# Properties

 Label 891.2.e.k 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: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{4} - \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 2) q^{7} +O(q^{10})$$ q + (-2*z + 2) * q^2 - 2*z * q^4 - z * q^5 + (-2*z + 2) * q^7 $$q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{4} - \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 2) q^{7} - 2 q^{10} + (\zeta_{6} - 1) q^{11} - 4 \zeta_{6} q^{13} - 4 \zeta_{6} q^{14} + ( - 4 \zeta_{6} + 4) q^{16} - 2 q^{17} + (2 \zeta_{6} - 2) q^{20} + 2 \zeta_{6} q^{22} + \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} - 8 q^{26} - 4 q^{28} - 7 \zeta_{6} q^{31} - 8 \zeta_{6} q^{32} + (4 \zeta_{6} - 4) q^{34} - 2 q^{35} + 3 q^{37} + 8 \zeta_{6} q^{41} + ( - 6 \zeta_{6} + 6) q^{43} + 2 q^{44} + 2 q^{46} + (8 \zeta_{6} - 8) q^{47} + 3 \zeta_{6} q^{49} - 8 \zeta_{6} q^{50} + (8 \zeta_{6} - 8) q^{52} - 6 q^{53} + q^{55} - 5 \zeta_{6} q^{59} + (12 \zeta_{6} - 12) q^{61} - 14 q^{62} - 8 q^{64} + (4 \zeta_{6} - 4) q^{65} + 7 \zeta_{6} q^{67} + 4 \zeta_{6} q^{68} + (4 \zeta_{6} - 4) q^{70} - 3 q^{71} + 4 q^{73} + ( - 6 \zeta_{6} + 6) q^{74} + 2 \zeta_{6} q^{77} + ( - 10 \zeta_{6} + 10) q^{79} - 4 q^{80} + 16 q^{82} + ( - 6 \zeta_{6} + 6) q^{83} + 2 \zeta_{6} q^{85} - 12 \zeta_{6} q^{86} + 15 q^{89} - 8 q^{91} + ( - 2 \zeta_{6} + 2) q^{92} + 16 \zeta_{6} q^{94} + ( - 7 \zeta_{6} + 7) q^{97} + 6 q^{98} +O(q^{100})$$ q + (-2*z + 2) * q^2 - 2*z * q^4 - z * q^5 + (-2*z + 2) * q^7 - 2 * q^10 + (z - 1) * q^11 - 4*z * q^13 - 4*z * q^14 + (-4*z + 4) * q^16 - 2 * q^17 + (2*z - 2) * q^20 + 2*z * q^22 + z * q^23 + (-4*z + 4) * q^25 - 8 * q^26 - 4 * q^28 - 7*z * q^31 - 8*z * q^32 + (4*z - 4) * q^34 - 2 * q^35 + 3 * q^37 + 8*z * q^41 + (-6*z + 6) * q^43 + 2 * q^44 + 2 * q^46 + (8*z - 8) * q^47 + 3*z * q^49 - 8*z * q^50 + (8*z - 8) * q^52 - 6 * q^53 + q^55 - 5*z * q^59 + (12*z - 12) * q^61 - 14 * q^62 - 8 * q^64 + (4*z - 4) * q^65 + 7*z * q^67 + 4*z * q^68 + (4*z - 4) * q^70 - 3 * q^71 + 4 * q^73 + (-6*z + 6) * q^74 + 2*z * q^77 + (-10*z + 10) * q^79 - 4 * q^80 + 16 * q^82 + (-6*z + 6) * q^83 + 2*z * q^85 - 12*z * q^86 + 15 * q^89 - 8 * q^91 + (-2*z + 2) * q^92 + 16*z * q^94 + (-7*z + 7) * q^97 + 6 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{4} - q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^4 - q^5 + 2 * q^7 $$2 q + 2 q^{2} - 2 q^{4} - q^{5} + 2 q^{7} - 4 q^{10} - q^{11} - 4 q^{13} - 4 q^{14} + 4 q^{16} - 4 q^{17} - 2 q^{20} + 2 q^{22} + q^{23} + 4 q^{25} - 16 q^{26} - 8 q^{28} - 7 q^{31} - 8 q^{32} - 4 q^{34} - 4 q^{35} + 6 q^{37} + 8 q^{41} + 6 q^{43} + 4 q^{44} + 4 q^{46} - 8 q^{47} + 3 q^{49} - 8 q^{50} - 8 q^{52} - 12 q^{53} + 2 q^{55} - 5 q^{59} - 12 q^{61} - 28 q^{62} - 16 q^{64} - 4 q^{65} + 7 q^{67} + 4 q^{68} - 4 q^{70} - 6 q^{71} + 8 q^{73} + 6 q^{74} + 2 q^{77} + 10 q^{79} - 8 q^{80} + 32 q^{82} + 6 q^{83} + 2 q^{85} - 12 q^{86} + 30 q^{89} - 16 q^{91} + 2 q^{92} + 16 q^{94} + 7 q^{97} + 12 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^4 - q^5 + 2 * q^7 - 4 * q^10 - q^11 - 4 * q^13 - 4 * q^14 + 4 * q^16 - 4 * q^17 - 2 * q^20 + 2 * q^22 + q^23 + 4 * q^25 - 16 * q^26 - 8 * q^28 - 7 * q^31 - 8 * q^32 - 4 * q^34 - 4 * q^35 + 6 * q^37 + 8 * q^41 + 6 * q^43 + 4 * q^44 + 4 * q^46 - 8 * q^47 + 3 * q^49 - 8 * q^50 - 8 * q^52 - 12 * q^53 + 2 * q^55 - 5 * q^59 - 12 * q^61 - 28 * q^62 - 16 * q^64 - 4 * q^65 + 7 * q^67 + 4 * q^68 - 4 * q^70 - 6 * q^71 + 8 * q^73 + 6 * q^74 + 2 * q^77 + 10 * q^79 - 8 * q^80 + 32 * q^82 + 6 * q^83 + 2 * q^85 - 12 * q^86 + 30 * q^89 - 16 * q^91 + 2 * q^92 + 16 * q^94 + 7 * q^97 + 12 * q^98

## 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
1.00000 1.73205i 0 −1.00000 1.73205i −0.500000 0.866025i 0 1.00000 1.73205i 0 0 −2.00000
595.1 1.00000 + 1.73205i 0 −1.00000 + 1.73205i −0.500000 + 0.866025i 0 1.00000 + 1.73205i 0 0 −2.00000
 $$n$$: 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.k 2
3.b odd 2 1 891.2.e.b 2
9.c even 3 1 11.2.a.a 1
9.c even 3 1 inner 891.2.e.k 2
9.d odd 6 1 99.2.a.d 1
9.d odd 6 1 891.2.e.b 2
36.f odd 6 1 176.2.a.b 1
36.h even 6 1 1584.2.a.g 1
45.h odd 6 1 2475.2.a.a 1
45.j even 6 1 275.2.a.b 1
45.k odd 12 2 275.2.b.a 2
45.l even 12 2 2475.2.c.a 2
63.g even 3 1 539.2.e.h 2
63.h even 3 1 539.2.e.h 2
63.k odd 6 1 539.2.e.g 2
63.l odd 6 1 539.2.a.a 1
63.o even 6 1 4851.2.a.t 1
63.t odd 6 1 539.2.e.g 2
72.j odd 6 1 6336.2.a.br 1
72.l even 6 1 6336.2.a.bu 1
72.n even 6 1 704.2.a.h 1
72.p odd 6 1 704.2.a.c 1
99.g even 6 1 1089.2.a.b 1
99.h odd 6 1 121.2.a.d 1
99.m even 15 4 121.2.c.e 4
99.o odd 30 4 121.2.c.a 4
