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Properties

Label 147.2.o.a
Level $147$
Weight $2$
Character orbit 147.o
Analytic conductor $1.174$
Analytic rank $0$
Dimension $12$
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] = [147,2,Mod(5,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 29]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.o (of order \(42\), degree \(12\), 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: \(1.17380090971\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{42}]$

$q$-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 \(\zeta_{21}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{21}^{8} + 2 \zeta_{21}) q^{3} + ( - 2 \zeta_{21}^{10} - 2 \zeta_{21}^{3}) q^{4} + ( - \zeta_{21}^{11} + \zeta_{21}^{9} + \cdots + 1) q^{7}+ \cdots + (3 \zeta_{21}^{9} + 3 \zeta_{21}^{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{21}^{8} + 2 \zeta_{21}) q^{3} + ( - 2 \zeta_{21}^{10} - 2 \zeta_{21}^{3}) q^{4} + ( - \zeta_{21}^{11} + \zeta_{21}^{9} + \cdots + 1) q^{7}+ \cdots + ( - 3 \zeta_{21}^{11} + 14 \zeta_{21}^{9} + \cdots + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} + 2 q^{4} - q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{3} + 2 q^{4} - q^{7} - 3 q^{9} - 6 q^{12} + 4 q^{16} + 9 q^{19} + 6 q^{21} - 5 q^{25} - 8 q^{28} - 15 q^{31} + 12 q^{36} - 76 q^{37} - 66 q^{39} + 10 q^{43} + 13 q^{49} + 34 q^{52} - 18 q^{57} + 79 q^{61} + 90 q^{63} - 16 q^{64} + 11 q^{67} + 27 q^{73} + 15 q^{75} - 13 q^{79} + 9 q^{81} + 6 q^{84} - 9 q^{91} + 15 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(\zeta_{21}^{2} + \zeta_{21}^{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.0747301 − 0.997204i
−0.733052 + 0.680173i
−0.733052 − 0.680173i
0.365341 − 0.930874i
−0.988831 + 0.149042i
0.0747301 + 0.997204i
0.365341 + 0.930874i
0.826239 + 0.563320i
0.955573 + 0.294755i
−0.988831 − 0.149042i
0.826239 − 0.563320i
0.955573 − 0.294755i
0 0.975699 − 1.43109i 1.91115 − 0.589510i 0 0 −1.71951 + 2.01079i 0 −1.09602 − 2.79262i 0
17.1 0 −0.510531 + 1.65510i −1.97766 + 0.298085i 0 0 −1.57775 + 2.12384i 0 −2.47872 − 1.68996i 0
26.1 0 −0.510531 − 1.65510i −1.97766 − 0.298085i 0 0 −1.57775 − 2.12384i 0 −2.47872 + 1.68996i 0
38.1 0 −0.258149 − 1.71271i 0.149460 − 1.99441i 0 0 −2.64420 + 0.0906624i 0 −2.86672 + 0.884266i 0
47.1 0 −1.61232 − 0.632789i 1.65248 + 1.12664i 0 0 2.42168 − 1.06559i 0 2.19916 + 2.04052i 0
59.1 0 0.975699 + 1.43109i 1.91115 + 0.589510i 0 0 −1.71951 − 2.01079i 0 −1.09602 + 2.79262i 0
89.1 0 −0.258149 + 1.71271i 0.149460 + 1.99441i 0 0 −2.64420 − 0.0906624i 0 −2.86672 − 0.884266i 0
101.1 0 1.72721 + 0.129436i −1.46610 − 1.36035i 0 0 2.34300 − 1.22896i 0 2.96649 + 0.447127i 0
110.1 0 1.17809 + 1.26968i 0.730682 − 1.86175i 0 0 0.676779 − 2.55773i 0 −0.224190 + 2.99161i 0
122.1 0 −1.61232 + 0.632789i 1.65248 − 1.12664i 0 0 2.42168 + 1.06559i 0 2.19916 − 2.04052i 0
131.1 0 1.72721 − 0.129436i −1.46610 + 1.36035i 0 0 2.34300 + 1.22896i 0 2.96649 − 0.447127i 0
143.1 0 1.17809 − 1.26968i 0.730682 + 1.86175i 0 0 0.676779 + 2.55773i 0 −0.224190 − 2.99161i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
49.h odd 42 1 inner
147.o even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.2.o.a ✓ 12
3.b odd 2 1 CM 147.2.o.a ✓ 12
49.h odd 42 1 inner 147.2.o.a ✓ 12
147.o even 42 1 inner 147.2.o.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.o.a ✓ 12 1.a even 1 1 trivial
147.2.o.a ✓ 12 3.b odd 2 1 CM
147.2.o.a ✓ 12 49.h odd 42 1 inner
147.2.o.a ✓ 12 147.o even 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 3 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - 3 T^{10} + \cdots + 82319329 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 124791241 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + 15 T^{11} + \cdots + 3108169 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 112669649569 \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 6707446201 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1476788041 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 163181257849 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 175555134049 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 120762505081 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 902334707569 \) Copy content Toggle raw display
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