# Properties

 Label 3360.1.cv.d Level $3360$ Weight $1$ Character orbit 3360.cv Analytic conductor $1.677$ Analytic rank $0$ Dimension $8$ Projective image $D_{8}$ CM discriminant -56 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.cv (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 840) Projective image: $$D_{8}$$ Projective field: Galois closure of 8.2.1778112000000.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{16}^{3} q^{3} + \zeta_{16} q^{5} -\zeta_{16}^{6} q^{7} + \zeta_{16}^{6} q^{9} +O(q^{10})$$ $$q -\zeta_{16}^{3} q^{3} + \zeta_{16} q^{5} -\zeta_{16}^{6} q^{7} + \zeta_{16}^{6} q^{9} + ( \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{13} -\zeta_{16}^{4} q^{15} + ( \zeta_{16}^{3} + \zeta_{16}^{5} ) q^{19} -\zeta_{16} q^{21} + ( 1 - \zeta_{16}^{4} ) q^{23} + \zeta_{16}^{2} q^{25} + \zeta_{16} q^{27} -\zeta_{16}^{7} q^{35} + ( 1 - \zeta_{16}^{2} ) q^{39} + \zeta_{16}^{7} q^{45} -\zeta_{16}^{4} q^{49} + ( 1 - \zeta_{16}^{6} ) q^{57} + ( -\zeta_{16} + \zeta_{16}^{7} ) q^{59} + ( \zeta_{16}^{3} - \zeta_{16}^{5} ) q^{61} + \zeta_{16}^{4} q^{63} + ( 1 + \zeta_{16}^{6} ) q^{65} + ( -\zeta_{16}^{3} + \zeta_{16}^{7} ) q^{69} + ( -\zeta_{16}^{2} - \zeta_{16}^{6} ) q^{71} -\zeta_{16}^{5} q^{75} + ( \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{79} -\zeta_{16}^{4} q^{81} + ( \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{83} + ( \zeta_{16}^{3} - \zeta_{16}^{5} ) q^{91} + ( \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{23} + 8q^{39} + 8q^{57} + 8q^{65} + O(q^{100})$$

