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Properties

Label 1890.2.bf.f
Level $1890$
Weight $2$
Character orbit 1890.bf
Analytic conductor $15.092$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1890,2,Mod(629,1890)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1890, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1890.629");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1890.bf (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.0917259820\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{2} - 16 q^{4} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{2} - 16 q^{4} - 32 q^{8} + 24 q^{11} - 16 q^{16} + 24 q^{22} + 24 q^{23} - 58 q^{25} + 36 q^{29} + 16 q^{32} + 48 q^{35} - 54 q^{43} + 48 q^{46} + 32 q^{49} - 50 q^{50} + 24 q^{53} + 36 q^{58} + 32 q^{64} + 90 q^{65} - 66 q^{67} + 36 q^{70} - 12 q^{74} - 18 q^{77} + 34 q^{79} + 4 q^{85} - 54 q^{86} - 24 q^{88} + 16 q^{91} + 24 q^{92} - 12 q^{95} + 64 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
629.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −2.23011 + 0.163074i 0 −2.62496 + 0.331028i −1.00000 0 −0.973830 + 2.01287i
629.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.55878 + 1.60318i 0 2.64557 − 0.0312377i −1.00000 0 0.609005 + 2.15154i
629.3 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.41003 − 1.73546i 0 −0.843817 + 2.50758i −1.00000 0 −2.20797 + 0.353395i
629.4 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.22443 − 1.87103i 0 2.45843 + 0.977807i −1.00000 0 −2.23258 + 0.124872i
629.5 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.07274 + 1.96195i 0 −0.858483 − 2.50260i −1.00000 0 1.16272 + 1.90999i
629.6 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.693615 + 2.12577i 0 −2.50039 − 0.864900i −1.00000 0 1.49416 + 1.66357i
629.7 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.452503 + 2.18980i 0 2.02132 + 1.70712i −1.00000 0 1.67017 + 1.48678i
629.8 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.150780 + 2.23098i 0 0.00309790 − 2.64575i −1.00000 0 1.85669 + 1.24607i
629.9 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.150780 − 2.23098i 0 −2.28974 + 1.32556i −1.00000 0 −1.85669 − 1.24607i
629.10 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.452503 − 2.18980i 0 2.48907 + 0.896960i −1.00000 0 −1.67017 − 1.48678i
629.11 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.693615 − 2.12577i 0 −1.99922 − 1.73295i −1.00000 0 −1.49416 − 1.66357i
629.12 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.07274 − 1.96195i 0 −2.59656 + 0.507832i −1.00000 0 −1.16272 − 1.90999i
629.13 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.22443 + 1.87103i 0 2.07602 + 1.64016i −1.00000 0 2.23258 − 0.124872i
629.14 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.41003 + 1.73546i 0 1.74972 − 1.98456i −1.00000 0 2.20797 − 0.353395i
629.15 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.55878 − 1.60318i 0 1.29573 + 2.30675i −1.00000 0 −0.609005 − 2.15154i
629.16 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 2.23011 − 0.163074i 0 −1.02580 − 2.43880i −1.00000 0 0.973830 − 2.01287i
1259.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.23011 − 0.163074i 0 −2.62496 − 0.331028i −1.00000 0 −0.973830 − 2.01287i
1259.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.55878 − 1.60318i 0 2.64557 + 0.0312377i −1.00000 0 0.609005 − 2.15154i
1259.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.41003 + 1.73546i 0 −0.843817 − 2.50758i −1.00000 0 −2.20797 − 0.353395i
1259.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.22443 + 1.87103i 0 2.45843 − 0.977807i −1.00000 0 −2.23258 − 0.124872i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 629.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
45.h odd 6 1 inner
315.z even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1890.2.bf.f 32
3.b odd 2 1 630.2.bf.e ✓ 32
5.b even 2 1 1890.2.bf.e 32
7.b odd 2 1 inner 1890.2.bf.f 32
9.c even 3 1 630.2.bf.f yes 32
9.d odd 6 1 1890.2.bf.e 32
15.d odd 2 1 630.2.bf.f yes 32
21.c even 2 1 630.2.bf.e ✓ 32
35.c odd 2 1 1890.2.bf.e 32
45.h odd 6 1 inner 1890.2.bf.f 32
45.j even 6 1 630.2.bf.e ✓ 32
63.l odd 6 1 630.2.bf.f yes 32
63.o even 6 1 1890.2.bf.e 32
105.g even 2 1 630.2.bf.f yes 32
315.z even 6 1 inner 1890.2.bf.f 32
315.bg odd 6 1 630.2.bf.e ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bf.e ✓ 32 3.b odd 2 1
630.2.bf.e ✓ 32 21.c even 2 1
630.2.bf.e ✓ 32 45.j even 6 1
630.2.bf.e ✓ 32 315.bg odd 6 1
630.2.bf.f yes 32 9.c even 3 1
630.2.bf.f yes 32 15.d odd 2 1
630.2.bf.f yes 32 63.l odd 6 1
630.2.bf.f yes 32 105.g even 2 1
1890.2.bf.e 32 5.b even 2 1
1890.2.bf.e 32 9.d odd 6 1
1890.2.bf.e 32 35.c odd 2 1
1890.2.bf.e 32 63.o even 6 1
1890.2.bf.f 32 1.a even 1 1 trivial
1890.2.bf.f 32 7.b odd 2 1 inner
1890.2.bf.f 32 45.h odd 6 1 inner
1890.2.bf.f 32 315.z even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1890, [\chi])\):

\( T_{11}^{16} - 12 T_{11}^{15} + 7 T_{11}^{14} + 492 T_{11}^{13} - 830 T_{11}^{12} - 21498 T_{11}^{11} + \cdots + 239754256 \) Copy content Toggle raw display
\( T_{13}^{32} + 131 T_{13}^{30} + 10509 T_{13}^{28} + 533244 T_{13}^{26} + 19752852 T_{13}^{24} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
\( T_{23}^{16} - 12 T_{23}^{15} + 177 T_{23}^{14} - 1156 T_{23}^{13} + 11169 T_{23}^{12} + \cdots + 2958924816 \) Copy content Toggle raw display