# Properties

 Label 891.2.e.g 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} - 2 \zeta_{6} q^{5} + (4 \zeta_{6} - 4) q^{7} + 3 q^{8} +O(q^{10})$$ q + (-z + 1) * q^2 + z * q^4 - 2*z * q^5 + (4*z - 4) * q^7 + 3 * q^8 $$q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} - 2 \zeta_{6} q^{5} + (4 \zeta_{6} - 4) q^{7} + 3 q^{8} - 2 q^{10} + ( - \zeta_{6} + 1) q^{11} + 2 \zeta_{6} q^{13} + 4 \zeta_{6} q^{14} + ( - \zeta_{6} + 1) q^{16} + 2 q^{17} + ( - 2 \zeta_{6} + 2) q^{20} - \zeta_{6} q^{22} + 8 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} + 2 q^{26} - 4 q^{28} + (6 \zeta_{6} - 6) q^{29} + 8 \zeta_{6} q^{31} + 5 \zeta_{6} q^{32} + ( - 2 \zeta_{6} + 2) q^{34} + 8 q^{35} + 6 q^{37} - 6 \zeta_{6} q^{40} - 2 \zeta_{6} q^{41} + q^{44} + 8 q^{46} + ( - 8 \zeta_{6} + 8) q^{47} - 9 \zeta_{6} q^{49} - \zeta_{6} q^{50} + (2 \zeta_{6} - 2) q^{52} - 6 q^{53} - 2 q^{55} + (12 \zeta_{6} - 12) q^{56} + 6 \zeta_{6} q^{58} - 4 \zeta_{6} q^{59} + (6 \zeta_{6} - 6) q^{61} + 8 q^{62} + 7 q^{64} + ( - 4 \zeta_{6} + 4) q^{65} + 4 \zeta_{6} q^{67} + 2 \zeta_{6} q^{68} + ( - 8 \zeta_{6} + 8) q^{70} - 14 q^{73} + ( - 6 \zeta_{6} + 6) q^{74} + 4 \zeta_{6} q^{77} + ( - 4 \zeta_{6} + 4) q^{79} - 2 q^{80} - 2 q^{82} + ( - 12 \zeta_{6} + 12) q^{83} - 4 \zeta_{6} q^{85} + ( - 3 \zeta_{6} + 3) q^{88} + 6 q^{89} - 8 q^{91} + (8 \zeta_{6} - 8) q^{92} - 8 \zeta_{6} q^{94} + (2 \zeta_{6} - 2) q^{97} - 9 q^{98} +O(q^{100})$$ q + (-z + 1) * q^2 + z * q^4 - 2*z * q^5 + (4*z - 4) * q^7 + 3 * q^8 - 2 * q^10 + (-z + 1) * q^11 + 2*z * q^13 + 4*z * q^14 + (-z + 1) * q^16 + 2 * q^17 + (-2*z + 2) * q^20 - z * q^22 + 8*z * q^23 + (-z + 1) * q^25 + 2 * q^26 - 4 * q^28 + (6*z - 6) * q^29 + 8*z * q^31 + 5*z * q^32 + (-2*z + 2) * q^34 + 8 * q^35 + 6 * q^37 - 6*z * q^40 - 2*z * q^41 + q^44 + 8 * q^46 + (-8*z + 8) * q^47 - 9*z * q^49 - z * q^50 + (2*z - 2) * q^52 - 6 * q^53 - 2 * q^55 + (12*z - 12) * q^56 + 6*z * q^58 - 4*z * q^59 + (6*z - 6) * q^61 + 8 * q^62 + 7 * q^64 + (-4*z + 4) * q^65 + 4*z * q^67 + 2*z * q^68 + (-8*z + 8) * q^70 - 14 * q^73 + (-6*z + 6) * q^74 + 4*z * q^77 + (-4*z + 4) * q^79 - 2 * q^80 - 2 * q^82 + (-12*z + 12) * q^83 - 4*z * q^85 + (-3*z + 3) * q^88 + 6 * q^89 - 8 * q^91 + (8*z - 8) * q^92 - 8*z * q^94 + (2*z - 2) * q^97 - 9 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} - 2 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + q^2 + q^4 - 2 * q^5 - 4 * q^7 + 6 * q^8 $$2 q + q^{2} + q^{4} - 2 q^{5} - 4 q^{7} + 6 q^{8} - 4 q^{10} + q^{11} + 2 q^{13} + 4 q^{14} + q^{16} + 4 q^{17} + 2 q^{20} - q^{22} + 8 q^{23} + q^{25} + 4 q^{26} - 8 q^{28} - 6 q^{29} + 8 q^{31} + 5 q^{32} + 2 q^{34} + 16 q^{35} + 12 q^{37} - 6 q^{40} - 2 q^{41} + 2 q^{44} + 16 q^{46} + 8 q^{47} - 9 q^{49} - q^{50} - 2 q^{52} - 12 q^{53} - 4 q^{55} - 12 q^{56} + 6 q^{58} - 4 q^{59} - 6 q^{61} + 16 q^{62} + 14 q^{64} + 4 q^{65} + 4 q^{67} + 2 q^{68} + 8 q^{70} - 28 q^{73} + 6 q^{74} + 4 q^{77} + 4 q^{79} - 4 q^{80} - 4 q^{82} + 12 q^{83} - 4 q^{85} + 3 q^{88} + 12 q^{89} - 16 q^{91} - 8 q^{92} - 8 q^{94} - 2 