# Properties

 Label 1116.1.x.a Level $1116$ Weight $1$ Character orbit 1116.x Analytic conductor $0.557$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1116 = 2^{2} \cdot 3^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1116.x (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.556956554098$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{3} q^{2} - q^{4} + \zeta_{12}^{2} q^{5} - \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ q - z^3 * q^2 - q^4 + z^2 * q^5 - z * q^7 + z^3 * q^8 $$q - \zeta_{12}^{3} q^{2} - q^{4} + \zeta_{12}^{2} q^{5} - \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{5} q^{10} - \zeta_{12}^{5} q^{11} + \zeta_{12}^{2} q^{13} + \zeta_{12}^{4} q^{14} + q^{16} - \zeta_{12}^{4} q^{17} + \zeta_{12} q^{19} - \zeta_{12}^{2} q^{20} - \zeta_{12}^{2} q^{22} - \zeta_{12}^{5} q^{26} + \zeta_{12} q^{28} - \zeta_{12}^{3} q^{31} - \zeta_{12}^{3} q^{32} - \zeta_{12} q^{34} - \zeta_{12}^{3} q^{35} + \zeta_{12}^{4} q^{37} - \zeta_{12}^{4} q^{38} + \zeta_{12}^{5} q^{40} + \zeta_{12}^{2} q^{41} + \zeta_{12} q^{43} + \zeta_{12}^{5} q^{44} - \zeta_{12}^{2} q^{52} - \zeta_{12}^{2} q^{53} + \zeta_{12} q^{55} - \zeta_{12}^{4} q^{56} + \zeta_{12} q^{59} - q^{62} - q^{64} + \zeta_{12}^{4} q^{65} + \zeta_{12}^{5} q^{67} + \zeta_{12}^{4} q^{68} - q^{70} + \zeta_{12}^{5} q^{71} - \zeta_{12}^{2} q^{73} + \zeta_{12} q^{74} - \zeta_{12} q^{76} - q^{77} - \zeta_{12} q^{79} + \zeta_{12}^{2} q^{80} - \zeta_{12}^{5} q^{82} + \zeta_{12}^{5} q^{83} + q^{85} - \zeta_{12}^{4} q^{86} + \zeta_{12}^{2} q^{88} - \zeta_{12}^{3} q^{91} + \zeta_{12}^{3} q^{95} +O(q^{100})$$ q - z^3 * q^2 - q^4 + z^2 * q^5 - z * q^7 + z^3 * q^8 - z^5 * q^10 - z^5 * q^11 + z^2 * q^13 + z^4 * q^14 + q^16 - z^4 * q^17 + z * q^19 - z^2 * q^20 - z^2 * q^22 - z^5 * q^26 + z * q^28 - z^3 * q^31 - z^3 * q^32 - z * q^34 - z^3 * q^35 + z^4 * q^37 - z^4 * q^38 + z^5 * q^40 + z^2 * q^41 + z * q^43 + z^5 * q^44 - z^2 * q^52 - z^2 * q^53 + z * q^55 - z^4 * q^56 + z * q^59 - q^62 - q^64 + z^4 * q^65 + z^5 * q^67 + z^4 * q^68 - q^70 + z^5 * q^71 - z^2 * q^73 + z * q^74 - z * q^76 - q^77 - z * q^79 + z^2 * q^80 - z^5 * q^82 + z^5 * q^83 + q^85 - z^4 * q^86 + z^2 * q^88 - z^3 * q^91 + z^3 * q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 2 q^{5}+O(q^{10})$$ 4 * q - 4 * q^4 + 2 * q^5 $$4 q - 4 q^{4} + 2 q^{5} + 2 q^{13} - 2 q^{14} + 4 q^{16} + 2 q^{17} - 2 q^{20} - 2 q^{22} - 2 q^{37} + 2 q^{38} + 2 q^{41} - 2 q^{52} - 2 q^{53} + 2 q^{56} - 4 q^{62} - 4 q^{64} - 2 q^{65} - 2 q^{68} - 4 q^{70} - 2 q^{73} - 4 q^{77} + 2 q^{80} + 4 q^{85} + 2 q^{86} + 2 q^{88}+O(q^{100})$$ 4 * q - 4 * q^4 + 2 * q^5 + 2 * q^13 - 2 * q^14 + 4 * q^16 + 2 * q^17 - 2 * q^20 - 2 * q^22 - 2 * q^37 + 2 * q^38 + 2 * q^41 - 2 * q^52 - 2 * q^53 + 2 * q^56 - 4 * q^62 - 4 * q^64 - 2 * q^65 - 2 * q^68 - 4 * q^70 - 2 * q^73 - 4 * q^77 + 2 * q^80 + 4 * q^85 + 2 * q^86 + 2 * q^88

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1116\mathbb{Z}\right)^\times$$.

