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Properties

Label 3136.1.n.a
Level $3136$
Weight $1$
Character orbit 3136.n
Analytic conductor $1.565$
Analytic rank $0$
Dimension $16$
Projective image $D_{8}$
CM discriminant -8
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3136,1,Mod(1697,3136)]
 
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([0, 3, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3136.1697");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3136.n (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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.421654016.1

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( - \zeta_{48}^{23} + \zeta_{48}^{17}) q^{3} + ( - \zeta_{48}^{22} + \cdots - \zeta_{48}^{10}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{48}^{23} + \zeta_{48}^{17}) q^{3} + ( - \zeta_{48}^{22} + \cdots - \zeta_{48}^{10}) q^{9}+ \cdots + (\zeta_{48}^{18} + \cdots + \zeta_{48}^{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{9} + 8 q^{25} - 8 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{48}^{16}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1697.1
−0.130526 − 0.991445i
−0.793353 − 0.608761i
0.608761 − 0.793353i
−0.991445 + 0.130526i
−0.608761 + 0.793353i
0.991445 − 0.130526i
0.130526 + 0.991445i
0.793353 + 0.608761i
−0.130526 + 0.991445i
−0.793353 + 0.608761i
0.608761 + 0.793353i
−0.991445 − 0.130526i
−0.608761 − 0.793353i
0.991445 + 0.130526i
0.130526 − 0.991445i
0.793353 − 0.608761i
0 −0.923880 + 1.60021i 0 0 0 0 0 −1.20711 − 2.09077i 0
1697.2 0 −0.923880 + 1.60021i 0 0 0 0 0 −1.20711 − 2.09077i 0
1697.3 0 −0.382683 + 0.662827i 0 0 0 0 0 0.207107 + 0.358719i 0
1697.4 0 −0.382683 + 0.662827i 0 0 0 0 0 0.207107 + 0.358719i 0
1697.5 0 0.382683 − 0.662827i 0 0 0 0 0 0.207107 + 0.358719i 0
1697.6 0 0.382683 − 0.662827i 0 0 0 0 0 0.207107 + 0.358719i 0
1697.7 0 0.923880 − 1.60021i 0 0 0 0 0 −1.20711 − 2.09077i 0
1697.8 0 0.923880 − 1.60021i 0 0 0 0 0 −1.20711 − 2.09077i 0
2273.1 0 −0.923880 − 1.60021i 0 0 0 0 0 −1.20711 + 2.09077i 0
2273.2 0 −0.923880 − 1.60021i 0 0 0 0 0 −1.20711 + 2.09077i 0
2273.3 0 −0.382683 − 0.662827i 0 0 0 0 0 0.207107 − 0.358719i 0
2273.4 0 −0.382683 − 0.662827i 0 0 0 0 0 0.207107 − 0.358719i 0
2273.5 0 0.382683 + 0.662827i 0 0 0 0 0 0.207107 − 0.358719i 0
2273.6 0 0.382683 + 0.662827i 0 0 0 0 0 0.207107 − 0.358719i 0
2273.7 0 0.923880 + 1.60021i 0 0 0 0 0 −1.20711 + 2.09077i 0
2273.8 0 0.923880 + 1.60021i 0 0 0 0 0 −1.20711 + 2.09077i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1697.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.1.n.a 16
4.b odd 2 1 inner 3136.1.n.a 16
7.b odd 2 1 inner 3136.1.n.a 16
7.c even 3 1 3136.1.h.a ✓ 8
7.c even 3 1 inner 3136.1.n.a 16
7.d odd 6 1 3136.1.h.a ✓ 8
7.d odd 6 1 inner 3136.1.n.a 16
8.b even 2 1 inner 3136.1.n.a 16
8.d odd 2 1 CM 3136.1.n.a 16
28.d even 2 1 inner 3136.1.n.a 16
28.f even 6 1 3136.1.h.a ✓ 8
28.f even 6 1 inner 3136.1.n.a 16
28.g odd 6 1 3136.1.h.a ✓ 8
28.g odd 6 1 inner 3136.1.n.a 16
56.e even 2 1 inner 3136.1.n.a 16
56.h odd 2 1 inner 3136.1.n.a 16
56.j odd 6 1 3136.1.h.a ✓ 8
56.j odd 6 1 inner 3136.1.n.a 16
56.k odd 6 1 3136.1.h.a ✓ 8
56.k odd 6 1 inner 3136.1.n.a 16
56.m even 6 1 3136.1.h.a ✓ 8
56.m even 6 1 inner 3136.1.n.a 16
56.p even 6 1 3136.1.h.a ✓ 8
56.p even 6 1 inner 3136.1.n.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3136.1.h.a ✓ 8 7.c even 3 1
3136.1.h.a ✓ 8 7.d odd 6 1
3136.1.h.a ✓ 8 28.f even 6 1
3136.1.h.a ✓ 8 28.g odd 6 1
3136.1.h.a ✓ 8 56.j odd 6 1
3136.1.h.a ✓ 8 56.k odd 6 1
3136.1.h.a ✓ 8 56.m even 6 1
3136.1.h.a ✓ 8 56.p even 6 1
3136.1.n.a 16 1.a even 1 1 trivial
3136.1.n.a 16 4.b odd 2 1 inner
3136.1.n.a 16 7.b odd 2 1 inner
3136.1.n.a 16 7.c even 3 1 inner
3136.1.n.a 16 7.d odd 6 1 inner
3136.1.n.a 16 8.b even 2 1 inner
3136.1.n.a 16 8.d odd 2 1 CM
3136.1.n.a 16 28.d even 2 1 inner
3136.1.n.a 16 28.f even 6 1 inner
3136.1.n.a 16 28.g odd 6 1 inner
3136.1.n.a 16 56.e even 2 1 inner
3136.1.n.a 16 56.h odd 2 1 inner
3136.1.n.a 16 56.j odd 6 1 inner
3136.1.n.a 16 56.k odd 6 1 inner
3136.1.n.a 16 56.m even 6 1 inner
3136.1.n.a 16 56.p even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3136, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( (T^{4} + 4 T^{2} + 2)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2)^{8} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( (T^{4} - 4 T^{2} + 2)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 4 T^{2} + 2)^{4} \) Copy content Toggle raw display
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