# Properties

 Label 3600.1.da.a Level 3600 Weight 1 Character orbit 3600.da Analytic conductor 1.797 Analytic rank 0 Dimension 8 Projective image $$D_{6}$$ CM discriminant -20 Inner twists 16

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3600.da (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{6}$$ Projective field Galois closure of 6.2.157464000.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{7} q^{3} + \zeta_{24}^{5} q^{7} -\zeta_{24}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{24}^{7} q^{3} + \zeta_{24}^{5} q^{7} -\zeta_{24}^{2} q^{9} + q^{21} + ( \zeta_{24}^{3} - \zeta_{24}^{11} ) q^{23} + \zeta_{24}^{9} q^{27} + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{29} + ( 1 - \zeta_{24}^{8} ) q^{41} + 2 \zeta_{24}^{11} q^{43} + ( \zeta_{24} - \zeta_{24}^{9} ) q^{47} -\zeta_{24}^{8} q^{61} -\zeta_{24}^{7} q^{63} + \zeta_{24} q^{67} + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{69} + \zeta_{24}^{4} q^{81} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{83} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{87} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{21} + 12q^{41} + 4q^{61} + 4q^{81} + O(q^{100})$$

## Character values

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

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$\zeta_{24}^{6}$$ $$1$$ $$\zeta_{24}^{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1343.1
 −0.258819 + 0.965926i 0.258819 − 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i
0 −0.965926 0.258819i 0 0 0 −0.965926 + 0.258819i 0 0.866025 + 0.500000i 0
1343.2 0 0.965926 + 0.258819i 0 0 0 0.965926 0.258819i 0 0.866025 + 0.500000i 0
2207.1 0 −0.258819 + 0.965926i 0 0 0 −0.258819 0.965926i 0 −0.866025 0.500000i 0
2207.2 0 0.258819 0.965926i 0 0 0 0.258819 + 0.965926i 0 −0.866025 0.500000i 0
2543.1 0 −0.258819 0.965926i 0 0 0 −0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0
2543.2 0 0.258819 + 0.965926i 0 0 0 0.258819 0.965926i 0 −0.866025 + 0.500000i 0
3407.1 0 −0.965926 + 0.258819i 0 0 0 −0.965926 0.258819i 0 0.866025 0.500000i 0
3407.2 0 0.965926 0.258819i 0 0 0 0.965926 + 0.258819i 0 0.866025 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3407.2 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 $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
9.d odd 6 1 inner
20.e even 4 2 inner
36.h even 6 1 inner
45.h odd 6 1 inner
45.l even 12 2 inner
180.n even 6 1 inner
180.v odd 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.da.a 8
4.b odd 2 1 inner 3600.1.da.a 8
5.b even 2 1 inner 3600.1.da.a 8
5.c odd 4 2 inner 3600.1.da.a 8
9.d odd 6 1 inner 3600.1.da.a 8
20.d odd 2 1 CM 3600.1.da.a 8
20.e even 4 2 inner 3600.1.da.a 8
36.h even 6 1 inner 3600.1.da.a 8
45.h odd 6 1 inner 3600.1.da.a 8
45.l even 12 2 inner 3600.1.da.a 8
180.n even 6 1 inner 3600.1.da.a 8
180.v odd 12 2 inner 3600.1.da.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3600.1.da.a 8 1.a even 1 1 trivial
3600.1.da.a 8 4.b odd 2 1 inner
3600.1.da.a 8 5.b even 2 1 inner
3600.1.da.a 8 5.c odd 4 2 inner
3600.1.da.a 8 9.d odd 6 1 inner
3600.1.da.a 8 20.d odd 2 1 CM
3600.1.da.a 8 20.e even 4 2 inner
3600.1.da.a 8 36.h even 6 1 inner
3600.1.da.a 8 45.h odd 6 1 inner
3600.1.da.a 8 45.l even 12 2 inner
3600.1.da.a 8 180.n even 6 1 inner
3600.1.da.a 8 180.v odd 12 2 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T^{4} + T^{8}$$
$5$ 1
$7$ $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
$11$ $$( 1 - T^{2} + T^{4} )^{4}$$
$13$ $$( 1 - T^{4} + T^{8} )^{2}$$
$17$ $$( 1 + T^{4} )^{4}$$
$19$ $$( 1 + T^{2} )^{8}$$
$23$ $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
$29$ $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}$$
$31$ $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
$37$ $$( 1 + T^{4} )^{4}$$
$41$ $$( 1 - T )^{8}( 1 - T + T^{2} )^{4}$$
$43$ $$( 1 - T^{4} + T^{8} )^{2}$$
$47$ $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
$53$ $$( 1 + T^{4} )^{4}$$
$59$ $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
$61$ $$( 1 - T )^{8}( 1 + T + T^{2} )^{4}$$
$67$ $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
$71$ $$( 1 + T^{2} )^{8}$$
$73$ $$( 1 + T^{4} )^{4}$$
$79$ $$( 1 - T^{2} + T^{4} )^{4}$$
$83$ $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
$89$ $$( 1 - T^{2} + T^{4} )^{4}$$
$97$ $$( 1 - T^{4} + T^{8} )^{2}$$