# Properties

 Label 1984.1.s.a Level $1984$ Weight $1$ Character orbit 1984.s Analytic conductor $0.990$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1984 = 2^{6} \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1984.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.990144985064$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 124) Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.15376.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{5} + \zeta_{12}^{5} q^{7} +O(q^{10})$$ q + z^5 * q^3 - z^4 * q^5 + z^5 * q^7 $$q + \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{5} + \zeta_{12}^{5} q^{7} + \zeta_{12} q^{11} + \zeta_{12}^{4} q^{13} + \zeta_{12}^{3} q^{15} - \zeta_{12}^{2} q^{17} + \zeta_{12}^{5} q^{19} - \zeta_{12}^{4} q^{21} - \zeta_{12}^{3} q^{27} + \zeta_{12}^{3} q^{31} - q^{33} + \zeta_{12}^{3} q^{35} + \zeta_{12}^{2} q^{37} - \zeta_{12}^{3} q^{39} + \zeta_{12}^{4} q^{41} + \zeta_{12}^{5} q^{43} + \zeta_{12} q^{51} + \zeta_{12}^{4} q^{53} - \zeta_{12}^{5} q^{55} - \zeta_{12}^{4} q^{57} - \zeta_{12}^{5} q^{59} + \zeta_{12}^{2} q^{65} + \zeta_{12} q^{67} + \zeta_{12} q^{71} + \zeta_{12}^{4} q^{73} - q^{77} + \zeta_{12}^{5} q^{79} + \zeta_{12}^{2} q^{81} - \zeta_{12} q^{83} - q^{85} - \zeta_{12}^{3} q^{91} - \zeta_{12}^{2} q^{93} + \zeta_{12}^{3} q^{95} +O(q^{100})$$ q + z^5 * q^3 - z^4 * q^5 + z^5 * q^7 + z * q^11 + z^4 * q^13 + z^3 * q^15 - z^2 * q^17 + z^5 * q^19 - z^4 * q^21 - z^3 * q^27 + z^3 * q^31 - q^33 + z^3 * q^35 + z^2 * q^37 - z^3 * q^39 + z^4 * q^41 + z^5 * q^43 + z * q^51 + z^4 * q^53 - z^5 * q^55 - z^4 * q^57 - z^5 * q^59 + z^2 * q^65 + z * q^67 + z * q^71 + z^4 * q^73 - q^77 + z^5 * q^79 + z^2 * q^81 - z * q^83 - q^85 - z^3 * q^91 - z^2 * q^93 + z^3 * q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{5}+O(q^{10})$$ 4 * q + 2 * q^5 $$4 q + 2 q^{5} - 2 q^{13} - 2 q^{17} + 2 q^{21} - 4 q^{33} + 2 q^{37} - 2 q^{41} - 2 q^{53} + 2 q^{57} + 2 q^{65} - 2 q^{73} - 4 q^{77} + 2 q^{81} - 4 q^{85} - 2 q^{93}+O(q^{100})$$ 4 * q + 2 * q^5 - 2 * q^13 - 2 * q^17 + 2 * q^21 - 4 * q^33 + 2 * q^37 - 2 * q^41 - 2 * q^53 + 2 * q^57 + 2 * q^65 - 2 * q^73 - 4 * q^77 + 2 * q^81 - 4 * q^85 - 2 * q^93

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1984\mathbb{Z}\right)^\times$$.

