Properties

Label 196.2.a.b
Level $196$
Weight $2$
Character orbit 196.a
Self dual yes
Analytic conductor $1.565$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [196,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(1.56506787962\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
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)
 
\(f(q)\) \(=\) \( q + q^{3} + 3 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 3 q^{5} - 2 q^{9} - 3 q^{11} + 2 q^{13} + 3 q^{15} + 3 q^{17} - q^{19} + 3 q^{23} + 4 q^{25} - 5 q^{27} - 6 q^{29} - 7 q^{31} - 3 q^{33} - q^{37} + 2 q^{39} + 6 q^{41} - 4 q^{43} - 6 q^{45} - 9 q^{47} + 3 q^{51} + 3 q^{53} - 9 q^{55} - q^{57} + 9 q^{59} - q^{61} + 6 q^{65} - 7 q^{67} + 3 q^{69} - q^{73} + 4 q^{75} - 13 q^{79} + q^{81} + 12 q^{83} + 9 q^{85} - 6 q^{87} + 15 q^{89} - 7 q^{93} - 3 q^{95} - 10 q^{97} + 6 q^{99}+O(q^{100}) \) 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 1.00000 0 3.00000 0 0 0 −2.00000 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 196.2.a.b 1
3.b odd 2 1 1764.2.a.a 1
4.b odd 2 1 784.2.a.d 1
5.b even 2 1 4900.2.a.g 1
5.c odd 4 2 4900.2.e.i 2
7.b odd 2 1 196.2.a.a 1
7.c even 3 2 28.2.e.a 2
7.d odd 6 2 196.2.e.a 2
8.b even 2 1 3136.2.a.h 1
8.d odd 2 1 3136.2.a.s 1
12.b even 2 1 7056.2.a.f 1
21.c even 2 1 1764.2.a.j 1
21.g even 6 2 1764.2.k.b 2
21.h odd 6 2 252.2.k.c 2
28.d even 2 1 784.2.a.g 1
28.f even 6 2 784.2.i.d 2
28.g odd 6 2 112.2.i.b 2
35.c odd 2 1 4900.2.a.n 1
35.f even 4 2 4900.2.e.h 2
35.j even 6 2 700.2.i.c 2
35.l odd 12 4 700.2.r.b 4
56.e even 2 1 3136.2.a.k 1
56.h odd 2 1 3136.2.a.v 1
56.k odd 6 2 448.2.i.c 2
56.p even 6 2 448.2.i.e 2
63.g even 3 2 2268.2.l.h 2
63.h even 3 2 2268.2.i.a 2
63.j odd 6 2 2268.2.i.h 2
63.n odd 6 2 2268.2.l.a 2
84.h odd 2 1 7056.2.a.bw 1
84.n even 6 2 1008.2.s.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.e.a 2 7.c even 3 2
112.2.i.b 2 28.g odd 6 2
196.2.a.a 1 7.b odd 2 1
196.2.a.b 1 1.a even 1 1 trivial
196.2.e.a 2 7.d odd 6 2
252.2.k.c 2 21.h odd 6 2
448.2.i.c 2 56.k odd 6 2
448.2.i.e 2 56.p even 6 2
700.2.i.c 2 35.j even 6 2
700.2.r.b 4 35.l odd 12 4
784.2.a.d 1 4.b odd 2 1
784.2.a.g 1 28.d even 2 1
784.2.i.d 2 28.f even 6 2
1008.2.s.p 2 84.n even 6 2
1764.2.a.a 1 3.b odd 2 1
1764.2.a.j 1 21.c even 2 1
1764.2.k.b 2 21.g even 6 2
2268.2.i.a 2 63.h even 3 2
2268.2.i.h 2 63.j odd 6 2
2268.2.l.a 2 63.n odd 6 2
2268.2.l.h 2 63.g even 3 2
3136.2.a.h 1 8.b even 2 1
3136.2.a.k 1 56.e even 2 1
3136.2.a.s 1 8.d odd 2 1
3136.2.a.v 1 56.h odd 2 1
4900.2.a.g 1 5.b even 2 1
4900.2.a.n 1 35.c odd 2 1
4900.2.e.h 2 35.f even 4 2
4900.2.e.i 2 5.c odd 4 2
7056.2.a.f 1 12.b even 2 1
7056.2.a.bw 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(196))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 3 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 3 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T + 1 \) Copy content Toggle raw display
$23$ \( T - 3 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 7 \) Copy content Toggle raw display
$37$ \( T + 1 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 9 \) Copy content Toggle raw display
$53$ \( T - 3 \) Copy content Toggle raw display
$59$ \( T - 9 \) Copy content Toggle raw display
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T + 7 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 1 \) Copy content Toggle raw display
$79$ \( T + 13 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T - 15 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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