# Properties

 Label 3360.1.ds.b Level $3360$ Weight $1$ Character orbit 3360.ds Analytic conductor $1.677$ Analytic rank $0$ Dimension $8$ Projective image $D_{12}$ CM discriminant -20 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3360.ds (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ 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: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{5} q^{3} + \zeta_{24}^{2} q^{5} -\zeta_{24} q^{7} + \zeta_{24}^{10} q^{9} +O(q^{10})$$ $$q -\zeta_{24}^{5} q^{3} + \zeta_{24}^{2} q^{5} -\zeta_{24} q^{7} + \zeta_{24}^{10} q^{9} -\zeta_{24}^{7} q^{15} + \zeta_{24}^{6} q^{21} + ( -\zeta_{24}^{7} - \zeta_{24}^{9} ) q^{23} + \zeta_{24}^{4} q^{25} + \zeta_{24}^{3} q^{27} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{29} -\zeta_{24}^{3} q^{35} -\zeta_{24}^{6} q^{41} + ( -\zeta_{24} - \zeta_{24}^{11} ) q^{43} - q^{45} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{47} + \zeta_{24}^{2} q^{49} + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{61} -\zeta_{24}^{11} q^{63} + ( \zeta_{24}^{9} + \zeta_{24}^{11} ) q^{67} + ( -1 - \zeta_{24}^{2} ) q^{69} -\zeta_{24}^{9} q^{75} -\zeta_{24}^{8} q^{81} + ( \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{83} + ( -\zeta_{24} + \zeta_{24}^{9} ) q^{87} + ( 1 + \zeta_{24}^{4} ) q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 4q^{25} - 8q^{45} - 8q^{69} + 4q^{81} + 12q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$$.

 $$n$$ $$421$$ $$1121$$ $$1471$$ $$1921$$ $$2017$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$\zeta_{24}^{8}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1409.1
 0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 − 0.965926i
0 −0.965926 + 0.258819i 0 −0.866025 0.500000i 0 −0.258819 + 0.965926i 0 0.866025 0.500000i 0
1409.2 0 −0.258819 0.965926i 0 0.866025 + 0.500000i 0 −0.965926 0.258819i 0 −0.866025 + 0.500000i 0
1409.3 0 0.258819 + 0.965926i 0 0.866025 + 0.500000i 0 0.965926 + 0.258819i 0 −0.866025 + 0.500000i 0
1409.4 0 0.965926 0.258819i 0 −0.866025 0.500000i 0 0.258819 0.965926i 0 0.866025 0.500000i 0
3329.1 0 −0.965926 0.258819i 0 −0.866025 + 0.500000i 0 −0.258819 0.965926i 0 0.866025 + 0.500000i 0
3329.2 0 −0.258819 + 0.965926i 0 0.866025 0.500000i 0 −0.965926 + 0.258819i 0 −0.866025 0.500000i 0
3329.3 0 0.258819 0.965926i 0 0.866025 0.500000i 0 0.965926 0.258819i 0 −0.866025 0.500000i 0
3329.4 0 0.965926 + 0.258819i 0 −0.866025 + 0.500000i 0 0.258819 + 0.965926i 0 0.866025 + 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3329.4 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
21.h odd 6 1 inner
84.n even 6 1 inner
105.o odd 6 1 inner
420.ba even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.1.ds.b yes 8
3.b odd 2 1 3360.1.ds.a 8
4.b odd 2 1 inner 3360.1.ds.b yes 8
5.b even 2 1 inner 3360.1.ds.b yes 8
7.c even 3 1 3360.1.ds.a 8
12.b even 2 1 3360.1.ds.a 8
15.d odd 2 1 3360.1.ds.a 8
20.d odd 2 1 CM 3360.1.ds.b yes 8
21.h odd 6 1 inner 3360.1.ds.b yes 8
28.g odd 6 1 3360.1.ds.a 8
35.j even 6 1 3360.1.ds.a 8
60.h even 2 1 3360.1.ds.a 8
84.n even 6 1 inner 3360.1.ds.b yes 8
105.o odd 6 1 inner 3360.1.ds.b yes 8
140.p odd 6 1 3360.1.ds.a 8
420.ba even 6 1 inner 3360.1.ds.b yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.1.ds.a 8 3.b odd 2 1
3360.1.ds.a 8 7.c even 3 1
3360.1.ds.a 8 12.b even 2 1
3360.1.ds.a 8 15.d odd 2 1
3360.1.ds.a 8 28.g odd 6 1
3360.1.ds.a 8 35.j even 6 1
3360.1.ds.a 8 60.h even 2 1
3360.1.ds.a 8 140.p odd 6 1
3360.1.ds.b yes 8 1.a even 1 1 trivial
3360.1.ds.b yes 8 4.b odd 2 1 inner
3360.1.ds.b yes 8 5.b even 2 1 inner
3360.1.ds.b yes 8 20.d odd 2 1 CM
3360.1.ds.b yes 8 21.h odd 6 1 inner
3360.1.ds.b yes 8 84.n even 6 1 inner
3360.1.ds.b yes 8 105.o odd 6 1 inner
3360.1.ds.b yes 8 420.ba even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{89}^{2} - 3 T_{89} + 3$$ acting on $$S_{1}^{\mathrm{new}}(3360, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$1 - T^{4} + T^{8}$$
$5$ $$( 1 - T^{2} + T^{4} )^{2}$$
$7$ $$1 - T^{4} + T^{8}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8}$$
$29$ $$( 3 + T^{2} )^{4}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$( 1 + T^{2} )^{4}$$
$43$ $$( 1 + 4 T^{2} + T^{4} )^{2}$$
$47$ $$( 4 + 2 T^{2} + T^{4} )^{2}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$( 9 + 3 T^{2} + T^{4} )^{2}$$
$67$ $$1 - 4 T^{2} + 15 T^{4} - 4 T^{6} + T^{8}$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$( 1 - 4 T^{2} + T^{4} )^{2}$$
$89$ $$( 3 - 3 T + T^{2} )^{4}$$
$97$ $$T^{8}$$