Properties

Label 189.2.p.d
Level $189$
Weight $2$
Character orbit 189.p
Analytic conductor $1.509$
Analytic rank $0$
Dimension $12$
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] = [189,2,Mod(26,189)]
 
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([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.p (of order \(6\), 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: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{9} - \beta_{7} - \beta_{3} + 1) q^{4} + (\beta_{6} + \beta_{5}) q^{5} + ( - \beta_{9} + \beta_{4} - 1) q^{7} + (\beta_{11} - \beta_{10} + \beta_{8}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{9} - \beta_{7} - \beta_{3} + 1) q^{4} + (\beta_{6} + \beta_{5}) q^{5} + ( - \beta_{9} + \beta_{4} - 1) q^{7} + (\beta_{11} - \beta_{10} + \beta_{8}) q^{8} + (\beta_{9} + \beta_{7} + \cdots - 2 \beta_{2}) q^{10}+ \cdots + (7 \beta_{11} - \beta_{10} + \cdots + 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{4} - 8 q^{7} - 6 q^{10} - 4 q^{16} - 6 q^{19} - 40 q^{22} - 24 q^{25} + 28 q^{28} - 12 q^{31} + 8 q^{37} + 12 q^{40} + 20 q^{43} + 14 q^{46} + 24 q^{49} + 78 q^{52} + 20 q^{58} + 18 q^{61} + 28 q^{64} + 36 q^{67} - 120 q^{70} - 42 q^{73} - 36 q^{79} - 54 q^{82} - 12 q^{85} - 74 q^{88} + 6 q^{91} - 114 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 81\nu^{11} - 531\nu^{9} + 3481\nu^{7} - 3627\nu^{5} + 1782\nu^{3} + 76298\nu ) / 21995 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 117\nu^{10} - 767\nu^{8} + 7472\nu^{6} - 27234\nu^{4} + 90554\nu^{2} - 60864 ) / 21995 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -1298\nu^{10} + 10953\nu^{8} - 71803\nu^{6} + 202311\nu^{4} - 490451\nu^{2} + 240966 ) / 197955 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 461\nu^{10} - 5466\nu^{8} + 28501\nu^{6} - 98847\nu^{4} + 142112\nu^{2} - 186237 ) / 65985 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1298\nu^{11} - 10953\nu^{9} + 71803\nu^{7} - 202311\nu^{5} + 490451\nu^{3} - 438921\nu ) / 197955 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1298\nu^{11} - 10953\nu^{9} + 71803\nu^{7} - 202311\nu^{5} + 490451\nu^{3} + 154944\nu ) / 197955 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -288\nu^{10} + 1888\nu^{8} - 9933\nu^{6} + 12896\nu^{4} - 6336\nu^{2} - 68439 ) / 21995 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -288\nu^{11} + 1888\nu^{9} - 9933\nu^{7} + 12896\nu^{5} - 6336\nu^{3} - 90434\nu ) / 21995 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1138\nu^{10} - 12348\nu^{8} + 80948\nu^{6} - 273351\nu^{4} + 552916\nu^{2} - 271656 ) / 65985 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4712\nu^{11} - 47997\nu^{9} + 314647\nu^{7} - 1022364\nu^{5} + 2149199\nu^{3} - 1055934\nu ) / 197955 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -5192\nu^{11} + 43812\nu^{9} - 287212\nu^{7} + 809244\nu^{5} - 1763849\nu^{3} + 172044\nu ) / 197955 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{9} + 2\beta_{7} + \beta_{4} - 9\beta_{3} - \beta_{2} + 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{11} + 8\beta_{6} + 4\beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + 2\beta_{7} - 2\beta_{4} - 12\beta_{3} - 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 18\beta_{11} + 3\beta_{10} + 17\beta_{6} + 34\beta_{5} + 18\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 32\beta_{9} - 5\beta_{7} - 64\beta_{4} - 32\beta_{2} - 153 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 27\beta_{8} - 74\beta_{6} + 74\beta_{5} + 96\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 51\beta_{9} - 51\beta_{7} - 55\beta_{4} + 222\beta_{3} + 55\beta_{2} - 222 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -495\beta_{11} - 177\beta_{10} + 177\beta_{8} - 656\beta_{6} - 328\beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -191\beta_{9} - 835\beta_{7} + 835\beta_{4} + 2952\beta_{3} + 1670\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -2505\beta_{11} - 1026\beta_{10} - 1477\beta_{6} - 2954\beta_{5} - 2505\beta_1 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.617942 + 0.356769i
−1.90412 1.09935i
1.65604 + 0.956115i
−1.65604 0.956115i
1.90412 + 1.09935i
−0.617942 0.356769i
0.617942 0.356769i
−1.90412 + 1.09935i
1.65604 0.956115i
−1.65604 + 0.956115i
1.90412 1.09935i
−0.617942 + 0.356769i
