Properties

Label 336.1.o.a
Level $336$
Weight $1$
Character orbit 336.o
Self dual yes
Analytic conductor $0.168$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -3, -84, 28
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,1,Mod(335,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.335");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 336.o (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.167685844245\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{7})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.1008.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{3} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{7} + q^{9} + 2 q^{19} - q^{21} - q^{25} - q^{27} - 2 q^{31} - 2 q^{37} + q^{49} - 2 q^{57} + q^{63} + q^{75} + q^{81} + 2 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
335.1
0
0 −1.00000 0 0 0 1.00000 0 1.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
28.d even 2 1 RM by \(\Q(\sqrt{7}) \)
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.1.o.a 1
3.b odd 2 1 CM 336.1.o.a 1
4.b odd 2 1 336.1.o.b yes 1
7.b odd 2 1 336.1.o.b yes 1
7.c even 3 2 2352.1.z.c 2
7.d odd 6 2 2352.1.z.b 2
8.b even 2 1 1344.1.o.b 1
8.d odd 2 1 1344.1.o.a 1
12.b even 2 1 336.1.o.b yes 1
21.c even 2 1 336.1.o.b yes 1
21.g even 6 2 2352.1.z.b 2
21.h odd 6 2 2352.1.z.c 2
24.f even 2 1 1344.1.o.a 1
24.h odd 2 1 1344.1.o.b 1
28.d even 2 1 RM 336.1.o.a 1
28.f even 6 2 2352.1.z.c 2
28.g odd 6 2 2352.1.z.b 2
56.e even 2 1 1344.1.o.b 1
56.h odd 2 1 1344.1.o.a 1
84.h odd 2 1 CM 336.1.o.a 1
84.j odd 6 2 2352.1.z.c 2
84.n even 6 2 2352.1.z.b 2
168.e odd 2 1 1344.1.o.b 1
168.i even 2 1 1344.1.o.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.1.o.a 1 1.a even 1 1 trivial
336.1.o.a 1 3.b odd 2 1 CM
336.1.o.a 1 28.d even 2 1 RM
336.1.o.a 1 84.h odd 2 1 CM
336.1.o.b yes 1 4.b odd 2 1
336.1.o.b yes 1 7.b odd 2 1
336.1.o.b yes 1 12.b even 2 1
336.1.o.b yes 1 21.c even 2 1
1344.1.o.a 1 8.d odd 2 1
1344.1.o.a 1 24.f even 2 1
1344.1.o.a 1 56.h odd 2 1
1344.1.o.a 1 168.i even 2 1
1344.1.o.b 1 8.b even 2 1
1344.1.o.b 1 24.h odd 2 1
1344.1.o.b 1 56.e even 2 1
1344.1.o.b 1 168.e odd 2 1
2352.1.z.b 2 7.d odd 6 2
2352.1.z.b 2 21.g even 6 2
2352.1.z.b 2 28.g odd 6 2
2352.1.z.b 2 84.n even 6 2
2352.1.z.c 2 7.c even 3 2
2352.1.z.c 2 21.h odd 6 2
2352.1.z.c 2 28.f even 6 2
2352.1.z.c 2 84.j odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{19} - 2 \) acting on \(S_{1}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 2 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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