Properties

 Label 3360.1.bn.b 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: $$x^{4} + 1$$ 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 4.0.294000.2

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{8} q^{3} + \zeta_{8} q^{5} -\zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{8} q^{3} + \zeta_{8} q^{5} -\zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} -\zeta_{8}^{2} q^{15} -2 \zeta_{8} q^{17} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{19} - q^{21} + ( -1 - \zeta_{8}^{2} ) q^{23} + \zeta_{8}^{2} q^{25} -\zeta_{8}^{3} q^{27} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{31} + q^{35} + ( 1 + \zeta_{8}^{2} ) q^{37} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{41} + \zeta_{8}^{3} q^{45} -\zeta_{8}^{2} q^{49} + 2 \zeta_{8}^{2} q^{51} + ( -1 - \zeta_{8}^{2} ) q^{57} + \zeta_{8} q^{63} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{69} + 2 \zeta_{8}^{2} q^{71} -\zeta_{8}^{3} q^{75} - q^{81} -2 \zeta_{8}^{2} q^{85} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{89} + ( -1 + \zeta_{8}^{2} ) q^{93} + ( 1 + \zeta_{8}^{2} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 4q^{21} - 4q^{23} + 4q^{35} + 4q^{37} - 4q^{57} - 4q^{81} - 4q^{93} + 4q^{95} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
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.b 4
3.b odd 2 1 3360.1.bn.c yes 4
4.b odd 2 1 3360.1.bn.c yes 4
5.c odd 4 1 3360.1.bn.c yes 4
7.b odd 2 1 inner 3360.1.bn.b 4
12.b even 2 1 inner 3360.1.bn.b 4
15.e even 4 1 inner 3360.1.bn.b 4
20.e even 4 1 inner 3360.1.bn.b 4
21.c even 2 1 3360.1.bn.c yes 4
28.d even 2 1 3360.1.bn.c yes 4
35.f even 4 1 3360.1.bn.c yes 4
60.l odd 4 1 3360.1.bn.c yes 4
84.h odd 2 1 CM 3360.1.bn.b 4
105.k odd 4 1 inner 3360.1.bn.b 4
140.j odd 4 1 inner 3360.1.bn.b 4
420.w even 4 1 3360.1.bn.c yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.1.bn.b 4 1.a even 1 1 trivial
3360.1.bn.b 4 7.b odd 2 1 inner
3360.1.bn.b 4 12.b even 2 1 inner
3360.1.bn.b 4 15.e even 4 1 inner
3360.1.bn.b 4 20.e even 4 1 inner
3360.1.bn.b 4 84.h odd 2 1 CM
3360.1.bn.b 4 105.k odd 4 1 inner
3360.1.bn.b 4 140.j odd 4 1 inner
3360.1.bn.c yes 4 3.b odd 2 1
3360.1.bn.c yes 4 4.b odd 2 1
3360.1.bn.c yes 4 5.c odd 4 1
3360.1.bn.c yes 4 21.c even 2 1
3360.1.bn.c yes 4 28.d even 2 1
3360.1.bn.c yes 4 35.f even 4 1
3360.1.bn.c yes 4 60.l odd 4 1
3360.1.bn.c yes 4 420.w even 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 + T^{4}$$
$5$ $$1 + T^{4}$$
$7$ $$1 + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$16 + T^{4}$$
$19$ $$( -2 + T^{2} )^{2}$$
$23$ $$( 2 + 2 T + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 2 + T^{2} )^{2}$$
$37$ $$( 2 - 2 T + T^{2} )^{2}$$
$41$ $$( -2 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$( 4 + T^{2} )^{2}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 2 + T^{2} )^{2}$$
$97$ $$T^{4}$$