# Properties

 Label 891.2.e.j Level $891$ Weight $2$ Character orbit 891.e Analytic conductor $7.115$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$891 = 3^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 891.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.11467082010$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 99) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} + \zeta_{6} q^{4} + 4 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{7} + 3 q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} + \zeta_{6} q^{4} + 4 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{7} + 3 q^{8} + 4 q^{10} + ( 1 - \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{13} -2 \zeta_{6} q^{14} + ( 1 - \zeta_{6} ) q^{16} + 2 q^{17} -6 q^{19} + ( -4 + 4 \zeta_{6} ) q^{20} -\zeta_{6} q^{22} -4 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + 2 q^{26} + 2 q^{28} + ( 6 - 6 \zeta_{6} ) q^{29} -4 \zeta_{6} q^{31} + 5 \zeta_{6} q^{32} + ( 2 - 2 \zeta_{6} ) q^{34} + 8 q^{35} -6 q^{37} + ( -6 + 6 \zeta_{6} ) q^{38} + 12 \zeta_{6} q^{40} + 10 \zeta_{6} q^{41} + ( -6 + 6 \zeta_{6} ) q^{43} + q^{44} -4 q^{46} + ( 8 - 8 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} + 11 \zeta_{6} q^{50} + ( -2 + 2 \zeta_{6} ) q^{52} + 4 q^{55} + ( 6 - 6 \zeta_{6} ) q^{56} -6 \zeta_{6} q^{58} -4 \zeta_{6} q^{59} + ( 6 - 6 \zeta_{6} ) q^{61} -4 q^{62} + 7 q^{64} + ( -8 + 8 \zeta_{6} ) q^{65} -8 \zeta_{6} q^{67} + 2 \zeta_{6} q^{68} + ( 8 - 8 \zeta_{6} ) q^{70} -2 q^{73} + ( -6 + 6 \zeta_{6} ) q^{74} -6 \zeta_{6} q^{76} -2 \zeta_{6} q^{77} + ( 10 - 10 \zeta_{6} ) q^{79} + 4 q^{80} + 10 q^{82} + ( -12 + 12 \zeta_{6} ) q^{83} + 8 \zeta_{6} q^{85} + 6 \zeta_{6} q^{86} + ( 3 - 3 \zeta_{6} ) q^{88} + 4 q^{91} + ( 4 - 4 \zeta_{6} ) q^{92} -8 \zeta_{6} q^{94} -24 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} + 3 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + q^{4} + 4q^{5} + 2q^{7} + 6q^{8} + O(q^{10})$$ $$2q + q^{2} + q^{4} + 4q^{5} + 2q^{7} + 6q^{8} + 8q^{10} + q^{11} + 2q^{13} - 2q^{14} + q^{16} + 4q^{17} - 12q^{19} - 4q^{20} - q^{22} - 4q^{23} - 11q^{25} + 4q^{26} + 4q^{28} + 6q^{29} - 4q^{31} + 5q^{32} + 2q^{34} + 16q^{35} - 12q^{37} - 6q^{38} + 12q^{40} + 10q^{41} - 6q^{43} + 2q^{44} - 8q^{46} + 8q^{47} + 3q^{49} + 11q^{50} - 2q^{52} + 8q^{55} + 6q^{56} - 6q^{58} - 4q^{59} + 6q^{61} - 8q^{62} + 14q^{64} - 8q^{65} - 8q^{67} + 2q^{68} + 8q^{70} - 4q^{73} - 6q^{74} - 6q^{76} - 2q^{77} + 10q^{79} + 8q^{80} + 20q^{82} - 12q^{83} + 8q^{85} + 6q^{86} + 3q^{88} + 8q^{91} + 4q^{92} - 8q^{94} - 24q^{95} - 2q^{97} + 6q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/891\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$650$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
298.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0 0.500000 + 0.866025i 2.00000 + 3.46410i 0 1.00000 1.73205i 3.00000 0 4.00000
595.1 0.500000 + 0.866025i 0 0.500000 0.866025i 2.00000 3.46410i 0 1.00000 + 1.73205i 3.00000 0 4.00000
 $$n$$: 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.e.j 2
3.b odd 2 1 891.2.e.c 2
9.c even 3 1 99.2.a.a 1
9.c even 3 1 inner 891.2.e.j 2
9.d odd 6 1 99.2.a.c yes 1
9.d odd 6 1 891.2.e.c 2
36.f odd 6 1 1584.2.a.b 1
36.h even 6 1 1584.2.a.r 1
45.h odd 6 1 2475.2.a.c 1
45.j even 6 1 2475.2.a.j 1
45.k odd 12 2 2475.2.c.b 2
45.l even 12 2 2475.2.c.g 2
63.l odd 6 1 4851.2.a.g 1
63.o even 6 1 4851.2.a.o 1
72.j odd 6 1 6336.2.a.b 1
72.l even 6 1 6336.2.a.f 1
72.n even 6 1 6336.2.a.cl 1
72.p odd 6 1 6336.2.a.cm 1
99.g even 6 1 1089.2.a.d 1
99.h odd 6 1 1089.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.a.a 1 9.c even 3 1
99.2.a.c yes 1 9.d odd 6 1
891.2.e.c 2 3.b odd 2 1
891.2.e.c 2 9.d odd 6 1
891.2.e.j 2 1.a even 1 1 trivial
891.2.e.j 2 9.c even 3 1 inner
1089.2.a.d 1 99.g even 6 1
1089.2.a.h 1 99.h odd 6 1
1584.2.a.b 1 36.f odd 6 1
1584.2.a.r 1 36.h even 6 1
2475.2.a.c 1 45.h odd 6 1
2475.2.a.j 1 45.j even 6 1
2475.2.c.b 2 45.k odd 12 2
2475.2.c.g 2 45.l even 12 2
4851.2.a.g 1 63.l odd 6 1
4851.2.a.o 1 63.o even 6 1
6336.2.a.b 1 72.j odd 6 1
6336.2.a.f 1 72.l even 6 1
6336.2.a.cl 1 72.n even 6 1
6336.2.a.cm 1 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(891, [\chi])$$:

 $$T_{2}^{2} - T_{2} + 1$$ $$T_{5}^{2} - 4 T_{5} + 16$$ $$T_{7}^{2} - 2 T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$16 - 4 T + T^{2}$$
$7$ $$4 - 2 T + T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$4 - 2 T + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$( 6 + T )^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$36 - 6 T + T^{2}$$
$31$ $$16 + 4 T + T^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$100 - 10 T + T^{2}$$
$43$ $$36 + 6 T + T^{2}$$
$47$ $$64 - 8 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$16 + 4 T + T^{2}$$
$61$ $$36 - 6 T + T^{2}$$
$67$ $$64 + 8 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$100 - 10 T + T^{2}$$
$83$ $$144 + 12 T + T^{2}$$
$89$ $$T^{2}$$
$97$ $$4 + 2 T + T^{2}$$