↑

Properties

Label 3136.1.r.c
Level $3136$
Weight $1$
Character orbit 3136.r
Analytic conductor $1.565$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -4
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3136,1,Mod(2431,3136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3136, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3136.2431");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3136.r (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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: no (minimal twist has level 784)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.1372.1
Artin image: $C_6\times D_8$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{48} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{5} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{5} + \beta_{2} q^{9} + \beta_{3} q^{13} - \beta_1 q^{17} + ( - \beta_{2} - 1) q^{25} + \beta_{3} q^{41} - \beta_1 q^{45} + ( - 2 \beta_{2} - 2) q^{53} + ( - \beta_{3} - \beta_1) q^{61} - 2 \beta_{2} q^{65} + \beta_1 q^{73} + ( - \beta_{2} - 1) q^{81} + 2 q^{85} + (\beta_{3} + \beta_1) q^{89} + \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{9} - 2 q^{25} - 4 q^{53} + 4 q^{65} - 2 q^{81} + 8 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

\(\nu\)\(=\) \( \beta_1 \) 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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
0.707107 + 1.22474i
−0.707107 − 1.22474i
0.707107 − 1.22474i
−0.707107 + 1.22474i
0 0 0 −0.707107 + 1.22474i 0 0 0 −0.500000 + 0.866025i 0
2431.2 0 0 0 0.707107 − 1.22474i 0 0 0 −0.500000 + 0.866025i 0
3007.1 0 0 0 −0.707107 − 1.22474i 0 0 0 −0.500000 − 0.866025i 0
3007.2 0 0 0 0.707107 + 1.22474i 0 0 0 −0.500000 − 0.866025i 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
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 3136.1.r.c 4
4.b odd 2 1 CM 3136.1.r.c 4
7.b odd 2 1 inner 3136.1.r.c 4
7.c even 3 1 3136.1.d.c 2
7.c even 3 1 inner 3136.1.r.c 4
7.d odd 6 1 3136.1.d.c 2
7.d odd 6 1 inner 3136.1.r.c 4
8.b even 2 1 784.1.r.a 4
8.d odd 2 1 784.1.r.a 4
28.d even 2 1 inner 3136.1.r.c 4
28.f even 6 1 3136.1.d.c 2
28.f even 6 1 inner 3136.1.r.c 4
28.g odd 6 1 3136.1.d.c 2
28.g odd 6 1 inner 3136.1.r.c 4
56.e even 2 1 784.1.r.a 4
56.h odd 2 1 784.1.r.a 4
56.j odd 6 1 784.1.d.a ✓ 2
56.j odd 6 1 784.1.r.a 4
56.k odd 6 1 784.1.d.a ✓ 2
56.k odd 6 1 784.1.r.a 4
56.m even 6 1 784.1.d.a ✓ 2
56.m even 6 1 784.1.r.a 4
56.p even 6 1 784.1.d.a ✓ 2
56.p even 6 1 784.1.r.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.1.d.a ✓ 2 56.j odd 6 1
784.1.d.a ✓ 2 56.k odd 6 1
784.1.d.a ✓ 2 56.m even 6 1
784.1.d.a ✓ 2 56.p even 6 1
784.1.r.a 4 8.b even 2 1
784.1.r.a 4 8.d odd 2 1
784.1.r.a 4 56.e even 2 1
784.1.r.a 4 56.h odd 2 1
784.1.r.a 4 56.j odd 6 1
784.1.r.a 4 56.k odd 6 1
784.1.r.a 4 56.m even 6 1
784.1.r.a 4 56.p even 6 1
3136.1.d.c 2 7.c even 3 1
3136.1.d.c 2 7.d odd 6 1
3136.1.d.c 2 28.f even 6 1
3136.1.d.c 2 28.g odd 6 1
3136.1.r.c 4 1.a even 1 1 trivial
3136.1.r.c 4 4.b odd 2 1 CM
3136.1.r.c 4 7.b odd 2 1 inner
3136.1.r.c 4 7.c even 3 1 inner
3136.1.r.c 4 7.d odd 6 1 inner
3136.1.r.c 4 28.d even 2 1 inner
3136.1.r.c 4 28.f even 6 1 inner
3136.1.r.c 4 28.g odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3136, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{4} + 2T_{5}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$97$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
show more
show less