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Properties

Label 1232.2.bi.c
Level $1232$
Weight $2$
Character orbit 1232.bi
Analytic conductor $9.838$
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] = [1232,2,Mod(527,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.527");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.bi (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: \(9.83756952902\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{5} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{5} + 24 q^{9} - 12 q^{25} + 2 q^{33} + 12 q^{37} + 20 q^{45} + 40 q^{49} + 24 q^{53} + 144 q^{69} - 34 q^{77} + 64 q^{89} - 4 q^{93} + 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
527.1 0 −2.51905 + 1.45437i 0 −1.30811 + 2.26571i 0 −2.58217 + 0.576531i 0 2.73040 − 4.72920i 0
527.2 0 −2.51905 + 1.45437i 0 −1.30811 + 2.26571i 0 2.58217 − 0.576531i 0 2.73040 − 4.72920i 0
527.3 0 −2.28410 + 1.31873i 0 1.68232 − 2.91387i 0 −1.12400 − 2.39512i 0 1.97809 − 3.42615i 0
527.4 0 −2.28410 + 1.31873i 0 1.68232 − 2.91387i 0 1.12400 + 2.39512i 0 1.97809 − 3.42615i 0
527.5 0 −1.35591 + 0.782834i 0 −0.711940 + 1.23312i 0 1.78850 + 1.94969i 0 −0.274341 + 0.475173i 0
527.6 0 −1.35591 + 0.782834i 0 −0.711940 + 1.23312i 0 −1.78850 − 1.94969i 0 −0.274341 + 0.475173i 0
527.7 0 −0.314280 + 0.181449i 0 0.837728 − 1.45099i 0 −2.31739 + 1.27660i 0 −1.43415 + 2.48402i 0
527.8 0 −0.314280 + 0.181449i 0 0.837728 − 1.45099i 0 2.31739 − 1.27660i 0 −1.43415 + 2.48402i 0
527.9 0 0.314280 − 0.181449i 0 0.837728 − 1.45099i 0 2.31739 − 1.27660i 0 −1.43415 + 2.48402i 0
527.10 0 0.314280 − 0.181449i 0 0.837728 − 1.45099i 0 −2.31739 + 1.27660i 0 −1.43415 + 2.48402i 0
527.11 0 1.35591 − 0.782834i 0 −0.711940 + 1.23312i 0 −1.78850 − 1.94969i 0 −0.274341 + 0.475173i 0
527.12 0 1.35591 − 0.782834i 0 −0.711940 + 1.23312i 0 1.78850 + 1.94969i 0 −0.274341 + 0.475173i 0
527.13 0 2.28410 − 1.31873i 0 1.68232 − 2.91387i 0 1.12400 + 2.39512i 0 1.97809 − 3.42615i 0
527.14 0 2.28410 − 1.31873i 0 1.68232 − 2.91387i 0 −1.12400 − 2.39512i 0 1.97809 − 3.42615i 0
527.15 0 2.51905 − 1.45437i 0 −1.30811 + 2.26571i 0 −2.58217 + 0.576531i 0 2.73040 − 4.72920i 0
527.16 0 2.51905 − 1.45437i 0 −1.30811 + 2.26571i 0 2.58217 − 0.576531i 0 2.73040 − 4.72920i 0
879.1 0 −2.51905 − 1.45437i 0 −1.30811 − 2.26571i 0 −2.58217 − 0.576531i 0 2.73040 + 4.72920i 0
879.2 0 −2.51905 − 1.45437i 0 −1.30811 − 2.26571i 0 2.58217 + 0.576531i 0 2.73040 + 4.72920i 0
879.3 0 −2.28410 − 1.31873i 0 1.68232 + 2.91387i 0 −1.12400 + 2.39512i 0 1.97809 + 3.42615i 0
879.4 0 −2.28410 − 1.31873i 0 1.68232 + 2.91387i 0 1.12400 − 2.39512i 0 1.97809 + 3.42615i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 527.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
11.b odd 2 1 inner
28.g odd 6 1 inner
44.c even 2 1 inner
77.h odd 6 1 inner
308.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.bi.c ✓ 32
4.b odd 2 1 inner 1232.2.bi.c ✓ 32
7.c even 3 1 inner 1232.2.bi.c ✓ 32
11.b odd 2 1 inner 1232.2.bi.c ✓ 32
28.g odd 6 1 inner 1232.2.bi.c ✓ 32
44.c even 2 1 inner 1232.2.bi.c ✓ 32
77.h odd 6 1 inner 1232.2.bi.c ✓ 32
308.n even 6 1 inner 1232.2.bi.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1232.2.bi.c ✓ 32 1.a even 1 1 trivial
1232.2.bi.c ✓ 32 4.b odd 2 1 inner
1232.2.bi.c ✓ 32 7.c even 3 1 inner
1232.2.bi.c ✓ 32 11.b odd 2 1 inner
1232.2.bi.c ✓ 32 28.g odd 6 1 inner
1232.2.bi.c ✓ 32 44.c even 2 1 inner
1232.2.bi.c ✓ 32 77.h odd 6 1 inner
1232.2.bi.c ✓ 32 308.n even 6 1 inner