Properties

Label 108.3.c.b
Level $108$
Weight $3$
Character orbit 108.c
Analytic conductor $2.943$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [108,3,Mod(53,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 108.c (of order \(2\), degree \(1\), 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: \(2.94278685509\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 9i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - 7 q^{7} + \beta q^{11} + 14 q^{13} + 2 \beta q^{17} + 8 q^{19} - 4 \beta q^{23} - 56 q^{25} - 2 \beta q^{29} + 35 q^{31} - 7 \beta q^{35} + 44 q^{37} + 4 \beta q^{41} - 22 q^{43} - 6 \beta q^{47} + \cdots + 11 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{7} + 28 q^{13} + 16 q^{19} - 112 q^{25} + 70 q^{31} + 88 q^{37} - 44 q^{43} - 162 q^{55} + 40 q^{61} + 28 q^{67} + 178 q^{73} + 220 q^{79} - 324 q^{85} - 196 q^{91} + 22 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
1.00000i
1.00000i
0 0 0 9.00000i 0 −7.00000 0 0 0
53.2 0 0 0 9.00000i 0 −7.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.3.c.b 2
3.b odd 2 1 inner 108.3.c.b 2
4.b odd 2 1 432.3.e.g 2
5.b even 2 1 2700.3.g.m 2
5.c odd 4 1 2700.3.b.a 2
5.c odd 4 1 2700.3.b.f 2
8.b even 2 1 1728.3.e.e 2
8.d odd 2 1 1728.3.e.n 2
9.c even 3 2 324.3.g.c 4
9.d odd 6 2 324.3.g.c 4
12.b even 2 1 432.3.e.g 2
15.d odd 2 1 2700.3.g.m 2
15.e even 4 1 2700.3.b.a 2
15.e even 4 1 2700.3.b.f 2
24.f even 2 1 1728.3.e.n 2
24.h odd 2 1 1728.3.e.e 2
36.f odd 6 2 1296.3.q.d 4
36.h even 6 2 1296.3.q.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.c.b 2 1.a even 1 1 trivial
108.3.c.b 2 3.b odd 2 1 inner
324.3.g.c 4 9.c even 3 2
324.3.g.c 4 9.d odd 6 2
432.3.e.g 2 4.b odd 2 1
432.3.e.g 2 12.b even 2 1
1296.3.q.d 4 36.f odd 6 2
1296.3.q.d 4 36.h even 6 2
1728.3.e.e 2 8.b even 2 1
1728.3.e.e 2 24.h odd 2 1
1728.3.e.n 2 8.d odd 2 1
1728.3.e.n 2 24.f even 2 1
2700.3.b.a 2 5.c odd 4 1
2700.3.b.a 2 15.e even 4 1
2700.3.b.f 2 5.c odd 4 1
2700.3.b.f 2 15.e even 4 1
2700.3.g.m 2 5.b even 2 1
2700.3.g.m 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 81 \) acting on \(S_{3}^{\mathrm{new}}(108, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 81 \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 81 \) Copy content Toggle raw display
$13$ \( (T - 14)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 324 \) Copy content Toggle raw display
$19$ \( (T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( T^{2} + 324 \) Copy content Toggle raw display
$31$ \( (T - 35)^{2} \) Copy content Toggle raw display
$37$ \( (T - 44)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 1296 \) Copy content Toggle raw display
$43$ \( (T + 22)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2916 \) Copy content Toggle raw display
$53$ \( T^{2} + 81 \) Copy content Toggle raw display
$59$ \( T^{2} + 324 \) Copy content Toggle raw display
$61$ \( (T - 20)^{2} \) Copy content Toggle raw display
$67$ \( (T - 14)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 15876 \) Copy content Toggle raw display
$73$ \( (T - 89)^{2} \) Copy content Toggle raw display
$79$ \( (T - 110)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 729 \) Copy content Toggle raw display
$89$ \( T^{2} + 324 \) Copy content Toggle raw display
$97$ \( (T - 11)^{2} \) Copy content Toggle raw display
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