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Properties

Label 171.2.f.b
Level $171$
Weight $2$
Character orbit 171.f
Analytic conductor $1.365$
Analytic rank $0$
Dimension $6$
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(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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4}) q^{2} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} + (\beta_{3} - \beta_1) q^{5} + \beta_{2} q^{7} + ( - \beta_{4} + \beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4}) q^{2} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} + (\beta_{3} - \beta_1) q^{5} + \beta_{2} q^{7} + ( - \beta_{4} + \beta_{2} + 1) q^{8} + ( - 2 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{10} + ( - 2 \beta_{4} - \beta_{2} - 1) q^{11} + (2 \beta_{5} - \beta_{3} + 1) q^{13} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{14} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{16} + ( - 2 \beta_{5} - \beta_{2}) q^{19} + (2 \beta_{4} - \beta_{2} - 7) q^{20} + (2 \beta_{5} - 2 \beta_{4} - 7 \beta_{3} + \beta_1) q^{22} + (2 \beta_{5} - 5 \beta_{3} - \beta_{2} - \beta_1 + 5) q^{23} + (2 \beta_{5} + \beta_{3} - 1) q^{25} + (\beta_{4} + 2 \beta_{2} + 8) q^{26} + (2 \beta_{5} + 5 \beta_{3} - 5) q^{28} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{29} + ( - 2 \beta_{4} - \beta_{2} + 4) q^{31} + ( - \beta_{5} + 6 \beta_{3} - 6) q^{32} + (2 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + \beta_1) q^{35} - q^{37} + (\beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 - 8) q^{38} + (4 \beta_{5} - 4 \beta_{4} + 7 \beta_{3} - \beta_1) q^{40} + ( - 2 \beta_{3} + 2 \beta_1) q^{41} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_1) q^{43} + (4 \beta_{5} - 5 \beta_{3} - \beta_{2} - \beta_1 + 5) q^{44} + (4 \beta_{4} + 3 \beta_{2} + 9) q^{46} + (6 \beta_{3} - 6) q^{47} + ( - 2 \beta_{4} - 2 \beta_{2} - 2) q^{49} + ( - \beta_{4} + 2 \beta_{2} + 8) q^{50} + ( - 6 \beta_{5} + 6 \beta_{4} + \beta_1) q^{52} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{53} + (2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 2 \beta_1) q^{55} + ( - 3 \beta_{4} + 6) q^{56} + (4 \beta_{4} - 2 \beta_{2} - 2) q^{58} + (2 \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_1) q^{59} + (2 \beta_{5} + 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 3) q^{61} + ( - 3 \beta_{5} + 3 \beta_{4} - 7 \beta_{3} + \beta_1) q^{62} + ( - 2 \beta_{4} - \beta_{2} - 6) q^{64} + (4 \beta_{4} - \beta_{2} - 1) q^{65} + ( - 4 \beta_{5} + 4 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{67} + ( - 4 \beta_{5} - 7 \beta_{3} + \beta_{2} + \beta_1 + 7) q^{70} + ( - 4 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} - 2 \beta_1) q^{71} + ( - 7 \beta_{3} + 2 \beta_1) q^{73} + (\beta_{5} - \beta_{4}) q^{74} + (4 \beta_{5} - 6 \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 5) q^{76} + (3 \beta_{2} - 3) q^{77} + (2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - \beta_1) q^{79} + ( - 4 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 3) q^{80} + (4 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{82} - 4 \beta_{4} q^{83} + ( - 3 \beta_{5} + 7 \beta_{3} - \beta_{2} - \beta_1 - 7) q^{86} + (3 \beta_{2} + 3) q^{88} + (5 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{89} + (2 \beta_{5} + 2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{91} + ( - 8 \beta_{5} + 8 \beta_{4} + 3 \beta_{3} - 3 \beta_1) q^{92} - 6 \beta_{4} q^{94} + ( - 2 \beta_{5} - 2 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{95} + ( - 6 \beta_{5} + 6 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{97} + (4 \beta_{5} - 4 \beta_{4} - 6 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 5 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} - 5 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 4 q^{10} + q^{13} - 3 q^{14} - 5 q^{16} + 4 q^{19} - 44 q^{20} - 18 q^{22} + 14 q^{23} - 5 q^{25} + 42 q^{26} - 17 q^{28} + 4 q^{29} + 30 q^{31} - 17 q^{32} + 18 q^{35} - 6 q^{37} - 41 q^{38} + 24 q^{40} - 4 q^{41} + 3 q^{43} + 12 q^{44} + 40 q^{46} - 18 q^{47} - 4 q^{49} + 46 q^{50} - 5 q^{52} - 6 q^{53} - 12 q^{55} + 42 q^{56} - 16 q^{58} - 13 q^{61} - 23 q^{62} - 30 q^{64} - 12 q^{65} - 9 q^{67} + 24 q^{70} - 18 q^{71} - 19 q^{73} + q^{74} + 27 q^{76} - 24 q^{77} - 11 q^{79} + 10 q^{80} - 8 q^{82} + 8 q^{83} - 17 q^{86} + 12 q^{88} - 16 q^{89} - 7 q^{91} - 2 q^{92} + 12 q^{94} - 