Properties

Label 945.2.m.a
Level $945$
Weight $2$
Character orbit 945.m
Analytic conductor $7.546$
Analytic rank $0$
Dimension $48$
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] = [945,2,Mod(323,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(4))
 
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("945.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.m (of order \(4\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 32 q^{10} + 8 q^{13} - 56 q^{16} + 16 q^{22} + 16 q^{25} + 48 q^{31} + 64 q^{37} - 16 q^{40} - 56 q^{43} - 8 q^{46} - 48 q^{52} - 64 q^{55} - 40 q^{58} - 32 q^{61} - 16 q^{70} + 16 q^{73} - 8 q^{76} - 16 q^{82} + 48 q^{85} - 24 q^{88} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1 −1.92327 1.92327i 0 5.39793i 1.20974 1.88057i 0 −0.707107 + 0.707107i 6.53512 6.53512i 0 −5.94349 + 1.29019i
323.2 −1.80466 1.80466i 0 4.51361i −2.22504 0.221835i 0 −0.707107 + 0.707107i 4.53621 4.53621i 0 3.61510 + 4.41578i
323.3 −1.71292 1.71292i 0 3.86818i 1.70013 1.45243i 0 0.707107 0.707107i 3.20004 3.20004i 0 −5.40008 0.424302i
323.4 −1.66804 1.66804i 0 3.56469i 0.420676 + 2.19614i 0 0.707107 0.707107i 2.60996 2.60996i 0 2.96154 4.36494i
323.5 −1.36262 1.36262i 0 1.71349i −1.41633 1.73032i 0 0.707107 0.707107i −0.390409 + 0.390409i 0 −0.427853 + 4.28770i
323.6 −1.20546 1.20546i 0 0.906269i 2.12218 + 0.704515i 0 −0.707107 + 0.707107i −1.31845 + 1.31845i 0 −1.70894 3.40747i
323.7 −1.05516 1.05516i 0 0.226725i 2.07957 0.821810i 0 0.707107 0.707107i −1.87109 + 1.87109i 0 −3.06142 1.32714i
323.8 −0.885953 0.885953i 0 0.430174i −0.684062 2.12886i 0 −0.707107 + 0.707107i −2.15302 + 2.15302i 0 −1.28003 + 2.49212i
323.9 −0.754986 0.754986i 0 0.859992i −2.23288 + 0.119338i 0 0.707107 0.707107i −2.15925 + 2.15925i 0 1.77589 + 1.59570i
323.10 −0.493412 0.493412i 0 1.51309i 0.449429 + 2.19044i 0 0.707107 0.707107i −1.73340 + 1.73340i 0 0.859034 1.30254i
323.11 −0.488896 0.488896i 0 1.52196i −2.07865 + 0.824144i 0 −0.707107 + 0.707107i −1.72187 + 1.72187i 0 1.41916 + 0.613323i
323.12 −0.259165 0.259165i 0 1.86567i 1.30674 1.81450i 0 −0.707107 + 0.707107i −1.00185 + 1.00185i 0 −0.808919 + 0.131594i
323.13 0.259165 + 0.259165i 0 1.86567i −1.30674 + 1.81450i 0 −0.707107 + 0.707107i 1.00185 1.00185i 0 −0.808919 + 0.131594i
323.14 0.488896 + 0.488896i 0 1.52196i 2.07865 0.824144i 0 −0.707107 + 0.707107i 1.72187 1.72187i 0 1.41916 + 0.613323i
323.15 0.493412 + 0.493412i 0 1.51309i −0.449429 2.19044i 0 0.707107 0.707107i 1.73340 1.73340i 0 0.859034 1.30254i
323.16 0.754986 + 0.754986i 0 0.859992i 2.23288 0.119338i 0 0.707107 0.707107i 2.15925 2.15925i 0 1.77589 + 1.59570i
323.17 0.885953 + 0.885953i 0 0.430174i 0.684062 + 2.12886i 0 −0.707107 + 0.707107i 2.15302 2.15302i 0 −1.28003 + 2.49212i
323.18 1.05516 + 1.05516i 0 0.226725i −2.07957 + 0.821810i 0 0.707107 0.707107i 1.87109 1.87109i 0 −3.06142 1.32714i
323.19 1.20546 + 1.20546i 0 0.906269i −2.12218 0.704515i 0 −0.707107 + 0.707107i 1.31845 1.31845i 0 −1.70894 3.40747i
323.20 1.36262 + 1.36262i 0 1.71349i 1.41633 + 1.73032i 0 0.707107 0.707107i 0.390409 0.390409i 0 −0.427853 + 4.28770i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.m.a 48
3.b odd 2 1 inner 945.2.m.a 48
5.c odd 4 1 inner 945.2.m.a 48
15.e even 4 1 inner 945.2.m.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.m.a 48 1.a even 1 1 trivial
945.2.m.a 48 3.b odd 2 1 inner
945.2.m.a 48 5.c odd 4 1 inner
945.2.m.a 48 15.e even 4 1 inner