Properties

Label 1089.1.s.a
Level $1089$
Weight $1$
Character orbit 1089.s
Analytic conductor $0.543$
Analytic rank $0$
Dimension $8$
Projective image $D_{3}$
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(40,1089)]
 
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([10, 21]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.40");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1089.s (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: no (minimal twist has level 99)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.891.1
Artin image: $S_3\times C_{15}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{30} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{30}^{7} q^{3} - \zeta_{30}^{13} q^{4} + \zeta_{30} q^{5} + \zeta_{30}^{14} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{30}^{7} q^{3} - \zeta_{30}^{13} q^{4} + \zeta_{30} q^{5} + \zeta_{30}^{14} q^{9} - \zeta_{30}^{5} q^{12} - \zeta_{30}^{8} q^{15} - \zeta_{30}^{11} q^{16} - \zeta_{30}^{14} q^{20} + \zeta_{30}^{10} q^{23} + \zeta_{30}^{6} q^{27} - \zeta_{30}^{4} q^{31} + \zeta_{30}^{12} q^{36} + \zeta_{30}^{3} q^{37} - q^{45} - \zeta_{30}^{2} q^{47} - \zeta_{30}^{3} q^{48} - \zeta_{30} q^{49} + \zeta_{30}^{9} q^{53} + \zeta_{30}^{13} q^{59} - \zeta_{30}^{6} q^{60} - \zeta_{30}^{9} q^{64} - \zeta_{30}^{10} q^{67} + 2 \zeta_{30}^{2} q^{69} - \zeta_{30}^{6} q^{71} - \zeta_{30}^{12} q^{80} - \zeta_{30}^{13} q^{81} + q^{89} + 2 \zeta_{30}^{8} q^{92} + \zeta_{30}^{11} q^{93} - \zeta_{30}^{14} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} + q^{4} - q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} + q^{4} - q^{5} + q^{9} - 4 q^{12} - q^{15} + q^{16} - q^{20} - 8 q^{23} - 2 q^{27} - q^{31} - 2 q^{36} + 2 q^{37} - 8 q^{45} - q^{47} - 2 q^{48} + q^{49} + 2 q^{53} - q^{59} + 2 q^{60} - 2 q^{64} + 4 q^{67} + 2 q^{69} + 2 q^{71} + 2 q^{80} + q^{81} + 16 q^{89} + 2 q^{92} - q^{93} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
−0.104528 0.994522i
0.669131 + 0.743145i
−0.978148 0.207912i
0.913545 + 0.406737i
−0.978148 + 0.207912i
0.669131 0.743145i
0.913545 0.406737i
−0.104528 + 0.994522i
0 0.669131 + 0.743145i −0.978148 0.207912i 0.104528 + 0.994522i 0 0 0 −0.104528 + 0.994522i 0
94.1 0 0.913545 0.406737i −0.104528 0.994522i −0.669131 0.743145i 0 0 0 0.669131 0.743145i 0
112.1 0 −0.104528 0.994522i 0.913545 0.406737i 0.978148 + 0.207912i 0 0 0 −0.978148 + 0.207912i 0
403.1 0 −0.978148 + 0.207912i 0.669131 0.743145i −0.913545 0.406737i 0 0 0 0.913545 0.406737i 0
457.1 0 −0.104528 + 0.994522i 0.913545 + 0.406737i 0.978148 0.207912i 0 0 0 −0.978148 0.207912i 0
475.1 0 0.913545 + 0.406737i −0.104528 + 0.994522i −0.669131 + 0.743145i 0 0 0 0.669131 + 0.743145i 0
481.1 0 −0.978148 0.207912i 0.669131 + 0.743145i −0.913545 + 0.406737i 0 0 0 0.913545 + 0.406737i 0
844.1 0 0.669131 0.743145i −0.978148 + 0.207912i 0.104528 0.994522i 0 0 0 −0.104528 0.994522i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.1
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.c even 3 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
99.h odd 6 1 inner
