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Properties

Label 189.2.e.e
Level $189$
Weight $2$
Character orbit 189.e
Analytic conductor $1.509$
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] = [189,2,Mod(109,189)]
 
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([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.e (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: \(1.50917259820\)
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: 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4} - \beta_1) q^{2} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} - 2) q^{4} + (\beta_{3} + \beta_{2}) q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{7} + (2 \beta_{3} + \beta_1 + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{4} - \beta_1) q^{2} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} - 2) q^{4} + (\beta_{3} + \beta_{2}) q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{7} + (2 \beta_{3} + \beta_1 + 4) q^{8} + (\beta_{5} - \beta_{4} - \beta_{2} + 1) q^{10} + (\beta_{5} + 2 \beta_{4} - 2) q^{11} + ( - \beta_{3} + 2 \beta_1 + 1) q^{13} + (2 \beta_{5} - 2 \beta_{4} - \beta_{2} + \cdots + 4) q^{14}+ \cdots + (5 \beta_{3} + \beta_{2} + \beta_1 + 10) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} - 4 q^{4} - q^{5} + 2 q^{7} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} - 4 q^{4} - q^{5} + 2 q^{7} + 18 q^{8} + q^{10} - 7 q^{11} + 4 q^{13} + 17 q^{14} - 10 q^{16} - 5 q^{19} - 26 q^{20} - 8 q^{22} - 6 q^{23} + 2 q^{25} - 17 q^{26} - 30 q^{28} + 26 q^{29} + 8 q^{31} - 25 q^{32} + 24 q^{34} + 10 q^{35} + 8 q^{37} + 7 q^{38} + 24 q^{40} + 4 q^{41} - 18 q^{43} + q^{44} + 3 q^{46} - 9 q^{47} + 12 q^{49} + 8 q^{50} - 9 q^{52} - 24 q^{53} + 8 q^{55} + 48 q^{56} - 14 q^{58} + 15 q^{59} + q^{61} - 42 q^{62} + 66 q^{64} - 10 q^{65} - 14 q^{67} - 39 q^{68} + 26 q^{70} - 6 q^{71} - 7 q^{73} - 36 q^{76} + q^{77} - 6 q^{79} + 16 q^{80} - 43 q^{82} + 6 q^{83} - 54 q^{85} + 32 q^{86} - 9 q^{88} + 5 q^{89} - 33 q^{91} + 24 q^{92} + 27 q^{94} - 16 q^{95} - 28 q^{97} + 49 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : 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 \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 − 0.224437i
0.500000 + 1.41036i
0.500000 − 2.05195i
0.500000 + 0.224437i
0.500000 − 1.41036i
0.500000 + 2.05195i
−1.34981 − 2.33795i 0 −2.64400 + 4.57954i 0.794182 + 1.37556i 0 1.23855 + 2.33795i 8.87636 0 2.14400 − 3.71351i
109.2 −0.380438 − 0.658939i 0 0.710533 − 1.23068i −1.59097 − 2.75564i 0 −2.56238 + 0.658939i −2.60301 0 −1.21053 + 2.09671i
109.3 0.730252 + 1.26483i 0 −0.0665372 + 0.115246i 0.296790 + 0.514055i 0 2.32383 − 1.26483i 2.72665 0 −0.433463 + 0.750780i
163.1 −1.34981 + 2.33795i 0 −2.64400 − 4.57954i 0.794182 − 1.37556i 0 1.23855 − 2.33795i 8.87636 0 2.14400 + 3.71351i
163.2 −0.380438 + 0.658939i 0 0.710533 + 1.23068i −1.59097 + 2.75564i 0 −2.56238 − 0.658939i −2.60301 0 −1.21053 − 2.09671i
163.3 0.730252 − 1.26483i 0 −0.0665372 − 0.115246i 0.296790 − 0.514055i 0 2.32383 + 1.26483i 2.72665 0 −0.433463 − 0.750780i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.e.e ✓ 6
3.b odd 2 1 189.2.e.f yes 6
7.c even 3 1 inner 189.2.e.e ✓ 6
7.c even 3 1 1323.2.a.ba 3
7.d odd 6 1 1323.2.a.z 3
9.c even 3 1 567.2.g.h 6
9.c even 3 1 567.2.h.i 6
9.d odd 6 1 567.2.g.i 6
9.d odd 6 1 567.2.h.h 6
21.g even 6 1 1323.2.a.y 3
21.h odd 6 1 189.2.e.f yes 6
21.h odd 6 1 1323.2.a.x 3
63.g even 3 1 567.2.h.i 6
63.h even 3 1 567.2.g.h 6
63.j odd 6 1 567.2.g.i 6
63.n odd 6 1 567.2.h.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.e ✓ 6 1.a even 1 1 trivial
189.2.e.e ✓ 6 7.c even 3 1 inner
189.2.e.f yes 6 3.b odd 2 1
189.2.e.f yes 6 21.h odd 6 1
567.2.g.h 6 9.c even 3 1
567.2.g.h 6 63.h even 3 1
567.2.g.i 6 9.d odd 6 1
567.2.g.i 6 63.j odd 6 1
567.2.h.h 6 9.d odd 6 1
567.2.h.h 6 63.n odd 6 1
567.2.h.i 6 9.c even 3 1
567.2.h.i 6 63.g even 3 1
1323.2.a.x 3 21.h odd 6 1
1323.2.a.y 3 21.g even 6 1
1323.2.a.z 3 7.d odd 6 1
1323.2.a.ba 3 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 2T_{2}^{5} + 7T_{2}^{4} + 15T_{2}^{2} + 9T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 2 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + T^{5} + 7 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} - 2 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} + 7 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( (T^{3} - 2 T^{2} - 19 T + 47)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 33 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{6} + 5 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( (T^{3} - 13 T^{2} + 30 T - 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 8 T^{5} + \cdots + 4761 \) Copy content Toggle raw display
$37$ \( T^{6} - 8 T^{5} + \cdots + 8649 \) Copy content Toggle raw display
$41$ \( (T^{3} - 2 T^{2} + \cdots + 387)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 9 T^{2} + \cdots - 101)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$53$ \( T^{6} + 24 T^{5} + \cdots + 59049 \) Copy content Toggle raw display
$59$ \( T^{6} - 15 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$61$ \( T^{6} - T^{5} + \cdots + 14641 \) Copy content Toggle raw display
$67$ \( T^{6} + 14 T^{5} + \cdots + 961 \) Copy content Toggle raw display
$71$ \( (T^{3} + 3 T^{2} + \cdots + 243)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 7 T^{5} + \cdots + 962361 \) Copy content Toggle raw display
$79$ \( T^{6} + 6 T^{5} + \cdots + 16129 \) Copy content Toggle raw display
$83$ \( (T^{3} - 3 T^{2} + \cdots - 729)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 5 T^{5} + \cdots + 239121 \) Copy content Toggle raw display
$97$ \( (T^{3} + 14 T^{2} + \cdots - 24)^{2} \) Copy content Toggle raw display
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