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Properties

Label 171.2.f.c
Level $171$
Weight $2$
Character orbit 171.f
Analytic conductor $1.365$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.764411904.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{6} + \beta_{4} - \beta_{3} - 1) q^{4} - \beta_1 q^{5} + ( - \beta_{6} - 1) q^{7} + ( - \beta_{7} + \beta_{5} + \beta_{2} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{6} + \beta_{4} - \beta_{3} - 1) q^{4} - \beta_1 q^{5} + ( - \beta_{6} - 1) q^{7} + ( - \beta_{7} + \beta_{5} + \beta_{2} + \beta_1) q^{8} + ( - \beta_{6} + \beta_{4}) q^{10} + (2 \beta_{7} - 2 \beta_{5} - \beta_{2} - \beta_1) q^{11} + (\beta_{3} + 1) q^{13} + (3 \beta_{5} + \beta_1) q^{14} + \beta_{3} q^{16} + 2 \beta_1 q^{17} + (\beta_{6} - 2 \beta_{4} + 1) q^{19} + ( - 2 \beta_{7} + 2 \beta_{5} - \beta_{2} - \beta_1) q^{20} + (3 \beta_{4} - 6 \beta_{3}) q^{22} + ( - 2 \beta_{7} - \beta_{2}) q^{23} + (2 \beta_{6} - 2 \beta_{4} - \beta_{3} - 1) q^{25} + (\beta_{7} - \beta_{5}) q^{26} + (2 \beta_{6} - 2 \beta_{4} + 7 \beta_{3} + 7) q^{28} + (4 \beta_{7} + 2 \beta_{2}) q^{29} + ( - \beta_{6} - 7) q^{31} + ( - \beta_{7} + 2 \beta_{2}) q^{32} + (2 \beta_{6} - 2 \beta_{4}) q^{34} + (2 \beta_{5} - \beta_1) q^{35} + (2 \beta_{6} - 1) q^{37} + (4 \beta_{7} - 3 \beta_{5} - 2 \beta_{2} - \beta_1) q^{38} + (\beta_{4} + 6 \beta_{3}) q^{40} - 4 \beta_{5} q^{41} + ( - 3 \beta_{4} - \beta_{3}) q^{43} + ( - 8 \beta_{7} + \beta_{2}) q^{44} + ( - \beta_{6} - 6) q^{46} - 4 \beta_{7} q^{47} + 2 \beta_{6} q^{49} + (3 \beta_{7} - 3 \beta_{5} - 2 \beta_{2} - 2 \beta_1) q^{50} + (\beta_{4} - \beta_{3}) q^{52} + \beta_{2} q^{53} - 6 \beta_{3} q^{55} + (5 \beta_{7} - 5 \beta_{5}) q^{56} + (2 \beta_{6} + 12) q^{58} + (2 \beta_{5} + 3 \beta_1) q^{59} + ( - 5 \beta_{3} - 5) q^{61} + (9 \beta_{5} + \beta_1) q^{62} + ( - 3 \beta_{6} - 5) q^{64} + ( - \beta_{2} - \beta_1) q^{65} + ( - 3 \beta_{6} + 3 \beta_{4} + 7 \beta_{3} + 7) q^{67} + (4 \beta_{7} - 4 \beta_{5} + 2 \beta_{2} + 2 \beta_1) q^{68} + (\beta_{6} - \beta_{4} + 6 \beta_{3} + 6) q^{70} + (4 \beta_{5} - 2 \beta_1) q^{71} + 5 \beta_{3} q^{73} + ( - 3 \beta_{5} - 2 \beta_1) q^{74} + (2 \beta_{4} - 7 \beta_{3} + 5) q^{76} + ( - 8 \beta_{7} + 8 \beta_{5} + \beta_{2} + \beta_1) q^{77} + ( - 3 \beta_{4} - 7 \beta_{3}) q^{79} - \beta_{2} q^{80} + ( - 4 \beta_{6} + 4 \beta_{4} - 12 \beta_{3} - 12) q^{82} + ( - 4 \beta_{7} + 4 \beta_{5} + 2 \beta_{2} + 2 \beta_1) q^{83} + ( - 4 \beta_{6} + 4 \beta_{4} + 12 \beta_{3} + 12) q^{85} + (5 \beta_{7} - 3 \beta_{2}) q^{86} + ( - 3 \beta_{6} - 12) q^{88} - 5 \beta_{2} q^{89} + ( - \beta_{6} + \beta_{4} - \beta_{3} - 1) q^{91} + (4 \beta_{5} + 3 \beta_1) q^{92} + ( - 4 \beta_{6} - 12) q^{94} + (4 \beta_{7} - 2 \beta_{5} + 4 \beta_{2} + \beta_1) q^{95} + (2 \beta_{4} + 8 \beta_{3}) q^{97} + ( - 4 \beta_{5} - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{4} - 8 q^{7} + 4 q^{13} - 4 q^{16} + 8 q^{19} + 24 q^{22} - 4 q^{25} + 28 q^{28} - 56 q^{31} - 8 q^{37} - 24 q^{40} + 4 q^{43} - 48 q^{46} + 4 q^{52} + 24 q^{55} + 96 q^{58} - 20 q^{61} - 40 q^{64} + 28 q^{67} + 24 q^{70} - 20 q^{73} + 68 q^{76} + 28 q^{79} - 48 q^{82} + 48 q^{85} - 96 q^{88} - 4 q^{91} - 96 q^{94} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 21\nu^{5} + 87\nu^{3} - 162\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 24\nu^{5} - 48\nu^{3} - 27\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} + 3\nu^{4} - 6\nu^{2} - 9 ) / 45 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{6} - 3\nu^{4} + 51\nu^{2} - 81 ) / 45 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{7} + 6\nu^{5} - 57\nu^{3} + 27\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} - 12\nu^{2} + 27 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -11\nu^{7} + 39\nu^{5} - 123\nu^{3} + 243\nu ) / 135 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{5} - 2\beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 3\beta_{5} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} + 2\beta_{4} - 3\beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{7} - 7\beta_{5} + 5\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{6} - 30\beta_{3} - 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -9\beta_{7} - 6\beta_{5} + 3\beta_{2} - 15\beta_1 \) 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
