# Properties

 Label 3600.1.cc.a Level $3600$ Weight $1$ Character orbit 3600.cc Analytic conductor $1.797$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -20 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.cc (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 720) Projective image $$D_{6}$$ Projective field Galois closure of 6.0.10497600.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{3} + ( -1 - \zeta_{6} ) q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{3} + ( -1 - \zeta_{6} ) q^{7} + \zeta_{6}^{2} q^{9} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{21} + ( -1 + \zeta_{6}^{2} ) q^{23} + q^{27} + \zeta_{6}^{2} q^{29} -\zeta_{6} q^{41} + ( 1 + \zeta_{6} ) q^{47} + ( 1 + \zeta_{6} + \zeta_{6}^{2} ) q^{49} + \zeta_{6}^{2} q^{61} + ( 1 - \zeta_{6}^{2} ) q^{63} + ( -1 + \zeta_{6}^{2} ) q^{67} + ( 1 + \zeta_{6} ) q^{69} -\zeta_{6} q^{81} + ( 1 + \zeta_{6} ) q^{83} + q^{87} + q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} - 3q^{7} - q^{9} + O(q^{10})$$ $$2q - q^{3} - 3q^{7} - q^{9} - 3q^{23} + 2q^{27} - q^{29} - q^{41} + 3q^{47} + 2q^{49} - q^{61} + 3q^{63} - 3q^{67} + 3q^{69} - q^{81} + 3q^{83} + 2q^{87} + 2q^{89} + O(q^{100})$$

## Character values

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

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
751.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 + 0.866025i 0 0 0 −1.50000 + 0.866025i 0 −0.500000 0.866025i 0
1951.1 0 −0.500000 0.866025i 0 0 0 −1.50000 0.866025i 0 −0.500000 + 0.866025i 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
36.f odd 6 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.cc.a 2
4.b odd 2 1 3600.1.cc.b 2
5.b even 2 1 3600.1.cc.b 2
5.c odd 4 2 720.1.bu.a 4
9.c even 3 1 3600.1.cc.b 2
15.e even 4 2 2160.1.bu.a 4
20.d odd 2 1 CM 3600.1.cc.a 2
20.e even 4 2 720.1.bu.a 4
36.f odd 6 1 inner 3600.1.cc.a 2
40.i odd 4 2 2880.1.bu.c 4
40.k even 4 2 2880.1.bu.c 4
45.j even 6 1 inner 3600.1.cc.a 2
45.k odd 12 2 720.1.bu.a 4
45.l even 12 2 2160.1.bu.a 4
60.l odd 4 2 2160.1.bu.a 4
180.p odd 6 1 3600.1.cc.b 2
180.v odd 12 2 2160.1.bu.a 4
180.x even 12 2 720.1.bu.a 4
360.bo even 12 2 2880.1.bu.c 4
360.bu odd 12 2 2880.1.bu.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.bu.a 4 5.c odd 4 2
720.1.bu.a 4 20.e even 4 2
720.1.bu.a 4 45.k odd 12 2
720.1.bu.a 4 180.x even 12 2
2160.1.bu.a 4 15.e even 4 2
2160.1.bu.a 4 45.l even 12 2
2160.1.bu.a 4 60.l odd 4 2
2160.1.bu.a 4 180.v odd 12 2
2880.1.bu.c 4 40.i odd 4 2
2880.1.bu.c 4 40.k even 4 2
2880.1.bu.c 4 360.bo even 12 2
2880.1.bu.c 4 360.bu odd 12 2
3600.1.cc.a 2 1.a even 1 1 trivial
3600.1.cc.a 2 20.d odd 2 1 CM
3600.1.cc.a 2 36.f odd 6 1 inner
3600.1.cc.a 2 45.j even 6 1 inner
3600.1.cc.b 2 4.b odd 2 1
3600.1.cc.b 2 5.b even 2 1
3600.1.cc.b 2 9.c even 3 1
3600.1.cc.b 2 180.p odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

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