 $$n$$ $$497$$ $$559$$ $$685$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
811.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
1.00000i 0 −1.00000 0.500000 + 0.866025i 0 −0.866025 0.500000i 1.00000i 0 0.866025 0.500000i
811.2 1.00000i 0 −1.00000 0.500000 + 0.866025i 0 0.866025 + 0.500000i 1.00000i 0 −0.866025 + 0.500000i
955.1 1.00000i 0 −1.00000 0.500000 0.866025i 0 0.866025 0.500000i 1.00000i 0 −0.866025 0.500000i
955.2 1.00000i 0 −1.00000 0.500000 0.866025i 0 −0.866025 + 0.500000i 1.00000i 0 0.866025 + 0.500000i
 $$n$$: 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 1116.1.x.a 4
3.b odd 2 1 124.1.i.a 4
4.b odd 2 1 inner 1116.1.x.a 4
12.b even 2 1 124.1.i.a 4
15.d odd 2 1 3100.1.z.a 4
15.e even 4 1 3100.1.t.a 4
15.e even 4 1 3100.1.t.b 4
24.f even 2 1 1984.1.s.a 4
24.h odd 2 1 1984.1.s.a 4
31.c even 3 1 inner 1116.1.x.a 4
60.h even 2 1 3100.1.z.a 4
60.l odd 4 1 3100.1.t.a 4
60.l odd 4 1 3100.1.t.b 4
93.c even 2 1 3844.1.i.d 4
93.g even 6 1 3844.1.b.c 2
93.g even 6 1 3844.1.i.d 4
93.h odd 6 1 124.1.i.a 4
93.h odd 6 1 3844.1.b.d 2
93.k even 10 4 3844.1.n.f 16
93.l odd 10 4 3844.1.n.e 16
93.o odd 30 4 3844.1.l.d 8
93.o odd 30 4 3844.1.n.e 16
93.p even 30 4 3844.1.l.c 8
93.p even 30 4 3844.1.n.f 16
124.i odd 6 1 inner 1116.1.x.a 4
372.b odd 2 1 3844.1.i.d 4
372.p even 6 1 124.1.i.a 4
372.p even 6 1 3844.1.b.d 2
372.q odd 6 1 3844.1.b.c 2
372.q odd 6 1 3844.1.i.d 4
372.t even 10 4 3844.1.n.e 16
372.u odd 10 4 3844.1.n.f 16
372.bc odd 30 4 3844.1.l.c 8
372.bc odd 30 4 3844.1.n.f 16
372.bd even 30 4 3844.1.l.d 8
372.bd even 30 4 3844.1.n.e 16
465.u odd 6 1 3100.1.z.a 4
465.be even 12 1 3100.1.t.a 4
465.be even 12 1 3100.1.t.b 4
744.s odd 6 1 1984.1.s.a 4
744.y even 6 1 1984.1.s.a 4
1860.bc even 6 1 3100.1.z.a 4
1860.cf odd 12 1 3100.1.t.a 4
1860.cf 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 3.b odd 2 1
124.1.i.a 4 12.b even 2 1
124.1.i.a 4 93.h odd 6 1
124.1.i.a 4 372.p even 6 1
1116.1.x.a 4 1.a even 1 1 trivial
1116.1.x.a 4 4.b odd 2 1 inner
1116.1.x.a 4 31.c even 3 1 inner
1116.1.x.a 4 124.i odd 6 1 inner
1984.1.s.a 4 24.f even 2 1
1984.1.s.a 4 24.h odd 2 1
1984.1.s.a 4 744.s odd 6 1
1984.1.s.a 4 744.y even 6 1
3100.1.t.a 4 15.e even 4 1
3100.1.t.a 4 60.l odd 4 1
3100.1.t.a 4 465.be even 12 1
3100.1.t.a 4 1860.cf odd 12 1
3100.1.t.b 4 15.e even 4 1
3100.1.t.b 4 60.l odd 4 1
3100.1.t.b 4 465.be even 12 1
3100.1.t.b 4 1860.cf odd 12 1
3100.1.z.a 4 15.d odd 2 1
3100.1.z.a 4 60.h even 2 1
3100.1.z.a 4 465.u odd 6 1
3100.1.z.a 4 1860.bc even 6 1
3844.1.b.c 2 93.g even 6 1
3844.1.b.c 2 372.q odd 6 1
3844.1.b.d 2 93.h odd 6 1
3844.1.b.d 2 372.p even 6 1
3844.1.i.d 4 93.c even 2 1
3844.1.i.d 4 93.g even 6 1
3844.1.i.d 4 372.b odd 2 1
3844.1.i.d 4 372.q odd 6 1
3844.1.l.c 8 93.p even 30 4
3844.1.l.c 8 372.bc odd 30 4
3844.1.l.d 8 93.o odd 30 4
3844.1.l.d 8 372.bd even 30 4
3844.1.n.e 16 93.l odd 10 4
3844.1.n.e 16 93.o odd 30 4
3844.1.n.e 16 372.t even 10 4
3844.1.n.e 16 372.bd even 30 4
3844.1.n.f 16 93.k even 10 4
3844.1.n.f 16 93.p even 30 4
3844.1.n.f 16 372.u odd 10 4
3844.1.n.f 16 372.bc odd 30 4

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1116, [\chi])$$.

## Hecke characteristic polynomials

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