Properties

Label 3600.2.a.bs
Level $3600$
Weight $2$
Character orbit 3600.a
Self dual yes
Analytic conductor $28.746$
Analytic rank $1$
Dimension $2$
CM discriminant -15
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.7461447277\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q -\beta q^{17} -4 q^{19} + 2 \beta q^{23} -8 q^{31} -2 \beta q^{47} -7 q^{49} + \beta q^{53} + 2 q^{61} -16 q^{79} -4 \beta q^{83} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 8q^{19} - 16q^{31} - 14q^{49} + 4q^{61} - 32q^{79} + O(q^{100}) \)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.61803
−0.618034
0 0 0 0 0 0 0 0 0
1.2 0 0 0 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.2.a.bs 2
3.b odd 2 1 inner 3600.2.a.bs 2
4.b odd 2 1 225.2.a.f 2
5.b even 2 1 inner 3600.2.a.bs 2
5.c odd 4 2 720.2.f.d 2
12.b even 2 1 225.2.a.f 2
15.d odd 2 1 CM 3600.2.a.bs 2
15.e even 4 2 720.2.f.d 2
20.d odd 2 1 225.2.a.f 2
20.e even 4 2 45.2.b.a 2
40.i odd 4 2 2880.2.f.j 2
40.k even 4 2 2880.2.f.k 2
60.h even 2 1 225.2.a.f 2
60.l odd 4 2 45.2.b.a 2
120.q odd 4 2 2880.2.f.k 2
120.w even 4 2 2880.2.f.j 2
140.j odd 4 2 2205.2.d.a 2
180.v odd 12 4 405.2.j.c 4
180.x even 12 4 405.2.j.c 4
420.w even 4 2 2205.2.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.b.a 2 20.e even 4 2
45.2.b.a 2 60.l odd 4 2
225.2.a.f 2 4.b odd 2 1
225.2.a.f 2 12.b even 2 1
225.2.a.f 2 20.d odd 2 1
225.2.a.f 2 60.h even 2 1
405.2.j.c 4 180.v odd 12 4
405.2.j.c 4 180.x even 12 4
720.2.f.d 2 5.c odd 4 2
720.2.f.d 2 15.e even 4 2
2205.2.d.a 2 140.j odd 4 2
2205.2.d.a 2 420.w even 4 2
2880.2.f.j 2 40.i odd 4 2
2880.2.f.j 2 120.w even 4 2
2880.2.f.k 2 40.k even 4 2
2880.2.f.k 2 120.q odd 4 2
3600.2.a.bs 2 1.a even 1 1 trivial
3600.2.a.bs 2 3.b odd 2 1 inner
3600.2.a.bs 2 5.b even 2 1 inner
3600.2.a.bs 2 15.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3600))\):

\( T_{7} \)
\( T_{11} \)
\( T_{13} \)
\( T_{17}^{2} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( -20 + T^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( -80 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( -80 + T^{2} \)
$53$ \( -20 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 16 + T )^{2} \)
$83$ \( -320 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
show more
show less