# Properties

 Label 756.2.e.b Level 756 Weight 2 Character orbit 756.e Analytic conductor 6.037 Analytic rank 0 Dimension 24 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.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 4q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 4q^{4} + 20q^{10} + 20q^{16} - 8q^{22} - 24q^{25} - 8q^{28} - 20q^{34} + 16q^{37} - 32q^{40} + 36q^{46} - 24q^{49} + 16q^{52} - 52q^{58} + 16q^{61} + 4q^{64} + 12q^{70} + 4q^{82} - 64q^{85} - 16q^{88} + 12q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
323.1 −1.40500 0.161188i 0 1.94804 + 0.452937i 4.14952i 0 1.00000i −2.66398 0.950375i 0 −0.668852 + 5.83006i
323.2 −1.40500 + 0.161188i 0 1.94804 0.452937i 4.14952i 0 1.00000i −2.66398 + 0.950375i 0 −0.668852 5.83006i
323.3 −1.38801 0.270968i 0 1.85315 + 0.752213i 1.41649i 0 1.00000i −2.36837 1.54622i 0 −0.383823 + 1.96611i
323.4 −1.38801 + 0.270968i 0 1.85315 0.752213i 1.41649i 0 1.00000i −2.36837 + 1.54622i 0 −0.383823 1.96611i
323.5 −1.21808 0.718530i 0 0.967429 + 1.75045i 1.77738i 0 1.00000i 0.0793484 2.82731i 0 1.27710 2.16499i
323.6 −1.21808 + 0.718530i 0 0.967429 1.75045i 1.77738i 0 1.00000i 0.0793484 + 2.82731i 0 1.27710 + 2.16499i
323.7 −0.932512 1.06321i 0 −0.260844 + 1.98292i 0.546170i 0 1.00000i 2.35150 1.57176i 0 −0.580695 + 0.509310i
323.8 −0.932512 + 1.06321i 0 −0.260844 1.98292i 0.546170i 0 1.00000i 2.35150 + 1.57176i 0 −0.580695 0.509310i
323.9 −0.492782 1.32558i 0 −1.51433 + 1.30645i 3.62883i 0 1.00000i 2.47803 + 1.36358i 0 4.81031 1.78822i
323.10 −0.492782 + 1.32558i 0 −1.51433 1.30645i 3.62883i 0 1.00000i 2.47803 1.36358i 0 4.81031 + 1.78822i
323.11 −0.0572568 1.41305i 0 −1.99344 + 0.161814i 0.386370i 0 1.00000i 0.342790 + 2.80758i 0 0.545962 0.0221223i
323.12 −0.0572568 + 1.41305i 0 −1.99344 0.161814i 0.386370i 0 1.00000i 0.342790 2.80758i 0 0.545962 + 0.0221223i
323.13 0.0572568 1.41305i 0 −1.99344 0.161814i 0.386370i 0 1.00000i −0.342790 + 2.80758i 0 0.545962 + 0.0221223i
323.14 0.0572568 + 1.41305i 0 −1.99344 + 0.161814i 0.386370i 0 1.00000i −0.342790 2.80758i 0 0.545962 0.0221223i
323.15 0.492782 1.32558i 0 −1.51433 1.30645i 3.62883i 0 1.00000i −2.47803 + 1.36358i 0 4.81031 + 1.78822i
323.16 0.492782 + 1.32558i 0 −1.51433 + 1.30645i 3.62883i 0 1.00000i −2.47803 1.36358i 0 4.81031 1.78822i
323.17 0.932512 1.06321i 0 −0.260844 1.98292i 0.546170i 0 1.00000i −2.35150 1.57176i 0 −0.580695 0.509310i
323.18 0.932512 + 1.06321i 0 −0.260844 + 1.98292i 0.546170i 0 1.00000i −2.35150 + 1.57176i 0 −0.580695 + 0.509310i
323.19 1.21808 0.718530i 0 0.967429 1.75045i 1.77738i 0 1.00000i −0.0793484 2.82731i 0 1.27710 + 2.16499i
323.20 1.21808 + 0.718530i 0 0.967429 + 1.75045i 1.77738i 0 1.00000i −0.0793484 + 2.82731i 0 1.27710 2.16499i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 323.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.e.b 24
3.b odd 2 1 inner 756.2.e.b 24
4.b odd 2 1 inner 756.2.e.b 24
12.b even 2 1 inner 756.2.e.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.e.b 24 1.a even 1 1 trivial
756.2.e.b 24 3.b odd 2 1 inner
756.2.e.b 24 4.b odd 2 1 inner
756.2.e.b 24 12.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} + 36 T_{5}^{10} + 406 T_{5}^{8} + 1540 T_{5}^{6} + 2065 T_{5}^{4} + 704 T_{5}^{2} + 64$$ acting on $$S_{2}^{\mathrm{new}}(756, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database