Properties

Label 3600.1.j.b
Level $3600$
Weight $1$
Character orbit 3600.j
Analytic conductor $1.797$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -3
Inner twists $8$

Related objects

Downloads

Learn more about

Newspace parameters

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

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.4320000.2

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{7} +O(q^{10})\) \( q + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{7} -\zeta_{12}^{3} q^{13} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{19} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{31} + 2 \zeta_{12}^{3} q^{37} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{43} + ( 1 + \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{49} - q^{61} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{67} + 2 \zeta_{12}^{3} q^{73} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{91} + \zeta_{12}^{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 8q^{49} - 4q^{61} + O(q^{100}) \)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1999.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0 0 0 0 0 −1.73205 0 0 0
1999.2 0 0 0 0 0 −1.73205 0 0 0
1999.3 0 0 0 0 0 1.73205 0 0 0
1999.4 0 0 0 0 0 1.73205 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 inner
5.b even 2 1 inner
12.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.b 4
3.b odd 2 1 CM 3600.1.j.b 4
4.b odd 2 1 inner 3600.1.j.b 4
5.b even 2 1 inner 3600.1.j.b 4
5.c odd 4 1 3600.1.e.c 2
5.c odd 4 1 3600.1.e.d yes 2
12.b even 2 1 inner 3600.1.j.b 4
15.d odd 2 1 inner 3600.1.j.b 4
15.e even 4 1 3600.1.e.c 2
15.e even 4 1 3600.1.e.d yes 2
20.d odd 2 1 inner 3600.1.j.b 4
20.e even 4 1 3600.1.e.c 2
20.e even 4 1 3600.1.e.d yes 2
60.h even 2 1 inner 3600.1.j.b 4
60.l odd 4 1 3600.1.e.c 2
60.l odd 4 1 3600.1.e.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3600.1.e.c 2 5.c odd 4 1
3600.1.e.c 2 15.e even 4 1
3600.1.e.c 2 20.e even 4 1
3600.1.e.c 2 60.l odd 4 1
3600.1.e.d yes 2 5.c odd 4 1
3600.1.e.d yes 2 15.e even 4 1
3600.1.e.d yes 2 20.e even 4 1
3600.1.e.d yes 2 60.l odd 4 1
3600.1.j.b 4 1.a even 1 1 trivial
3600.1.j.b 4 3.b odd 2 1 CM
3600.1.j.b 4 4.b odd 2 1 inner
3600.1.j.b 4 5.b even 2 1 inner
3600.1.j.b 4 12.b even 2 1 inner
3600.1.j.b 4 15.d odd 2 1 inner
3600.1.j.b 4 20.d odd 2 1 inner
3600.1.j.b 4 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 3 \) acting on \(S_{1}^{\mathrm{new}}(3600, [\chi])\).