Properties

Label 3600.1.bb.a
Level $3600$
Weight $1$
Character orbit 3600.bb
Analytic conductor $1.797$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -15
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3600,1,Mod(757,3600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("3600.757");
 
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.bb (of order \(4\), 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: \(1.79663404548\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.153600.2

$q$-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.

\(f(q)\) \(=\) \( q - i q^{2} - q^{4} + i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - q^{4} + i q^{8} + q^{16} + (i - 1) q^{17} + (i - 1) q^{19} + (i - 1) q^{23} - i q^{32} + (i + 1) q^{34} + (i + 1) q^{38} + (i + 1) q^{46} + (i - 1) q^{47} - i q^{49} - 2 q^{53} + (i + 1) q^{61} - q^{64} + ( - i + 1) q^{68} + ( - i + 1) q^{76} + 2 i q^{79} - 2 i q^{83} + ( - i + 1) q^{92} + (i + 1) q^{94} - q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{16} - 2 q^{17} - 2 q^{19} - 2 q^{23} + 2 q^{34} + 2 q^{38} + 2 q^{46} - 2 q^{47} - 4 q^{53} + 2 q^{61} - 2 q^{64} + 2 q^{68} + 2 q^{76} + 2 q^{92} + 2 q^{94} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 0 1.00000i 0 0
1693.1 1.00000i 0 −1.00000 0 0 0 1.00000i 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
80.i odd 4 1 inner
240.bf even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.bb.a 2
3.b odd 2 1 3600.1.bb.b yes 2
5.b even 2 1 3600.1.bb.b yes 2
5.c odd 4 1 3600.1.bf.a yes 2
5.c odd 4 1 3600.1.bf.b yes 2
15.d odd 2 1 CM 3600.1.bb.a 2
15.e even 4 1 3600.1.bf.a yes 2
15.e even 4 1 3600.1.bf.b yes 2
16.e even 4 1 3600.1.bf.b yes 2
48.i odd 4 1 3600.1.bf.a yes 2
80.i odd 4 1 inner 3600.1.bb.a 2
80.q even 4 1 3600.1.bf.a yes 2
80.t odd 4 1 3600.1.bb.b yes 2
240.bb even 4 1 3600.1.bb.b yes 2
240.bf even 4 1 inner 3600.1.bb.a 2
240.bm odd 4 1 3600.1.bf.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3600.1.bb.a 2 1.a even 1 1 trivial
3600.1.bb.a 2 15.d odd 2 1 CM
3600.1.bb.a 2 80.i odd 4 1 inner
3600.1.bb.a 2 240.bf even 4 1 inner
3600.1.bb.b yes 2 3.b odd 2 1
3600.1.bb.b yes 2 5.b even 2 1
3600.1.bb.b yes 2 80.t odd 4 1
3600.1.bb.b yes 2 240.bb even 4 1
3600.1.bf.a yes 2 5.c odd 4 1
3600.1.bf.a yes 2 15.e even 4 1
3600.1.bf.a yes 2 48.i odd 4 1
3600.1.bf.a yes 2 80.q even 4 1
3600.1.bf.b yes 2 5.c odd 4 1
3600.1.bf.b yes 2 15.e even 4 1
3600.1.bf.b yes 2 16.e even 4 1
3600.1.bf.b yes 2 240.bm odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{2} + 2T_{17} + 2 \) acting on \(S_{1}^{\mathrm{new}}(3600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) 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
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 4 \) Copy content Toggle raw display
$83$ \( T^{2} + 4 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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