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Properties

Label 189.2.f.a
Level $189$
Weight $2$
Character orbit 189.f
Analytic conductor $1.509$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [189,2,Mod(64,189)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(189, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.f (of order \(3\), degree \(2\), not 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: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_1) q^{2} + (\beta_{5} + \beta_{4} + \beta_{2} - 1) q^{4} + (2 \beta_{4} + \beta_{2} - 2) q^{5} + \beta_{4} q^{7} + (\beta_{3} - \beta_1 + 2) q^{8} + ( - 3 \beta_1 - 1) q^{10}+ \cdots - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 3 q^{4} - 5 q^{5} + 3 q^{7} + 12 q^{8} - 2 q^{11} - 3 q^{13} + q^{14} - 3 q^{16} + 24 q^{17} - 6 q^{19} - 16 q^{20} + 15 q^{22} - 6 q^{25} + 2 q^{26} - 6 q^{28} + q^{29} + 3 q^{31} - 8 q^{32}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\)

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 − 2.05195i
0.500000 + 1.41036i
0.500000 − 0.224437i
0.500000 + 2.05195i
0.500000 − 1.41036i
0.500000 + 0.224437i
−1.23025 − 2.13086i 0 −2.02704 + 3.51094i −1.29679 + 2.24611i 0 0.500000 + 0.866025i 5.05408 0 6.38151
64.2 −0.119562 − 0.207087i 0 0.971410 − 1.68253i 0.590972 − 1.02359i 0 0.500000 + 0.866025i −0.942820 0 −0.282630
64.3 0.849814 + 1.47192i 0 −0.444368 + 0.769668i −1.79418 + 3.10761i 0 0.500000 + 0.866025i 1.88874 0 −6.09888
127.1 −1.23025 + 2.13086i 0 −2.02704 − 3.51094i −1.29679 − 2.24611i 0 0.500000 − 0.866025i 5.05408 0 6.38151
127.2 −0.119562 + 0.207087i 0 0.971410 + 1.68253i 0.590972 + 1.02359i 0 0.500000 − 0.866025i −0.942820 0 −0.282630
127.3 0.849814 − 1.47192i 0 −0.444368 − 0.769668i −1.79418 − 3.10761i 0 0.500000 − 0.866025i 1.88874 0 −6.09888
\(n\): 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.f.a 6
3.b odd 2 1 63.2.f.b ✓ 6
4.b odd 2 1 3024.2.r.g 6
7.b odd 2 1 1323.2.f.c 6
7.c even 3 1 1323.2.g.c 6
7.c even 3 1 1323.2.h.d 6
7.d odd 6 1 1323.2.g.b 6
7.d odd 6 1 1323.2.h.e 6
9.c even 3 1 inner 189.2.f.a 6
9.c even 3 1 567.2.a.g 3
9.d odd 6 1 63.2.f.b ✓ 6
9.d odd 6 1 567.2.a.d 3
12.b even 2 1 1008.2.r.k 6
21.c even 2 1 441.2.f.d 6
21.g even 6 1 441.2.g.d 6
21.g even 6 1 441.2.h.b 6
21.h odd 6 1 441.2.g.e 6
21.h odd 6 1 441.2.h.c 6
36.f odd 6 1 3024.2.r.g 6
36.f odd 6 1 9072.2.a.cd 3
36.h even 6 1 1008.2.r.k 6
36.h even 6 1 9072.2.a.bq 3
63.g even 3 1 1323.2.h.d 6
63.h even 3 1 1323.2.g.c 6
63.i even 6 1 441.2.g.d 6
63.j odd 6 1 441.2.g.e 6
63.k odd 6 1 1323.2.h.e 6
63.l odd 6 1 1323.2.f.c 6
63.l odd 6 1 3969.2.a.p 3
63.n odd 6 1 441.2.h.c 6
63.o even 6 1 441.2.f.d 6
63.o even 6 1 3969.2.a.m 3
63.s even 6 1 441.2.h.b 6
63.t odd 6 1 1323.2.g.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b ✓ 6 3.b odd 2 1
63.2.f.b ✓ 6 9.d odd 6 1
189.2.f.a 6 1.a even 1 1 trivial
189.2.f.a 6 9.c even 3 1 inner
441.2.f.d 6 21.c even 2 1
441.2.f.d 6 63.o even 6 1
441.2.g.d 6 21.g even 6 1
441.2.g.d 6 63.i even 6 1
441.2.g.e 6 21.h odd 6 1
441.2.g.e 6 63.j odd 6 1
441.2.h.b 6 21.g even 6 1
441.2.h.b 6 63.s even 6 1
441.2.h.c 6 21.h odd 6 1
441.2.h.c 6 63.n odd 6 1
567.2.a.d 3 9.d odd 6 1
567.2.a.g 3 9.c even 3 1
1008.2.r.k 6 12.b even 2 1
1008.2.r.k 6 36.h even 6 1
1323.2.f.c 6 7.b odd 2 1
1323.2.f.c 6 63.l odd 6 1
1323.2.g.b 6 7.d odd 6 1
1323.2.g.b 6 63.t odd 6 1
1323.2.g.c 6 7.c even 3 1
1323.2.g.c 6 63.h even 3 1
1323.2.h.d 6 7.c even 3 1
1323.2.h.d 6 63.g even 3 1
1323.2.h.e 6 7.d odd 6 1
1323.2.h.e 6 63.k odd 6 1
3024.2.r.g 6 4.b odd 2 1
3024.2.r.g 6 36.f odd 6 1
3969.2.a.m 3 63.o even 6 1
3969.2.a.p 3 63.l odd 6 1
9072.2.a.bq 3 36.h even 6 1
9072.2.a.cd 3 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + T_{2}^{5} + 5T_{2}^{4} - 2T_{2}^{3} + 17T_{2}^{2} + 4T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{5} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 5 T^{5} + \cdots + 121 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 2 T^{5} + \cdots + 2209 \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$17$ \( (T^{3} - 12 T^{2} + \cdots - 27)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 3 T^{2} - 6 T - 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 33 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{6} - T^{5} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{6} - 3 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$37$ \( (T^{3} + 3 T^{2} - 54 T + 81)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 22 T^{5} + \cdots + 124609 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} + \cdots + 14641 \) Copy content Toggle raw display
$47$ \( T^{6} + 9 T^{5} + \cdots + 35721 \) Copy content Toggle raw display
$53$ \( (T^{3} - 18 T^{2} + 75 T - 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 9 T^{5} + \cdots + 3969 \) Copy content Toggle raw display
$61$ \( T^{6} - 6 T^{5} + \cdots + 4489 \) Copy content Toggle raw display
$67$ \( T^{6} + 207 T^{4} + \cdots + 466489 \) Copy content Toggle raw display
$71$ \( (T^{3} + 9 T^{2} - 6 T - 81)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 3 T^{2} + \cdots - 243)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 15 T^{5} + \cdots + 591361 \) Copy content Toggle raw display
$83$ \( T^{6} + 12 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$89$ \( (T^{3} - 2 T^{2} + \cdots - 379)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 3 T^{5} + \cdots + 363609 \) Copy content Toggle raw display
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