# Properties

 Label 840.1.cg.a Level $840$ Weight $1$ Character orbit 840.cg Analytic conductor $0.419$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$840 = 2^{3} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 840.cg (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.419214610612$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.518616000.10

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{12} q^{2} - \zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{5} - q^{6} - \zeta_{12}^{2} q^{7} - \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} +O(q^{10})$$ q - z * q^2 - z^5 * q^3 + z^2 * q^4 + z^5 * q^5 - q^6 - z^2 * q^7 - z^3 * q^8 - z^4 * q^9 $$q - \zeta_{12} q^{2} - \zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{5} - q^{6} - \zeta_{12}^{2} q^{7} - \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} + q^{10} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{11} + \zeta_{12} q^{12} + \zeta_{12}^{3} q^{14} + \zeta_{12}^{4} q^{15} + \zeta_{12}^{4} q^{16} + \zeta_{12}^{5} q^{18} - \zeta_{12} q^{20} - \zeta_{12} q^{21} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{22} - \zeta_{12}^{2} q^{24} - \zeta_{12}^{4} q^{25} - \zeta_{12}^{3} q^{27} - \zeta_{12}^{4} q^{28} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{29} - \zeta_{12}^{5} q^{30} - \zeta_{12}^{2} q^{31} - \zeta_{12}^{5} q^{32} + ( - \zeta_{12}^{2} - 1) q^{33} + \zeta_{12} q^{35} + q^{36} + \zeta_{12}^{2} q^{40} + \zeta_{12}^{2} q^{42} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{44} + \zeta_{12}^{3} q^{45} + \zeta_{12}^{3} q^{48} + \zeta_{12}^{4} q^{49} + \zeta_{12}^{5} q^{50} + \zeta_{12}^{5} q^{53} + \zeta_{12}^{4} q^{54} + (\zeta_{12}^{2} + 1) q^{55} + \zeta_{12}^{5} q^{56} + ( - \zeta_{12}^{2} - 1) q^{58} + (\zeta_{12}^{3} + \zeta_{12}) q^{59} - q^{60} + \zeta_{12}^{3} q^{62} - q^{63} - q^{64} + (\zeta_{12}^{3} + \zeta_{12}) q^{66} - \zeta_{12}^{2} q^{70} - \zeta_{12} q^{72} - \zeta_{12}^{3} q^{75} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{77} - \zeta_{12}^{4} q^{79} - \zeta_{12}^{3} q^{80} - \zeta_{12}^{2} q^{81} + \zeta_{12}^{3} q^{83} - \zeta_{12}^{3} q^{84} + ( - \zeta_{12}^{4} + 1) q^{87} + (\zeta_{12}^{4} - 1) q^{88} - \zeta_{12}^{4} q^{90} - \zeta_{12} q^{93} - \zeta_{12}^{4} q^{96} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{97} - \zeta_{12}^{5} q^{98} + (\zeta_{12}^{5} - \zeta_{12}) q^{99} +O(q^{100})$$ q - z * q^2 - z^5 * q^3 + z^2 * q^4 + z^5 * q^5 - q^6 - z^2 * q^7 - z^3 * q^8 - z^4 * q^9 + q^10 + (-z^3 - z) * q^11 + z * q^12 + z^3 * q^14 + z^4 * q^15 + z^4 * q^16 + z^5 * q^18 - z * q^20 - z * q^21 + (z^4 + z^2) * q^22 - z^2 * q^24 - z^4 * q^25 - z^3 * q^27 - z^4 * q^28 + (-z^5 + z) * q^29 - z^5 * q^30 - z^2 * q^31 - z^5 * q^32 + (-z^2 - 1) * q^33 + z * q^35 + q^36 + z^2 * q^40 + z^2 * q^42 + (-z^5 - z^3) * q^44 + z^3 * q^45 + z^3 * q^48 + z^4 * q^49 + z^5 * q^50 + z^5 * q^53 + z^4 * q^54 + (z^2 + 1) * q^55 + z^5 * q^56 + (-z^2 - 1) * q^58 + (z^3 + z) * q^59 - q^60 + z^3 * q^62 - q^63 - q^64 + (z^3 + z) * q^66 - z^2 * q^70 - z * q^72 - z^3 * q^75 + (z^5 + z^3) * q^77 - z^4 * q^79 - z^3 * q^80 - z^2 * q^81 + z^3 * q^83 - z^3 * q^84 + (-z^4 + 1) * q^87 + (z^4 - 1) * q^88 - z^4 * q^90 - z * q^93 - z^4 * q^96 + (z^4 + z^2) * q^97 - z^5 * q^98 + (z^5 - z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 4 q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 - 4 * q^6 - 2 * q^7 + 2 * q^9 $$4 q + 2 q^{4} - 4 q^{6} - 2 q^{7} + 2 q^{9} + 4 q^{10} - 2 q^{15} - 2 q^{16} - 2 q^{24} + 2 q^{25} + 2 q^{28} - 2 q^{31} - 6 