## Character values

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

 $$n$$ $$421$$ $$1121$$ $$1471$$ $$1921$$ $$2017$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-\zeta_{16}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1553.1
 −0.382683 + 0.923880i 0.923880 + 0.382683i −0.923880 − 0.382683i 0.382683 − 0.923880i −0.382683 − 0.923880i 0.923880 − 0.382683i −0.923880 + 0.382683i 0.382683 + 0.923880i
0 −0.923880 + 0.382683i 0 −0.382683 + 0.923880i 0 −0.707107 + 0.707107i 0 0.707107 0.707107i 0
1553.2 0 −0.382683 0.923880i 0 0.923880 + 0.382683i 0 0.707107 0.707107i 0 −0.707107 + 0.707107i 0
1553.3 0 0.382683 + 0.923880i 0 −0.923880 0.382683i 0 0.707107 0.707107i 0 −0.707107 + 0.707107i 0
1553.4 0 0.923880 0.382683i 0 0.382683 0.923880i 0 −0.707107 + 0.707107i 0 0.707107 0.707107i 0
2897.1 0 −0.923880 0.382683i 0 −0.382683 0.923880i 0 −0.707107 0.707107i 0 0.707107 + 0.707107i 0
2897.2 0 −0.382683 + 0.923880i 0 0.923880 0.382683i 0 0.707107 + 0.707107i 0 −0.707107 0.707107i 0
2897.3 0 0.382683 0.923880i 0 −0.923880 + 0.382683i 0 0.707107 + 0.707107i 0 −0.707107 0.707107i 0
2897.4 0 0.923880 + 0.382683i 0 0.382683 + 0.923880i 0 −0.707107 0.707107i 0 0.707107 + 0.707107i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2897.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
7.b odd 2 1 inner
8.b even 2 1 inner
15.e even 4 1 inner
105.k odd 4 1 inner
120.w even 4 1 inner
840.bp odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.1.cv.d 8
3.b odd 2 1 3360.1.cv.c 8
4.b odd 2 1 840.1.bp.d yes 8
5.c odd 4 1 3360.1.cv.c 8
7.b odd 2 1 inner 3360.1.cv.d 8
8.b even 2 1 inner 3360.1.cv.d 8
8.d odd 2 1 840.1.bp.d yes 8
12.b even 2 1 840.1.bp.c 8
15.e even 4 1 inner 3360.1.cv.d 8
20.e even 4 1 840.1.bp.c 8
21.c even 2 1 3360.1.cv.c 8
24.f even 2 1 840.1.bp.c 8
24.h odd 2 1 3360.1.cv.c 8
28.d even 2 1 840.1.bp.d yes 8
35.f even 4 1 3360.1.cv.c 8
40.i odd 4 1 3360.1.cv.c 8
40.k even 4 1 840.1.bp.c 8
56.e even 2 1 840.1.bp.d yes 8
56.h odd 2 1 CM 3360.1.cv.d 8
60.l odd 4 1 840.1.bp.d yes 8
84.h odd 2 1 840.1.bp.c 8
105.k odd 4 1 inner 3360.1.cv.d 8
120.q odd 4 1 840.1.bp.d yes 8
120.w even 4 1 inner 3360.1.cv.d 8
140.j odd 4 1 840.1.bp.c 8
168.e odd 2 1 840.1.bp.c 8
168.i even 2 1 3360.1.cv.c 8
280.s even 4 1 3360.1.cv.c 8
280.y odd 4 1 840.1.bp.c 8
420.w even 4 1 840.1.bp.d yes 8
840.bm even 4 1 840.1.bp.d yes 8
840.bp odd 4 1 inner 3360.1.cv.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.1.bp.c 8 12.b even 2 1
840.1.bp.c 8 20.e even 4 1
840.1.bp.c 8 24.f even 2 1
840.1.bp.c 8 40.k even 4 1
840.1.bp.c 8 84.h odd 2 1
840.1.bp.c 8 140.j odd 4 1
840.1.bp.c 8 168.e odd 2 1
840.1.bp.c 8 280.y odd 4 1
840.1.bp.d yes 8 4.b odd 2 1
840.1.bp.d yes 8 8.d odd 2 1
840.1.bp.d yes 8 28.d even 2 1
840.1.bp.d yes 8 56.e even 2 1
840.1.bp.d yes 8 60.l odd 4 1
840.1.bp.d yes 8 120.q odd 4 1
840.1.bp.d yes 8 420.w even 4 1
840.1.bp.d yes 8 840.bm even 4 1
3360.1.cv.c 8 3.b odd 2 1
3360.1.cv.c 8 5.c odd 4 1
3360.1.cv.c 8 21.c even 2 1
3360.1.cv.c 8 24.h odd 2 1
3360.1.cv.c 8 35.f even 4 1
3360.1.cv.c 8 40.i odd 4 1
3360.1.cv.c 8 168.i even 2 1
3360.1.cv.c 8 280.s even 4 1
3360.1.cv.d 8 1.a even 1 1 trivial
3360.1.cv.d 8 7.b odd 2 1 inner
3360.1.cv.d 8 8.b even 2 1 inner
3360.1.cv.d 8 15.e even 4 1 inner
3360.1.cv.d 8 56.h odd 2 1 CM
3360.1.cv.d 8 105.k odd 4 1 inner
3360.1.cv.d 8 120.w even 4 1 inner
3360.1.cv.d 8 840.bp odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3360, [\chi])$$:

 $$T_{11}$$ $$T_{23}^{2} - 2 T_{23} + 2$$ $$T_{73}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$1 + T^{8}$$
$5$ $$1 + T^{8}$$
$7$ $$( 1 + T^{4} )^{2}$$
$11$ $$T^{8}$$
$13$ $$4 + 12 T^{4} + T^{8}$$
$17$ $$T^{8}$$
$19$ $$( 2 + 4 T^{2} + T^{4} )^{2}$$
$23$ $$( 2 - 2 T + T^{2} )^{4}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$( 2 - 4 T^{2} + T^{4} )^{2}$$
$61$ $$( 2 - 4 T^{2} + T^{4} )^{2}$$
$67$ $$T^{8}$$
$71$ $$( 2 + T^{2} )^{4}$$
$73$ $$T^{8}$$
$79$ $$( 2 + T^{2} )^{4}$$
$83$ $$4 + 12 T^{4} + T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$