Properties

Label 171.2.f.a
Level $171$
Weight $2$
Character orbit 171.f
Analytic conductor $1.365$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [171,2,Mod(64,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.f (of order \(3\), degree \(2\), 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: \(1.36544187456\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + q^{7} + 3 q^{8} + 2 q^{11} - 5 \zeta_{6} q^{13} + ( - \zeta_{6} + 1) q^{14} + ( - \zeta_{6} + 1) q^{16} + (4 \zeta_{6} - 4) q^{17} + (2 \zeta_{6} - 5) q^{19} + ( - 2 \zeta_{6} + 2) q^{22} - 4 \zeta_{6} q^{23} + 5 \zeta_{6} q^{25} - 5 q^{26} + \zeta_{6} q^{28} - 8 \zeta_{6} q^{29} - 3 q^{31} + 5 \zeta_{6} q^{32} + 4 \zeta_{6} q^{34} + 3 q^{37} + (5 \zeta_{6} - 3) q^{38} + (12 \zeta_{6} - 12) q^{41} + ( - \zeta_{6} + 1) q^{43} + 2 \zeta_{6} q^{44} - 4 q^{46} - 6 \zeta_{6} q^{47} - 6 q^{49} + 5 q^{50} + ( - 5 \zeta_{6} + 5) q^{52} + 4 \zeta_{6} q^{53} + 3 q^{56} - 8 q^{58} + ( - 10 \zeta_{6} + 10) q^{59} + 13 \zeta_{6} q^{61} + (3 \zeta_{6} - 3) q^{62} + 7 q^{64} - 11 \zeta_{6} q^{67} - 4 q^{68} + ( - 6 \zeta_{6} + 6) q^{71} + ( - 11 \zeta_{6} + 11) q^{73} + ( - 3 \zeta_{6} + 3) q^{74} + ( - 3 \zeta_{6} - 2) q^{76} + 2 q^{77} + (\zeta_{6} - 1) q^{79} + 12 \zeta_{6} q^{82} - \zeta_{6} q^{86} + 6 q^{88} - 6 \zeta_{6} q^{89} - 5 \zeta_{6} q^{91} + ( - 4 \zeta_{6} + 4) q^{92} - 6 q^{94} + (2 \zeta_{6} - 2) q^{97} + (6 \zeta_{6} - 6) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{4} + 2 q^{7} + 6 q^{8} + 4 q^{11} - 5 q^{13} + q^{14} + q^{16} - 4 q^{17} - 8 q^{19} + 2 q^{22} - 4 q^{23} + 5 q^{25} - 10 q^{26} + q^{28} - 8 q^{29} - 6 q^{31} + 5 q^{32} + 4 q^{34} + 6 q^{37} - q^{38} - 12 q^{41} + q^{43} + 2 q^{44} - 8 q^{46} - 6 q^{47} - 12 q^{49} + 10 q^{50} + 5 q^{52} + 4 q^{53} + 6 q^{56} - 16 q^{58} + 10 q^{59} + 13 q^{61} - 3 q^{62} + 14 q^{64} - 11 q^{67} - 8 q^{68} + 6 q^{71} + 11 q^{73} + 3 q^{74} - 7 q^{76} + 4 q^{77} - q^{79} + 12 q^{82} - q^{86} + 12 q^{88} - 6 q^{89} - 5 q^{91} + 4 q^{92} - 12 q^{94} - 2 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i 0 0 1.00000 3.00000 0 0
163.1 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0 1.00000 3.00000 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.f.a 2
3.b odd 2 1 57.2.e.a 2
4.b odd 2 1 2736.2.s.j 2
12.b even 2 1 912.2.q.a 2
19.c even 3 1 inner 171.2.f.a 2
19.c even 3 1 3249.2.a.c 1
19.d odd 6 1 3249.2.a.f 1
57.f even 6 1 1083.2.a.b 1
57.h odd 6 1 57.2.e.a 2
57.h odd 6 1 1083.2.a.c 1
76.g odd 6 1 2736.2.s.j 2
228.m even 6 1 912.2.q.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.e.a 2 3.b odd 2 1
57.2.e.a 2 57.h odd 6 1
171.2.f.a 2 1.a even 1 1 trivial
171.2.f.a 2 19.c even 3 1 inner
912.2.q.a 2 12.b even 2 1
912.2.q.a 2 228.m even 6 1
1083.2.a.b 1 57.f even 6 1
1083.2.a.c 1 57.h odd 6 1
2736.2.s.j 2 4.b odd 2 1
2736.2.s.j 2 76.g odd 6 1
3249.2.a.c 1 19.c even 3 1
3249.2.a.f 1 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(171, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$31$ \( (T + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T - 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$79$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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