Properties

Label 196.6.a.b
Level $196$
Weight $6$
Character orbit 196.a
Self dual yes
Analytic conductor $31.435$
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] = [196,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(31.4352286833\)
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)
 
\(f(q)\) \(=\) \( q - 19 q^{3} + 19 q^{5} + 118 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 19 q^{3} + 19 q^{5} + 118 q^{9} - 559 q^{11} + 282 q^{13} - 361 q^{15} + 1259 q^{17} - 1957 q^{19} - 2977 q^{23} - 2764 q^{25} + 2375 q^{27} - 62 q^{29} + 2037 q^{31} + 10621 q^{33} + 6023 q^{37} - 5358 q^{39} - 2178 q^{41} + 23180 q^{43} + 2242 q^{45} + 26235 q^{47} - 23921 q^{51} + 30267 q^{53} - 10621 q^{55} + 37183 q^{57} + 44965 q^{59} + 27639 q^{61} + 5358 q^{65} - 58667 q^{67} + 56563 q^{69} - 9520 q^{71} - 6785 q^{73} + 52516 q^{75} - 16929 q^{79} - 73799 q^{81} - 59572 q^{83} + 23921 q^{85} + 1178 q^{87} - 51873 q^{89} - 38703 q^{93} - 37183 q^{95} + 134110 q^{97} - 65962 q^{99}+O(q^{100}) \) 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 −19.0000 0 19.0000 0 0 0 118.000 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 196.6.a.b 1
4.b odd 2 1 784.6.a.k 1
7.b odd 2 1 196.6.a.g 1
7.c even 3 2 28.6.e.a 2
7.d odd 6 2 196.6.e.b 2
21.h odd 6 2 252.6.k.c 2
28.d even 2 1 784.6.a.a 1
28.g odd 6 2 112.6.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.e.a 2 7.c even 3 2
112.6.i.a 2 28.g odd 6 2
196.6.a.b 1 1.a even 1 1 trivial
196.6.a.g 1 7.b odd 2 1
196.6.e.b 2 7.d odd 6 2
252.6.k.c 2 21.h odd 6 2
784.6.a.a 1 28.d even 2 1
784.6.a.k 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 19 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(196))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 19 \) Copy content Toggle raw display
$5$ \( T - 19 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 559 \) Copy content Toggle raw display
$13$ \( T - 282 \) Copy content Toggle raw display
$17$ \( T - 1259 \) Copy content Toggle raw display
$19$ \( T + 1957 \) Copy content Toggle raw display
$23$ \( T + 2977 \) Copy content Toggle raw display
$29$ \( T + 62 \) Copy content Toggle raw display
$31$ \( T - 2037 \) Copy content Toggle raw display
$37$ \( T - 6023 \) Copy content Toggle raw display
$41$ \( T + 2178 \) Copy content Toggle raw display
$43$ \( T - 23180 \) Copy content Toggle raw display
$47$ \( T - 26235 \) Copy content Toggle raw display
$53$ \( T - 30267 \) Copy content Toggle raw display
$59$ \( T - 44965 \) Copy content Toggle raw display
$61$ \( T - 27639 \) Copy content Toggle raw display
$67$ \( T + 58667 \) Copy content Toggle raw display
$71$ \( T + 9520 \) Copy content Toggle raw display
$73$ \( T + 6785 \) Copy content Toggle raw display
$79$ \( T + 16929 \) Copy content Toggle raw display
$83$ \( T + 59572 \) Copy content Toggle raw display
$89$ \( T + 51873 \) Copy content Toggle raw display
$97$ \( T - 134110 \) Copy content Toggle raw display
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