Properties

Label 1089.6.a.c
Level $1089$
Weight $6$
Character orbit 1089.a
Self dual yes
Analytic conductor $174.658$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,6,Mod(1,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1089.a (trivial)

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: \(174.657979776\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 4 q^{2} - 16 q^{4} + 19 q^{5} - 10 q^{7} + 192 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} - 16 q^{4} + 19 q^{5} - 10 q^{7} + 192 q^{8} - 76 q^{10} + 1148 q^{13} + 40 q^{14} - 256 q^{16} + 686 q^{17} + 384 q^{19} - 304 q^{20} - 3709 q^{23} - 2764 q^{25} - 4592 q^{26} + 160 q^{28} - 5424 q^{29} - 6443 q^{31} - 5120 q^{32} - 2744 q^{34} - 190 q^{35} + 12063 q^{37} - 1536 q^{38} + 3648 q^{40} - 1528 q^{41} + 4026 q^{43} + 14836 q^{46} - 7168 q^{47} - 16707 q^{49} + 11056 q^{50} - 18368 q^{52} + 29862 q^{53} - 1920 q^{56} + 21696 q^{58} + 6461 q^{59} + 16980 q^{61} + 25772 q^{62} + 28672 q^{64} + 21812 q^{65} + 29999 q^{67} - 10976 q^{68} + 760 q^{70} - 31023 q^{71} - 1924 q^{73} - 48252 q^{74} - 6144 q^{76} - 65138 q^{79} - 4864 q^{80} + 6112 q^{82} - 102714 q^{83} + 13034 q^{85} - 16104 q^{86} - 17415 q^{89} - 11480 q^{91} + 59344 q^{92} + 28672 q^{94} + 7296 q^{95} + 66905 q^{97} + 66828 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−4.00000 0 −16.0000 19.0000 0 −10.0000 192.000 0 −76.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.6.a.c 1
3.b odd 2 1 121.6.a.b 1
11.b odd 2 1 99.6.a.c 1
33.d even 2 1 11.6.a.a 1
132.d odd 2 1 176.6.a.c 1
165.d even 2 1 275.6.a.a 1
165.l odd 4 2 275.6.b.a 2
231.h odd 2 1 539.6.a.c 1
264.m even 2 1 704.6.a.h 1
264.p odd 2 1 704.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.a 1 33.d even 2 1
99.6.a.c 1 11.b odd 2 1
121.6.a.b 1 3.b odd 2 1
176.6.a.c 1 132.d odd 2 1
275.6.a.a 1 165.d even 2 1
275.6.b.a 2 165.l odd 4 2
539.6.a.c 1 231.h odd 2 1
704.6.a.c 1 264.p odd 2 1
704.6.a.h 1 264.m even 2 1
1089.6.a.c 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\):

\( T_{2} + 4 \) Copy content Toggle raw display
\( T_{5} - 19 \) Copy content Toggle raw display
\( T_{7} + 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 19 \) Copy content Toggle raw display
$7$ \( T + 10 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 1148 \) Copy content Toggle raw display
$17$ \( T - 686 \) Copy content Toggle raw display
$19$ \( T - 384 \) Copy content Toggle raw display
$23$ \( T + 3709 \) Copy content Toggle raw display
$29$ \( T + 5424 \) Copy content Toggle raw display
$31$ \( T + 6443 \) Copy content Toggle raw display
$37$ \( T - 12063 \) Copy content Toggle raw display
$41$ \( T + 1528 \) Copy content Toggle raw display
$43$ \( T - 4026 \) Copy content Toggle raw display
$47$ \( T + 7168 \) Copy content Toggle raw display
$53$ \( T - 29862 \) Copy content Toggle raw display
$59$ \( T - 6461 \) Copy content Toggle raw display
$61$ \( T - 16980 \) Copy content Toggle raw display
$67$ \( T - 29999 \) Copy content Toggle raw display
$71$ \( T + 31023 \) Copy content Toggle raw display
$73$ \( T + 1924 \) Copy content Toggle raw display
$79$ \( T + 65138 \) Copy content Toggle raw display
$83$ \( T + 102714 \) Copy content Toggle raw display
$89$ \( T + 17415 \) Copy content Toggle raw display
$97$ \( T - 66905 \) Copy content Toggle raw display
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