# Properties

 Label 3150.2.bf.e Level 3150 Weight 2 Character orbit 3150.bf Analytic conductor 25.153 Analytic rank 0 Dimension 24 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.bf (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 12q^{4} + 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 12q^{4} + 4q^{7} - 12q^{16} + 12q^{19} - 4q^{28} - 28q^{37} - 96q^{43} - 8q^{46} - 52q^{49} + 12q^{52} - 8q^{58} - 12q^{61} - 24q^{64} + 4q^{67} + 12q^{73} + 4q^{79} + 68q^{91} - 24q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 1.52781 2.16005i 1.00000i 0 0
1151.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −2.34325 1.22849i 1.00000i 0 0
1151.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −0.295801 + 2.62916i 1.00000i 0 0
1151.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 0.397202 + 2.61577i 1.00000i 0 0
1151.5 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 2.43194 1.04195i 1.00000i 0 0
1151.6 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −0.717905 2.54649i 1.00000i 0 0
1151.7 0.866025 0.500000i 0 0.500000 0.866025i 0 0 −0.717905 2.54649i 1.00000i 0 0
1151.8 0.866025 0.500000i 0 0.500000 0.866025i 0 0 2.43194 1.04195i 1.00000i 0 0
1151.9 0.866025 0.500000i 0 0.500000 0.866025i 0 0 0.397202 + 2.61577i 1.00000i 0 0
1151.10 0.866025 0.500000i 0 0.500000 0.866025i 0 0 −0.295801 + 2.62916i 1.00000i 0 0
1151.11 0.866025 0.500000i 0 0.500000 0.866025i 0 0 −2.34325 1.22849i 1.00000i 0 0
1151.12 0.866025 0.500000i 0 0.500000 0.866025i 0 0 1.52781 2.16005i 1.00000i 0 0
1601.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 1.52781 + 2.16005i 1.00000i 0 0
1601.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 −2.34325 + 1.22849i 1.00000i 0 0
1601.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 −0.295801 2.62916i 1.00000i 0 0
1601.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 0.397202 2.61577i 1.00000i 0 0
1601.5 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 2.43194 + 1.04195i 1.00000i 0 0
1601.6 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 −0.717905 + 2.54649i 1.00000i 0 0
1601.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 −0.717905 + 2.54649i 1.00000i 0 0
1601.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 2.43194 + 1.04195i 1.00000i 0 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1601.12 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
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.2.bf.e yes 24
3.b odd 2 1 inner 3150.2.bf.e yes 24
5.b even 2 1 3150.2.bf.d 24
5.c odd 4 1 3150.2.bp.g 24
5.c odd 4 1 3150.2.bp.h 24
7.d odd 6 1 inner 3150.2.bf.e yes 24
15.d odd 2 1 3150.2.bf.d 24
15.e even 4 1 3150.2.bp.g 24
15.e even 4 1 3150.2.bp.h 24
21.g even 6 1 inner 3150.2.bf.e yes 24
35.i odd 6 1 3150.2.bf.d 24
35.k even 12 1 3150.2.bp.g 24
35.k even 12 1 3150.2.bp.h 24
105.p even 6 1 3150.2.bf.d 24
105.w odd 12 1 3150.2.bp.g 24
105.w odd 12 1 3150.2.bp.h 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3150.2.bf.d 24 5.b even 2 1
3150.2.bf.d 24 15.d odd 2 1
3150.2.bf.d 24 35.i odd 6 1
3150.2.bf.d 24 105.p even 6 1
3150.2.bf.e yes 24 1.a even 1 1 trivial
3150.2.bf.e yes 24 3.b odd 2 1 inner
3150.2.bf.e yes 24 7.d odd 6 1 inner
3150.2.bf.e yes 24 21.g even 6 1 inner
3150.2.bp.g 24 5.c odd 4 1
3150.2.bp.g 24 15.e even 4 1
3150.2.bp.g 24 35.k even 12 1
3150.2.bp.g 24 105.w odd 12 1
3150.2.bp.h 24 5.c odd 4 1
3150.2.bp.h 24 15.e even 4 1
3150.2.bp.h 24 35.k even 12 1
3150.2.bp.h 24 105.w odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3150, [\chi])$$:

 $$T_{11}^{24} - \cdots$$ $$T_{37}^{12} + \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database