Properties

Label 891.1.c.a
Level $891$
Weight $1$
Character orbit 891.c
Self dual yes
Analytic conductor $0.445$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -11
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [891,1,Mod(406,891)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(891, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.406");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 891.c (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: \(0.444666926256\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.891.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.891.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{4} - q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{4} - q^{5} + q^{11} + q^{16} - q^{20} + 2 q^{23} - q^{31} - q^{37} + q^{44} - q^{47} + q^{49} - q^{53} - q^{55} - q^{59} + q^{64} - q^{67} - q^{71} - q^{80} + 2 q^{89} + 2 q^{92} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(650\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
406.1
0
0 0 1.00000 −1.00000 0 0 0 0 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.1.c.a 1
3.b odd 2 1 891.1.c.b 1
9.c even 3 2 99.1.h.a 2
9.d odd 6 2 297.1.h.a 2
11.b odd 2 1 CM 891.1.c.a 1
33.d even 2 1 891.1.c.b 1
36.f odd 6 2 1584.1.bf.b 2
45.j even 6 2 2475.1.y.a 2
45.k odd 12 4 2475.1.t.a 4
99.g even 6 2 297.1.h.a 2
99.h odd 6 2 99.1.h.a 2
99.m even 15 8 1089.1.s.a 8
99.n odd 30 8 3267.1.w.a 8
99.o odd 30 8 1089.1.s.a 8
99.p even 30 8 3267.1.w.a 8
396.k even 6 2 1584.1.bf.b 2
495.o odd 6 2 2475.1.y.a 2
495.bf even 12 4 2475.1.t.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.1.h.a 2 9.c even 3 2
99.1.h.a 2 99.h odd 6 2
297.1.h.a 2 9.d odd 6 2
297.1.h.a 2 99.g even 6 2
891.1.c.a 1 1.a even 1 1 trivial
891.1.c.a 1 11.b odd 2 1 CM
891.1.c.b 1 3.b odd 2 1
891.1.c.b 1 33.d even 2 1
1089.1.s.a 8 99.m even 15 8
1089.1.s.a 8 99.o odd 30 8
1584.1.bf.b 2 36.f odd 6 2
1584.1.bf.b 2 396.k even 6 2
2475.1.t.a 4 45.k odd 12 4
2475.1.t.a 4 495.bf even 12 4
2475.1.y.a 2 45.j even 6 2
2475.1.y.a 2 495.o odd 6 2
3267.1.w.a 8 99.n odd 30 8
3267.1.w.a 8 99.p even 30 8

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(891, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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