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

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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:

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