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Properties

Label 1170.2.bp.i
Level $1170$
Weight $2$
Character orbit 1170.bp
Analytic conductor $9.342$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1170,2,Mod(289,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1170.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.bp (of order \(6\), degree \(2\), 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: \(9.34249703649\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{4} + 2 q^{10} - 12 q^{16} + 20 q^{19} - 28 q^{25} - 24 q^{31} + 40 q^{34} + 4 q^{40} - 12 q^{46} - 8 q^{49} - 12 q^{55} + 68 q^{61} - 24 q^{64} + 48 q^{70} - 20 q^{76} + 40 q^{79} + 70 q^{85} + 44 q^{91} - 40 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −2.19196 − 0.441924i 0 0.790182 + 0.456212i 1.00000i 0 2.11926 − 0.713264i
289.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 2.19196 − 0.441924i 0 −0.790182 − 0.456212i 1.00000i 0 −1.67733 + 1.47870i
289.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.590837 − 2.15660i 0 2.93152 + 1.69251i 1.00000i 0 1.58998 + 1.57225i
289.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.590837 − 2.15660i 0 −2.93152 − 1.69251i 1.00000i 0 0.566619 + 2.16309i
289.5 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.772145 + 2.09852i 0 2.24317 + 1.29509i 1.00000i 0 −0.380563 − 2.20345i
289.6 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.772145 + 2.09852i 0 −2.24317 − 1.29509i 1.00000i 0 −1.71796 − 1.43130i
289.7 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.772145 − 2.09852i 0 −2.24317 − 1.29509i − 1.00000i 0 −1.71796 − 1.43130i
289.8 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.772145 − 2.09852i 0 2.24317 + 1.29509i − 1.00000i 0 −0.380563 − 2.20345i
289.9 0.866025 − 0.500000i 0 0.500000 − 0.866025i −2.19196 + 0.441924i 0 −0.790182 − 0.456212i − 1.00000i 0 −1.67733 + 1.47870i
289.10 0.866025 − 0.500000i 0 0.500000 − 0.866025i 2.19196 + 0.441924i 0 0.790182 + 0.456212i − 1.00000i 0 2.11926 − 0.713264i
289.11 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.590837 + 2.15660i 0 −2.93152 − 1.69251i − 1.00000i 0 0.566619 + 2.16309i
289.12 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.590837 + 2.15660i 0 2.93152 + 1.69251i − 1.00000i 0 1.58998 + 1.57225i
919.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −2.19196 + 0.441924i 0 0.790182 − 0.456212i − 1.00000i 0 2.11926 + 0.713264i
919.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 2.19196 + 0.441924i 0 −0.790182 + 0.456212i − 1.00000i 0 −1.67733 − 1.47870i
919.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.590837 + 2.15660i 0 2.93152 − 1.69251i − 1.00000i 0 1.58998 − 1.57225i
919.4 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.590837 + 2.15660i 0 −2.93152 + 1.69251i − 1.00000i 0 0.566619 − 2.16309i
919.5 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.772145 − 2.09852i 0 2.24317 − 1.29509i − 1.00000i 0 −0.380563 + 2.20345i
919.6 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.772145 − 2.09852i 0 −2.24317 + 1.29509i − 1.00000i 0 −1.71796 + 1.43130i
919.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.772145 + 2.09852i 0 −2.24317 + 1.29509i 1.00000i 0 −1.71796 + 1.43130i
919.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.772145 + 2.09852i 0 2.24317 − 1.29509i 1.00000i 0 −0.380563 + 2.20345i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.12
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
13.c even 3 1 inner
15.d odd 2 1 inner
39.i odd 6 1 inner
65.n even 6 1 inner
195.x odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.bp.i ✓ 24
3.b odd 2 1 inner 1170.2.bp.i ✓ 24
5.b even 2 1 inner 1170.2.bp.i ✓ 24
13.c even 3 1 inner 1170.2.bp.i ✓ 24
15.d odd 2 1 inner 1170.2.bp.i ✓ 24
39.i odd 6 1 inner 1170.2.bp.i ✓ 24
65.n even 6 1 inner 1170.2.bp.i ✓ 24
195.x odd 6 1 inner 1170.2.bp.i ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.2.bp.i ✓ 24 1.a even 1 1 trivial
1170.2.bp.i ✓ 24 3.b odd 2 1 inner
1170.2.bp.i ✓ 24 5.b even 2 1 inner
1170.2.bp.i ✓ 24 13.c even 3 1 inner
1170.2.bp.i ✓ 24 15.d odd 2 1 inner
1170.2.bp.i ✓ 24 39.i odd 6 1 inner
1170.2.bp.i ✓ 24 65.n even 6 1 inner
1170.2.bp.i ✓ 24 195.x odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} - 19T_{7}^{10} + 269T_{7}^{8} - 1620T_{7}^{6} + 7248T_{7}^{4} - 5888T_{7}^{2} + 4096 \) acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\). Copy content Toggle raw display