# Properties

 Label 2891.1.g.d Level $2891$ Weight $1$ Character orbit 2891.g Analytic conductor $1.443$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -59 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2891 = 7^{2} \cdot 59$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2891.g (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.44279695152$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 59) Projective image $$D_{3}$$ Projective field Galois closure of 3.1.59.1 Artin image $C_3\times S_3$ Artin field Galois closure of 6.0.493114979.5

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{3} -\zeta_{6} q^{4} -\zeta_{6}^{2} q^{5} +O(q^{10})$$ $$q + \zeta_{6} q^{3} -\zeta_{6} q^{4} -\zeta_{6}^{2} q^{5} -\zeta_{6}^{2} q^{12} + q^{15} + \zeta_{6}^{2} q^{16} -2 \zeta_{6} q^{17} -\zeta_{6}^{2} q^{19} - q^{20} + q^{27} - q^{29} - q^{41} - q^{48} -2 \zeta_{6}^{2} q^{51} + \zeta_{6} q^{53} + q^{57} -\zeta_{6} q^{59} -\zeta_{6} q^{60} + q^{64} + 2 \zeta_{6}^{2} q^{68} + 2 q^{71} - q^{76} -\zeta_{6}^{2} q^{79} + \zeta_{6} q^{80} + \zeta_{6} q^{81} -2 q^{85} -\zeta_{6} q^{87} -\zeta_{6} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - q^{4} + q^{5} + O(q^{10})$$ $$2q + q^{3} - q^{4} + q^{5} + q^{12} + 2q^{15} - q^{16} - 2q^{17} + q^{19} - 2q^{20} + 2q^{27} - 2q^{29} - 2q^{41} - 2q^{48} + 2q^{51} + q^{53} + 2q^{57} - q^{59} - q^{60} + 2q^{64} - 2q^{68} + 4q^{71} - 2q^{76} + q^{79} + q^{80} + q^{81} - 4q^{85} - q^{87} - q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2891\mathbb{Z}\right)^\times$$.

 $$n$$ $$591$$ $$2598$$ $$\chi(n)$$ $$\zeta_{6}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
471.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 + 0.866025i −0.500000 0.866025i 0.500000 0.866025i 0 0 0 0 0
2713.1 0 0.500000 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0 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
59.b odd 2 1 CM by $$\Q(\sqrt{-59})$$
7.c even 3 1 inner
413.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2891.1.g.d 2
7.b odd 2 1 2891.1.g.b 2
7.c even 3 1 59.1.b.a 1
7.c even 3 1 inner 2891.1.g.d 2
7.d odd 6 1 2891.1.c.e 1
7.d odd 6 1 2891.1.g.b 2
21.h odd 6 1 531.1.c.a 1
28.g odd 6 1 944.1.h.a 1
35.j even 6 1 1475.1.c.b 1
35.l odd 12 2 1475.1.d.a 2
56.k odd 6 1 3776.1.h.a 1
56.p even 6 1 3776.1.h.b 1
59.b odd 2 1 CM 2891.1.g.d 2
413.b even 2 1 2891.1.g.b 2
413.g odd 6 1 59.1.b.a 1
413.g odd 6 1 inner 2891.1.g.d 2
413.h even 6 1 2891.1.c.e 1
413.h even 6 1 2891.1.g.b 2
413.m even 87 28 3481.1.d.a 28
413.o odd 174 28 3481.1.d.a 28
1239.k even 6 1 531.1.c.a 1
1652.m even 6 1 944.1.h.a 1
2065.p odd 6 1 1475.1.c.b 1
2065.v even 12 2 1475.1.d.a 2
3304.x even 6 1 3776.1.h.a 1
3304.bd odd 6 1 3776.1.h.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.1.b.a 1 7.c even 3 1
59.1.b.a 1 413.g odd 6 1
531.1.c.a 1 21.h odd 6 1
531.1.c.a 1 1239.k even 6 1
944.1.h.a 1 28.g odd 6 1
944.1.h.a 1 1652.m even 6 1
1475.1.c.b 1 35.j even 6 1
1475.1.c.b 1 2065.p odd 6 1
1475.1.d.a 2 35.l odd 12 2
1475.1.d.a 2 2065.v even 12 2
2891.1.c.e 1 7.d odd 6 1
2891.1.c.e 1 413.h even 6 1
2891.1.g.b 2 7.b odd 2 1
2891.1.g.b 2 7.d odd 6 1
2891.1.g.b 2 413.b even 2 1
2891.1.g.b 2 413.h even 6 1
2891.1.g.d 2 1.a even 1 1 trivial
2891.1.g.d 2 7.c even 3 1 inner
2891.1.g.d 2 59.b odd 2 1 CM
2891.1.g.d 2 413.g odd 6 1 inner
3481.1.d.a 28 413.m even 87 28
3481.1.d.a 28 413.o odd 174 28
3776.1.h.a 1 56.k odd 6 1
3776.1.h.a 1 3304.x even 6 1
3776.1.h.b 1 56.p even 6 1
3776.1.h.b 1 3304.bd odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2891, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{5}^{2} - T_{5} + 1$$ $$T_{17}^{2} + 2 T_{17} + 4$$

## Hecke characteristic polynomials

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