Properties

Label 171.2.a.e
Level $171$
Weight $2$
Character orbit 171.a
Self dual yes
Analytic conductor $1.365$
Analytic rank $0$
Dimension $4$
CM no
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(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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13068.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{3} + 2) q^{4} + (\beta_{2} + \beta_1) q^{5} + ( - \beta_{3} + 1) q^{7} + ( - 2 \beta_{2} - 3 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{3} + 2) q^{4} + (\beta_{2} + \beta_1) q^{5} + ( - \beta_{3} + 1) q^{7} + ( - 2 \beta_{2} - 3 \beta_1) q^{8} + ( - 2 \beta_{3} - 2) q^{10} + ( - \beta_{2} + \beta_1) q^{11} + 2 q^{13} + (2 \beta_{2} + 2 \beta_1) q^{14} + (3 \beta_{3} + 4) q^{16} + (\beta_{2} + \beta_1) q^{17} + q^{19} + (2 \beta_{2} + 6 \beta_1) q^{20} - 6 q^{22} - 2 \beta_{2} q^{23} + (\beta_{3} + 2) q^{25} - 2 \beta_1 q^{26} + ( - 2 \beta_{3} - 6) q^{28} - 2 \beta_{2} q^{29} + (2 \beta_{3} - 2) q^{31} + ( - 2 \beta_{2} - 7 \beta_1) q^{32} + ( - 2 \beta_{3} - 2) q^{34} + (\beta_{2} - 3 \beta_1) q^{35} + (2 \beta_{3} + 4) q^{37} - \beta_1 q^{38} + ( - 4 \beta_{3} - 16) q^{40} + 2 \beta_1 q^{41} + ( - 3 \beta_{3} - 1) q^{43} + (2 \beta_{2} + 4 \beta_1) q^{44} + (2 \beta_{3} - 4) q^{46} + (3 \beta_{2} + \beta_1) q^{47} + ( - \beta_{3} + 2) q^{49} + ( - 2 \beta_{2} - 5 \beta_1) q^{50} + (2 \beta_{3} + 4) q^{52} + (2 \beta_{2} - 4 \beta_1) q^{53} + (3 \beta_{3} - 3) q^{55} + 8 \beta_1 q^{56} + (2 \beta_{3} - 4) q^{58} - 4 \beta_1 q^{59} + ( - 3 \beta_{3} + 5) q^{61} + ( - 4 \beta_{2} - 4 \beta_1) q^{62} + (3 \beta_{3} + 16) q^{64} + (2 \beta_{2} + 2 \beta_1) q^{65} - 4 q^{67} + (2 \beta_{2} + 6 \beta_1) q^{68} + (2 \beta_{3} + 14) q^{70} - 4 \beta_{2} q^{71} + ( - 3 \beta_{3} + 5) q^{73} + ( - 4 \beta_{2} - 10 \beta_1) q^{74} + (\beta_{3} + 2) q^{76} + ( - 5 \beta_{2} - \beta_1) q^{77} - 4 q^{79} + (4 \beta_{2} + 16 \beta_1) q^{80} + ( - 2 \beta_{3} - 8) q^{82} + (2 \beta_{2} + 4 \beta_1) q^{83} + (\beta_{3} + 7) q^{85} + (6 \beta_{2} + 10 \beta_1) q^{86} - 6 \beta_{3} q^{88} + (2 \beta_{2} - 4 \beta_1) q^{89} + ( - 2 \beta_{3} + 2) q^{91} - 2 \beta_1 q^{92} + ( - 4 \beta_{3} + 2) q^{94} + (\beta_{2} + \beta_1) q^{95} + (4 \beta_{3} - 6) q^{97} + (2 \beta_{2} + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{4} + 2 q^{7} - 12 q^{10} + 8 q^{13} + 22 q^{16} + 4 q^{19} - 24 q^{22} + 10 q^{25} - 28 q^{28} - 4 q^{31} - 12 q^{34} + 20 q^{37} - 72 q^{40} - 10 q^{43} - 12 q^{46} + 6 q^{49} + 20 q^{52} - 6 q^{55} - 12 q^{58} + 14 q^{61} + 70 q^{64} - 16 q^{67} + 60 q^{70} + 14 q^{73} + 10 q^{76} - 16 q^{79} - 36 q^{82} + 30 q^{85} - 12 q^{88} + 4 q^{91} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 6x^{2} - x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + 2\nu^{2} + 4\nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 7\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 3\beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + \beta_{2} + 5\beta _1 + 4 \) 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
3.04374
−0.548230
−1.82405
0.328543
−2.71519 0 5.37228 3.22060 0 −2.37228 −9.15640 0 −8.74456
1.2 −1.27582 0 −0.372281 −2.15121 0 3.37228 3.02661 0 2.74456
1.3 1.27582 0 −0.372281 2.15121 0 3.37228 −3.02661 0 2.74456
1.4 2.71519 0 5.37228 −3.22060 0 −2.37228 9.15640 0 −8.74456
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.a.e 4
3.b odd 2 1 inner 171.2.a.e 4
4.b odd 2 1 2736.2.a.bf 4
5.b even 2 1 4275.2.a.bp 4
7.b odd 2 1 8379.2.a.bw 4
12.b even 2 1 2736.2.a.bf 4
15.d odd 2 1 4275.2.a.bp 4
19.b odd 2 1 3249.2.a.bf 4
21.c even 2 1 8379.2.a.bw 4
57.d even 2 1 3249.2.a.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.a.e 4 1.a even 1 1 trivial
171.2.a.e 4 3.b odd 2 1 inner
2736.2.a.bf 4 4.b odd 2 1
2736.2.a.bf 4 12.b even 2 1
3249.2.a.bf 4 19.b odd 2 1
3249.2.a.bf 4 57.d even 2 1
4275.2.a.bp 4 5.b even 2 1
4275.2.a.bp 4 15.d odd 2 1
8379.2.a.bw 4 7.b odd 2 1
8379.2.a.bw 4 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(171))\):

\( T_{2}^{4} - 9T_{2}^{2} + 12 \) Copy content Toggle raw display
\( T_{5}^{4} - 15T_{5}^{2} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 9T^{2} + 12 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 15T^{2} + 48 \) Copy content Toggle raw display
$7$ \( (T^{2} - T - 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 27T^{2} + 108 \) Copy content Toggle raw display
$13$ \( (T - 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 15T^{2} + 48 \) Copy content Toggle raw display
$19$ \( (T - 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 48T^{2} + 48 \) Copy content Toggle raw display
$29$ \( T^{4} - 48T^{2} + 48 \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 10 T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 36T^{2} + 192 \) Copy content Toggle raw display
$43$ \( (T^{2} + 5 T - 68)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 99T^{2} + 1452 \) Copy content Toggle raw display
$53$ \( T^{4} - 240 T^{2} + 13872 \) Copy content Toggle raw display
$59$ \( T^{4} - 144T^{2} + 3072 \) Copy content Toggle raw display
$61$ \( (T^{2} - 7 T - 62)^{2} \) Copy content Toggle raw display
$67$ \( (T + 4)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 192T^{2} + 768 \) Copy content Toggle raw display
$73$ \( (T^{2} - 7 T - 62)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 144T^{2} + 432 \) Copy content Toggle raw display
$89$ \( T^{4} - 240 T^{2} + 13872 \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 116)^{2} \) Copy content Toggle raw display
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