# Properties

 Label 3150.2.bf.f Level 3150 Weight 2 Character orbit 3150.bf Analytic conductor 25.153 Analytic rank 0 Dimension 32 CM no Inner twists 8

# 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: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 630) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{4} - 16q^{16} - 48q^{19} + 24q^{31} - 16q^{46} + 56q^{49} + 48q^{61} - 32q^{64} - 8q^{79} - 56q^{91} + 120q^{94} + O(q^{100})$$

## 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 −2.54232 0.732536i 1.00000i 0 0
1151.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 2.54232 + 0.732536i 1.00000i 0 0
1151.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −2.07665 1.63937i 1.00000i 0 0
1151.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 2.07665 + 1.63937i 1.00000i 0 0
1151.5 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −2.24547 + 1.39924i 1.00000i 0 0
1151.6 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 2.24547 1.39924i 1.00000i 0 0
1151.7 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −1.29693 + 2.30608i 1.00000i 0 0
1151.8 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 1.29693 2.30608i 1.00000i 0 0
1151.9 0.866025 0.500000i 0 0.500000 0.866025i 0 0 −1.29693 + 2.30608i 1.00000i 0 0
1151.10 0.866025 0.500000i 0 0.500000 0.866025i 0 0 1.29693 2.30608i 1.00000i 0 0
1151.11 0.866025 0.500000i 0 0.500000 0.866025i 0 0 −2.54232 0.732536i 1.00000i 0 0
1151.12 0.866025 0.500000i 0 0.500000 0.866025i 0 0 2.54232 + 0.732536i 1.00000i 0 0
1151.13 0.866025 0.500000i 0 0.500000 0.866025i 0 0 −2.07665 1.63937i 1.00000i 0 0
1151.14 0.866025 0.500000i 0 0.500000 0.866025i 0 0 2.07665 + 1.63937i 1.00000i 0 0
1151.15 0.866025 0.500000i 0 0.500000 0.866025i 0 0 −2.24547 + 1.39924i 1.00000i 0 0
1151.16 0.866025 0.500000i 0 0.500000 0.866025i 0 0 2.24547 1.39924i 1.00000i 0 0
1601.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 −2.54232 + 0.732536i 1.00000i 0 0
1601.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 2.54232 0.732536i 1.00000i 0 0
1601.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 −2.07665 + 1.63937i 1.00000i 0 0
1601.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 2.07665 1.63937i 1.00000i 0 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1601.16 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
5.b even 2 1 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 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.bf.f 32
3.b odd 2 1 inner 3150.2.bf.f 32
5.b even 2 1 inner 3150.2.bf.f 32
5.c odd 4 1 630.2.bo.a 16
5.c odd 4 1 630.2.bo.b yes 16
7.d odd 6 1 inner 3150.2.bf.f 32
15.d odd 2 1 inner 3150.2.bf.f 32
15.e even 4 1 630.2.bo.a 16
15.e even 4 1 630.2.bo.b yes 16
21.g even 6 1 inner 3150.2.bf.f 32
35.i odd 6 1 inner 3150.2.bf.f 32
35.k even 12 1 630.2.bo.a 16
35.k even 12 1 630.2.bo.b yes 16
35.k even 12 1 4410.2.d.a 16
35.k even 12 1 4410.2.d.b 16
35.l odd 12 1 4410.2.d.a 16
35.l odd 12 1 4410.2.d.b 16
105.p even 6 1 inner 3150.2.bf.f 32
105.w odd 12 1 630.2.bo.a 16
105.w odd 12 1 630.2.bo.b yes 16
105.w odd 12 1 4410.2.d.a 16
105.w odd 12 1 4410.2.d.b 16
105.x even 12 1 4410.2.d.a 16
105.x even 12 1 4410.2.d.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bo.a 16 5.c odd 4 1
630.2.bo.a 16 15.e even 4 1
630.2.bo.a 16 35.k even 12 1
630.2.bo.a 16 105.w odd 12 1
630.2.bo.b yes 16 5.c odd 4 1
630.2.bo.b yes 16 15.e even 4 1
630.2.bo.b yes 16 35.k even 12 1
630.2.bo.b yes 16 105.w odd 12 1
3150.2.bf.f 32 1.a even 1 1 trivial
3150.2.bf.f 32 3.b odd 2 1 inner
3150.2.bf.f 32 5.b even 2 1 inner
3150.2.bf.f 32 7.d odd 6 1 inner
3150.2.bf.f 32 15.d odd 2 1 inner
3150.2.bf.f 32 21.g even 6 1 inner
3150.2.bf.f 32 35.i odd 6 1 inner
3150.2.bf.f 32 105.p even 6 1 inner
4410.2.d.a 16 35.k even 12 1
4410.2.d.a 16 35.l odd 12 1
4410.2.d.a 16 105.w odd 12 1
4410.2.d.a 16 105.x even 12 1
4410.2.d.b 16 35.k even 12 1
4410.2.d.b 16 35.l odd 12 1
4410.2.d.b 16 105.w odd 12 1
4410.2.d.b 16 105.x even 12 1

## Hecke kernels

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

 $$T_{11}^{16} - \cdots$$ $$T_{37}^{16} + \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database