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Properties

Label 3150.2.bp.g
Level $3150$
Weight $2$
Character orbit 3150.bp
Analytic conductor $25.153$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3150,2,Mod(899,3150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3150, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3150.899"); S:= CuspForms(chi, 2); N := Newforms(S);
 
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

Copy content comment:select newform
 
Copy content sage:traces = [24,-12,0,-12,0,0,0,24,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{2} - 12 q^{4} + 24 q^{8} - 12 q^{16} + 24 q^{17} - 12 q^{19} - 8 q^{23} - 12 q^{32} + 12 q^{38} - 8 q^{46} - 24 q^{47} + 52 q^{49} - 32 q^{53} - 12 q^{61} + 24 q^{64} - 24 q^{68} - 16 q^{77}+ \cdots - 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
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 899.12
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(3150, [\chi])\):

\( T_{11}^{24} - 68 T_{11}^{22} + 3232 T_{11}^{20} - 77312 T_{11}^{18} + 1330444 T_{11}^{16} + \cdots + 1679616 \) Copy content Toggle raw display
\( T_{13}^{12} - 114T_{13}^{10} + 4937T_{13}^{8} - 100896T_{13}^{6} + 969088T_{13}^{4} - 3581952T_{13}^{2} + 1806336 \) Copy content Toggle raw display
\( T_{17}^{12} - 12 T_{17}^{11} + 32 T_{17}^{10} + 192 T_{17}^{9} - 861 T_{17}^{8} - 2832 T_{17}^{7} + \cdots + 142884 \) Copy content Toggle raw display