# Properties

 Label 3360.1.ft.b Level $3360$ Weight $1$ Character orbit 3360.ft Analytic conductor $1.677$ Analytic rank $0$ Dimension $8$ Projective image $D_{12}$ 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.ft (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - 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: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{24}^{5} q^{3} + \zeta_{24}^{3} q^{5} + \zeta_{24}^{2} q^{7} + \zeta_{24}^{10} q^{9} +O(q^{10})$$ q - z^5 * q^3 + z^3 * q^5 + z^2 * q^7 + z^10 * q^9 $$q - \zeta_{24}^{5} q^{3} + \zeta_{24}^{3} q^{5} + \zeta_{24}^{2} q^{7} + \zeta_{24}^{10} q^{9} + ( - \zeta_{24}^{9} - \zeta_{24}^{7}) q^{11} - \zeta_{24}^{8} q^{15} - \zeta_{24}^{7} q^{21} + \zeta_{24}^{6} q^{25} + \zeta_{24}^{3} q^{27} + ( - \zeta_{24}^{11} - \zeta_{24}) q^{29} - \zeta_{24}^{2} q^{31} + ( - \zeta_{24}^{2} - 1) q^{33} + \zeta_{24}^{5} q^{35} - \zeta_{24} q^{45} + \zeta_{24}^{4} q^{49} + ( - \zeta_{24}^{9} + \zeta_{24}) q^{53} + ( - \zeta_{24}^{10} + 1) q^{55} + (\zeta_{24}^{3} - \zeta_{24}) q^{59} - q^{63} + (\zeta_{24}^{8} + \zeta_{24}^{2}) q^{73} - \zeta_{24}^{11} q^{75} + ( - \zeta_{24}^{11} - \zeta_{24}^{9}) q^{77} + (\zeta_{24}^{8} - 1) q^{79} - \zeta_{24}^{8} q^{81} + ( - \zeta_{24}^{11} - \zeta_{24}^{7}) q^{83} + (\zeta_{24}^{6} - \zeta_{24}^{4}) q^{87} + \zeta_{24}^{7} q^{93} + ( - \zeta_{24}^{10} + \zeta_{24}^{8}) q^{97} + (\zeta_{24}^{7} + \zeta_{24}^{5}) q^{99} +O(q^{100})$$ q - z^5 * q^3 + z^3 * q^5 + z^2 * q^7 + z^10 * q^9 + (-z^9 - z^7) * q^11 - z^8 * q^15 - z^7 * q^21 + z^6 * q^25 + z^3 * q^27 + (-z^11 - z) * q^29 - z^2 * q^31 + (-z^2 - 1) * q^33 + z^5 * q^35 - z * q^45 + z^4 * q^49 + (-z^9 + z) * q^53 + (-z^10 + 1) * q^55 + (z^3 - z) * q^59 - q^63 + (z^8 + z^2) * q^73 - z^11 * q^75 + (-z^11 - z^9) * q^77 + (z^8 - 1) * q^79 - z^8 * q^81 + (-z^11 - z^7) * q^83 + (z^6 - z^4) * q^87 + z^7 * q^93 + (-z^10 + z^8) * q^97 + (z^7 + z^5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 4 q^{15} - 8 q^{33} + 4 q^{49} + 8 q^{55} - 8 q^{63} - 4 q^{73} - 12 q^{79} + 4 q^{81} - 4 q^{87} - 4 q^{97}+O(q^{100})$$ 8 * q + 4 * q^15 - 8 * q^33 + 4 * q^49 + 8 * q^55 - 8 * q^63 - 4 * q^73 - 12 * q^79 + 4 * q^81 - 4 * q^87 - 4 * q^97

