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Properties

Label 147.4.g.e
Level $147$
Weight $4$
Character orbit 147.g
Analytic conductor $8.673$
Analytic rank $0$
Dimension $48$
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] = [147,4,Mod(68,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.68");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.g (of order \(6\), degree \(2\), not 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: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 96 q^{4} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 96 q^{4} + 64 q^{9} + 512 q^{15} - 864 q^{16} + 32 q^{18} - 768 q^{22} - 744 q^{25} + 1704 q^{30} + 1168 q^{36} - 432 q^{37} + 2368 q^{39} - 1248 q^{43} - 3744 q^{46} + 2160 q^{51} + 4064 q^{57} - 6384 q^{58} + 5832 q^{60} - 7008 q^{64} - 3792 q^{67} + 7472 q^{72} + 4496 q^{78} - 2784 q^{79} + 1968 q^{81} - 7488 q^{85} + 624 q^{88} + 3232 q^{93} + 2640 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 −4.65551 − 2.68786i −5.16437 + 0.573836i 10.4492 + 18.0985i 1.39408 − 2.41461i 25.5852 + 11.2096i 0 − 69.3382i 26.3414 − 5.92701i −12.9803 + 7.49417i
68.2 −4.65551 − 2.68786i 5.16437 − 0.573836i 10.4492 + 18.0985i −1.39408 + 2.41461i −25.5852 − 11.2096i 0 − 69.3382i 26.3414 − 5.92701i 12.9803 − 7.49417i
68.3 −4.19576 − 2.42242i −1.77458 − 4.88374i 7.73627 + 13.3996i −8.65059 + 14.9833i −4.38478 + 24.7898i 0 − 36.2033i −20.7018 + 17.3331i 72.5916 − 41.9108i
68.4 −4.19576 − 2.42242i 1.77458 + 4.88374i 7.73627 + 13.3996i 8.65059 − 14.9833i 4.38478 − 24.7898i 0 − 36.2033i −20.7018 + 17.3331i −72.5916 + 41.9108i
68.5 −2.77147 − 1.60011i −4.89236 + 1.75065i 1.12071 + 1.94113i 5.98813 − 10.3717i 16.3603 + 2.97644i 0 18.4287i 20.8704 − 17.1296i −33.1919 + 19.1634i
68.6 −2.77147 − 1.60011i 4.89236 − 1.75065i 1.12071 + 1.94113i −5.98813 + 10.3717i −16.3603 − 2.97644i 0 18.4287i 20.8704 − 17.1296i 33.1919 − 19.1634i
68.7 −2.48522 − 1.43484i −2.42981 + 4.59304i 0.117558 + 0.203616i 2.78530 − 4.82427i 12.6289 − 7.92832i 0 22.2828i −15.1920 − 22.3205i −13.8442 + 7.99293i
68.8 −2.48522 − 1.43484i 2.42981 − 4.59304i 0.117558 + 0.203616i −2.78530 + 4.82427i −12.6289 + 7.92832i 0 22.2828i −15.1920 − 22.3205i 13.8442 − 7.99293i
68.9 −0.909459 − 0.525077i −0.0170622 − 5.19612i −3.44859 − 5.97313i −5.11525 + 8.85988i −2.71285 + 4.73462i 0 15.6443i −26.9994 + 0.177315i 9.30423 − 5.37180i
68.10 −0.909459 − 0.525077i 0.0170622 + 5.19612i −3.44859 − 5.97313i 5.11525 − 8.85988i 2.71285 − 4.73462i 0 15.6443i −26.9994 + 0.177315i −9.30423 + 5.37180i
68.11 −0.193079 − 0.111474i −1.31765 + 5.02631i −3.97515 − 6.88516i −9.35107 + 16.1965i 0.814713 − 0.823589i 0 3.55609i −23.5276 − 13.2458i 3.61098 − 2.08480i
