Properties

Label 3600.1.j.a
Level $3600$
Weight $1$
Character orbit 3600.j
Analytic conductor $1.797$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -3, -4, 12
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3600,1,Mod(1999,3600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3600.1999");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3600.j (of order \(2\), degree \(1\), 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: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 144)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\zeta_{12})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.2916000000.5

$q$-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.

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q - i q^{13} - i q^{37} - q^{49} + q^{61} - i q^{73} + i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{49} + 4 q^{61}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1999.1
1.00000i
1.00000i
0 0 0 0 0 0 0 0 0
1999.2 0 0 0 0 0 0 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 CM by \(\Q(\sqrt{-3}) \)
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
12.b even 2 1 RM by \(\Q(\sqrt{3}) \)
5.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.j.a 2
3.b odd 2 1 CM 3600.1.j.a 2
4.b odd 2 1 CM 3600.1.j.a 2
5.b even 2 1 inner 3600.1.j.a 2
5.c odd 4 1 144.1.g.a 1
5.c odd 4 1 3600.1.e.b 1
12.b even 2 1 RM 3600.1.j.a 2
15.d odd 2 1 inner 3600.1.j.a 2
15.e even 4 1 144.1.g.a 1
15.e even 4 1 3600.1.e.b 1
20.d odd 2 1 inner 3600.1.j.a 2
20.e even 4 1 144.1.g.a 1
20.e even 4 1 3600.1.e.b 1
40.i odd 4 1 576.1.g.a 1
40.k even 4 1 576.1.g.a 1
45.k odd 12 2 1296.1.o.b 2
45.l even 12 2 1296.1.o.b 2
60.h even 2 1 inner 3600.1.j.a 2
60.l odd 4 1 144.1.g.a 1
60.l odd 4 1 3600.1.e.b 1
80.i odd 4 1 2304.1.b.a 2
80.j even 4 1 2304.1.b.a 2
80.s even 4 1 2304.1.b.a 2
80.t odd 4 1 2304.1.b.a 2
120.q odd 4 1 576.1.g.a 1
120.w even 4 1 576.1.g.a 1
180.v odd 12 2 1296.1.o.b 2
180.x even 12 2 1296.1.o.b 2
240.z odd 4 1 2304.1.b.a 2
240.bb even 4 1 2304.1.b.a 2
240.bd odd 4 1 2304.1.b.a 2
240.bf even 4 1 2304.1.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.1.g.a 1 5.c odd 4 1
144.1.g.a 1 15.e even 4 1
144.1.g.a 1 20.e even 4 1
144.1.g.a 1 60.l odd 4 1
576.1.g.a 1 40.i odd 4 1
576.1.g.a 1 40.k even 4 1
576.1.g.a 1 120.q odd 4 1
576.1.g.a 1 120.w even 4 1
1296.1.o.b 2 45.k odd 12 2
1296.1.o.b 2 45.l even 12 2
1296.1.o.b 2 180.v odd 12 2
1296.1.o.b 2 180.x even 12 2
2304.1.b.a 2 80.i odd 4 1
2304.1.b.a 2 80.j even 4 1
2304.1.b.a 2 80.s even 4 1
2304.1.b.a 2 80.t odd 4 1
2304.1.b.a 2 240.z odd 4 1
2304.1.b.a 2 240.bb even 4 1
2304.1.b.a 2 240.bd odd 4 1
2304.1.b.a 2 240.bf even 4 1
3600.1.e.b 1 5.c odd 4 1
3600.1.e.b 1 15.e even 4 1
3600.1.e.b 1 20.e even 4 1
3600.1.e.b 1 60.l odd 4 1
3600.1.j.a 2 1.a even 1 1 trivial
3600.1.j.a 2 3.b odd 2 1 CM
3600.1.j.a 2 4.b odd 2 1 CM
3600.1.j.a 2 5.b even 2 1 inner
3600.1.j.a 2 12.b even 2 1 RM
3600.1.j.a 2 15.d odd 2 1 inner
3600.1.j.a 2 20.d odd 2 1 inner
3600.1.j.a 2 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{1}^{\mathrm{new}}(3600, [\chi])\). 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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