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Properties

Label 147.1.l.a
Level $147$
Weight $1$
Character orbit 147.l
Analytic conductor $0.073$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
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] = [147,1,Mod(8,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 12]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.8");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 147.l (of order \(14\), 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: \(0.0733625568571\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.373714754427.1

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{14}^{2} q^{3} + \zeta_{14}^{6} q^{4} - \zeta_{14}^{3} q^{7} + \zeta_{14}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{14}^{2} q^{3} + \zeta_{14}^{6} q^{4} - \zeta_{14}^{3} q^{7} + \zeta_{14}^{4} q^{9} - \zeta_{14} q^{12} + ( - \zeta_{14}^{5} - \zeta_{14}) q^{13} - \zeta_{14}^{5} q^{16} + (\zeta_{14}^{4} - \zeta_{14}^{3}) q^{19} - \zeta_{14}^{5} q^{21} + \zeta_{14}^{4} q^{25} + \zeta_{14}^{6} q^{27} + \zeta_{14}^{2} q^{28} + (\zeta_{14}^{6} - \zeta_{14}) q^{31} - \zeta_{14}^{3} q^{36} + (\zeta_{14}^{2} + 1) q^{37} + ( - \zeta_{14}^{3} + 1) q^{39} + (\zeta_{14}^{2} - \zeta_{14}) q^{43} + q^{48} + \zeta_{14}^{6} q^{49} + (\zeta_{14}^{4} + 1) q^{52} + (\zeta_{14}^{6} - \zeta_{14}^{5}) q^{57} + (\zeta_{14}^{2} + 1) q^{61} + q^{63} + \zeta_{14}^{4} q^{64} + ( - \zeta_{14}^{5} + \zeta_{14}^{2}) q^{67} + ( - \zeta_{14}^{5} - \zeta_{14}^{3}) q^{73} + \zeta_{14}^{6} q^{75} + ( - \zeta_{14}^{3} + \zeta_{14}^{2}) q^{76} + (\zeta_{14}^{6} - \zeta_{14}) q^{79} - \zeta_{14} q^{81} + \zeta_{14}^{4} q^{84} + (\zeta_{14}^{4} - \zeta_{14}) q^{91} + ( - \zeta_{14}^{3} - \zeta_{14}) q^{93} + (\zeta_{14}^{4} - \zeta_{14}^{3}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{3} - q^{4} - q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{3} - q^{4} - q^{7} - q^{9} - q^{12} - 2 q^{13} - q^{16} - 2 q^{19} - q^{21} - q^{25} - q^{27} - q^{28} - 2 q^{31} - q^{36} + 5 q^{37} + 5 q^{39} - 2 q^{43} + 6 q^{48} - q^{49} + 5 q^{52} - 2 q^{57} + 5 q^{61} + 6 q^{63} - q^{64} - 2 q^{67} - 2 q^{73} - q^{75} - 2 q^{76} - 2 q^{79} - q^{81} - q^{84} - 2 q^{91} - 2 q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(\zeta_{14}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
0.222521 + 0.974928i
−0.623490 + 0.781831i
−0.623490 − 0.781831i
0.222521 − 0.974928i
0.900969 − 0.433884i
0.900969 + 0.433884i
0 −0.900969 + 0.433884i −0.222521 + 0.974928i 0 0 0.623490 + 0.781831i 0 0.623490 − 0.781831i 0
29.1 0 −0.222521 − 0.974928i 0.623490 + 0.781831i 0 0 −0.900969 − 0.433884i 0 −0.900969 + 0.433884i 0
71.1 0 −0.222521 + 0.974928i 0.623490 − 0.781831i 0 0 −0.900969 + 0.433884i 0 −0.900969 − 0.433884i 0
