Properties

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

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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: 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)
 
\(f(q)\) \(=\) \( q - 4 q^{3} - 6 q^{5} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{3} - 6 q^{5} - 11 q^{9} - 12 q^{11} + 82 q^{13} + 24 q^{15} + 30 q^{17} - 68 q^{19} + 216 q^{23} - 89 q^{25} + 152 q^{27} + 246 q^{29} + 112 q^{31} + 48 q^{33} + 110 q^{37} - 328 q^{39} + 246 q^{41} - 172 q^{43} + 66 q^{45} - 192 q^{47} - 120 q^{51} + 558 q^{53} + 72 q^{55} + 272 q^{57} - 540 q^{59} - 110 q^{61} - 492 q^{65} + 140 q^{67} - 864 q^{69} - 840 q^{71} + 550 q^{73} + 356 q^{75} - 208 q^{79} - 311 q^{81} - 516 q^{83} - 180 q^{85} - 984 q^{87} + 1398 q^{89} - 448 q^{93} + 408 q^{95} - 1586 q^{97} + 132 q^{99}+O(q^{100}) \) 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 −6.00000 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.b 1
3.b odd 2 1 1764.4.a.k 1
4.b odd 2 1 784.4.a.n 1
7.b odd 2 1 28.4.a.b 1
7.c even 3 2 196.4.e.d 2
7.d odd 6 2 196.4.e.c 2
21.c even 2 1 252.4.a.c 1
21.g even 6 2 1764.4.k.k 2
21.h odd 6 2 1764.4.k.e 2
28.d even 2 1 112.4.a.c 1
35.c odd 2 1 700.4.a.e 1
35.f even 4 2 700.4.e.f 2
56.e even 2 1 448.4.a.m 1
56.h odd 2 1 448.4.a.d 1
84.h odd 2 1 1008.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.b 1 7.b odd 2 1
112.4.a.c 1 28.d even 2 1
196.4.a.b 1 1.a even 1 1 trivial
196.4.e.c 2 7.d odd 6 2
196.4.e.d 2 7.c even 3 2
252.4.a.c 1 21.c even 2 1
448.4.a.d 1 56.h odd 2 1
448.4.a.m 1 56.e even 2 1
700.4.a.e 1 35.c odd 2 1
700.4.e.f 2 35.f even 4 2
784.4.a.n 1 4.b odd 2 1
1008.4.a.f 1 84.h odd 2 1
1764.4.a.k 1 3.b odd 2 1
1764.4.k.e 2 21.h odd 6 2
1764.4.k.k 2 21.g even 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
\( T_{5} + 6 \) 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
$5$ \( T + 6 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 12 \) Copy content Toggle raw display
$13$ \( T - 82 \) Copy content Toggle raw display
$17$ \( T - 30 \) Copy content Toggle raw display
$19$ \( T + 68 \) Copy content Toggle raw display
$23$ \( T - 216 \) Copy content Toggle raw display
$29$ \( T - 246 \) Copy content Toggle raw display
$31$ \( T - 112 \) Copy content Toggle raw display
$37$ \( T - 110 \) Copy content Toggle raw display
$41$ \( T - 246 \) Copy content Toggle raw display
$43$ \( T + 172 \) Copy content Toggle raw display
$47$ \( T + 192 \) Copy content Toggle raw display
$53$ \( T - 558 \) Copy content Toggle raw display
$59$ \( T + 540 \) Copy content Toggle raw display
$61$ \( T + 110 \) Copy content Toggle raw display
$67$ \( T - 140 \) Copy content Toggle raw display
$71$ \( T + 840 \) Copy content Toggle raw display
$73$ \( T - 550 \) Copy content Toggle raw display
$79$ \( T + 208 \) Copy content Toggle raw display
$83$ \( T + 516 \) Copy content Toggle raw display
$89$ \( T - 1398 \) Copy content Toggle raw display
$97$ \( T + 1586 \) Copy content Toggle raw display
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