# 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{1}^{\mathrm{new}}(3600, [\chi])$$.