↑

Properties

Label 6192.2.l.g
Level $6192$
Weight $2$
Character orbit 6192.l
Analytic conductor $49.443$
Analytic rank $0$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6192,2,Mod(2321,6192)] 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("6192.2321"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6192, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6192 = 2^{4} \cdot 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6192.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0, -12,0,0,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(65)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(49.4433689316\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.30233088.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{3} + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 774)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{5} - \beta_{2} q^{7} + (\beta_{2} - \beta_1) q^{11} + (\beta_{4} + 2) q^{13} + \beta_1 q^{17} + (\beta_{5} - \beta_{2}) q^{19} + ( - \beta_{5} - \beta_1) q^{23} + (\beta_{4} + \beta_{3} + 1) q^{25}+ \cdots + (3 \beta_{4} + \beta_{3} - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 12 q^{13} + 6 q^{25} - 12 q^{31} - 6 q^{43} + 6 q^{49} + 36 q^{65} - 12 q^{67} + 36 q^{77} - 12 q^{79} + 36 q^{89} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 6x^{3} + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 6\nu^{2} - 9\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 3\nu^{4} + 3\nu^{2} + 9\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 6\nu^{2} + 9\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 3\nu^{4} + 3\nu^{2} - 9\nu ) / 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{5} + 3\beta_{4} - 3\beta_{3} - 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{5} - 3\beta_{4} + 6\beta_{3} - 3\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6192\mathbb{Z}\right)^\times\).

\(n\) \(433\) \(3871\) \(4645\) \(4817\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2321.1
−1.29211 − 1.15345i
−1.29211 + 1.15345i
−0.352860 − 1.69573i
−0.352860 + 1.69573i
1.64497 − 0.542278i
1.64497 + 0.542278i
0 0 0 −2.58423 0 − 2.30690i 0 0 0
2321.2 0 0 0 −2.58423 0 2.30690i 0 0 0
2321.3 0 0 0 −0.705720 0 − 3.39145i 0 0 0
2321.4 0 0 0 −0.705720 0 3.39145i 0 0 0
2321.5 0 0 0 3.28995 0 − 1.08456i 0 0 0
2321.6 0 0 0 3.28995 0 1.08456i 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 2321.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
129.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6192.2.l.g 6
3.b odd 2 1 6192.2.l.f 6
4.b odd 2 1 774.2.d.a ✓ 6
12.b even 2 1 774.2.d.b yes 6
43.b odd 2 1 6192.2.l.f 6
129.d even 2 1 inner 6192.2.l.g 6
172.d even 2 1 774.2.d.b yes 6
516.h odd 2 1 774.2.d.a ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
774.2.d.a ✓ 6 4.b odd 2 1
774.2.d.a ✓ 6 516.h odd 2 1
774.2.d.b yes 6 12.b even 2 1
774.2.d.b yes 6 172.d even 2 1
6192.2.l.f 6 3.b odd 2 1
6192.2.l.f 6 43.b odd 2 1
6192.2.l.g 6 1.a even 1 1 trivial
6192.2.l.g 6 129.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6192, [\chi])\):

\( T_{5}^{3} - 9T_{5} - 6 \) Copy content Toggle raw display
\( T_{7}^{6} + 18T_{7}^{4} + 81T_{7}^{2} + 72 \) Copy content Toggle raw display
\( T_{11}^{6} + 24T_{11}^{4} + 165T_{11}^{2} + 338 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} - 9 T - 6)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + 18 T^{4} + \cdots + 72 \) Copy content Toggle raw display
$11$ \( T^{6} + 24 T^{4} + \cdots + 338 \) Copy content Toggle raw display
$13$ \( (T^{3} - 6 T^{2} + 3 T + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2)^{3} \) Copy content Toggle raw display
$19$ \( T^{6} + 54 T^{4} + \cdots + 1152 \) Copy content Toggle raw display
$23$ \( T^{6} + 42 T^{4} + \cdots + 32 \) Copy content Toggle raw display
$29$ \( (T^{3} - 81 T - 18)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 6 T^{2} - 42 T + 44)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 144 T^{4} + \cdots + 18432 \) Copy content Toggle raw display
$41$ \( T^{6} + 114 T^{4} + \cdots + 36992 \) Copy content Toggle raw display
$43$ \( T^{6} + 6 T^{5} + \cdots + 79507 \) Copy content Toggle raw display
$47$ \( T^{6} + 60 T^{4} + \cdots + 242 \) Copy content Toggle raw display
$53$ \( T^{6} + 114 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$59$ \( T^{6} + 114 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$61$ \( T^{6} + 252 T^{4} + \cdots + 225792 \) Copy content Toggle raw display
$67$ \( (T^{3} + 6 T^{2} - 6 T - 52)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 18 T - 24)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 108 T^{4} + \cdots + 14112 \) Copy content Toggle raw display
$79$ \( (T^{3} + 6 T^{2} + \cdots - 352)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 168 T^{4} + \cdots + 2 \) Copy content Toggle raw display
$89$ \( (T^{3} - 18 T^{2} + 576)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 12 T^{2} + \cdots - 686)^{2} \) Copy content Toggle raw display
show more
show less