 $$n$$ $$63$$ $$65$$ $$1861$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 −0.866025 0.500000i 0 0 0
191.2 0 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0.866025 + 0.500000i 0 0 0
831.1 0 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 −0.866025 + 0.500000i 0 0 0
831.2 0 0.866025 0.500000i 0 0.500000 0.866025i 0 0.866025 0.500000i 0 0 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
4.b odd 2 1 inner
31.c even 3 1 inner
124.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1984.1.s.a 4
4.b odd 2 1 inner 1984.1.s.a 4
8.b even 2 1 124.1.i.a 4
8.d odd 2 1 124.1.i.a 4
24.f even 2 1 1116.1.x.a 4
24.h odd 2 1 1116.1.x.a 4
31.c even 3 1 inner 1984.1.s.a 4
40.e odd 2 1 3100.1.z.a 4
40.f even 2 1 3100.1.z.a 4
40.i odd 4 1 3100.1.t.a 4
40.i odd 4 1 3100.1.t.b 4
40.k even 4 1 3100.1.t.a 4
40.k even 4 1 3100.1.t.b 4
124.i odd 6 1 inner 1984.1.s.a 4
248.b even 2 1 3844.1.i.d 4
248.g odd 2 1 3844.1.i.d 4
248.l odd 6 1 3844.1.b.c 2
248.l odd 6 1 3844.1.i.d 4
248.m odd 6 1 124.1.i.a 4
248.m odd 6 1 3844.1.b.d 2
248.p even 6 1 124.1.i.a 4
248.p even 6 1 3844.1.b.d 2
248.q even 6 1 3844.1.b.c 2
248.q even 6 1 3844.1.i.d 4
248.r odd 10 4 3844.1.n.f 16
248.s odd 10 4 3844.1.n.e 16
248.u even 10 4 3844.1.n.e 16
248.v even 10 4 3844.1.n.f 16
248.bb even 30 4 3844.1.l.c 8
248.bb even 30 4 3844.1.n.f 16
248.bc even 30 4 3844.1.l.d 8
248.bc even 30 4 3844.1.n.e 16
248.be odd 30 4 3844.1.l.d 8
248.be odd 30 4 3844.1.n.e 16
248.bf odd 30 4 3844.1.l.c 8
248.bf odd 30 4 3844.1.n.f 16
744.s odd 6 1 1116.1.x.a 4
744.y even 6 1 1116.1.x.a 4
1240.bg even 6 1 3100.1.z.a 4
1240.bi odd 6 1 3100.1.z.a 4
1240.ch even 12 1 3100.1.t.a 4
1240.ch even 12 1 3100.1.t.b 4
1240.cj odd 12 1 3100.1.t.a 4
1240.cj odd 12 1 3100.1.t.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.1.i.a 4 8.b even 2 1
124.1.i.a 4 8.d odd 2 1
124.1.i.a 4 248.m odd 6 1
124.1.i.a 4 248.p even 6 1
1116.1.x.a 4 24.f even 2 1
1116.1.x.a 4 24.h odd 2 1
1116.1.x.a 4 744.s odd 6 1
1116.1.x.a 4 744.y even 6 1
1984.1.s.a 4 1.a even 1 1 trivial
1984.1.s.a 4 4.b odd 2 1 inner
1984.1.s.a 4 31.c even 3 1 inner
1984.1.s.a 4 124.i odd 6 1 inner
3100.1.t.a 4 40.i odd 4 1
3100.1.t.a 4 40.k even 4 1
3100.1.t.a 4 1240.ch even 12 1
3100.1.t.a 4 1240.cj odd 12 1
3100.1.t.b 4 40.i odd 4 1
3100.1.t.b 4 40.k even 4 1
3100.1.t.b 4 1240.ch even 12 1
3100.1.t.b 4 1240.cj odd 12 1
3100.1.z.a 4 40.e odd 2 1
3100.1.z.a 4 40.f even 2 1
3100.1.z.a 4 1240.bg even 6 1
3100.1.z.a 4 1240.bi odd 6 1
3844.1.b.c 2 248.l odd 6 1
3844.1.b.c 2 248.q even 6 1
3844.1.b.d 2 248.m odd 6 1
3844.1.b.d 2 248.p even 6 1
3844.1.i.d 4 248.b even 2 1
3844.1.i.d 4 248.g odd 2 1
3844.1.i.d 4 248.l odd 6 1
3844.1.i.d 4 248.q even 6 1
3844.1.l.c 8 248.bb even 30 4
3844.1.l.c 8 248.bf odd 30 4
3844.1.l.d 8 248.bc even 30 4
3844.1.l.d 8 248.be odd 30 4
3844.1.n.e 16 248.s odd 10 4
3844.1.n.e 16 248.u even 10 4
3844.1.n.e 16 248.bc even 30 4
3844.1.n.e 16 248.be odd 30 4
3844.1.n.f 16 248.r odd 10 4
3844.1.n.f 16 248.v even 10 4
3844.1.n.f 16 248.bb even 30 4
3844.1.n.f 16 248.bf odd 30 4

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1984, [\chi])$$.

## Hecke characteristic polynomials

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