# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 840) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.441000.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{3} q^{3} -\zeta_{8} q^{5} + q^{7} -\zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{8}^{3} q^{3} -\zeta_{8} q^{5} + q^{7} -\zeta_{8}^{2} q^{9} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{11} - q^{15} -\zeta_{8}^{3} q^{21} + \zeta_{8}^{2} q^{25} -\zeta_{8} q^{27} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{29} + 2 \zeta_{8}^{2} q^{31} + ( 1 - \zeta_{8}^{2} ) q^{33} -\zeta_{8} q^{35} + \zeta_{8}^{3} q^{45} + q^{49} + ( -1 - \zeta_{8}^{2} ) q^{55} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{59} -\zeta_{8}^{2} q^{63} + ( -1 + \zeta_{8}^{2} ) q^{73} + \zeta_{8} q^{75} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{77} - q^{81} + ( -1 - \zeta_{8}^{2} ) q^{87} + 2 \zeta_{8} q^{93} + ( -1 - \zeta_{8}^{2} ) q^{97} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{7} + O(q^{10})$$ $$4 q + 4 q^{7} - 4 q^{15} + 4 q^{33} + 4 q^{49} - 4 q^{55} - 4 q^{73} - 4 q^{81} - 4 q^{87} - 4 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$421$$ $$1121$$ $$1471$$ $$1921$$ $$2017$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-\zeta_{8}^{2}$$

## 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
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 −0.707107 + 0.707107i 0 0.707107 + 0.707107i 0 1.00000 0 1.00000i 0
1553.2 0 0.707107 0.707107i 0 −0.707107 0.707107i 0 1.00000 0 1.00000i 0
2897.1 0 −0.707107 0.707107i 0 0.707107 0.707107i 0 1.00000 0 1.00000i 0
2897.2 0 0.707107 + 0.707107i 0 −0.707107 + 0.707107i 0 1.00000 0 1.00000i 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.f even 4 1 inner
105.k odd 4 1 inner
280.s 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.b 4
3.b odd 2 1 inner 3360.1.cv.b 4
4.b odd 2 1 840.1.bp.a 4
5.c odd 4 1 3360.1.cv.a 4
7.b odd 2 1 3360.1.cv.a 4
8.b even 2 1 inner 3360.1.cv.b 4
8.d odd 2 1 840.1.bp.a 4
12.b even 2 1 840.1.bp.a 4
15.e even 4 1 3360.1.cv.a 4
20.e even 4 1 840.1.bp.b yes 4
21.c even 2 1 3360.1.cv.a 4
24.f even 2 1 840.1.bp.a 4
24.h odd 2 1 CM 3360.1.cv.b 4
28.d even 2 1 840.1.bp.b yes 4
35.f even 4 1 inner 3360.1.cv.b 4
40.i odd 4 1 3360.1.cv.a 4
40.k even 4 1 840.1.bp.b yes 4
56.e even 2 1 840.1.bp.b yes 4
56.h odd 2 1 3360.1.cv.a 4
60.l odd 4 1 840.1.bp.b yes 4
84.h odd 2 1 840.1.bp.b yes 4
105.k odd 4 1 inner 3360.1.cv.b 4
120.q odd 4 1 840.1.bp.b yes 4
120.w even 4 1 3360.1.cv.a 4
140.j odd 4 1 840.1.bp.a 4
168.e odd 2 1 840.1.bp.b yes 4
168.i even 2 1 3360.1.cv.a 4
280.s even 4 1 inner 3360.1.cv.b 4
280.y odd 4 1 840.1.bp.a 4
420.w even 4 1 840.1.bp.a 4
840.bm even 4 1 840.1.bp.a 4
840.bp odd 4 1 inner 3360.1.cv.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.1.bp.a 4 4.b odd 2 1
840.1.bp.a 4 8.d odd 2 1
840.1.bp.a 4 12.b even 2 1
840.1.bp.a 4 24.f even 2 1
840.1.bp.a 4 140.j odd 4 1
840.1.bp.a 4 280.y odd 4 1
840.1.bp.a 4 420.w even 4 1
840.1.bp.a 4 840.bm even 4 1
840.1.bp.b yes 4 20.e even 4 1
840.1.bp.b yes 4 28.d even 2 1
840.1.bp.b yes 4 40.k even 4 1
840.1.bp.b yes 4 56.e even 2 1
840.1.bp.b yes 4 60.l odd 4 1
840.1.bp.b yes 4 84.h odd 2 1
840.1.bp.b yes 4 120.q odd 4 1
840.1.bp.b yes 4 168.e odd 2 1
3360.1.cv.a 4 5.c odd 4 1
3360.1.cv.a 4 7.b odd 2 1
3360.1.cv.a 4 15.e even 4 1
3360.1.cv.a 4 21.c even 2 1
3360.1.cv.a 4 40.i odd 4 1
3360.1.cv.a 4 56.h odd 2 1
3360.1.cv.a 4 120.w even 4 1
3360.1.cv.a 4 168.i even 2 1
3360.1.cv.b 4 1.a even 1 1 trivial
3360.1.cv.b 4 3.b odd 2 1 inner
3360.1.cv.b 4 8.b even 2 1 inner
3360.1.cv.b 4 24.h odd 2 1 CM
3360.1.cv.b 4 35.f even 4 1 inner
3360.1.cv.b 4 105.k odd 4 1 inner
3360.1.cv.b 4 280.s even 4 1 inner
3360.1.cv.b 4 840.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 $$S_{1}^{\mathrm{new}}(3360, [\chi])$$:

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

## Hecke characteristic polynomials

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