Properties

Label 1089.2.a.o
Level $1089$
Weight $2$
Character orbit 1089.a
Self dual yes
Analytic conductor $8.696$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(8.69570878012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 363)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{4} + 3 q^{5} + 2 \beta q^{7} - \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + q^{4} + 3 q^{5} + 2 \beta q^{7} - \beta q^{8} + 3 \beta q^{10} + \beta q^{13} + 6 q^{14} - 5 q^{16} + \beta q^{17} - 4 \beta q^{19} + 3 q^{20} + 6 q^{23} + 4 q^{25} + 3 q^{26} + 2 \beta q^{28} - \beta q^{29} + 4 q^{31} - 3 \beta q^{32} + 3 q^{34} + 6 \beta q^{35} - 11 q^{37} - 12 q^{38} - 3 \beta q^{40} - \beta q^{41} - 2 \beta q^{43} + 6 \beta q^{46} + 5 q^{49} + 4 \beta q^{50} + \beta q^{52} + 9 q^{53} - 6 q^{56} - 3 q^{58} + 6 q^{59} + 4 \beta q^{62} + q^{64} + 3 \beta q^{65} - 2 q^{67} + \beta q^{68} + 18 q^{70} + 6 q^{71} - 4 \beta q^{73} - 11 \beta q^{74} - 4 \beta q^{76} - 15 q^{80} - 3 q^{82} + 3 \beta q^{85} - 6 q^{86} - 9 q^{89} + 6 q^{91} + 6 q^{92} - 12 \beta q^{95} - 7 q^{97} + 5 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 6 q^{5} + 12 q^{14} - 10 q^{16} + 6 q^{20} + 12 q^{23} + 8 q^{25} + 6 q^{26} + 8 q^{31} + 6 q^{34} - 22 q^{37} - 24 q^{38} + 10 q^{49} + 18 q^{53} - 12 q^{56} - 6 q^{58} + 12 q^{59} + 2 q^{64} - 4 q^{67} + 36 q^{70} + 12 q^{71} - 30 q^{80} - 6 q^{82} - 12 q^{86} - 18 q^{89} + 12 q^{91} + 12 q^{92} - 14 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
−1.73205 0 1.00000 3.00000 0 −3.46410 1.73205 0 −5.19615
1.2 1.73205 0 1.00000 3.00000 0 3.46410 −1.73205 0 5.19615
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.2.a.o 2
3.b odd 2 1 363.2.a.f 2
11.b odd 2 1 inner 1089.2.a.o 2
12.b even 2 1 5808.2.a.ca 2
15.d odd 2 1 9075.2.a.bo 2
33.d even 2 1 363.2.a.f 2
33.f even 10 4 363.2.e.m 8
33.h odd 10 4 363.2.e.m 8
132.d odd 2 1 5808.2.a.ca 2
165.d even 2 1 9075.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.f 2 3.b odd 2 1
363.2.a.f 2 33.d even 2 1
363.2.e.m 8 33.f even 10 4
363.2.e.m 8 33.h odd 10 4
1089.2.a.o 2 1.a even 1 1 trivial
1089.2.a.o 2 11.b odd 2 1 inner
5808.2.a.ca 2 12.b even 2 1
5808.2.a.ca 2 132.d odd 2 1
9075.2.a.bo 2 15.d odd 2 1
9075.2.a.bo 2 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1089))\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{7}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 12 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 3 \) Copy content Toggle raw display
$17$ \( T^{2} - 3 \) Copy content Toggle raw display
$19$ \( T^{2} - 48 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 3 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( (T + 11)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 3 \) Copy content Toggle raw display
$43$ \( T^{2} - 12 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 9)^{2} \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T + 2)^{2} \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 48 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T + 9)^{2} \) Copy content Toggle raw display
$97$ \( (T + 7)^{2} \) Copy content Toggle raw display
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