Properties

Label 171.4.a.d
Level $171$
Weight $4$
Character orbit 171.a
Self dual yes
Analytic conductor $10.089$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [171,4,Mod(1,171)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 171.a (trivial)

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: \(10.0893266110\)
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)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 3 q^{2} + q^{4} + 12 q^{5} + 11 q^{7} - 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} + q^{4} + 12 q^{5} + 11 q^{7} - 21 q^{8} + 36 q^{10} + 54 q^{11} + 11 q^{13} + 33 q^{14} - 71 q^{16} + 93 q^{17} + 19 q^{19} + 12 q^{20} + 162 q^{22} - 183 q^{23} + 19 q^{25} + 33 q^{26} + 11 q^{28} + 249 q^{29} + 56 q^{31} - 45 q^{32} + 279 q^{34} + 132 q^{35} - 250 q^{37} + 57 q^{38} - 252 q^{40} - 240 q^{41} - 196 q^{43} + 54 q^{44} - 549 q^{46} + 168 q^{47} - 222 q^{49} + 57 q^{50} + 11 q^{52} - 435 q^{53} + 648 q^{55} - 231 q^{56} + 747 q^{58} - 195 q^{59} - 358 q^{61} + 168 q^{62} + 433 q^{64} + 132 q^{65} - 961 q^{67} + 93 q^{68} + 396 q^{70} + 246 q^{71} + 353 q^{73} - 750 q^{74} + 19 q^{76} + 594 q^{77} - 34 q^{79} - 852 q^{80} - 720 q^{82} - 234 q^{83} + 1116 q^{85} - 588 q^{86} - 1134 q^{88} + 168 q^{89} + 121 q^{91} - 183 q^{92} + 504 q^{94} + 228 q^{95} + 758 q^{97} - 666 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
3.00000 0 1.00000 12.0000 0 11.0000 −21.0000 0 36.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.4.a.d 1
3.b odd 2 1 19.4.a.a 1
12.b even 2 1 304.4.a.b 1
15.d odd 2 1 475.4.a.e 1
15.e even 4 2 475.4.b.c 2
21.c even 2 1 931.4.a.a 1
24.f even 2 1 1216.4.a.a 1
24.h odd 2 1 1216.4.a.f 1
33.d even 2 1 2299.4.a.b 1
57.d even 2 1 361.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.a 1 3.b odd 2 1
171.4.a.d 1 1.a even 1 1 trivial
304.4.a.b 1 12.b even 2 1
361.4.a.b 1 57.d even 2 1
475.4.a.e 1 15.d odd 2 1
475.4.b.c 2 15.e even 4 2
931.4.a.a 1 21.c even 2 1
1216.4.a.a 1 24.f even 2 1
1216.4.a.f 1 24.h odd 2 1
2299.4.a.b 1 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(171))\):

\( T_{2} - 3 \) Copy content Toggle raw display
\( T_{5} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 3 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 12 \) Copy content Toggle raw display
$7$ \( T - 11 \) Copy content Toggle raw display
$11$ \( T - 54 \) Copy content Toggle raw display
$13$ \( T - 11 \) Copy content Toggle raw display
$17$ \( T - 93 \) Copy content Toggle raw display
$19$ \( T - 19 \) Copy content Toggle raw display
$23$ \( T + 183 \) Copy content Toggle raw display
$29$ \( T - 249 \) Copy content Toggle raw display
$31$ \( T - 56 \) Copy content Toggle raw display
$37$ \( T + 250 \) Copy content Toggle raw display
$41$ \( T + 240 \) Copy content Toggle raw display
$43$ \( T + 196 \) Copy content Toggle raw display
$47$ \( T - 168 \) Copy content Toggle raw display
$53$ \( T + 435 \) Copy content Toggle raw display
$59$ \( T + 195 \) Copy content Toggle raw display
$61$ \( T + 358 \) Copy content Toggle raw display
$67$ \( T + 961 \) Copy content Toggle raw display
$71$ \( T - 246 \) Copy content Toggle raw display
$73$ \( T - 353 \) Copy content Toggle raw display
$79$ \( T + 34 \) Copy content Toggle raw display
$83$ \( T + 234 \) Copy content Toggle raw display
$89$ \( T - 168 \) Copy content Toggle raw display
$97$ \( T - 758 \) Copy content Toggle raw display
show more
show less