Properties

Label 196.2.e.b
Level $196$
Weight $2$
Character orbit 196.e
Analytic conductor $1.565$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [196,2,Mod(165,196)]
 
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([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.165");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 196.e (of order \(3\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{5} + 5 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{5} + 5 \beta_{2} q^{9} + ( - 4 \beta_{2} - 4) q^{11} - 3 \beta_{3} q^{13} - 4 q^{15} - \beta_1 q^{17} + (2 \beta_{3} + 2 \beta_1) q^{19} - 4 \beta_{2} q^{23} + (3 \beta_{2} + 3) q^{25} + 4 \beta_{3} q^{27} + 8 q^{29} + ( - 8 \beta_{3} - 8 \beta_1) q^{33} - 8 \beta_{2} q^{37} + (12 \beta_{2} + 12) q^{39} + 5 \beta_{3} q^{41} - 4 q^{43} - 5 \beta_1 q^{45} + (4 \beta_{3} + 4 \beta_1) q^{47} - 4 \beta_{2} q^{51} + ( - 10 \beta_{2} - 10) q^{53} - 4 \beta_{3} q^{55} - 8 q^{57} - 10 \beta_1 q^{59} + ( - 5 \beta_{3} - 5 \beta_1) q^{61} + 6 \beta_{2} q^{65} - 8 \beta_{3} q^{69} + 5 \beta_1 q^{73} + (6 \beta_{3} + 6 \beta_1) q^{75} + 8 \beta_{2} q^{79} + ( - \beta_{2} - 1) q^{81} + 10 \beta_{3} q^{83} + 2 q^{85} + 16 \beta_1 q^{87} + (5 \beta_{3} + 5 \beta_1) q^{89} + ( - 4 \beta_{2} - 4) q^{95} + \beta_{3} q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{9} - 8 q^{11} - 16 q^{15} + 8 q^{23} + 6 q^{25} + 32 q^{29} + 16 q^{37} + 24 q^{39} - 16 q^{43} + 8 q^{51} - 20 q^{53} - 32 q^{57} - 12 q^{65} - 16 q^{79} - 2 q^{81} + 8 q^{85} - 8 q^{95} + 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(99\) \(101\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 −1.41421 + 2.44949i 0 0.707107 + 1.22474i 0 0 0 −2.50000 4.33013i 0
165.2 0 1.41421 2.44949i 0 −0.707107 1.22474i 0 0 0 −2.50000 4.33013i 0
177.1 0 −1.41421 2.44949i 0 0.707107 1.22474i 0 0 0 −2.50000 + 4.33013i 0
177.2 0 1.41421 + 2.44949i 0 −0.707107 + 1.22474i 0 0 0 −2.50000 + 4.33013i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d 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.e.b 4
3.b odd 2 1 1764.2.k.l 4
4.b odd 2 1 784.2.i.l 4
7.b odd 2 1 inner 196.2.e.b 4
7.c even 3 1 196.2.a.c 2
7.c even 3 1 inner 196.2.e.b 4
7.d odd 6 1 196.2.a.c 2
7.d odd 6 1 inner 196.2.e.b 4
21.c even 2 1 1764.2.k.l 4
21.g even 6 1 1764.2.a.l 2
21.g even 6 1 1764.2.k.l 4
21.h odd 6 1 1764.2.a.l 2
21.h odd 6 1 1764.2.k.l 4
28.d even 2 1 784.2.i.l 4
28.f even 6 1 784.2.a.m 2
28.f even 6 1 784.2.i.l 4
28.g odd 6 1 784.2.a.m 2
28.g odd 6 1 784.2.i.l 4
35.i odd 6 1 4900.2.a.y 2
35.j even 6 1 4900.2.a.y 2
35.k even 12 2 4900.2.e.p 4
35.l odd 12 2 4900.2.e.p 4
56.j odd 6 1 3136.2.a.br 2
56.k odd 6 1 3136.2.a.bs 2
56.m even 6 1 3136.2.a.bs 2
56.p even 6 1 3136.2.a.br 2
84.j odd 6 1 7056.2.a.cr 2
84.n even 6 1 7056.2.a.cr 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.a.c 2 7.c even 3 1
196.2.a.c 2 7.d odd 6 1
196.2.e.b 4 1.a even 1 1 trivial
196.2.e.b 4 7.b odd 2 1 inner
196.2.e.b 4 7.c even 3 1 inner
196.2.e.b 4 7.d odd 6 1 inner
784.2.a.m 2 28.f even 6 1
784.2.a.m 2 28.g odd 6 1
784.2.i.l 4 4.b odd 2 1
784.2.i.l 4 28.d even 2 1
784.2.i.l 4 28.f even 6 1
784.2.i.l 4 28.g odd 6 1
1764.2.a.l 2 21.g even 6 1
1764.2.a.l 2 21.h odd 6 1
1764.2.k.l 4 3.b odd 2 1
1764.2.k.l 4 21.c even 2 1
1764.2.k.l 4 21.g even 6 1
1764.2.k.l 4 21.h odd 6 1
3136.2.a.br 2 56.j odd 6 1
3136.2.a.br 2 56.p even 6 1
3136.2.a.bs 2 56.k odd 6 1
3136.2.a.bs 2 56.m even 6 1
4900.2.a.y 2 35.i odd 6 1
4900.2.a.y 2 35.j even 6 1
4900.2.e.p 4 35.k even 12 2
4900.2.e.p 4 35.l odd 12 2
7056.2.a.cr 2 84.j odd 6 1
7056.2.a.cr 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 8T_{3}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(196, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$23$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T - 8)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$53$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 200 T^{2} + 40000 \) Copy content Toggle raw display
$61$ \( T^{4} + 50T^{2} + 2500 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 50T^{2} + 2500 \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 200)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 50T^{2} + 2500 \) Copy content Toggle raw display
$97$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
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