# Properties

 Label 3360.1.dm.a Level $3360$ Weight $1$ Character orbit 3360.dm Analytic conductor $1.677$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -24 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.dm (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 840) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.518616000.10

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{12} q^{3} + \zeta_{12}^{3} q^{5} + \zeta_{12}^{4} q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{12} q^{3} + \zeta_{12}^{3} q^{5} + \zeta_{12}^{4} q^{7} + \zeta_{12}^{2} q^{9} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{11} -\zeta_{12}^{4} q^{15} -\zeta_{12}^{5} q^{21} - q^{25} -\zeta_{12}^{3} q^{27} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{29} -\zeta_{12}^{4} q^{31} + ( 1 - \zeta_{12}^{4} ) q^{33} -\zeta_{12} q^{35} + \zeta_{12}^{5} q^{45} -\zeta_{12}^{2} q^{49} -\zeta_{12} q^{53} + ( -1 - \zeta_{12}^{2} ) q^{55} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{59} - q^{63} + \zeta_{12} q^{75} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{77} -\zeta_{12}^{2} q^{79} + \zeta_{12}^{4} q^{81} + \zeta_{12}^{3} q^{83} + ( 1 + \zeta_{12}^{2} ) q^{87} + \zeta_{12}^{5} q^{93} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{97} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$4 q - 2 q^{7} + 2 q^{9} + 2 q^{15} - 4 q^{25} + 2 q^{31} + 6 q^{33} - 2 q^{49} - 6 q^{55} - 4 q^{63} - 2 q^{79} - 2 q^{81} + 6 q^{87} + O(q^{100})$$

## Character values

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

 $$n$$ $$421$$ $$1121$$ $$1471$$ $$1921$$ $$2017$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$-\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1649.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.866025 0.500000i 0 1.00000i 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0
1649.2 0 0.866025 + 0.500000i 0 1.00000i 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0
3089.1 0 −0.866025 + 0.500000i 0 1.00000i 0 −0.500000 0.866025i 0 0.500000 0.866025i 0
3089.2 0 0.866025 0.500000i 0 1.00000i 0 −0.500000 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
8.b even 2 1 inner
35.j even 6 1 inner
105.o odd 6 1 inner
280.bf even 6 1 inner
840.cg odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.1.dm.a 4
3.b odd 2 1 inner 3360.1.dm.a 4
4.b odd 2 1 840.1.cg.b yes 4
5.b even 2 1 3360.1.dm.b 4
7.c even 3 1 3360.1.dm.b 4
8.b even 2 1 inner 3360.1.dm.a 4
8.d odd 2 1 840.1.cg.b yes 4
12.b even 2 1 840.1.cg.b yes 4
15.d odd 2 1 3360.1.dm.b 4
20.d odd 2 1 840.1.cg.a 4
21.h odd 6 1 3360.1.dm.b 4
24.f even 2 1 840.1.cg.b yes 4
24.h odd 2 1 CM 3360.1.dm.a 4
28.g odd 6 1 840.1.cg.a 4
35.j even 6 1 inner 3360.1.dm.a 4
40.e odd 2 1 840.1.cg.a 4
40.f even 2 1 3360.1.dm.b 4
56.k odd 6 1 840.1.cg.a 4
56.p even 6 1 3360.1.dm.b 4
60.h even 2 1 840.1.cg.a 4
84.n even 6 1 840.1.cg.a 4
105.o odd 6 1 inner 3360.1.dm.a 4
120.i odd 2 1 3360.1.dm.b 4
120.m even 2 1 840.1.cg.a 4
140.p odd 6 1 840.1.cg.b yes 4
168.s odd 6 1 3360.1.dm.b 4
168.v even 6 1 840.1.cg.a 4
280.bf even 6 1 inner 3360.1.dm.a 4
280.bi odd 6 1 840.1.cg.b yes 4
420.ba even 6 1 840.1.cg.b yes 4
840.cg odd 6 1 inner 3360.1.dm.a 4
840.cv even 6 1 840.1.cg.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.1.cg.a 4 20.d odd 2 1
840.1.cg.a 4 28.g odd 6 1
840.1.cg.a 4 40.e odd 2 1
840.1.cg.a 4 56.k odd 6 1
840.1.cg.a 4 60.h even 2 1
840.1.cg.a 4 84.n even 6 1
840.1.cg.a 4 120.m even 2 1
840.1.cg.a 4 168.v even 6 1
840.1.cg.b yes 4 4.b odd 2 1
840.1.cg.b yes 4 8.d odd 2 1
840.1.cg.b yes 4 12.b even 2 1
840.1.cg.b yes 4 24.f even 2 1
840.1.cg.b yes 4 140.p odd 6 1
840.1.cg.b yes 4 280.bi odd 6 1
840.1.cg.b yes 4 420.ba even 6 1
840.1.cg.b yes 4 840.cv even 6 1
3360.1.dm.a 4 1.a even 1 1 trivial
3360.1.dm.a 4 3.b odd 2 1 inner
3360.1.dm.a 4 8.b even 2 1 inner
3360.1.dm.a 4 24.h odd 2 1 CM
3360.1.dm.a 4 35.j even 6 1 inner
3360.1.dm.a 4 105.o odd 6 1 inner
3360.1.dm.a 4 280.bf even 6 1 inner
3360.1.dm.a 4 840.cg odd 6 1 inner
3360.1.dm.b 4 5.b even 2 1
3360.1.dm.b 4 7.c even 3 1
3360.1.dm.b 4 15.d odd 2 1
3360.1.dm.b 4 21.h odd 6 1
3360.1.dm.b 4 40.f even 2 1
3360.1.dm.b 4 56.p even 6 1
3360.1.dm.b 4 120.i odd 2 1
3360.1.dm.b 4 168.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{193}^{2} + 3 T_{193} + 3$$ acting on $$S_{1}^{\mathrm{new}}(3360, [\chi])$$.

## Hecke characteristic polynomials

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