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Properties

Label 189.3.q.b
Level $189$
Weight $3$
Character orbit 189.q
Analytic conductor $5.150$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [189,3,Mod(53,189)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("189.53"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(189, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 189.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.14987699641\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \zeta_{6} q^{4} + (3 \zeta_{6} - 8) q^{7} - 22 q^{13} + (16 \zeta_{6} - 16) q^{16} + (11 \zeta_{6} - 11) q^{19} - 25 \zeta_{6} q^{25} + (20 \zeta_{6} + 12) q^{28} - 59 \zeta_{6} q^{31} + (26 \zeta_{6} - 26) q^{37} + \cdots - 169 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 13 q^{7} - 44 q^{13} - 16 q^{16} - 11 q^{19} - 25 q^{25} + 44 q^{28} - 59 q^{31} - 26 q^{37} + 166 q^{43} + 71 q^{49} + 88 q^{52} - 47 q^{61} + 128 q^{64} - 122 q^{67} + 97 q^{73} + 88 q^{76}+ \cdots - 338 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1\) \(-1 + \zeta_{6}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 −2.00000 + 3.46410i 0 0 −6.50000 − 2.59808i 0 0 0
107.1 0 0 −2.00000 − 3.46410i 0 0 −6.50000 + 2.59808i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.3.q.b ✓ 2
3.b odd 2 1 CM 189.3.q.b ✓ 2
7.c even 3 1 inner 189.3.q.b ✓ 2
21.h odd 6 1 inner 189.3.q.b ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.3.q.b ✓ 2 1.a even 1 1 trivial
189.3.q.b ✓ 2 3.b odd 2 1 CM
189.3.q.b ✓ 2 7.c even 3 1 inner
189.3.q.b ✓ 2 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 13T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 22)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 59T + 3481 \) Copy content Toggle raw display
$37$ \( T^{2} + 26T + 676 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 83)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 47T + 2209 \) Copy content Toggle raw display
$67$ \( T^{2} + 122T + 14884 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 97T + 9409 \) Copy content Toggle raw display
$79$ \( T^{2} - 142T + 20164 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 169)^{2} \) Copy content Toggle raw display
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