33358 1.49049i −3.11886 1.80067i 0 −0.838804 2.50926i −2.69409 + 0.861330i 0 −4.12508 + 2.98715i
271.12 0.691316 + 1.23373i 0 −1.04416 + 1.70579i −0.766614 0.442605i 0 −2.19153 1.48229i −2.82633 0.108975i 0 0.0160812 1.25177i
271.13 1.15218 0.820056i 0 0.655017 1.88970i 2.47070 + 1.42646i 0 2.58673 0.555723i −0.794961 2.71441i 0 4.01646 0.382580i
271.14 1.28874 + 0.582370i 0 1.32169 + 1.50104i −1.99770 1.15337i 0 0.0928632 + 2.64412i 0.829152 + 2.70416i 0 −1.90282 2.64979i
271.15 1.38947 + 0.263404i 0 1.86124 + 0.731982i 3.59250 + 2.07413i 0 −2.54686 + 0.716577i 2.39332 + 1.50732i 0 4.44533 + 3.82822i
271.16 1.41359 0.0418553i 0 1.99650 0.118333i −1.51137 0.872592i 0 1.00609 2.44699i 2.81728 0.250838i 0 −2.17299 1.17023i
703.1 −1.41359 0.0418553i 0 1.99650 + 0.118333i 1.51137 0.872592i 0 1.00609 + 2.44699i −2.81728 0.250838i 0 −2.17299 + 1.17023i
703.2 −1.38947 + 0.263404i 0 1.86124 0.731982i −3.59250 + 2.07413i 0 −2.54686 0.716577i −2.39332 + 1.50732i 0 4.44533 3.82822i
703.3 −1.28874 + 0.582370i 0 1.32169 1.50104i 1.99770 1.15337i 0 0.0928632 2.64412i −0.829152 + 2.70416i 0 −1.90282 + 2.64979i
703.4 −1.15218 0.820056i 0 0.655017 + 1.88970i −2.47070 + 1.42646i 0 2.58673 + 0.555723i 0.794961 2.71441i 0 4.01646 + 0.382580i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 703.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.f even 6 1 inner
84.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.bf.d yes 32
3.b odd 2 1 inner 756.2.bf.d yes 32
4.b odd 2 1 756.2.bf.a 32
7.d odd 6 1 756.2.bf.a 32
12.b even 2 1 756.2.bf.a 32
21.g even 6 1 756.2.bf.a 32
28.f even 6 1 inner 756.2.bf.d yes 32
84.j odd 6 1 inner 756.2.bf.d yes 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.bf.a 32 4.b odd 2 1
756.2.bf.a 32 7.d odd 6 1
756.2.bf.a 32 12.b even 2 1
756.2.bf.a 32 21.g even 6 1
756.2.bf.d yes 32 1.a even 1 1 trivial
756.2.bf.d yes 32 3.b odd 2 1 inner
756.2.bf.d yes 32 28.f even 6 1 inner
756.2.bf.d yes 32 84.j odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(756, [\chi])$$:

 $$T_{5}^{32} - \cdots$$ $$T_{11}^{32} - \cdots$$ $$T_{19}^{16} + \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database