117.t even 6 1 1859.2.a.b 1
144.v odd 12 2 2816.2.c.f 2
144.x even 12 2 2816.2.c.j 2
153.h even 6 1 3179.2.a.a 1
171.o odd 6 1 3971.2.a.b 1
180.p odd 6 1 4400.2.a.i 1
180.x even 12 2 4400.2.b.h 2
207.f odd 6 1 5819.2.a.a 1
252.bi even 6 1 8624.2.a.j 1
261.i even 6 1 9251.2.a.d 1
396.k even 6 1 1936.2.a.i 1
495.o odd 6 1 3025.2.a.a 1
693.bj even 6 1 5929.2.a.h 1
792.z even 6 1 7744.2.a.k 1
792.be odd 6 1 7744.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 9.c even 3 1
99.2.a.d 1 9.d odd 6 1
121.2.a.d 1 99.h odd 6 1
121.2.c.a 4 99.o odd 30 4
121.2.c.e 4 99.m even 15 4
176.2.a.b 1 36.f odd 6 1
275.2.a.b 1 45.j even 6 1
275.2.b.a 2 45.k odd 12 2
539.2.a.a 1 63.l odd 6 1
539.2.e.g 2 63.k odd 6 1
539.2.e.g 2 63.t odd 6 1
539.2.e.h 2 63.g even 3 1
539.2.e.h 2 63.h even 3 1
704.2.a.c 1 72.p odd 6 1
704.2.a.h 1 72.n even 6 1
891.2.e.b 2 3.b odd 2 1
891.2.e.b 2 9.d odd 6 1
891.2.e.k 2 1.a even 1 1 trivial
891.2.e.k 2 9.c even 3 1 inner
1089.2.a.b 1 99.g even 6 1
1584.2.a.g 1 36.h even 6 1
1859.2.a.b 1 117.t even 6 1
1936.2.a.i 1 396.k even 6 1
2475.2.a.a 1 45.h odd 6 1
2475.2.c.a 2 45.l even 12 2
2816.2.c.f 2 144.v odd 12 2
2816.2.c.j 2 144.x even 12 2
3025.2.a.a 1 495.o odd 6 1
3179.2.a.a 1 153.h even 6 1
3971.2.a.b 1 171.o odd 6 1
4400.2.a.i 1 180.p odd 6 1
4400.2.b.h 2 180.x even 12 2
4851.2.a.t 1 63.o even 6 1
5819.2.a.a 1 207.f odd 6 1
5929.2.a.h 1 693.bj even 6 1
6336.2.a.br 1 72.j odd 6 1
6336.2.a.bu 1 72.l even 6 1
7744.2.a.k 1 792.z even 6 1
7744.2.a.x 1 792.be odd 6 1
8624.2.a.j 1 252.bi even 6 1
9251.2.a.d 1 261.i even 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 2T_{2} + 4$$ T2^2 - 2*T2 + 4 $$T_{5}^{2} + T_{5} + 1$$ T5^2 + T5 + 1 $$T_{7}^{2} - 2T_{7} + 4$$ T7^2 - 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} - 2T + 4$$
$11$ $$T^{2} + T + 1$$
$13$ $$T^{2} + 4T + 16$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} - T + 1$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 7T + 49$$
$37$ $$(T - 3)^{2}$$
$41$ $$T^{2} - 8T + 64$$
$43$ $$T^{2} - 6T + 36$$
$47$ $$T^{2} + 8T + 64$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 5T + 25$$
$61$ $$T^{2} + 12T + 144$$
$67$ $$T^{2} - 7T + 49$$
$71$ $$(T + 3)^{2}$$
$73$ $$(T - 4)^{2}$$
$79$ $$T^{2} - 10T + 100$$
$83$ $$T^{2} - 6T + 36$$
$89$ $$(T - 15)^{2}$$
$97$ $$T^{2} - 7T + 49$$