# Properties

 Label 840.1.bp.b Level $840$ Weight $1$ Character orbit 840.bp Analytic conductor $0.419$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ 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.bp (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.419214610612$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.441000.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{8}^{3} q^{2} - \zeta_{8}^{3} q^{3} - \zeta_{8}^{2} q^{4} + \zeta_{8} q^{5} - \zeta_{8}^{2} q^{6} + \zeta_{8}^{2} q^{7} - \zeta_{8} q^{8} - \zeta_{8}^{2} q^{9} +O(q^{10})$$ q - z^3 * q^2 - z^3 * q^3 - z^2 * q^4 + z * q^5 - z^2 * q^6 + z^2 * q^7 - z * q^8 - z^2 * q^9 $$q - \zeta_{8}^{3} q^{2} - \zeta_{8}^{3} q^{3} - \zeta_{8}^{2} q^{4} + \zeta_{8} q^{5} - \zeta_{8}^{2} q^{6} + \zeta_{8}^{2} q^{7} - \zeta_{8} q^{8} - \zeta_{8}^{2} q^{9} + q^{10} + (\zeta_{8}^{3} - \zeta_{8}) q^{11} - \zeta_{8} q^{12} + \zeta_{8} q^{14} + q^{15} - q^{16} - \zeta_{8} q^{18} - \zeta_{8}^{3} q^{20} + \zeta_{8} q^{21} + (\zeta_{8}^{2} - 1) q^{22} - q^{24} + \zeta_{8}^{2} q^{25} - \zeta_{8} q^{27} + q^{28} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{29} - \zeta_{8}^{3} q^{30} + \zeta_{8}^{2} q^{31} + \zeta_{8}^{3} q^{32} + (\zeta_{8}^{2} - 1) q^{33} + \zeta_{8}^{3} q^{35} - q^{36} - \zeta_{8}^{2} q^{40} + q^{42} + (\zeta_{8}^{3} + \zeta_{8}) q^{44} - \zeta_{8}^{3} q^{45} + \zeta_{8}^{3} q^{48} - q^{49} + \zeta_{8} q^{50} - q^{54} + ( - \zeta_{8}^{2} - 1) q^{55} - \zeta_{8}^{3} q^{56} + ( - \zeta_{8}^{2} - 1) q^{58} + (\zeta_{8}^{3} - \zeta_{8}) q^{59} - \zeta_{8}^{2} q^{60} + 2 \zeta_{8} q^{62} + q^{63} + \zeta_{8}^{2} q^{64} + (\zeta_{8}^{3} + \zeta_{8}) q^{66} + \zeta_{8}^{2} q^{70} + \zeta_{8}^{3} q^{72} + ( - \zeta_{8}^{2} + 1) q^{73} + \zeta_{8} q^{75} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{77} - \zeta_{8} q^{80} - q^{81} - \zeta_{8}^{3} q^{84} + ( - \zeta_{8}^{2} - 1) q^{87} + (\zeta_{8}^{2} + 1) q^{88} - \zeta_{8}^{2} q^{90} + 2 \zeta_{8} q^{93} + \zeta_{8}^{2} q^{96} + (\zeta_{8}^{2} + 1) q^{97} + \zeta_{8}^{3} q^{98} + (\zeta_{8}^{3} + \zeta_{8}) q^{99} +O(q^{100})$$ q - z^3 * q^2 - z^3 * q^3 - z^2 * q^4 + z * q^5 - z^2 * q^6 + z^2 * q^7 - z * q^8 - z^2 * q^9 + q^10 + (z^3 - z) * q^11 - z * q^12 + z * q^14 + q^15 - q^16 - z * q^18 - z^3 * q^20 + z * q^21 + (z^2 - 1) * q^22 - q^24 + z^2 * q^25 - z * q^27 + q^28 + (-z^3 - z) * q^29 - z^3 * q^30 + z^2 * q^31 + z^3 * q^32 + (z^2 - 1) * q^33 + z^3 * q^35 - q^36 - z^2 * q^40 + q^42 + (z^3 + z) * q^44 - z^3 * q^45 + z^3 * q^48 - q^49 + z * q^50 - q^54 + (-z^2 - 1) * q^55 - z^3 * q^56 + (-z^2 - 1) * q^58 + (z^3 - z) * q^59 - z^2 * q^60 + 2*z * q^62 + q^63 + z^2 * q^64 + (z^3 + z) * q^66 + z^2 * q^70 + z^3 * q^72 + (-z^2 + 1) * q^73 + z * q^75 + (-z^3 - z) * q^77 - z * q^80 - q^81 - z^3 * q^84 + (-z^2 - 1) * q^87 + (z^2 + 1) * q^88 - z^2 * q^90 + 2*z * q^93 + z^2 * q^96 + (z^2 + 1) * q^97 + z^3 * q^98 + (z^3 + z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 4 q^{10} + 4 q^{15} - 4 q^{16} - 4 q^{22} - 4 q^{24} + 4 q^{28} - 4 q^{33} - 4 q^{36} + 4 q^{42} - 4 q^{49} - 4 q^{54} - 4 q^{55} - 4 q^{58} + 4 q^{63} + 4 q^{73} - 4 q^{81} - 4 q^{87} + 4 q^{88} + 4 q^{97}+O(q^{100})$$ 4 * q + 4 * q^10 + 4 * q^15 - 4 * q^16 - 4 * q^22 - 4 * q^24 + 4 * q^28 - 4 * q^33 - 4 * q^36 + 4 * q^42 - 4 * q^49 - 4 * q^54 - 4 * q^55 - 4 * q^58 + 4 * q^63 + 4 * q^73 - 4 * q^81 - 4 * q^87 + 4 * q^88 + 4 * q^97

