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Properties

Label 148.2.k.a
Level $148$
Weight $2$
Character orbit 148.k
Analytic conductor $1.182$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [148,2,Mod(9,148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(148, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("148.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 148 = 2^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 148.k (of order \(9\), degree \(6\), 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: \(1.18178594991\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{3} - 3 q^{5} - 9 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{3} - 3 q^{5} - 9 q^{7} + 3 q^{9} - 6 q^{11} + 9 q^{13} - 3 q^{15} - 3 q^{17} - 12 q^{19} + 12 q^{21} + 18 q^{23} + 15 q^{25} - 6 q^{27} - 3 q^{29} - 48 q^{31} - 27 q^{33} - 42 q^{35} - 24 q^{37} + 36 q^{39} - 6 q^{41} - 24 q^{43} - 69 q^{45} - 3 q^{47} + 39 q^{49} + 3 q^{53} - 9 q^{55} - 69 q^{57} + 6 q^{59} + 60 q^{61} + 42 q^{63} + 30 q^{65} + 30 q^{67} + 39 q^{69} - 15 q^{71} + 18 q^{73} + 150 q^{75} + 90 q^{77} - 12 q^{79} + 48 q^{81} + 21 q^{85} + 138 q^{87} + 24 q^{89} - 27 q^{91} - 54 q^{93} - 81 q^{95} + 18 q^{97} - 69 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −0.573501 + 3.25248i 0 1.73739 + 1.45784i 0 −2.44417 − 2.05091i 0 −7.43067 − 2.70454i 0
9.2 0 −0.151541 + 0.859431i 0 −0.674805 − 0.566229i 0 3.25999 + 2.73545i 0 2.10342 + 0.765582i 0
9.3 0 0.244003 − 1.38381i 0 −2.48747 − 2.08724i 0 −3.29163 − 2.76200i 0 0.963679 + 0.350751i 0
9.4 0 0.388642 − 2.20410i 0 2.45698 + 2.06165i 0 −0.137524 − 0.115397i 0 −1.88793 − 0.687149i 0
33.1 0 −0.573501 − 3.25248i 0 1.73739 − 1.45784i 0 −2.44417 + 2.05091i 0 −7.43067 + 2.70454i 0
33.2 0 −0.151541 − 0.859431i 0 −0.674805 + 0.566229i 0 3.25999 − 2.73545i 0 2.10342 − 0.765582i 0
33.3 0 0.244003 + 1.38381i 0 −2.48747 + 2.08724i 0 −3.29163 + 2.76200i 0 0.963679 − 0.350751i 0
33.4 0 0.388642 + 2.20410i 0 2.45698 − 2.06165i 0 −0.137524 + 0.115397i 0 −1.88793 + 0.687149i 0
49.1 0 −1.54703 − 1.29812i 0 −2.79940 + 1.01890i 0 −2.38826 + 0.869255i 0 0.187265 + 1.06203i 0
49.2 0 −0.136857 − 0.114836i 0 0.929194 − 0.338199i 0 2.13067 − 0.775502i 0 −0.515402 − 2.92299i 0
49.3 0 1.57471 + 1.32134i 0 3.27213 − 1.19096i 0 −4.36270 + 1.58789i 0 0.212835 + 1.20704i 0
49.4 0 2.31491 + 1.94244i 0 −3.78131 + 1.37628i 0 2.52789 − 0.920076i 0 1.06480 + 6.03875i 0
53.1 0 −3.21865 + 1.17149i 0 0.328408 − 1.86250i 0 0.284135 − 1.61141i 0 6.68919 − 5.61290i 0
53.2 0 −0.792270 + 0.288363i 0 −0.686629 + 3.89407i 0 0.201984 − 1.14551i 0 −1.75359 + 1.47144i 0
53.3 0 0.782886 − 0.284947i 0 0.453648 − 2.57277i 0 0.343039 − 1.94547i 0 −1.76642 + 1.48220i 0
53.4 0 2.61470 − 0.951671i 0 −0.248132 + 1.40722i 0 −0.623421 + 3.53560i 0 3.63282 − 3.04830i 0
81.1 0 −3.21865 − 1.17149i 0 0.328408 + 1.86250i 0 0.284135 + 1.61141i 0 6.68919 + 5.61290i 0
81.2 0 −0.792270 − 0.288363i 0 −0.686629 − 3.89407i 0 0.201984 + 1.14551i 0 −1.75359 − 1.47144i 0
81.3 0 0.782886 + 0.284947i 0 0.453648 + 2.57277i 0 0.343039 + 1.94547i 0 −1.76642 − 1.48220i 0
81.4 0 2.61470 + 0.951671i 0 −0.248132 − 1.40722i 0 −0.623421 − 3.53560i 0 3.63282 + 3.04830i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 148.2.k.a ✓ 24
3.b odd 2 1 1332.2.bt.d 24
4.b odd 2 1 592.2.bc.e 24
37.f even 9 1 inner 148.2.k.a ✓ 24
37.f even 9 1 5476.2.a.k 12
37.h even 18 1 5476.2.a.j 12
111.p odd 18 1 1332.2.bt.d 24
148.p odd 18 1 592.2.bc.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.2.k.a ✓ 24 1.a even 1 1 trivial
148.2.k.a ✓ 24 37.f even 9 1 inner
592.2.bc.e 24 4.b odd 2 1
592.2.bc.e 24 148.p odd 18 1
1332.2.bt.d 24 3.b odd 2 1
1332.2.bt.d 24 111.p odd 18 1
5476.2.a.j 12 37.h even 18 1
5476.2.a.k 12 37.f even 9 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(148, [\chi])\).