−1.69185 − 0.370982i
1.27970 − 1.16721i
−1.27970 + 1.16721i
1.69185 + 0.370982i
−1.69185 + 0.370982i
1.27970 + 1.16721i
−1.27970 − 1.16721i
1.69185 − 0.370982i
−1.16721 − 2.02166i 0 −1.72474 + 2.98735i −0.524648 − 0.908716i 0 −3.44949 3.38371 0 −1.22474 + 2.12132i
64.2 −0.370982 − 0.642559i 0 0.724745 − 1.25529i 1.65068 + 2.85906i 0 1.44949 −2.55940 0 1.22474 − 2.12132i
64.3 0.370982 + 0.642559i 0 0.724745 − 1.25529i −1.65068 − 2.85906i 0 1.44949 2.55940 0 1.22474 − 2.12132i
64.4 1.16721 + 2.02166i 0 −1.72474 + 2.98735i 0.524648 + 0.908716i 0 −3.44949 −3.38371 0 −1.22474 + 2.12132i
163.1 −1.16721 + 2.02166i 0 −1.72474 − 2.98735i −0.524648 + 0.908716i 0 −3.44949 3.38371 0 −1.22474 − 2.12132i
163.2 −0.370982 + 0.642559i 0 0.724745 + 1.25529i 1.65068 − 2.85906i 0 1.44949 −2.55940 0 1.22474 + 2.12132i
163.3 0.370982 − 0.642559i 0 0.724745 + 1.25529i −1.65068 + 2.85906i 0 1.44949 2.55940 0 1.22474 + 2.12132i
163.4 1.16721 − 2.02166i 0 −1.72474 − 2.98735i 0.524648 − 0.908716i 0 −3.44949 −3.38371 0 −1.22474 − 2.12132i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.f.c ✓ 8
3.b odd 2 1 inner 171.2.f.c ✓ 8
4.b odd 2 1 2736.2.s.bb 8
12.b even 2 1 2736.2.s.bb 8
19.c even 3 1 inner 171.2.f.c ✓ 8
19.c even 3 1 3249.2.a.bd 4
19.d odd 6 1 3249.2.a.be 4
57.f even 6 1 3249.2.a.be 4
57.h odd 6 1 inner 171.2.f.c ✓ 8
57.h odd 6 1 3249.2.a.bd 4
76.g odd 6 1 2736.2.s.bb 8
228.m even 6 1 2736.2.s.bb 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.f.c ✓ 8 1.a even 1 1 trivial
171.2.f.c ✓ 8 3.b odd 2 1 inner
171.2.f.c ✓ 8 19.c even 3 1 inner
171.2.f.c ✓ 8 57.h odd 6 1 inner
2736.2.s.bb 8 4.b odd 2 1
2736.2.s.bb 8 12.b even 2 1
2736.2.s.bb 8 76.g odd 6 1
2736.2.s.bb 8 228.m even 6 1
3249.2.a.bd 4 19.c even 3 1
3249.2.a.bd 4 57.h odd 6 1
3249.2.a.be 4 19.d odd 6 1
3249.2.a.be 4 57.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 6T_{2}^{6} + 33T_{2}^{4} + 18T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(171, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 6 T^{6} + 33 T^{4} + 18 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 12 T^{6} + 132 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T - 5)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 36 T^{2} + 108)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + 48 T^{6} + 2112 T^{4} + \cdots + 36864 \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T + 19)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 36 T^{6} + 996 T^{4} + \cdots + 90000 \) Copy content Toggle raw display
$29$ \( T^{8} + 144 T^{6} + \cdots + 23040000 \) Copy content Toggle raw display
$31$ \( (T^{2} + 14 T + 43)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T - 23)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + 96 T^{6} + 8448 T^{4} + \cdots + 589824 \) Copy content Toggle raw display
$43$ \( (T^{4} - 2 T^{3} + 57 T^{2} + 106 T + 2809)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 96 T^{6} + 8448 T^{4} + \cdots + 589824 \) Copy content Toggle raw display
$53$ \( T^{8} + 12 T^{6} + 132 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$59$ \( T^{8} + 132 T^{6} + \cdots + 18766224 \) Copy content Toggle raw display
$61$ \( (T^{2} + 5 T + 25)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 14 T^{3} + 201 T^{2} + 70 T + 25)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 144 T^{6} + \cdots + 23040000 \) Copy content Toggle raw display
$73$ \( (T^{2} + 5 T + 25)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 14 T^{3} + 201 T^{2} + 70 T + 25)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 144 T^{2} + 1728)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 300 T^{6} + \cdots + 56250000 \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} + 216 T^{2} + 640 T + 1600)^{2} \) Copy content Toggle raw display
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