# 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

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ 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

The $$q$$-expansion and trace form are shown below.

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

## Character values

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

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

## 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.

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{17}^{2} + 2 T_{17} + 2$$ acting on $$S_{1}^{\mathrm{new}}(3600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ 1
$5$ 1
$7$ $$1 + T^{4}$$
$11$ $$1 + T^{4}$$
$13$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$17$ $$( 1 + T )^{2}( 1 + T^{2} )$$
$19$ $$( 1 + T )^{2}( 1 + T^{2} )$$
$23$ $$( 1 + T )^{2}( 1 + T^{2} )$$
$29$ $$1 + T^{4}$$
$31$ $$( 1 + T^{2} )^{2}$$
$37$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$41$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$43$ $$( 1 + T^{2} )^{2}$$
$47$ $$( 1 + T )^{2}( 1 + T^{2} )$$
$53$ $$( 1 + T )^{4}$$
$59$ $$1 + T^{4}$$
$61$ $$( 1 - T )^{2}( 1 + T^{2} )$$
$67$ $$( 1 + T^{2} )^{2}$$
$71$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$73$ $$1 + T^{4}$$
$79$ $$( 1 + T^{2} )^{2}$$
$83$ $$( 1 + T^{2} )^{2}$$
$89$ $$( 1 + T^{2} )^{2}$$
$97$ $$1 + T^{4}$$