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Properties

Label 1089.1.r.a
Level $1089$
Weight $1$
Character orbit 1089.r
Analytic conductor $0.543$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -11
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,1,Mod(245,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([5, 24]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.245");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1089.r (of order \(30\), degree \(8\), 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: \(0.543481798757\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.26198073.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{30}^{7} q^{3} - \zeta_{30}^{13} q^{4} + (\zeta_{30}^{11} + \zeta_{30}^{6}) q^{5} + \zeta_{30}^{14} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{30}^{7} q^{3} - \zeta_{30}^{13} q^{4} + (\zeta_{30}^{11} + \zeta_{30}^{6}) q^{5} + \zeta_{30}^{14} q^{9} + \zeta_{30}^{5} q^{12} + (\zeta_{30}^{13} - \zeta_{30}^{3}) q^{15} - \zeta_{30}^{11} q^{16} + (\zeta_{30}^{9} + \zeta_{30}^{4}) q^{20} + (\zeta_{30}^{12} + \cdots - \zeta_{30}^{2}) q^{25} + \cdots - \zeta_{30}^{14} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} + q^{4} - 3 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{3} + q^{4} - 3 q^{5} + q^{9} + 4 q^{12} - 3 q^{15} + q^{16} + 3 q^{20} - 2 q^{25} + 2 q^{27} + q^{31} - 2 q^{36} - 2 q^{37} + 3 q^{47} + 2 q^{48} - q^{49} + 3 q^{59} - 2 q^{64} - 4 q^{67} - 4 q^{75} + q^{81} - q^{93} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-\zeta_{30}^{3}\) \(-\zeta_{30}^{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1
−0.104528 + 0.994522i
0.913545 − 0.406737i
0.669131 − 0.743145i
−0.978148 + 0.207912i
0.913545 + 0.406737i
−0.978148 − 0.207912i
0.669131 + 0.743145i
−0.104528 − 0.994522i
0 −0.669131 + 0.743145i −0.978148 + 0.207912i −1.72256 − 0.181049i 0 0 0 −0.104528 − 0.994522i 0
608.1 0 0.978148 + 0.207912i 0.669131 + 0.743145i −0.704489 − 1.58231i 0 0 0 0.913545 + 0.406737i 0
614.1 0 −0.913545 − 0.406737i −0.104528 + 0.994522i 1.28716 + 1.15897i 0 0 0 0.669131 + 0.743145i 0
632.1 0 0.104528 − 0.994522i 0.913545 + 0.406737i −0.360114 − 1.69420i 0 0 0 −0.978148 − 0.207912i 0
686.1 0 0.978148 − 0.207912i 0.669131 − 0.743145i −0.704489 + 1.58231i 0 0 0 0.913545 − 0.406737i 0
977.1 0 0.104528 + 0.994522i 0.913545 − 0.406737i −0.360114 + 1.69420i 0 0 0 −0.978148 + 0.207912i 0
995.1 0 −0.913545 + 0.406737i −0.104528 − 0.994522i 1.28716 − 1.15897i 0 0 0 0.669131 − 0.743145i 0
1049.1 0 −0.669131 − 0.743145i −0.978148 − 0.207912i −1.72256 + 0.181049i 0 0 0 −0.104528 + 0.994522i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 245.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
9.d odd 6 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
99.g even 6 1 inner
99.n odd 30 3 inner
99.p even 30 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.1.r.a 8
3.b odd 2 1 3267.1.v.a 8
9.c even 3 1 3267.1.v.a 8
9.d odd 6 1 inner 1089.1.r.a 8
11.b odd 2 1 CM 1089.1.r.a 8
11.c even 5 1 1089.1.i.a ✓ 2
11.c even 5 3 inner 1089.1.r.a 8
11.d odd 10 1 1089.1.i.a ✓ 2
11.d odd 10 3 inner 1089.1.r.a 8
33.d even 2 1 3267.1.v.a 8
33.f even 10 1 3267.1.i.a 2
33.f even 10 3 3267.1.v.a 8
33.h odd 10 1 3267.1.i.a 2
33.h odd 10 3 3267.1.v.a 8
99.g even 6 1 inner 1089.1.r.a 8
99.h odd 6 1 3267.1.v.a 8
99.m even 15 1 3267.1.i.a 2
99.m even 15 3 3267.1.v.a 8
99.n odd 30 1 1089.1.i.a ✓ 2
99.n odd 30 3 inner 1089.1.r.a 8
99.o odd 30 1 3267.1.i.a 2
99.o odd 30 3 3267.1.v.a 8
99.p even 30 1 1089.1.i.a ✓ 2
99.p even 30 3 inner 1089.1.r.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.1.i.a ✓ 2 11.c even 5 1
1089.1.i.a ✓ 2 11.d odd 10 1
1089.1.i.a ✓ 2 99.n odd 30 1
1089.1.i.a ✓ 2 99.p even 30 1
1089.1.r.a 8 1.a even 1 1 trivial
1089.1.r.a 8 9.d odd 6 1 inner
1089.1.r.a 8 11.b odd 2 1 CM
1089.1.r.a 8 11.c even 5 3 inner
1089.1.r.a 8 11.d odd 10 3 inner
1089.1.r.a 8 99.g even 6 1 inner
1089.1.r.a 8 99.n odd 30 3 inner
1089.1.r.a 8 99.p even 30 3 inner
3267.1.i.a 2 33.f even 10 1
3267.1.i.a 2 33.h odd 10 1
3267.1.i.a 2 99.m even 15 1
3267.1.i.a 2 99.o odd 30 1
3267.1.v.a 8 3.b odd 2 1
3267.1.v.a 8 9.c even 3 1
3267.1.v.a 8 33.d even 2 1
3267.1.v.a 8 33.f even 10 3
3267.1.v.a 8 33.h odd 10 3
3267.1.v.a 8 99.h odd 6 1
3267.1.v.a 8 99.m even 15 3
3267.1.v.a 8 99.o odd 30 3

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1089, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} + 3 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} - T^{7} + T^{5} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} - 3 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$53$ \( T^{8} - 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{8} - 3 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} - 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
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