q^{97} - 18 q^{98}+O(q^{100})$$ 2 * q + q^2 + q^4 - 2 * q^5 - 4 * q^7 + 6 * q^8 - 4 * q^10 + q^11 + 2 * q^13 + 4 * q^14 + q^16 + 4 * q^17 + 2 * q^20 - q^22 + 8 * q^23 + q^25 + 4 * q^26 - 8 * q^28 - 6 * q^29 + 8 * q^31 + 5 * q^32 + 2 * q^34 + 16 * q^35 + 12 * q^37 - 6 * q^40 - 2 * q^41 + 2 * q^44 + 16 * q^46 + 8 * q^47 - 9 * q^49 - q^50 - 2 * q^52 - 12 * q^53 - 4 * q^55 - 12 * q^56 + 6 * q^58 - 4 * q^59 - 6 * q^61 + 16 * q^62 + 14 * q^64 + 4 * q^65 + 4 * q^67 + 2 * q^68 + 8 * q^70 - 28 * q^73 + 6 * q^74 + 4 * q^77 + 4 * q^79 - 4 * q^80 - 4 * q^82 + 12 * q^83 - 4 * q^85 + 3 * q^88 + 12 * q^89 - 16 * q^91 - 8 * q^92 - 8 * q^94 - 2 * q^97 - 18 * 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
0.500000 0.866025i 0 0.500000 + 0.866025i −1.00000 1.73205i 0 −2.00000 + 3.46410i 3.00000 0 −2.00000
595.1 0.500000 + 0.866025i 0 0.500000 0.866025i −1.00000 + 1.73205i 0 −2.00000 3.46410i 3.00000 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.g 2
3.b odd 2 1 891.2.e.e 2
9.c even 3 1 99.2.a.b 1
9.c even 3 1 inner 891.2.e.g 2
9.d odd 6 1 33.2.a.a 1
9.d odd 6 1 891.2.e.e 2
36.f odd 6 1 1584.2.a.o 1
36.h even 6 1 528.2.a.g 1
45.h odd 6 1 825.2.a.a 1
45.j even 6 1 2475.2.a.g 1
45.k odd 12 2 2475.2.c.d 2
45.l even 12 2 825.2.c.a 2
63.l odd 6 1 4851.2.a.b 1
63.o even 6 1 1617.2.a.j 1
72.j odd 6 1 2112.2.a.bb 1
72.l even 6 1 2112.2.a.j 1
72.n even 6 1 6336.2.a.x 1
72.p odd 6 1 6336.2.a.n 1
99.g even 6 1 363.2.a.b 1
99.h odd 6 1 1089.2.a.j 1
99.n odd 30 4 363.2.e.e 4
99.p even 30 4 363.2.e.g 4
117.n odd 6 1 5577.2.a.a 1
153.i odd 6 1 9537.2.a.m 1
396.o odd 6 1 5808.2.a.t 1
495.r even 6 1 9075.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 9.d odd 6 1
99.2.a.b 1 9.c even 3 1
363.2.a.b 1 99.g even 6 1
363.2.e.e 4 99.n odd 30 4
363.2.e.g 4 99.p even 30 4
528.2.a.g 1 36.h even 6 1
825.2.a.a 1 45.h odd 6 1
825.2.c.a 2 45.l even 12 2
891.2.e.e 2 3.b odd 2 1
891.2.e.e 2 9.d odd 6 1
891.2.e.g 2 1.a even 1 1 trivial
891.2.e.g 2 9.c even 3 1 inner
1089.2.a.j 1 99.h odd 6 1
1584.2.a.o 1 36.f odd 6 1
1617.2.a.j 1 63.o even 6 1
2112.2.a.j 1 72.l even 6 1
2112.2.a.bb 1 72.j odd 6 1
2475.2.a.g 1 45.j even 6 1
2475.2.c.d 2 45.k odd 12 2
4851.2.a.b 1 63.l odd 6 1
5577.2.a.a 1 117.n odd 6 1
5808.2.a.t 1 396.o odd 6 1
6336.2.a.n 1 72.p odd 6 1
6336.2.a.x 1 72.n even 6 1
9075.2.a.q 1 495.r even 6 1
9537.2.a.m 1 153.i odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} + 1$$ T2^2 - T2 + 1 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{7}^{2} + 4T_{7} + 16$$ T7^2 + 4*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2} + 4T + 16$$
$11$ $$T^{2} - T + 1$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 8T + 64$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} - 8T + 64$$
$37$ $$(T - 6)^{2}$$
$41$ $$T^{2} + 2T + 4$$
$43$ $$T^{2}$$
$47$ $$T^{2} - 8T + 64$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 4T + 16$$
$61$ $$T^{2} + 6T + 36$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$T^{2}$$
$73$ $$(T + 14)^{2}$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 2T + 4$$