# Properties

 Label 3150.2.bp.g Level 3150 Weight 2 Character orbit 3150.bp 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.bp (of order $$6$$, degree $$2$$, not 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 12q^{2} - 12q^{4} + 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 12q^{2} - 12q^{4} + 24q^{8} - 12q^{16} + 24q^{17} - 12q^{19} - 8q^{23} - 12q^{32} + 12q^{38} - 8q^{46} - 24q^{47} + 52q^{49} - 32q^{53} - 12q^{61} + 24q^{64} - 24q^{68} - 16q^{77} - 4q^{79} + 68q^{91} + 16q^{92} + 24q^{94} - 20q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
899.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −2.62916 0.295801i 1.00000 0 0
899.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −2.61577 + 0.397202i 1.00000 0 0
899.3 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −2.54649 + 0.717905i 1.00000 0 0
899.4 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −2.16005 1.52781i 1.00000 0 0
899.5 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −1.22849 + 2.34325i 1.00000 0 0
899.6 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −1.04195 2.43194i 1.00000 0 0
899.7 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 1.04195 + 2.43194i 1.00000 0 0
899.8 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 1.22849 2.34325i 1.00000 0 0
899.9 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 2.16005 + 1.52781i 1.00000 0 0
899.10 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 2.54649 0.717905i 1.00000 0 0
899.11 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 2.61577 0.397202i 1.00000 0 0
899.12 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 2.62916 + 0.295801i 1.00000 0 0
1349.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −2.62916 + 0.295801i 1.00000 0 0
1349.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −2.61577 0.397202i 1.00000 0 0
1349.3 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −2.54649 0.717905i 1.00000 0 0
1349.4 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −2.16005 + 1.52781i 1.00000 0 0
1349.5 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −1.22849 2.34325i 1.00000 0 0
1349.6 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −1.04195 + 2.43194i 1.00000 0 0
1349.7 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 1.04195 2.43194i 1.00000 0 0
1349.8 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 1.22849 + 2.34325i 1.00000 0 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1349.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

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

## Hecke kernels

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

 $$T_{11}^{24} - \cdots$$ $$T_{13}^{12} - 114 T_{13}^{10} + 4937 T_{13}^{8} - 100896 T_{13}^{6} + 969088 T_{13}^{4} - 3581952 T_{13}^{2} + 1806336$$ $$T_{17}^{12} - \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database