92.1 0 −0.900969 − 0.433884i −0.222521 − 0.974928i 0 0 0.623490 − 0.781831i 0 0.623490 + 0.781831i 0
113.1 0 0.623490 − 0.781831i −0.900969 − 0.433884i 0 0 −0.222521 + 0.974928i 0 −0.222521 − 0.974928i 0
134.1 0 0.623490 + 0.781831i −0.900969 + 0.433884i 0 0 −0.222521 − 0.974928i 0 −0.222521 + 0.974928i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.1
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.e even 7 1 inner
147.l odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.1.l.a ✓ 6
3.b odd 2 1 CM 147.1.l.a ✓ 6
4.b odd 2 1 2352.1.cj.a 6
5.b even 2 1 3675.1.bm.a 6
5.c odd 4 2 3675.1.bj.a 12
7.b odd 2 1 1029.1.l.a 6
7.c even 3 2 1029.1.n.b 12
7.d odd 6 2 1029.1.n.a 12
9.c even 3 2 3969.1.bt.a 12
9.d odd 6 2 3969.1.bt.a 12
12.b even 2 1 2352.1.cj.a 6
15.d odd 2 1 3675.1.bm.a 6
15.e even 4 2 3675.1.bj.a 12
21.c even 2 1 1029.1.l.a 6
21.g even 6 2 1029.1.n.a 12
21.h odd 6 2 1029.1.n.b 12
49.e even 7 1 inner 147.1.l.a ✓ 6
49.f odd 14 1 1029.1.l.a 6
49.g even 21 2 1029.1.n.b 12
49.h odd 42 2 1029.1.n.a 12
147.k even 14 1 1029.1.l.a 6
147.l odd 14 1 inner 147.1.l.a ✓ 6
147.n odd 42 2 1029.1.n.b 12
147.o even 42 2 1029.1.n.a 12
196.k odd 14 1 2352.1.cj.a 6
245.p even 14 1 3675.1.bm.a 6
245.r odd 28 2 3675.1.bj.a 12
441.ba even 21 2 3969.1.bt.a 12
441.be odd 42 2 3969.1.bt.a 12
588.u even 14 1 2352.1.cj.a 6
735.bb odd 14 1 3675.1.bm.a 6
735.bj even 28 2 3675.1.bj.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.1.l.a ✓ 6 1.a even 1 1 trivial
147.1.l.a ✓ 6 3.b odd 2 1 CM
147.1.l.a ✓ 6 49.e even 7 1 inner
147.1.l.a ✓ 6 147.l odd 14 1 inner
1029.1.l.a 6 7.b odd 2 1
1029.1.l.a 6 21.c even 2 1
1029.1.l.a 6 49.f odd 14 1
1029.1.l.a 6 147.k even 14 1
1029.1.n.a 12 7.d odd 6 2
1029.1.n.a 12 21.g even 6 2
1029.1.n.a 12 49.h odd 42 2
1029.1.n.a 12 147.o even 42 2
1029.1.n.b 12 7.c even 3 2
1029.1.n.b 12 21.h odd 6 2
1029.1.n.b 12 49.g even 21 2
1029.1.n.b 12 147.n odd 42 2
2352.1.cj.a 6 4.b odd 2 1
2352.1.cj.a 6 12.b even 2 1
2352.1.cj.a 6 196.k odd 14 1
2352.1.cj.a 6 588.u even 14 1
3675.1.bj.a 12 5.c odd 4 2
3675.1.bj.a 12 15.e even 4 2
3675.1.bj.a 12 245.r odd 28 2
3675.1.bj.a 12 735.bj even 28 2
3675.1.bm.a 6 5.b even 2 1
3675.1.bm.a 6 15.d odd 2 1
3675.1.bm.a 6 245.p even 14 1
3675.1.bm.a 6 735.bb odd 14 1
3969.1.bt.a 12 9.c even 3 2
3969.1.bt.a 12 9.d odd 6 2
3969.1.bt.a 12 441.ba even 21 2
3969.1.bt.a 12 441.be odd 42 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(147, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + T^{5} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 2 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 2 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} - 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} + 2 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
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