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Properties

Label 891.1.q.a
Level $891$
Weight $1$
Character orbit 891.q
Analytic conductor $0.445$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [891,1,Mod(10,891)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(891, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([8, 9])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("891.10"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 891.q (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.444666926256\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 297)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.459450093735369.1

$q$-expansion

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

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

\(f(q)\) \(=\) \( q - \zeta_{18}^{5} q^{4} + ( - \zeta_{18}^{4} + \zeta_{18}^{3}) q^{5} + \zeta_{18} q^{11} - \zeta_{18} q^{16} + ( - \zeta_{18}^{8} - 1) q^{20} - \zeta_{18}^{5} q^{23} + (\zeta_{18}^{8} + \cdots + \zeta_{18}^{6}) q^{25} + \cdots + (\zeta_{18}^{2} + 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} - 6 q^{20} - 3 q^{25} - 3 q^{31} + 3 q^{44} + 3 q^{47} + 3 q^{59} - 3 q^{64} + 6 q^{67} - 3 q^{89} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(244\) \(650\)
\(\chi(n)\) \(-1\) \(-\zeta_{18}^{5}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−0.766044 + 0.642788i
0.939693 − 0.342020i
−0.173648 − 0.984808i
−0.173648 + 0.984808i
0.939693 + 0.342020i
−0.766044 − 0.642788i
0 0 −0.939693 + 0.342020i 1.43969 + 1.20805i 0 0 0 0 0
208.1 0 0 0.173648 + 0.984808i 0.326352 + 0.118782i 0 0 0 0 0
307.1 0 0 0.766044 + 0.642788i −0.266044 + 1.50881i 0 0 0 0 0
505.1 0 0 0.766044 − 0.642788i −0.266044 − 1.50881i 0 0 0 0 0
604.1 0 0 0.173648 − 0.984808i 0.326352 − 0.118782i 0 0 0 0 0
802.1 0 0 −0.939693 − 0.342020i 1.43969 − 1.20805i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
27.e even 9 1 inner
297.q odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.1.q.a 6
3.b odd 2 1 297.1.q.a ✓ 6
9.c even 3 1 2673.1.q.a 6
9.c even 3 1 2673.1.q.c 6
9.d odd 6 1 2673.1.q.b 6
9.d odd 6 1 2673.1.q.d 6
11.b odd 2 1 CM 891.1.q.a 6
27.e even 9 1 inner 891.1.q.a 6
27.e even 9 1 2673.1.q.a 6
27.e even 9 1 2673.1.q.c 6
27.f odd 18 1 297.1.q.a ✓ 6
27.f odd 18 1 2673.1.q.b 6
27.f odd 18 1 2673.1.q.d 6
33.d even 2 1 297.1.q.a ✓ 6
33.f even 10 4 3267.1.bf.a 24
33.h odd 10 4 3267.1.bf.a 24
99.g even 6 1 2673.1.q.b 6
99.g even 6 1 2673.1.q.d 6
99.h odd 6 1 2673.1.q.a 6
99.h odd 6 1 2673.1.q.c 6
297.o even 18 1 297.1.q.a ✓ 6
297.o even 18 1 2673.1.q.b 6
297.o even 18 1 2673.1.q.d 6
297.q odd 18 1 inner 891.1.q.a 6
297.q odd 18 1 2673.1.q.a 6
297.q odd 18 1 2673.1.q.c 6
297.v odd 90 4 3267.1.bf.a 24
297.x even 90 4 3267.1.bf.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.1.q.a ✓ 6 3.b odd 2 1
297.1.q.a ✓ 6 27.f odd 18 1
297.1.q.a ✓ 6 33.d even 2 1
297.1.q.a ✓ 6 297.o even 18 1
891.1.q.a 6 1.a even 1 1 trivial
891.1.q.a 6 11.b odd 2 1 CM
891.1.q.a 6 27.e even 9 1 inner
891.1.q.a 6 297.q odd 18 1 inner
2673.1.q.a 6 9.c even 3 1
2673.1.q.a 6 27.e even 9 1
2673.1.q.a 6 99.h odd 6 1
2673.1.q.a 6 297.q odd 18 1
2673.1.q.b 6 9.d odd 6 1
2673.1.q.b 6 27.f odd 18 1
2673.1.q.b 6 99.g even 6 1
2673.1.q.b 6 297.o even 18 1
2673.1.q.c 6 9.c even 3 1
2673.1.q.c 6 27.e even 9 1
2673.1.q.c 6 99.h odd 6 1
2673.1.q.c 6 297.q odd 18 1
2673.1.q.d 6 9.d odd 6 1
2673.1.q.d 6 27.f odd 18 1
2673.1.q.d 6 99.g even 6 1
2673.1.q.d 6 297.o even 18 1
3267.1.bf.a 24 33.f even 10 4
3267.1.bf.a 24 33.h odd 10 4
3267.1.bf.a 24 297.v odd 90 4
3267.1.bf.a 24 297.x even 90 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(891, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} + 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} \) Copy content Toggle raw display
$47$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( (T^{3} - 3 T - 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} - 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$97$ \( T^{6} - 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
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