Properties

Label 171.9.c.a
Level $171$
Weight $9$
Character orbit 171.c
Self dual yes
Analytic conductor $69.662$
Analytic rank $0$
Dimension $1$
CM discriminant -19
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [171,9,Mod(37,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.37");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 171.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: yes
Analytic conductor: \(69.6617423205\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 256 q^{4} + 289 q^{5} + 527 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 256 q^{4} + 289 q^{5} + 527 q^{7} - 25007 q^{11} + 65536 q^{16} + 42433 q^{17} + 130321 q^{19} + 73984 q^{20} + 534718 q^{23} - 307104 q^{25} + 134912 q^{28} + 152303 q^{35} + 5602127 q^{43} - 6401792 q^{44} + 8302513 q^{47} - 5487072 q^{49} - 7227023 q^{55} - 17661793 q^{61} + 16777216 q^{64} + 10862848 q^{68} + 43864607 q^{73} + 33362176 q^{76} - 13178689 q^{77} + 18939904 q^{80} + 62676958 q^{83} + 12263137 q^{85} + 136887808 q^{92} + 37662769 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0
0 0 256.000 289.000 0 527.000 0 0 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.9.c.a 1
3.b odd 2 1 19.9.b.a 1
12.b even 2 1 304.9.e.a 1
19.b odd 2 1 CM 171.9.c.a 1
57.d even 2 1 19.9.b.a 1
228.b odd 2 1 304.9.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.9.b.a 1 3.b odd 2 1
19.9.b.a 1 57.d even 2 1
171.9.c.a 1 1.a even 1 1 trivial
171.9.c.a 1 19.b odd 2 1 CM
304.9.e.a 1 12.b even 2 1
304.9.e.a 1 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(171, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} - 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 289 \) Copy content Toggle raw display
$7$ \( T - 527 \) Copy content Toggle raw display
$11$ \( T + 25007 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 42433 \) Copy content Toggle raw display
$19$ \( T - 130321 \) Copy content Toggle raw display
$23$ \( T - 534718 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 5602127 \) Copy content Toggle raw display
$47$ \( T - 8302513 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 17661793 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 43864607 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 62676958 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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