q^{33} + 4 q^{36} + 2 q^{40} + 2 q^{42} - 2 q^{49} - 2 q^{54} + 6 q^{55} - 6 q^{58} - 4 q^{60} - 4 q^{63} - 4 q^{64} - 2 q^{70} + 2 q^{79} - 2 q^{81} + 6 q^{87} - 6 q^{88} + 2 q^{90} + 2 q^{96}+O(q^{100})$$ 4 * q + 2 * q^4 - 4 * q^6 - 2 * q^7 + 2 * q^9 + 4 * q^10 - 2 * q^15 - 2 * q^16 - 2 * q^24 + 2 * q^25 + 2 * q^28 - 2 * q^31 - 6 * q^33 + 4 * q^36 + 2 * q^40 + 2 * q^42 - 2 * q^49 - 2 * q^54 + 6 * q^55 - 6 * q^58 - 4 * q^60 - 4 * q^63 - 4 * q^64 - 2 * q^70 + 2 * q^79 - 2 * q^81 + 6 * q^87 - 6 * q^88 + 2 * q^90 + 2 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$281$$ $$337$$ $$421$$ $$631$$ $$\chi(n)$$ $$\zeta_{12}^{4}$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.866025 + 0.500000i −1.00000 −0.500000 0.866025i 1.00000i 0.500000 0.866025i 1.00000
149.2 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.866025 0.500000i −1.00000 −0.500000 0.866025i 1.00000i 0.500000 0.866025i 1.00000
389.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −0.866025 0.500000i −1.00000 −0.500000 + 0.866025i 1.00000i 0.500000 + 0.866025i 1.00000
389.2 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0.866025 + 0.500000i −1.00000 −0.500000 + 0.866025i 1.00000i 0.500000 + 0.866025i 1.00000
 $$n$$: 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.j even 6 1 inner
105.o odd 6 1 inner
280.bf even 6 1 inner
840.cg odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.1.cg.a 4
3.b odd 2 1 inner 840.1.cg.a 4
4.b odd 2 1 3360.1.dm.b 4
5.b even 2 1 840.1.cg.b yes 4
7.c even 3 1 840.1.cg.b yes 4
8.b even 2 1 inner 840.1.cg.a 4
8.d odd 2 1 3360.1.dm.b 4
12.b even 2 1 3360.1.dm.b 4
15.d odd 2 1 840.1.cg.b yes 4
20.d odd 2 1 3360.1.dm.a 4
21.h odd 6 1 840.1.cg.b yes 4
24.f even 2 1 3360.1.dm.b 4
24.h odd 2 1 CM 840.1.cg.a 4
28.g odd 6 1 3360.1.dm.a 4
35.j even 6 1 inner 840.1.cg.a 4
40.e odd 2 1 3360.1.dm.a 4
40.f even 2 1 840.1.cg.b yes 4
56.k odd 6 1 3360.1.dm.a 4
56.p even 6 1 840.1.cg.b yes 4
60.h even 2 1 3360.1.dm.a 4
84.n even 6 1 3360.1.dm.a 4
105.o odd 6 1 inner 840.1.cg.a 4
120.i odd 2 1 840.1.cg.b yes 4
120.m even 2 1 3360.1.dm.a 4
140.p odd 6 1 3360.1.dm.b 4
168.s odd 6 1 840.1.cg.b yes 4
168.v even 6 1 3360.1.dm.a 4
280.bf even 6 1 inner 840.1.cg.a 4
280.bi odd 6 1 3360.1.dm.b 4
420.ba even 6 1 3360.1.dm.b 4
840.cg odd 6 1 inner 840.1.cg.a 4
840.cv even 6 1 3360.1.dm.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.1.cg.a 4 1.a even 1 1 trivial
840.1.cg.a 4 3.b odd 2 1 inner
840.1.cg.a 4 8.b even 2 1 inner
840.1.cg.a 4 24.h odd 2 1 CM
840.1.cg.a 4 35.j even 6 1 inner
840.1.cg.a 4 105.o odd 6 1 inner
840.1.cg.a 4 280.bf even 6 1 inner
840.1.cg.a 4 840.cg odd 6 1 inner
840.1.cg.b yes 4 5.b even 2 1
840.1.cg.b yes 4 7.c even 3 1
840.1.cg.b yes 4 15.d odd 2 1
840.1.cg.b yes 4 21.h odd 6 1
840.1.cg.b yes 4 40.f even 2 1
840.1.cg.b yes 4 56.p even 6 1
840.1.cg.b yes 4 120.i odd 2 1
840.1.cg.b yes 4 168.s odd 6 1
3360.1.dm.a 4 20.d odd 2 1
3360.1.dm.a 4 28.g odd 6 1
3360.1.dm.a 4 40.e odd 2 1
3360.1.dm.a 4 56.k odd 6 1
3360.1.dm.a 4 60.h even 2 1
3360.1.dm.a 4 84.n even 6 1
3360.1.dm.a 4 120.m even 2 1
3360.1.dm.a 4 168.v even 6 1
3360.1.dm.b 4 4.b odd 2 1
3360.1.dm.b 4 8.d odd 2 1
3360.1.dm.b 4 12.b even 2 1
3360.1.dm.b 4 24.f even 2 1
3360.1.dm.b 4 140.p odd 6 1
3360.1.dm.b 4 280.bi odd 6 1
3360.1.dm.b 4 420.ba even 6 1
3360.1.dm.b 4 840.cv even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{193}^{2} - 3T_{193} + 3$$ acting on $$S_{1}^{\mathrm{new}}(840, [\chi])$$.

## Hecke characteristic polynomials

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