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Properties

Label 1890.2.z.b
Level $1890$
Weight $2$
Character orbit 1890.z
Analytic conductor $15.092$
Analytic rank $0$
Dimension $24$
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(1009,1890)]
 
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([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1890.1009");
 
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.z (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: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{4} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{4} + 2 q^{5} - 4 q^{10} - 4 q^{11} + 12 q^{14} - 12 q^{16} + 32 q^{19} - 2 q^{20} + 16 q^{26} - 20 q^{29} + 36 q^{31} - 4 q^{35} - 2 q^{40} + 20 q^{41} - 8 q^{44} - 32 q^{46} + 12 q^{49} + 28 q^{55} - 12 q^{56} + 16 q^{61} - 24 q^{64} - 16 q^{65} - 2 q^{70} - 24 q^{71} - 44 q^{74} + 16 q^{76} - 68 q^{79} - 4 q^{80} + 26 q^{85} + 24 q^{86} - 40 q^{89} + 16 q^{91} - 48 q^{94} + 50 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.482247 − 2.18345i 0 −0.866025 + 0.500000i 1.00000i 0 1.50936 + 1.64980i
1009.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.95488 − 1.08556i 0 −0.866025 + 0.500000i 1.00000i 0 2.23576 − 0.0373163i
1009.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.674085 + 2.13204i 0 −0.866025 + 0.500000i 1.00000i 0 −1.64980 − 1.50936i
1009.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 2.06058 − 0.868341i 0 −0.866025 + 0.500000i 1.00000i 0 −1.35034 + 1.78229i
1009.5 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.15020 + 1.91756i 0 −0.866025 + 0.500000i 1.00000i 0 0.0373163 − 2.23576i
1009.6 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 2.21868 − 0.278284i 0 −0.866025 + 0.500000i 1.00000i 0 −1.78229 + 1.35034i
1009.7 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.278284 − 2.21868i 0 0.866025 − 0.500000i − 1.00000i 0 −1.35034 − 1.78229i
1009.8 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.08556 + 1.95488i 0 0.866025 − 0.500000i − 1.00000i 0 0.0373163 + 2.23576i
1009.9 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.91756 + 1.15020i 0 0.866025 − 0.500000i − 1.00000i 0 2.23576 + 0.0373163i
1009.10 0.866025 − 0.500000i 0 0.500000 − 0.866025i −2.18345 + 0.482247i 0 0.866025 − 0.500000i − 1.00000i 0 −1.64980 + 1.50936i
1009.11 0.866025 − 0.500000i 0 0.500000 − 0.866025i 2.13204 − 0.674085i 0 0.866025 − 0.500000i − 1.00000i 0 1.50936 − 1.64980i
1009.12 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.868341 − 2.06058i 0 0.866025 − 0.500000i − 1.00000i 0 −1.78229 − 1.35034i
1639.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.482247 + 2.18345i 0 −0.866025 − 0.500000i − 1.00000i 0 1.50936 − 1.64980i
1639.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.95488 + 1.08556i 0 −0.866025 − 0.500000i − 1.00000i 0 2.23576 + 0.0373163i
1639.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.674085 − 2.13204i 0 −0.866025 − 0.500000i − 1.00000i 0 −1.64980 + 1.50936i
1639.4 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 2.06058 + 0.868341i 0 −0.866025 − 0.500000i − 1.00000i 0 −1.35034 − 1.78229i
1639.5 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.15020 − 1.91756i 0 −0.866025 − 0.500000i − 1.00000i 0 0.0373163 + 2.23576i
1639.6 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 2.21868 + 0.278284i 0 −0.866025 − 0.500000i − 1.00000i 0 −1.78229 − 1.35034i
1639.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.278284 + 2.21868i 0 0.866025 + 0.500000i 1.00000i 0 −1.35034 + 1.78229i
1639.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.08556 − 1.95488i 0 0.866025 + 0.500000i 1.00000i 0 0.0373163 − 2.23576i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1009.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j 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.z.b 24
3.b odd 2 1 630.2.z.b ✓ 24
5.b even 2 1 inner 1890.2.z.b 24
9.c even 3 1 inner 1890.2.z.b 24
9.d odd 6 1 630.2.z.b ✓ 24
15.d odd 2 1 630.2.z.b ✓ 24
45.h odd 6 1 630.2.z.b ✓ 24
45.j even 6 1 inner 1890.2.z.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.z.b ✓ 24 3.b odd 2 1
630.2.z.b ✓ 24 9.d odd 6 1
630.2.z.b ✓ 24 15.d odd 2 1
630.2.z.b ✓ 24 45.h odd 6 1
1890.2.z.b 24 1.a even 1 1 trivial
1890.2.z.b 24 5.b even 2 1 inner
1890.2.z.b 24 9.c even 3 1 inner
1890.2.z.b 24 45.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{12} + 2 T_{11}^{11} + 27 T_{11}^{10} + 98 T_{11}^{9} + 700 T_{11}^{8} + 1788 T_{11}^{7} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(1890, [\chi])\). Copy content Toggle raw display