Properties

Label 105.2.i.d
Level $105$
Weight $2$
Character orbit 105.i
Analytic conductor $0.838$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [105,2,Mod(16,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 105.i (of order \(3\), degree \(2\), 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: \(0.838429221223\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{3} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \cdots - 2) q^{4} - \zeta_{12}^{2} q^{5} + (\zeta_{12}^{3} - 2 \zeta_{12} + 1) q^{6} + \cdots + (\zeta_{12}^{3} - 2 \zeta_{12} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{3} - 4 q^{4} - 2 q^{5} + 4 q^{6} - 24 q^{8} - 2 q^{9} + 2 q^{10} + 2 q^{11} + 4 q^{12} + 16 q^{13} + 18 q^{14} - 4 q^{15} - 16 q^{16} - 10 q^{17} + 2 q^{18} - 2 q^{19} + 8 q^{20} - 8 q^{22}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(-\zeta_{12}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.366025 + 0.633975i 0.500000 + 0.866025i 0.732051 + 1.26795i −0.500000 + 0.866025i −0.732051 −0.866025 2.50000i −2.53590 −0.500000 + 0.866025i −0.366025 0.633975i
16.2 1.36603 2.36603i 0.500000 + 0.866025i −2.73205 4.73205i −0.500000 + 0.866025i 2.73205 0.866025 + 2.50000i −9.46410 −0.500000 + 0.866025i 1.36603 + 2.36603i
46.1 −0.366025 0.633975i 0.500000 0.866025i 0.732051 1.26795i −0.500000 0.866025i −0.732051 −0.866025 + 2.50000i −2.53590 −0.500000 0.866025i −0.366025 + 0.633975i
46.2 1.36603 + 2.36603i 0.500000 0.866025i −2.73205 + 4.73205i −0.500000 0.866025i 2.73205 0.866025 2.50000i −9.46410 −0.500000 0.866025i 1.36603 2.36603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.2.i.d 4
3.b odd 2 1 315.2.j.c 4
4.b odd 2 1 1680.2.bg.o 4
5.b even 2 1 525.2.i.f 4
5.c odd 4 1 525.2.r.a 4
5.c odd 4 1 525.2.r.f 4
7.b odd 2 1 735.2.i.l 4
7.c even 3 1 inner 105.2.i.d 4
7.c even 3 1 735.2.a.g 2
7.d odd 6 1 735.2.a.h 2
7.d odd 6 1 735.2.i.l 4
21.g even 6 1 2205.2.a.ba 2
21.h odd 6 1 315.2.j.c 4
21.h odd 6 1 2205.2.a.z 2
28.g odd 6 1 1680.2.bg.o 4
35.i odd 6 1 3675.2.a.be 2
35.j even 6 1 525.2.i.f 4
35.j even 6 1 3675.2.a.bg 2
35.l odd 12 1 525.2.r.a 4
35.l odd 12 1 525.2.r.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d 4 1.a even 1 1 trivial
105.2.i.d 4 7.c even 3 1 inner
315.2.j.c 4 3.b odd 2 1
315.2.j.c 4 21.h odd 6 1
525.2.i.f 4 5.b even 2 1
525.2.i.f 4 35.j even 6 1
525.2.r.a 4 5.c odd 4 1
525.2.r.a 4 35.l odd 12 1
525.2.r.f 4 5.c odd 4 1
525.2.r.f 4 35.l odd 12 1
735.2.a.g 2 7.c even 3 1
735.2.a.h 2 7.d odd 6 1
735.2.i.l 4 7.b odd 2 1
735.2.i.l 4 7.d odd 6 1
1680.2.bg.o 4 4.b odd 2 1
1680.2.bg.o 4 28.g odd 6 1
2205.2.a.z 2 21.h odd 6 1
2205.2.a.ba 2 21.g even 6 1
3675.2.a.be 2 35.i odd 6 1
3675.2.a.bg 2 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} + 6T_{2}^{2} + 4T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(105, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 11T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} - 8 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 10 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T - 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 23)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} + \cdots + 10816 \) Copy content Toggle raw display
$59$ \( T^{4} + 10 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 12 T^{3} + \cdots + 1521 \) Copy content Toggle raw display
$71$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 8 T^{3} + \cdots + 3481 \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} + \cdots + 9801 \) Copy content Toggle raw display
$83$ \( (T^{2} - 6 T - 138)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} + \cdots + 19044 \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T + 16)^{2} \) Copy content Toggle raw display
show more
show less