99.m even 15 3 inner
99.o odd 30 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.1.s.a 8
3.b odd 2 1 3267.1.w.a 8
9.c even 3 1 inner 1089.1.s.a 8
9.d odd 6 1 3267.1.w.a 8
11.b odd 2 1 CM 1089.1.s.a 8
11.c even 5 1 99.1.h.a 2
11.c even 5 3 inner 1089.1.s.a 8
11.d odd 10 1 99.1.h.a 2
11.d odd 10 3 inner 1089.1.s.a 8
33.d even 2 1 3267.1.w.a 8
33.f even 10 1 297.1.h.a 2
33.f even 10 3 3267.1.w.a 8
33.h odd 10 1 297.1.h.a 2
33.h odd 10 3 3267.1.w.a 8
44.g even 10 1 1584.1.bf.b 2
44.h odd 10 1 1584.1.bf.b 2
55.h odd 10 1 2475.1.y.a 2
55.j even 10 1 2475.1.y.a 2
55.k odd 20 2 2475.1.t.a 4
55.l even 20 2 2475.1.t.a 4
99.g even 6 1 3267.1.w.a 8
99.h odd 6 1 inner 1089.1.s.a 8
99.m even 15 1 99.1.h.a 2
99.m even 15 1 891.1.c.a 1
99.m even 15 3 inner 1089.1.s.a 8
99.n odd 30 1 297.1.h.a 2
99.n odd 30 1 891.1.c.b 1
99.n odd 30 3 3267.1.w.a 8
99.o odd 30 1 99.1.h.a 2
99.o odd 30 1 891.1.c.a 1
99.o odd 30 3 inner 1089.1.s.a 8
99.p even 30 1 297.1.h.a 2
99.p even 30 1 891.1.c.b 1
99.p even 30 3 3267.1.w.a 8
396.be odd 30 1 1584.1.bf.b 2
396.bf even 30 1 1584.1.bf.b 2
495.bl even 30 1 2475.1.y.a 2
495.br odd 30 1 2475.1.y.a 2
495.bs even 60 2 2475.1.t.a 4
495.bt odd 60 2 2475.1.t.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.1.h.a 2 11.c even 5 1
99.1.h.a 2 11.d odd 10 1
99.1.h.a 2 99.m even 15 1
99.1.h.a 2 99.o odd 30 1
297.1.h.a 2 33.f even 10 1
297.1.h.a 2 33.h odd 10 1
297.1.h.a 2 99.n odd 30 1
297.1.h.a 2 99.p even 30 1
891.1.c.a 1 99.m even 15 1
891.1.c.a 1 99.o odd 30 1
891.1.c.b 1 99.n odd 30 1
891.1.c.b 1 99.p even 30 1
1089.1.s.a 8 1.a even 1 1 trivial
1089.1.s.a 8 9.c even 3 1 inner
1089.1.s.a 8 11.b odd 2 1 CM
1089.1.s.a 8 11.c even 5 3 inner
1089.1.s.a 8 11.d odd 10 3 inner
1089.1.s.a 8 99.h odd 6 1 inner
1089.1.s.a 8 99.m even 15 3 inner
1089.1.s.a 8 99.o odd 30 3 inner
1584.1.bf.b 2 44.g even 10 1
1584.1.bf.b 2 44.h odd 10 1
1584.1.bf.b 2 396.be odd 30 1
1584.1.bf.b 2 396.bf even 30 1
2475.1.t.a 4 55.k odd 20 2
2475.1.t.a 4 55.l even 20 2
2475.1.t.a 4 495.bs even 60 2
2475.1.t.a 4 495.bt odd 60 2
2475.1.y.a 2 55.h odd 10 1
2475.1.y.a 2 55.j even 10 1
2475.1.y.a 2 495.bl even 30 1
2475.1.y.a 2 495.br odd 30 1
3267.1.w.a 8 3.b odd 2 1
3267.1.w.a 8 9.d odd 6 1
3267.1.w.a 8 33.d even 2 1
3267.1.w.a 8 33.f even 10 3
3267.1.w.a 8 33.h odd 10 3
3267.1.w.a 8 99.g even 6 1
3267.1.w.a 8 99.n odd 30 3
3267.1.w.a 8 99.p even 30 3

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{1}^{\mathrm{new}}(1089, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \) 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
$23$ \( (T^{2} + 2 T + 4)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \) Copy content Toggle raw display
$37$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \) Copy content Toggle raw display
$53$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) 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
$89$ \( (T - 2)^{8} \) Copy content Toggle raw display
$97$ \( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \) Copy content Toggle raw display
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