68.12 −0.193079 − 0.111474i 1.31765 − 5.02631i −3.97515 − 6.88516i 9.35107 − 16.1965i −0.814713 + 0.823589i 0 3.55609i −23.5276 − 13.2458i −3.61098 + 2.08480i
68.13 0.193079 + 0.111474i −5.01174 − 1.37204i −3.97515 − 6.88516i 9.35107 − 16.1965i −0.814713 − 0.823589i 0 − 3.55609i 23.2350 + 13.7526i 3.61098 − 2.08480i
68.14 0.193079 + 0.111474i 5.01174 + 1.37204i −3.97515 − 6.88516i −9.35107 + 16.1965i 0.814713 + 0.823589i 0 − 3.55609i 23.2350 + 13.7526i −3.61098 + 2.08480i
68.15 0.909459 + 0.525077i −4.49144 − 2.61284i −3.44859 − 5.97313i −5.11525 + 8.85988i −2.71285 − 4.73462i 0 − 15.6443i 13.3461 + 23.4708i −9.30423 + 5.37180i
68.16 0.909459 + 0.525077i 4.49144 + 2.61284i −3.44859 − 5.97313i 5.11525 − 8.85988i 2.71285 + 4.73462i 0 − 15.6443i 13.3461 + 23.4708i 9.30423 − 5.37180i
68.17 2.48522 + 1.43484i −5.19260 − 0.192239i 0.117558 + 0.203616i −2.78530 + 4.82427i −12.6289 − 7.92832i 0 − 22.2828i 26.9261 + 1.99644i −13.8442 + 7.99293i
68.18 2.48522 + 1.43484i 5.19260 + 0.192239i 0.117558 + 0.203616i 2.78530 − 4.82427i 12.6289 + 7.92832i 0 − 22.2828i 26.9261 + 1.99644i 13.8442 − 7.99293i
68.19 2.77147 + 1.60011i −3.96229 + 3.36159i 1.12071 + 1.94113i −5.98813 + 10.3717i −16.3603 + 2.97644i 0 − 18.4287i 4.39949 − 26.6392i −33.1919 + 19.1634i
68.20 2.77147 + 1.60011i 3.96229 − 3.36159i 1.12071 + 1.94113i 5.98813 − 10.3717i 16.3603 − 2.97644i 0 − 18.4287i 4.39949 − 26.6392i 33.1919 − 19.1634i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.g.e 48
3.b odd 2 1 inner 147.4.g.e 48
7.b odd 2 1 inner 147.4.g.e 48
7.c even 3 1 147.4.c.b ✓ 24
7.c even 3 1 inner 147.4.g.e 48
7.d odd 6 1 147.4.c.b ✓ 24
7.d odd 6 1 inner 147.4.g.e 48
21.c even 2 1 inner 147.4.g.e 48
21.g even 6 1 147.4.c.b ✓ 24
21.g even 6 1 inner 147.4.g.e 48
21.h odd 6 1 147.4.c.b ✓ 24
21.h odd 6 1 inner 147.4.g.e 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.c.b ✓ 24 7.c even 3 1
147.4.c.b ✓ 24 7.d odd 6 1
147.4.c.b ✓ 24 21.g even 6 1
147.4.c.b ✓ 24 21.h odd 6 1
147.4.g.e 48 1.a even 1 1 trivial
147.4.g.e 48 3.b odd 2 1 inner
147.4.g.e 48 7.b odd 2 1 inner
147.4.g.e 48 7.c even 3 1 inner
147.4.g.e 48 7.d odd 6 1 inner
147.4.g.e 48 21.c even 2 1 inner
147.4.g.e 48 21.g even 6 1 inner
147.4.g.e 48 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{24} - 72 T_{2}^{22} + 3372 T_{2}^{20} - 92568 T_{2}^{18} + 1842249 T_{2}^{16} - 23335824 T_{2}^{14} + \cdots + 9834496 \) Copy content Toggle raw display
\( T_{19}^{24} - 43812 T_{19}^{22} + 1243604430 T_{19}^{20} - 20016213369800 T_{19}^{18} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display