## Character values

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

 $$n$$ $$421$$ $$1121$$ $$1471$$ $$1921$$ $$2017$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$\zeta_{24}^{4}$$ $$-\zeta_{24}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.258819 − 0.965926i −0.258819 + 0.965926i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i 0.965926 + 0.258819i −0.965926 − 0.258819i
0 −0.965926 + 0.258819i 0 −0.707107 + 0.707107i 0 −0.866025 0.500000i 0 0.866025 0.500000i 0
17.2 0 0.965926 0.258819i 0 0.707107 0.707107i 0 −0.866025 0.500000i 0 0.866025 0.500000i 0
593.1 0 −0.965926 0.258819i 0 −0.707107 0.707107i 0 −0.866025 + 0.500000i 0 0.866025 + 0.500000i 0
593.2 0 0.965926 + 0.258819i 0 0.707107 + 0.707107i 0 −0.866025 + 0.500000i 0 0.866025 + 0.500000i 0
1937.1 0 −0.258819 + 0.965926i 0 0.707107 0.707107i 0 0.866025 0.500000i 0 −0.866025 0.500000i 0
1937.2 0 0.258819 0.965926i 0 −0.707107 + 0.707107i 0 0.866025 0.500000i 0 −0.866025 0.500000i 0
2033.1 0 −0.258819 0.965926i 0 0.707107 + 0.707107i 0 0.866025 + 0.500000i 0 −0.866025 + 0.500000i 0
2033.2 0 0.258819 + 0.965926i 0 −0.707107 0.707107i 0 0.866025 + 0.500000i 0 −0.866025 + 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2033.2 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.k even 12 1 inner
105.w odd 12 1 inner
280.bv even 12 1 inner
840.dh odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.1.ft.b 8
3.b odd 2 1 inner 3360.1.ft.b 8
4.b odd 2 1 840.1.dh.a 8
5.c odd 4 1 3360.1.ft.a 8
7.d odd 6 1 3360.1.ft.a 8
8.b even 2 1 inner 3360.1.ft.b 8
8.d odd 2 1 840.1.dh.a 8
12.b even 2 1 840.1.dh.a 8
15.e even 4 1 3360.1.ft.a 8
20.e even 4 1 840.1.dh.b yes 8
21.g even 6 1 3360.1.ft.a 8
24.f even 2 1 840.1.dh.a 8
24.h odd 2 1 CM 3360.1.ft.b 8
28.f even 6 1 840.1.dh.b yes 8
35.k even 12 1 inner 3360.1.ft.b 8
40.i odd 4 1 3360.1.ft.a 8
40.k even 4 1 840.1.dh.b yes 8
56.j odd 6 1 3360.1.ft.a 8
56.m even 6 1 840.1.dh.b yes 8
60.l odd 4 1 840.1.dh.b yes 8
84.j odd 6 1 840.1.dh.b yes 8
105.w odd 12 1 inner 3360.1.ft.b 8
120.q odd 4 1 840.1.dh.b yes 8
120.w even 4 1 3360.1.ft.a 8
140.x odd 12 1 840.1.dh.a 8
168.ba even 6 1 3360.1.ft.a 8
168.be odd 6 1 840.1.dh.b yes 8
280.bp odd 12 1 840.1.dh.a 8
280.bv even 12 1 inner 3360.1.ft.b 8
420.br even 12 1 840.1.dh.a 8
840.dh odd 12 1 inner 3360.1.ft.b 8
840.dk even 12 1 840.1.dh.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.1.dh.a 8 4.b odd 2 1
840.1.dh.a 8 8.d odd 2 1
840.1.dh.a 8 12.b even 2 1
840.1.dh.a 8 24.f even 2 1
840.1.dh.a 8 140.x odd 12 1
840.1.dh.a 8 280.bp odd 12 1
840.1.dh.a 8 420.br even 12 1
840.1.dh.a 8 840.dk even 12 1
840.1.dh.b yes 8 20.e even 4 1
840.1.dh.b yes 8 28.f even 6 1
840.1.dh.b yes 8 40.k even 4 1
840.1.dh.b yes 8 56.m even 6 1
840.1.dh.b yes 8 60.l odd 4 1
840.1.dh.b yes 8 84.j odd 6 1
840.1.dh.b yes 8 120.q odd 4 1
840.1.dh.b yes 8 168.be odd 6 1
3360.1.ft.a 8 5.c odd 4 1
3360.1.ft.a 8 7.d odd 6 1
3360.1.ft.a 8 15.e even 4 1
3360.1.ft.a 8 21.g even 6 1
3360.1.ft.a 8 40.i odd 4 1
3360.1.ft.a 8 56.j odd 6 1
3360.1.ft.a 8 120.w even 4 1
3360.1.ft.a 8 168.ba even 6 1
3360.1.ft.b 8 1.a even 1 1 trivial
3360.1.ft.b 8 3.b odd 2 1 inner
3360.1.ft.b 8 8.b even 2 1 inner
3360.1.ft.b 8 24.h odd 2 1 CM
3360.1.ft.b 8 35.k even 12 1 inner
3360.1.ft.b 8 105.w odd 12 1 inner
3360.1.ft.b 8 280.bv even 12 1 inner
3360.1.ft.b 8 840.dh odd 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{73}^{4} + 2T_{73}^{3} + 2T_{73}^{2} + 4T_{73} + 4$$ acting on $$S_{1}^{\mathrm{new}}(3360, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - T^{4} + 1$$
$5$ $$(T^{4} + 1)^{2}$$
$7$ $$(T^{4} - T^{2} + 1)^{2}$$
$11$ $$T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$(T^{4} + 4 T^{2} + 1)^{2}$$
$31$ $$(T^{4} - T^{2} + 1)^{2}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$T^{8} - 9T^{4} + 81$$
$59$ $$T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1$$
$61$ $$T^{8}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$(T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4)^{2}$$
$79$ $$(T^{2} + 3 T + 3)^{4}$$
$83$ $$(T^{4} + 9)^{2}$$
$89$ $$T^{8}$$
$97$ $$(T^{4} + 2 T^{3} + 2 T^{2} - 2 T + 1)^{2}$$