## Character values

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

 $$n$$ $$241$$ $$281$$ $$337$$ $$421$$ $$631$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{8}^{2}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
293.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
−0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i −0.707107 0.707107i 1.00000i 1.00000i 0.707107 + 0.707107i 1.00000i 1.00000
293.2 0.707107 0.707107i 0.707107 0.707107i 1.00000i 0.707107 + 0.707107i 1.00000i 1.00000i −0.707107 0.707107i 1.00000i 1.00000
797.1 −0.707107 0.707107i −0.707107 0.707107i 1.00000i −0.707107 + 0.707107i 1.00000i 1.00000i 0.707107 0.707107i 1.00000i 1.00000
797.2 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0.707107 0.707107i 1.00000i 1.00000i −0.707107 + 0.707107i 1.00000i 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.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 840.1.bp.b yes 4
3.b odd 2 1 inner 840.1.bp.b yes 4
4.b odd 2 1 3360.1.cv.a 4
5.c odd 4 1 840.1.bp.a 4
7.b odd 2 1 840.1.bp.a 4
8.b even 2 1 inner 840.1.bp.b yes 4
8.d odd 2 1 3360.1.cv.a 4
12.b even 2 1 3360.1.cv.a 4
15.e even 4 1 840.1.bp.a 4
20.e even 4 1 3360.1.cv.b 4
21.c even 2 1 840.1.bp.a 4
24.f even 2 1 3360.1.cv.a 4
24.h odd 2 1 CM 840.1.bp.b yes 4
28.d even 2 1 3360.1.cv.b 4
35.f even 4 1 inner 840.1.bp.b yes 4
40.i odd 4 1 840.1.bp.a 4
40.k even 4 1 3360.1.cv.b 4
56.e even 2 1 3360.1.cv.b 4
56.h odd 2 1 840.1.bp.a 4
60.l odd 4 1 3360.1.cv.b 4
84.h odd 2 1 3360.1.cv.b 4
105.k odd 4 1 inner 840.1.bp.b yes 4
120.q odd 4 1 3360.1.cv.b 4
120.w even 4 1 840.1.bp.a 4
140.j odd 4 1 3360.1.cv.a 4
168.e odd 2 1 3360.1.cv.b 4
168.i even 2 1 840.1.bp.a 4
280.s even 4 1 inner 840.1.bp.b yes 4
280.y odd 4 1 3360.1.cv.a 4
420.w even 4 1 3360.1.cv.a 4
840.bm even 4 1 3360.1.cv.a 4
840.bp odd 4 1 inner 840.1.bp.b yes 4

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(840, [\chi])$$:

 $$T_{11}^{2} - 2$$ T11^2 - 2 $$T_{23}$$ T23 $$T_{73}^{2} - 2T_{73} + 2$$ T73^2 - 2*T73 + 2

## Hecke characteristic polynomials

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