Properties

Label 147.4.g.a
Level $147$
Weight $4$
Character orbit 147.g
Analytic conductor $8.673$
Analytic rank $0$
Dimension $2$
CM discriminant -3
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(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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \zeta_{6} - 3) q^{3} + (8 \zeta_{6} - 8) q^{4} + 27 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \zeta_{6} - 3) q^{3} + (8 \zeta_{6} - 8) q^{4} + 27 \zeta_{6} q^{9} + ( - 24 \zeta_{6} + 48) q^{12} + ( - 72 \zeta_{6} + 36) q^{13} - 64 \zeta_{6} q^{16} + ( - 90 \zeta_{6} + 180) q^{19} + ( - 125 \zeta_{6} + 125) q^{25} + ( - 162 \zeta_{6} + 81) q^{27} + ( - 90 \zeta_{6} - 90) q^{31} - 216 q^{36} + 110 \zeta_{6} q^{37} + (324 \zeta_{6} - 324) q^{39} + 520 q^{43} + (384 \zeta_{6} - 192) q^{48} + (288 \zeta_{6} + 288) q^{52} - 810 q^{57} + (540 \zeta_{6} - 1080) q^{61} + 512 q^{64} + ( - 880 \zeta_{6} + 880) q^{67} + ( - 216 \zeta_{6} - 216) q^{73} + (375 \zeta_{6} - 750) q^{75} + (1440 \zeta_{6} - 720) q^{76} - 884 \zeta_{6} q^{79} + (729 \zeta_{6} - 729) q^{81} + 810 \zeta_{6} q^{93} + ( - 1584 \zeta_{6} + 792) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{3} - 8 q^{4} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{3} - 8 q^{4} + 27 q^{9} + 72 q^{12} - 64 q^{16} + 270 q^{19} + 125 q^{25} - 270 q^{31} - 432 q^{36} + 110 q^{37} - 324 q^{39} + 1040 q^{43} + 864 q^{52} - 1620 q^{57} - 1620 q^{61} + 1024 q^{64} + 880 q^{67} - 648 q^{73} - 1125 q^{75} - 884 q^{79} - 729 q^{81} + 810 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −4.50000 + 2.59808i −4.00000 6.92820i 0 0 0 0 13.5000 23.3827i 0
80.1 0 −4.50000 2.59808i −4.00000 + 6.92820i 0 0 0 0 13.5000 + 23.3827i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.d odd 6 1 inner
21.g even 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.a 2
3.b odd 2 1 CM 147.4.g.a 2
7.b odd 2 1 147.4.g.b 2
7.c even 3 1 21.4.c.a 2
7.c even 3 1 147.4.g.b 2
7.d odd 6 1 21.4.c.a 2
7.d odd 6 1 inner 147.4.g.a 2
21.c even 2 1 147.4.g.b 2
21.g even 6 1 21.4.c.a 2
21.g even 6 1 inner 147.4.g.a 2
21.h odd 6 1 21.4.c.a 2
21.h odd 6 1 147.4.g.b 2
28.f even 6 1 336.4.k.a 2
28.g odd 6 1 336.4.k.a 2
84.j odd 6 1 336.4.k.a 2
84.n even 6 1 336.4.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.c.a 2 7.c even 3 1
21.4.c.a 2 7.d odd 6 1
21.4.c.a 2 21.g even 6 1
21.4.c.a 2 21.h odd 6 1
147.4.g.a 2 1.a even 1 1 trivial
147.4.g.a 2 3.b odd 2 1 CM
147.4.g.a 2 7.d odd 6 1 inner
147.4.g.a 2 21.g even 6 1 inner
147.4.g.b 2 7.b odd 2 1
147.4.g.b 2 7.c even 3 1
147.4.g.b 2 21.c even 2 1
147.4.g.b 2 21.h odd 6 1
336.4.k.a 2 28.f even 6 1
336.4.k.a 2 28.g odd 6 1
336.4.k.a 2 84.j odd 6 1
336.4.k.a 2 84.n even 6 1

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{19}^{2} - 270T_{19} + 24300 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3888 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 270T + 24300 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 270T + 24300 \) Copy content Toggle raw display
$37$ \( T^{2} - 110T + 12100 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 520)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 1620 T + 874800 \) Copy content Toggle raw display
$67$ \( T^{2} - 880T + 774400 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 648T + 139968 \) Copy content Toggle raw display
$79$ \( T^{2} + 884T + 781456 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1881792 \) Copy content Toggle raw display
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