Properties

Label 196.4.a.c
Level $196$
Weight $4$
Character orbit 196.a
Self dual yes
Analytic conductor $11.564$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [196,4,Mod(1,196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(196, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 196.a (trivial)

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: yes
Analytic conductor: \(11.5643743611\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 4 q^{3} + 20 q^{5} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{3} + 20 q^{5} - 11 q^{9} + 44 q^{11} + 44 q^{13} + 80 q^{15} - 72 q^{17} - 100 q^{19} - 120 q^{23} + 275 q^{25} - 152 q^{27} + 218 q^{29} + 280 q^{31} + 176 q^{33} - 30 q^{37} + 176 q^{39} - 120 q^{41} + 220 q^{43} - 220 q^{45} - 88 q^{47} - 288 q^{51} + 110 q^{53} + 880 q^{55} - 400 q^{57} - 580 q^{59} - 380 q^{61} + 880 q^{65} - 980 q^{67} - 480 q^{69} - 112 q^{71} + 640 q^{73} + 1100 q^{75} - 488 q^{79} - 311 q^{81} - 660 q^{83} - 1440 q^{85} + 872 q^{87} - 320 q^{89} + 1120 q^{93} - 2000 q^{95} - 248 q^{97} - 484 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 4.00000 0 20.0000 0 0 0 −11.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.4.a.c yes 1
3.b odd 2 1 1764.4.a.a 1
4.b odd 2 1 784.4.a.f 1
7.b odd 2 1 196.4.a.a 1
7.c even 3 2 196.4.e.b 2
7.d odd 6 2 196.4.e.e 2
21.c even 2 1 1764.4.a.m 1
21.g even 6 2 1764.4.k.a 2
21.h odd 6 2 1764.4.k.p 2
28.d even 2 1 784.4.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.4.a.a 1 7.b odd 2 1
196.4.a.c yes 1 1.a even 1 1 trivial
196.4.e.b 2 7.c even 3 2
196.4.e.e 2 7.d odd 6 2
784.4.a.f 1 4.b odd 2 1
784.4.a.m 1 28.d even 2 1
1764.4.a.a 1 3.b odd 2 1
1764.4.a.m 1 21.c even 2 1
1764.4.k.a 2 21.g even 6 2
1764.4.k.p 2 21.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(196))\):

\( T_{3} - 4 \) Copy content Toggle raw display
\( T_{5} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 4 \) Copy content Toggle raw display
$5$ \( T - 20 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 44 \) Copy content Toggle raw display
$13$ \( T - 44 \) Copy content Toggle raw display
$17$ \( T + 72 \) Copy content Toggle raw display
$19$ \( T + 100 \) Copy content Toggle raw display
$23$ \( T + 120 \) Copy content Toggle raw display
$29$ \( T - 218 \) Copy content Toggle raw display
$31$ \( T - 280 \) Copy content Toggle raw display
$37$ \( T + 30 \) Copy content Toggle raw display
$41$ \( T + 120 \) Copy content Toggle raw display
$43$ \( T - 220 \) Copy content Toggle raw display
$47$ \( T + 88 \) Copy content Toggle raw display
$53$ \( T - 110 \) Copy content Toggle raw display
$59$ \( T + 580 \) Copy content Toggle raw display
$61$ \( T + 380 \) Copy content Toggle raw display
$67$ \( T + 980 \) Copy content Toggle raw display
$71$ \( T + 112 \) Copy content Toggle raw display
$73$ \( T - 640 \) Copy content Toggle raw display
$79$ \( T + 488 \) Copy content Toggle raw display
$83$ \( T + 660 \) Copy content Toggle raw display
$89$ \( T + 320 \) Copy content Toggle raw display
$97$ \( T + 248 \) Copy content Toggle raw display
show more
show less