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Properties

Label 1260.2.dc.a
Level $1260$
Weight $2$
Character orbit 1260.dc
Analytic conductor $10.061$
Analytic rank $0$
Dimension $32$
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] = [1260,2,Mod(89,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.dc (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: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 24 q^{19} - 12 q^{25} + 24 q^{31} + 40 q^{49} - 24 q^{61} + 32 q^{79} + 32 q^{85} - 56 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1 0 0 0 −2.14792 + 0.621638i 0 −2.32992 + 1.25359i 0 0 0
89.2 0 0 0 −2.03329 − 0.930453i 0 −2.30504 − 1.29877i 0 0 0
89.3 0 0 0 −1.84826 + 1.25855i 0 −0.906070 − 2.48577i 0 0 0
89.4 0 0 0 −1.82244 − 1.29565i 0 2.30504 + 1.29877i 0 0 0
89.5 0 0 0 −1.48314 + 1.67341i 0 2.22201 − 1.43620i 0 0 0
89.6 0 0 0 −0.707646 + 2.12114i 0 −2.22201 + 1.43620i 0 0 0
89.7 0 0 0 −0.535606 − 2.17097i 0 2.32992 − 1.25359i 0 0 0
89.8 0 0 0 −0.165802 + 2.22991i 0 0.906070 + 2.48577i 0 0 0
89.9 0 0 0 0.165802 − 2.22991i 0 0.906070 + 2.48577i 0 0 0
89.10 0 0 0 0.535606 + 2.17097i 0 2.32992 − 1.25359i 0 0 0
89.11 0 0 0 0.707646 − 2.12114i 0 −2.22201 + 1.43620i 0 0 0
89.12 0 0 0 1.48314 − 1.67341i 0 2.22201 − 1.43620i 0 0 0
89.13 0 0 0 1.82244 + 1.29565i 0 2.30504 + 1.29877i 0 0 0
89.14 0 0 0 1.84826 − 1.25855i 0 −0.906070 − 2.48577i 0 0 0
89.15 0 0 0 2.03329 + 0.930453i 0 −2.30504 − 1.29877i 0 0 0
89.16 0 0 0 2.14792 − 0.621638i 0 −2.32992 + 1.25359i 0 0 0
269.1 0 0 0 −2.14792 − 0.621638i 0 −2.32992 − 1.25359i 0 0 0
269.2 0 0 0 −2.03329 + 0.930453i 0 −2.30504 + 1.29877i 0 0 0
269.3 0 0 0 −1.84826 − 1.25855i 0 −0.906070 + 2.48577i 0 0 0
269.4 0 0 0 −1.82244 + 1.29565i 0 2.30504 − 1.29877i 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.16
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 1260.2.dc.a ✓ 32
3.b odd 2 1 inner 1260.2.dc.a ✓ 32
5.b even 2 1 inner 1260.2.dc.a ✓ 32
5.c odd 4 2 6300.2.ch.f 32
7.c even 3 1 8820.2.f.a 32
7.d odd 6 1 inner 1260.2.dc.a ✓ 32
7.d odd 6 1 8820.2.f.a 32
15.d odd 2 1 inner 1260.2.dc.a ✓ 32
15.e even 4 2 6300.2.ch.f 32
21.g even 6 1 inner 1260.2.dc.a ✓ 32
21.g even 6 1 8820.2.f.a 32
21.h odd 6 1 8820.2.f.a 32
35.i odd 6 1 inner 1260.2.dc.a ✓ 32
35.i odd 6 1 8820.2.f.a 32
35.j even 6 1 8820.2.f.a 32
35.k even 12 2 6300.2.ch.f 32
105.o odd 6 1 8820.2.f.a 32
105.p even 6 1 inner 1260.2.dc.a ✓ 32
105.p even 6 1 8820.2.f.a 32
105.w odd 12 2 6300.2.ch.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.dc.a ✓ 32 1.a even 1 1 trivial
1260.2.dc.a ✓ 32 3.b odd 2 1 inner
1260.2.dc.a ✓ 32 5.b even 2 1 inner
1260.2.dc.a ✓ 32 7.d odd 6 1 inner
1260.2.dc.a ✓ 32 15.d odd 2 1 inner
1260.2.dc.a ✓ 32 21.g even 6 1 inner
1260.2.dc.a ✓ 32 35.i odd 6 1 inner
1260.2.dc.a ✓ 32 105.p even 6 1 inner
6300.2.ch.f 32 5.c odd 4 2
6300.2.ch.f 32 15.e even 4 2
6300.2.ch.f 32 35.k even 12 2
6300.2.ch.f 32 105.w odd 12 2
8820.2.f.a 32 7.c even 3 1
8820.2.f.a 32 7.d odd 6 1
8820.2.f.a 32 21.g even 6 1
8820.2.f.a 32 21.h odd 6 1
8820.2.f.a 32 35.i odd 6 1
8820.2.f.a 32 35.j even 6 1
8820.2.f.a 32 105.o odd 6 1
8820.2.f.a 32 105.p even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1260, [\chi])\).