# Properties

 Label 3360.1.bn.d Level $3360$ Weight $1$ Character orbit 3360.bn Analytic conductor $1.677$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -84 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.bn (of order $$4$$, degree $$2$$, 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: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.294000.4

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{3} - \zeta_{8}^{3} q^{5} - \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ q + z * q^3 - z^3 * q^5 - z^3 * q^7 + z^2 * q^9 $$q + \zeta_{8} q^{3} - \zeta_{8}^{3} q^{5} - \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} + \zeta_{8}^{2} q^{11} + q^{15} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{19} + q^{21} + ( - \zeta_{8}^{2} - 1) q^{23} - \zeta_{8}^{2} q^{25} + \zeta_{8}^{3} q^{27} + (\zeta_{8}^{3} + \zeta_{8}) q^{31} + 2 \zeta_{8}^{3} q^{33} - \zeta_{8}^{2} q^{35} + ( - \zeta_{8}^{2} - 1) q^{37} + (\zeta_{8}^{3} - \zeta_{8}) q^{41} + \zeta_{8} q^{45} - \zeta_{8}^{2} q^{49} + 2 \zeta_{8} q^{55} + (\zeta_{8}^{2} + 1) q^{57} + \zeta_{8} q^{63} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{69} - \zeta_{8}^{3} q^{75} + 2 \zeta_{8} q^{77} - q^{81} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{89} + (\zeta_{8}^{2} - 1) q^{93} + ( - \zeta_{8}^{2} + 1) q^{95} - 2 q^{99} +O(q^{100})$$ q + z * q^3 - z^3 * q^5 - z^3 * q^7 + z^2 * q^9 + z^2 * q^11 + q^15 + (-z^3 + z) * q^19 + q^21 + (-z^2 - 1) * q^23 - z^2 * q^25 + z^3 * q^27 + (z^3 + z) * q^31 + 2*z^3 * q^33 - z^2 * q^35 + (-z^2 - 1) * q^37 + (z^3 - z) * q^41 + z * q^45 - z^2 * q^49 + 2*z * q^55 + (z^2 + 1) * q^57 + z * q^63 + (-z^3 - z) * q^69 - z^3 * q^75 + 2*z * q^77 - q^81 + (-z^3 - z) * q^89 + (z^2 - 1) * q^93 + (-z^2 + 1) * q^95 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 4 q^{15} + 4 q^{21} - 4 q^{23} - 4 q^{37} + 4 q^{57} - 4 q^{81} - 4 q^{93} + 4 q^{95} - 8 q^{99}+O(q^{100})$$ 4 * q + 4 * q^15 + 4 * q^21 - 4 * q^23 - 4 * q^37 + 4 * q^57 - 4 * q^81 - 4 * q^93 + 4 * q^95 - 8 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1217.1
 −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 −0.707107 + 0.707107i 0 1.00000i 0
1217.2 0 0.707107 + 0.707107i 0 0.707107 0.707107i 0 0.707107 0.707107i 0 1.00000i 0
3233.1 0 −0.707107 + 0.707107i 0 −0.707107 0.707107i 0 −0.707107 0.707107i 0 1.00000i 0
3233.2 0 0.707107 0.707107i 0 0.707107 + 0.707107i 0 0.707107 + 0.707107i 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
84.h odd 2 1 CM by $$\Q(\sqrt{-21})$$
7.b odd 2 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
105.k odd 4 1 inner
140.j 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.bn.d yes 4
3.b odd 2 1 3360.1.bn.a 4
4.b odd 2 1 3360.1.bn.a 4
5.c odd 4 1 3360.1.bn.a 4
7.b odd 2 1 inner 3360.1.bn.d yes 4
12.b even 2 1 inner 3360.1.bn.d yes 4
15.e even 4 1 inner 3360.1.bn.d yes 4
20.e even 4 1 inner 3360.1.bn.d yes 4
21.c even 2 1 3360.1.bn.a 4
28.d even 2 1 3360.1.bn.a 4
35.f even 4 1 3360.1.bn.a 4
60.l odd 4 1 3360.1.bn.a 4
84.h odd 2 1 CM 3360.1.bn.d yes 4
105.k odd 4 1 inner 3360.1.bn.d yes 4
140.j odd 4 1 inner 3360.1.bn.d yes 4
420.w even 4 1 3360.1.bn.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.1.bn.a 4 3.b odd 2 1
3360.1.bn.a 4 4.b odd 2 1
3360.1.bn.a 4 5.c odd 4 1
3360.1.bn.a 4 21.c even 2 1
3360.1.bn.a 4 28.d even 2 1
3360.1.bn.a 4 35.f even 4 1
3360.1.bn.a 4 60.l odd 4 1
3360.1.bn.a 4 420.w even 4 1
3360.1.bn.d yes 4 1.a even 1 1 trivial
3360.1.bn.d yes 4 7.b odd 2 1 inner
3360.1.bn.d yes 4 12.b even 2 1 inner
3360.1.bn.d yes 4 15.e even 4 1 inner
3360.1.bn.d yes 4 20.e even 4 1 inner
3360.1.bn.d yes 4 84.h odd 2 1 CM
3360.1.bn.d yes 4 105.k odd 4 1 inner
3360.1.bn.d yes 4 140.j 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 $$S_{1}^{\mathrm{new}}(3360, [\chi])$$:

 $$T_{11}^{2} + 4$$ T11^2 + 4 $$T_{23}^{2} + 2T_{23} + 2$$ T23^2 + 2*T23 + 2

## Hecke characteristic polynomials

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