↑

Properties

Label 196.6.e.h
Level $196$
Weight $6$
Character orbit 196.e
Analytic conductor $31.435$
Analytic rank $0$
Dimension $2$
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] = [196,6,Mod(165,196)] 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("196.165"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(196, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 196.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(31.4352286833\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 16 \zeta_{6} + 16) q^{3} - 16 \zeta_{6} q^{5} - 13 \zeta_{6} q^{9} + ( - 76 \zeta_{6} + 76) q^{11} + 880 q^{13} - 256 q^{15} + ( - 1056 \zeta_{6} + 1056) q^{17} - 1936 \zeta_{6} q^{19} - 936 \zeta_{6} q^{23} + \cdots - 988 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{3} - 16 q^{5} - 13 q^{9} + 76 q^{11} + 1760 q^{13} - 512 q^{15} + 1056 q^{17} - 1936 q^{19} - 936 q^{23} + 2869 q^{25} + 7360 q^{27} - 7964 q^{29} - 1568 q^{31} - 1216 q^{33} - 4938 q^{37} + 14080 q^{39}+ \cdots - 1976 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\) \(99\) \(101\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 8.00000 − 13.8564i 0 −8.00000 − 13.8564i 0 0 0 −6.50000 − 11.2583i 0
177.1 0 8.00000 + 13.8564i 0 −8.00000 + 13.8564i 0 0 0 −6.50000 + 11.2583i 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
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 196.6.e.h 2
7.b odd 2 1 196.6.e.c 2
7.c even 3 1 196.6.a.c ✓ 1
7.c even 3 1 inner 196.6.e.h 2
7.d odd 6 1 196.6.a.f yes 1
7.d odd 6 1 196.6.e.c 2
28.f even 6 1 784.6.a.b 1
28.g odd 6 1 784.6.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.6.a.c ✓ 1 7.c even 3 1
196.6.a.f yes 1 7.d odd 6 1
196.6.e.c 2 7.b odd 2 1
196.6.e.c 2 7.d odd 6 1
196.6.e.h 2 1.a even 1 1 trivial
196.6.e.h 2 7.c even 3 1 inner
784.6.a.b 1 28.f even 6 1
784.6.a.j 1 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 16T_{3} + 256 \) acting on \(S_{6}^{\mathrm{new}}(196, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$5$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 76T + 5776 \) Copy content Toggle raw display
$13$ \( (T - 880)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1056 T + 1115136 \) Copy content Toggle raw display
$19$ \( T^{2} + 1936 T + 3748096 \) Copy content Toggle raw display
$23$ \( T^{2} + 936T + 876096 \) Copy content Toggle raw display
$29$ \( (T + 3982)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1568 T + 2458624 \) Copy content Toggle raw display
$37$ \( T^{2} + 4938 T + 24383844 \) Copy content Toggle raw display
$41$ \( (T + 15840)^{2} \) Copy content Toggle raw display
$43$ \( (T + 16412)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 20768 T + 431309824 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 1398909604 \) Copy content Toggle raw display
$59$ \( T^{2} + 21136 T + 446730496 \) Copy content Toggle raw display
$61$ \( T^{2} - 2992 T + 8952064 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 2100938896 \) Copy content Toggle raw display
$71$ \( (T + 49840)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 3171942400 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 1660073536 \) Copy content Toggle raw display
$83$ \( (T - 112464)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 4128833536 \) Copy content Toggle raw display
$97$ \( (T + 2272)^{2} \) Copy content Toggle raw display
show more
show less