2 q^{95} + 2 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - \nu^{3} + 9\nu^{2} - 21\nu - 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - \nu^{3} - 18\nu^{2} + 33\nu - 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} - \nu^{4} + 7\nu^{3} + 9\nu^{2} + 12\nu + 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} + 3\nu - 72 ) / 27 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 7\beta_{4} - 11\beta_{3} + \beta_{2} - \beta _1 + 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} - 7\beta_{3} + 8\beta_{2} + 10\beta _1 + 20 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{5} - 2\beta_{4} - 20\beta_{3} + \beta_{2} - 10\beta _1 + 43 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(-1 + \beta_{3}\)

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
0.403374 + 1.68443i
−1.62241 − 0.606458i
1.71903 − 0.211943i
0.403374 − 1.68443i
−1.62241 + 0.606458i
1.71903 + 0.211943i
−1.25707 − 2.17731i 0 −2.16044 + 3.74200i 1.66044 + 2.87597i 0 2.32088 5.83502 0 4.17458 − 7.23058i
64.2 −0.285997 − 0.495361i 0 0.836412 − 1.44871i −1.33641 − 2.31473i 0 −3.67282 −2.10083 0 −0.764419 + 1.32401i
64.3 1.04307 + 1.80664i 0 −1.17597 + 2.03684i 0.675970 + 1.17081i 0 0.351939 −0.734191 0 −1.41016 + 2.44247i
163.1 −1.25707 + 2.17731i 0 −2.16044 − 3.74200i 1.66044 − 2.87597i 0 2.32088 5.83502 0 4.17458 + 7.23058i
163.2 −0.285997 + 0.495361i 0 0.836412 + 1.44871i −1.33641 + 2.31473i 0 −3.67282 −2.10083 0 −0.764419 − 1.32401i
163.3 1.04307 − 1.80664i 0 −1.17597 − 2.03684i 0.675970 − 1.17081i 0 0.351939 −0.734191 0 −1.41016 − 2.44247i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.3
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.b 6
3.b odd 2 1 57.2.e.b ✓ 6
4.b odd 2 1 2736.2.s.z 6
12.b even 2 1 912.2.q.l 6
19.c even 3 1 inner 171.2.f.b 6
19.c even 3 1 3249.2.a.y 3
19.d odd 6 1 3249.2.a.t 3
57.f even 6 1 1083.2.a.o 3
57.h odd 6 1 57.2.e.b ✓ 6
57.h odd 6 1 1083.2.a.l 3
76.g odd 6 1 2736.2.s.z 6
228.m even 6 1 912.2.q.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.e.b ✓ 6 3.b odd 2 1
57.2.e.b ✓ 6 57.h odd 6 1
171.2.f.b 6 1.a even 1 1 trivial
171.2.f.b 6 19.c even 3 1 inner
912.2.q.l 6 12.b even 2 1
912.2.q.l 6 228.m even 6 1
1083.2.a.l 3 57.h odd 6 1
1083.2.a.o 3 57.f even 6 1
2736.2.s.z 6 4.b odd 2 1
2736.2.s.z 6 76.g odd 6 1
3249.2.a.t 3 19.d odd 6 1
3249.2.a.y 3 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + T_{2}^{5} + 6T_{2}^{4} + T_{2}^{3} + 28T_{2}^{2} + 15T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(171, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{5} + 6 T^{4} + T^{3} + 28 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + 12 T^{4} - 8 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( (T^{3} + T^{2} - 9 T + 3)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 24 T + 36)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - T^{5} + 22 T^{4} + 27 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} - 4 T^{5} + 17 T^{4} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} - 14 T^{5} + 168 T^{4} + \cdots + 24336 \) Copy content Toggle raw display
$29$ \( T^{6} - 4 T^{5} + 48 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$31$ \( (T^{3} - 15 T^{2} + 51 T + 31)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{6} \) Copy content Toggle raw display
$41$ \( T^{6} + 4 T^{5} + 48 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} + 30 T^{4} + 89 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T + 36)^{3} \) Copy content Toggle raw display
$53$ \( T^{6} + 6 T^{5} + 108 T^{4} + \cdots + 104976 \) Copy content Toggle raw display
$59$ \( T^{6} + 24 T^{4} - 72 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$61$ \( T^{6} + 13 T^{5} + 158 T^{4} + \cdots + 5329 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} + 162 T^{4} + \cdots + 292681 \) Copy content Toggle raw display
$71$ \( T^{6} + 18 T^{5} + 312 T^{4} + \cdots + 419904 \) Copy content Toggle raw display
$73$ \( T^{6} + 19 T^{5} + 278 T^{4} + \cdots + 961 \) Copy content Toggle raw display
$79$ \( T^{6} + 11 T^{5} + 118 T^{4} + \cdots + 29241 \) Copy content Toggle raw display
$83$ \( (T^{3} - 4 T^{2} - 80 T + 192)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 16 T^{5} + 180 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$97$ \( T^{6} - 2 T^{5} + 272 T^{4} + \cdots + 2096704 \) Copy content Toggle raw display
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