−2.15715 1.24543i 0 2.10220 + 3.64112i −0.617942 + 1.07031i 0 −1.53189 2.15715i 5.49086i 0 2.66599 1.53921i
26.2 −1.58850 0.917122i 0 0.682224 + 1.18165i 1.90412 3.29804i 0 2.11581 + 1.58850i 1.16576i 0 −6.04940 + 3.49262i
26.3 −0.568650 0.328310i 0 −0.784425 1.35866i −1.65604 + 2.86834i 0 −2.58392 + 0.568650i 2.34338i 0 1.88341 1.08739i
26.4 0.568650 + 0.328310i 0 −0.784425 1.35866i 1.65604 2.86834i 0 −2.58392 + 0.568650i 2.34338i 0 1.88341 1.08739i
26.5 1.58850 + 0.917122i 0 0.682224 + 1.18165i −1.90412 + 3.29804i 0 2.11581 + 1.58850i 1.16576i 0 −6.04940 + 3.49262i
26.6 2.15715 + 1.24543i 0 2.10220 + 3.64112i 0.617942 1.07031i 0 −1.53189 2.15715i 5.49086i 0 2.66599 1.53921i
80.1 −2.15715 + 1.24543i 0 2.10220 3.64112i −0.617942 1.07031i 0 −1.53189 + 2.15715i 5.49086i 0 2.66599 + 1.53921i
80.2 −1.58850 + 0.917122i 0 0.682224 1.18165i 1.90412 + 3.29804i 0 2.11581 1.58850i 1.16576i 0 −6.04940 3.49262i
80.3 −0.568650 + 0.328310i 0 −0.784425 + 1.35866i −1.65604 2.86834i 0 −2.58392 0.568650i 2.34338i 0 1.88341 + 1.08739i
80.4 0.568650 0.328310i 0 −0.784425 + 1.35866i 1.65604 + 2.86834i 0 −2.58392 0.568650i 2.34338i 0 1.88341 + 1.08739i
80.5 1.58850 0.917122i 0 0.682224 1.18165i −1.90412 3.29804i 0 2.11581 1.58850i 1.16576i 0 −6.04940 3.49262i
80.6 2.15715 1.24543i 0 2.10220 3.64112i 0.617942 + 1.07031i 0 −1.53189 + 2.15715i 5.49086i 0 2.66599 + 1.53921i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.p.d 12
3.b odd 2 1 inner 189.2.p.d 12
7.c even 3 1 1323.2.c.d 12
7.d odd 6 1 inner 189.2.p.d 12
7.d odd 6 1 1323.2.c.d 12
9.c even 3 1 567.2.i.f 12
9.c even 3 1 567.2.s.f 12
9.d odd 6 1 567.2.i.f 12
9.d odd 6 1 567.2.s.f 12
21.g even 6 1 inner 189.2.p.d 12
21.g even 6 1 1323.2.c.d 12
21.h odd 6 1 1323.2.c.d 12
63.i even 6 1 567.2.s.f 12
63.k odd 6 1 567.2.i.f 12
63.s even 6 1 567.2.i.f 12
63.t odd 6 1 567.2.s.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.d 12 1.a even 1 1 trivial
189.2.p.d 12 3.b odd 2 1 inner
189.2.p.d 12 7.d odd 6 1 inner
189.2.p.d 12 21.g even 6 1 inner
567.2.i.f 12 9.c even 3 1
567.2.i.f 12 9.d odd 6 1
567.2.i.f 12 63.k odd 6 1
567.2.i.f 12 63.s even 6 1
567.2.s.f 12 9.c even 3 1
567.2.s.f 12 9.d odd 6 1
567.2.s.f 12 63.i even 6 1
567.2.s.f 12 63.t odd 6 1
1323.2.c.d 12 7.c even 3 1
1323.2.c.d 12 7.d odd 6 1
1323.2.c.d 12 21.g even 6 1
1323.2.c.d 12 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 10T_{2}^{10} + 75T_{2}^{8} - 232T_{2}^{6} + 535T_{2}^{4} - 225T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 10 T^{10} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 27 T^{10} + \cdots + 59049 \) Copy content Toggle raw display
$7$ \( (T^{6} + 4 T^{5} + \cdots + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 37 T^{10} + \cdots + 50625 \) Copy content Toggle raw display
$13$ \( (T^{6} + 42 T^{4} + \cdots + 675)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 30 T^{10} + \cdots + 59049 \) Copy content Toggle raw display
$19$ \( (T^{6} + 3 T^{5} + \cdots + 243)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 94 T^{10} + \cdots + 531441 \) Copy content Toggle raw display
$29$ \( (T^{6} + 37 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 6 T^{5} + \cdots + 64827)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 4 T^{5} + \cdots + 4489)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 114 T^{4} + \cdots - 243)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 5 T^{2} - 16 T - 1)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 387420489 \) Copy content Toggle raw display
$53$ \( T^{12} - 118 T^{10} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{12} + 63 T^{10} + \cdots + 4782969 \) Copy content Toggle raw display
$61$ \( (T^{6} - 9 T^{5} + \cdots + 49923)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 18 T^{5} + \cdots + 458329)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 85 T^{4} + \cdots + 19881)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 21 T^{5} + \cdots + 177147)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 18 T^{5} + \cdots + 49)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 159 T^{4} + \cdots - 6075)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 7695324729 \) Copy content Toggle raw display
$97$ \( (T^{6} + 204 T^{4} + \cdots + 1728)^{2} \) Copy content Toggle raw display
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