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Properties

Label 1232.2.bi.b
Level $1232$
Weight $2$
Character orbit 1232.bi
Analytic conductor $9.838$
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] = [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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{5} + 12 q^{9} + 9 q^{11} + 18 q^{23} - 6 q^{25} + 5 q^{33} - 6 q^{37} - 10 q^{45} + 36 q^{47} - 32 q^{49} + 42 q^{59} + 18 q^{67} - 24 q^{69} + 78 q^{75} - 19 q^{77} - 24 q^{81} - 8 q^{89} + 18 q^{91} + 2 q^{93} + 12 q^{97}+O(q^{100}) \) 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 −2.36660 + 1.36635i 0 −0.921875 + 1.59674i 0 0.396390 − 2.61589i 0 2.23385 − 3.86914i 0
527.2 0 −2.36660 + 1.36635i 0 −0.921875 + 1.59674i 0 −0.396390 + 2.61589i 0 2.23385 − 3.86914i 0
527.3 0 −2.08822 + 1.20564i 0 1.26539 − 2.19172i 0 2.64572 + 0.0128382i 0 1.40712 − 2.43720i 0
527.4 0 −2.08822 + 1.20564i 0 1.26539 − 2.19172i 0 −2.64572 − 0.0128382i 0 1.40712 − 2.43720i 0
527.5 0 −0.888754 + 0.513123i 0 −1.89666 + 3.28512i 0 1.88305 − 1.85853i 0 −0.973411 + 1.68600i 0
527.6 0 −0.888754 + 0.513123i 0 −1.89666 + 3.28512i 0 −1.88305 + 1.85853i 0 −0.973411 + 1.68600i 0
527.7 0 −0.497216 + 0.287068i 0 0.0931719 − 0.161379i 0 1.17817 + 2.36895i 0 −1.33518 + 2.31261i 0
527.8 0 −0.497216 + 0.287068i 0 0.0931719 − 0.161379i 0 −1.17817 − 2.36895i 0 −1.33518 + 2.31261i 0
527.9 0 0.378296 − 0.218409i 0 1.77235 − 3.06979i 0 −0.0782028 + 2.64460i 0 −1.40459 + 2.43283i 0
527.10 0 0.378296 − 0.218409i 0 1.77235 − 3.06979i 0 0.0782028 − 2.64460i 0 −1.40459 + 2.43283i 0
527.11 0 1.31308 − 0.758109i 0 −0.222329 + 0.385085i 0 −2.33979 + 1.23506i 0 −0.350541 + 0.607155i 0
527.12 0 1.31308 − 0.758109i 0 −0.222329 + 0.385085i 0 2.33979 − 1.23506i 0 −0.350541 + 0.607155i 0
527.13 0 1.35869 − 0.784440i 0 −1.10994 + 1.92248i 0 2.33384 + 1.24627i 0 −0.269308 + 0.466454i 0
527.14 0 1.35869 − 0.784440i 0 −1.10994 + 1.92248i 0 −2.33384 − 1.24627i 0 −0.269308 + 0.466454i 0
527.15 0 2.79072 − 1.61122i 0 0.519904 − 0.900500i 0 −0.990726 + 2.45326i 0 3.69207 − 6.39486i 0
527.16 0 2.79072 − 1.61122i 0 0.519904 − 0.900500i 0 0.990726 − 2.45326i 0 3.69207 − 6.39486i 0
879.1 0 −2.36660 − 1.36635i 0 −0.921875 − 1.59674i 0 0.396390 + 2.61589i 0 2.23385 + 3.86914i 0
879.2 0 −2.36660 − 1.36635i 0 −0.921875 − 1.59674i 0 −0.396390 − 2.61589i 0 2.23385 + 3.86914i 0
879.3 0 −2.08822 − 1.20564i 0 1.26539 + 2.19172i 0 2.64572 − 0.0128382i 0 1.40712 + 2.43720i 0
879.4 0 −2.08822 − 1.20564i 0 1.26539 + 2.19172i 0 −2.64572 + 0.0128382i 0 1.40712 + 2.43720i 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
11.b odd 2 1 inner
28.g 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.b yes 32
4.b odd 2 1 1232.2.bi.a ✓ 32
7.c even 3 1 1232.2.bi.a ✓ 32
11.b odd 2 1 inner 1232.2.bi.b yes 32
28.g odd 6 1 inner 1232.2.bi.b yes 32
44.c even 2 1 1232.2.bi.a ✓ 32
77.h odd 6 1 1232.2.bi.a ✓ 32
308.n even 6 1 inner 1232.2.bi.b yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1232.2.bi.a ✓ 32 4.b odd 2 1
1232.2.bi.a ✓ 32 7.c even 3 1
1232.2.bi.a ✓ 32 44.c even 2 1
1232.2.bi.a ✓ 32 77.h odd 6 1
1232.2.bi.b yes 32 1.a even 1 1 trivial
1232.2.bi.b yes 32 11.b odd 2 1 inner
1232.2.bi.b yes 32 28.g odd 6 1 inner
1232.2.bi.b yes 32 308.n even 6 1 inner