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Properties

Label 147.4.c.b
Level $147$
Weight $4$
Character orbit 147.c
Analytic conductor $8.673$
Analytic rank $0$
Dimension $24$
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] = [147,4,Mod(146,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.146");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.c (of order \(2\), degree \(1\), 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: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 96 q^{4} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 96 q^{4} - 64 q^{9} + 256 q^{15} + 864 q^{16} - 32 q^{18} - 384 q^{22} + 744 q^{25} - 1704 q^{30} + 584 q^{36} + 432 q^{37} - 2368 q^{39} - 624 q^{43} + 3744 q^{46} - 2160 q^{51} + 2032 q^{57} + 6384 q^{58} - 5832 q^{60} - 3504 q^{64} + 3792 q^{67} - 7472 q^{72} + 2248 q^{78} + 2784 q^{79} - 1968 q^{81} - 3744 q^{85} - 624 q^{88} - 3232 q^{93} + 1320 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1 − 5.37572i −2.08523 − 4.75939i −20.8984 2.78815 −25.5852 + 11.2096i 0 69.3382i −18.3036 + 19.8488i − 14.9883i
146.2 − 5.37572i 2.08523 + 4.75939i −20.8984 −2.78815 25.5852 − 11.2096i 0 69.3382i −18.3036 + 19.8488i 14.9883i
146.3 − 4.84485i −5.11673 + 0.905040i −15.4725 −17.3012 4.38478 + 24.7898i 0 36.2033i 25.3618 − 9.26169i 83.8216i
146.4 − 4.84485i 5.11673 − 0.905040i −15.4725 17.3012 −4.38478 − 24.7898i 0 36.2033i 25.3618 − 9.26169i − 83.8216i
146.5 − 3.20022i −0.930073 − 5.11224i −2.24142 11.9763 −16.3603 + 2.97644i 0 − 18.4287i −25.2699 + 9.50951i − 38.3267i
146.6 − 3.20022i 0.930073 + 5.11224i −2.24142 −11.9763 16.3603 − 2.97644i 0 − 18.4287i −25.2699 + 9.50951i 38.3267i
146.7 − 2.86969i −2.76278 + 4.40080i −0.235115 −5.57059 12.6289 + 7.92832i 0 − 22.2828i −11.7341 − 24.3169i 15.9859i
146.8 − 2.86969i 2.76278 − 4.40080i −0.235115 5.57059 −12.6289 − 7.92832i 0 − 22.2828i −11.7341 − 24.3169i − 15.9859i
146.9 − 1.05015i −4.50851 + 2.58329i 6.89718 −10.2305 2.71285 + 4.73462i 0 − 15.6443i 13.6533 − 23.2935i 10.7436i
146.10 − 1.05015i 4.50851 − 2.58329i 6.89718 10.2305 −2.71285 − 4.73462i 0 − 15.6443i 13.6533 − 23.2935i − 10.7436i
146.11 − 0.222948i −3.69409 + 3.65427i 7.95029 18.7021 0.814713 + 0.823589i 0 − 3.55609i 0.292585 − 26.9984i − 4.16960i
146.12 − 0.222948i 3.69409 − 3.65427i 7.95029 −18.7021 −0.814713 − 0.823589i 0 − 3.55609i 0.292585 − 26.9984i 4.16960i
146.13 0.222948i −3.69409 − 3.65427i 7.95029 18.7021 0.814713 − 0.823589i 0 3.55609i 0.292585 + 26.9984i 4.16960i
146.14 0.222948i 3.69409 + 3.65427i 7.95029 −18.7021 −0.814713 + 0.823589i 0 3.55609i 0.292585 + 26.9984i − 4.16960i
146.15 1.05015i −4.50851 − 2.58329i 6.89718 −10.2305 2.71285 − 4.73462i 0 15.6443i 13.6533 + 23.2935i − 10.7436i
146.16 1.05015i 4.50851 + 2.58329i 6.89718 10.2305 −2.71285 + 4.73462i 0 15.6443i 13.6533 + 23.2935i 10.7436i
146.17 2.86969i −2.76278 − 4.40080i −0.235115 −5.57059 12.6289 − 7.92832i 0 22.2828i −11.7341 + 24.3169i − 15.9859i
146.18 2.86969i 2.76278 + 4.40080i −0.235115 5.57059 −12.6289 + 7.92832i 0 22.2828i −11.7341 + 24.3169i 15.9859i
146.19 3.20022i −0.930073 + 5.11224i −2.24142 11.9763 −16.3603 − 2.97644i 0 18.4287i −25.2699 − 9.50951i 38.3267i
146.20 3.20022i 0.930073 − 5.11224i −2.24142 −11.9763 16.3603 + 2.97644i 0 18.4287i −25.2699 − 9.50951i − 38.3267i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.24
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
21.c even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 72T_{2}^{10} + 1812T_{2}^{8} + 18948T_{2}^{6} + 76839T_{2}^{4} + 66864T_{2}^{2} + 3136 \) acting on \(S_{4}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display