Properties

Label 192.3.h.a
Level $192$
Weight $3$
Character orbit 192.h
Analytic conductor $5.232$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,3,Mod(161,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 192.h (of order \(2\), degree \(1\), 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: \(5.23162107572\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{3} - \beta_{3} q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_1 q^{3} - \beta_{3} q^{7} - 9 q^{9} - \beta_{2} q^{13} + 26 \beta_1 q^{19} - 3 \beta_{2} q^{21} - 25 q^{25} - 27 \beta_1 q^{27} - 3 \beta_{3} q^{31} + 5 \beta_{2} q^{37} + 3 \beta_{3} q^{39} + 22 \beta_1 q^{43} + 143 q^{49} - 78 q^{57} + 7 \beta_{2} q^{61} + 9 \beta_{3} q^{63} - 122 \beta_1 q^{67} - 46 q^{73} - 75 \beta_1 q^{75} + 5 \beta_{3} q^{79} + 81 q^{81} + 192 \beta_1 q^{91} - 9 \beta_{2} q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{9} - 100 q^{25} + 572 q^{49} - 312 q^{57} - 184 q^{73} + 324 q^{81} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 16\zeta_{12}^{2} - 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -8\zeta_{12}^{3} + 16\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 8\beta_1 ) / 16 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 8 ) / 16 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 3.00000i 0 0 0 −13.8564 0 −9.00000 0
161.2 0 3.00000i 0 0 0 13.8564 0 −9.00000 0
161.3 0 3.00000i 0 0 0 −13.8564 0 −9.00000 0
161.4 0 3.00000i 0 0 0 13.8564 0 −9.00000 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 \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.3.h.a 4
3.b odd 2 1 CM 192.3.h.a 4
4.b odd 2 1 inner 192.3.h.a 4
8.b even 2 1 inner 192.3.h.a 4
8.d odd 2 1 inner 192.3.h.a 4
12.b even 2 1 inner 192.3.h.a 4
16.e even 4 1 768.3.e.a 2
16.e even 4 1 768.3.e.f 2
16.f odd 4 1 768.3.e.a 2
16.f odd 4 1 768.3.e.f 2
24.f even 2 1 inner 192.3.h.a 4
24.h odd 2 1 inner 192.3.h.a 4
48.i odd 4 1 768.3.e.a 2
48.i odd 4 1 768.3.e.f 2
48.k even 4 1 768.3.e.a 2
48.k even 4 1 768.3.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.3.h.a 4 1.a even 1 1 trivial
192.3.h.a 4 3.b odd 2 1 CM
192.3.h.a 4 4.b odd 2 1 inner
192.3.h.a 4 8.b even 2 1 inner
192.3.h.a 4 8.d odd 2 1 inner
192.3.h.a 4 12.b even 2 1 inner
192.3.h.a 4 24.f even 2 1 inner
192.3.h.a 4 24.h odd 2 1 inner
768.3.e.a 2 16.e even 4 1
768.3.e.a 2 16.f odd 4 1
768.3.e.a 2 48.i odd 4 1
768.3.e.a 2 48.k even 4 1
768.3.e.f 2 16.e even 4 1
768.3.e.f 2 16.f odd 4 1
768.3.e.f 2 48.i odd 4 1
768.3.e.f 2 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(192, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} - 192 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 676)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 1728)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4800)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 484)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 9408)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 14884)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T + 46)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4800)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T + 2)^{4} \) Copy content Toggle raw display
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