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Properties

Label 196.2.f.c
Level $196$
Weight $2$
Character orbit 196.f
Analytic conductor $1.565$
Analytic rank $0$
Dimension $8$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [196,2,Mod(19,196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(196, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 196.f (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(1.56506787962\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{2}) q^{2} + 2 \beta_{4} q^{4} - \beta_{3} q^{5} + 2 \beta_{2} q^{8} + (3 \beta_{4} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{2}) q^{2} + 2 \beta_{4} q^{4} - \beta_{3} q^{5} + 2 \beta_{2} q^{8} + (3 \beta_{4} + 3) q^{9} + \beta_{7} q^{10} + (\beta_{7} - \beta_{6} - \beta_{3} + \beta_1) q^{13} + ( - 4 \beta_{4} - 4) q^{16} + ( - \beta_{7} - \beta_{6}) q^{17} - 3 \beta_{5} q^{18} + (2 \beta_{6} + 2 \beta_{3}) q^{20} + (\beta_{5} - 5 \beta_{4}) q^{25} + (2 \beta_{3} - \beta_1) q^{26} - 3 \beta_{2} q^{29} + 4 \beta_{5} q^{32} + ( - \beta_{7} - 2 \beta_{6} + \cdots - \beta_1) q^{34}+ \cdots + (2 \beta_{7} + 3 \beta_{6} + \cdots + 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 12 q^{9} - 16 q^{16} + 20 q^{25} - 48 q^{36} + 16 q^{50} - 16 q^{53} + 24 q^{58} + 64 q^{64} - 32 q^{65} + 40 q^{74} - 36 q^{81} - 96 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 76\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 27\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 2 ) / 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 21\nu^{2} + 2 ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 35\nu^{5} - 112\nu^{3} + 64\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\nu^{7} - 42\nu^{5} + 154\nu^{3} - 88\nu ) / 14 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} + 6\beta_{6} + 6\beta_{3} + 5\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{5} - 6\beta_{4} + 4\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16\beta_{7} + 22\beta_{6} ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{2} - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -76\beta_{3} - 54\beta_1 ) / 7 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(99\) \(101\)
\(\chi(n)\) \(-1\) \(1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.662827 − 0.382683i
0.662827 + 0.382683i
1.60021 + 0.923880i
−1.60021 − 0.923880i
−0.662827 + 0.382683i
0.662827 − 0.382683i
1.60021 − 0.923880i
−1.60021 + 0.923880i
−0.707107 + 1.22474i 0 −1.00000 − 1.73205i −2.53759 − 1.46508i 0 0 2.82843 1.50000 − 2.59808i 3.58869 − 2.07193i
19.2 −0.707107 + 1.22474i 0 −1.00000 − 1.73205i 2.53759 + 1.46508i 0 0 2.82843 1.50000 − 2.59808i −3.58869 + 2.07193i
19.3 0.707107 − 1.22474i 0 −1.00000 − 1.73205i −2.92586 − 1.68925i 0 0 −2.82843 1.50000 − 2.59808i −4.13779 + 2.38896i
19.4 0.707107 − 1.22474i 0 −1.00000 − 1.73205i 2.92586 + 1.68925i 0 0 −2.82843 1.50000 − 2.59808i 4.13779 − 2.38896i
31.1 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i −2.53759 + 1.46508i 0 0 2.82843 1.50000 + 2.59808i 3.58869 + 2.07193i
31.2 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i 2.53759 − 1.46508i 0 0 2.82843 1.50000 + 2.59808i −3.58869 − 2.07193i
31.3 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −2.92586 + 1.68925i 0 0 −2.82843 1.50000 + 2.59808i −4.13779 − 2.38896i
31.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 2.92586 − 1.68925i 0 0 −2.82843 1.50000 + 2.59808i 4.13779 + 2.38896i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.2.f.c 8
4.b odd 2 1 CM 196.2.f.c 8
7.b odd 2 1 inner 196.2.f.c 8
7.c even 3 1 196.2.d.a ✓ 4
7.c even 3 1 inner 196.2.f.c 8
7.d odd 6 1 196.2.d.a ✓ 4
7.d odd 6 1 inner 196.2.f.c 8
21.g even 6 1 1764.2.b.f 4
21.h odd 6 1 1764.2.b.f 4
28.d even 2 1 inner 196.2.f.c 8
28.f even 6 1 196.2.d.a ✓ 4
28.f even 6 1 inner 196.2.f.c 8
28.g odd 6 1 196.2.d.a ✓ 4
28.g odd 6 1 inner 196.2.f.c 8
56.j odd 6 1 3136.2.f.c 4
56.k odd 6 1 3136.2.f.c 4
56.m even 6 1 3136.2.f.c 4
56.p even 6 1 3136.2.f.c 4
84.j odd 6 1 1764.2.b.f 4
84.n even 6 1 1764.2.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.d.a ✓ 4 7.c even 3 1
196.2.d.a ✓ 4 7.d odd 6 1
196.2.d.a ✓ 4 28.f even 6 1
196.2.d.a ✓ 4 28.g odd 6 1
196.2.f.c 8 1.a even 1 1 trivial
196.2.f.c 8 4.b odd 2 1 CM
196.2.f.c 8 7.b odd 2 1 inner
196.2.f.c 8 7.c even 3 1 inner
196.2.f.c 8 7.d odd 6 1 inner
196.2.f.c 8 28.d even 2 1 inner
196.2.f.c 8 28.f even 6 1 inner
196.2.f.c 8 28.g odd 6 1 inner
1764.2.b.f 4 21.g even 6 1
1764.2.b.f 4 21.h odd 6 1
1764.2.b.f 4 84.j odd 6 1
1764.2.b.f 4 84.n even 6 1
3136.2.f.c 4 56.j odd 6 1
3136.2.f.c 4 56.k odd 6 1
3136.2.f.c 4 56.m even 6 1
3136.2.f.c 4 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(196, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{8} - 20T_{5}^{6} + 302T_{5}^{4} - 1960T_{5}^{2} + 9604 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 20 T^{6} + \cdots + 9604 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 52 T^{2} + 98)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 68 T^{6} + \cdots + 9604 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 50 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 164 T^{2} + 4802)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} - 244 T^{6} + \cdots + 23059204 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} - 292 T^{6} + \cdots + 9604 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} - 356 T^{6} + \cdots + 802135684 \) Copy content Toggle raw display
$97$ \( (T^{4} + 388 T^{2} + 98)^{2} \) Copy content Toggle raw display
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