Properties

 Label 891.2.g.a Level 891 Weight 2 Character orbit 891.g Analytic conductor 7.115 Analytic rank 0 Dimension 4 CM discriminant -11 Inner twists 8

Related objects

Newspace parameters

 Level: $$N$$ = $$891 = 3^{4} \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 891.g (of order $$6$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$7.11467082010$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \beta_{2} ) q^{4} + \beta_{1} q^{5} +O(q^{10})$$ $$q + ( 2 - 2 \beta_{2} ) q^{4} + \beta_{1} q^{5} + \beta_{3} q^{11} -4 \beta_{2} q^{16} + 2 \beta_{3} q^{20} + \beta_{1} q^{23} + 6 \beta_{2} q^{25} + ( -5 + 5 \beta_{2} ) q^{31} -7 q^{37} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{44} + 2 \beta_{3} q^{47} + ( -7 + 7 \beta_{2} ) q^{49} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{53} + 11 q^{55} + \beta_{1} q^{59} -8 q^{64} + ( 13 - 13 \beta_{2} ) q^{67} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{71} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{80} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{89} + 2 \beta_{3} q^{92} -17 \beta_{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} + O(q^{10})$$ $$4q + 4q^{4} - 8q^{16} + 12q^{25} - 10q^{31} - 28q^{37} - 14q^{49} + 44q^{55} - 32q^{64} + 26q^{67} - 34q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} + 10 \nu - 9$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{3} + 2 \nu^{2} + 10 \nu + 15$$$$)/6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 5 \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{3} + \beta_{1} + 4$$

Character values

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

 $$n$$ $$244$$ $$650$$ $$\chi(n)$$ $$-1$$ $$1 - \beta_{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
296.1
 −1.18614 − 1.26217i 1.68614 + 0.396143i −1.18614 + 1.26217i 1.68614 − 0.396143i
0 0 1.00000 1.73205i −2.87228 1.65831i 0 0 0 0 0
296.2 0 0 1.00000 1.73205i 2.87228 + 1.65831i 0 0 0 0 0
593.1 0 0 1.00000 + 1.73205i −2.87228 + 1.65831i 0 0 0 0 0
593.2 0 0 1.00000 + 1.73205i 2.87228 1.65831i 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
33.d even 2 1 inner
99.g even 6 1 inner
99.h odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.g.a 4
3.b odd 2 1 inner 891.2.g.a 4
9.c even 3 1 33.2.d.a 2
9.c even 3 1 inner 891.2.g.a 4
9.d odd 6 1 33.2.d.a 2
9.d odd 6 1 inner 891.2.g.a 4
11.b odd 2 1 CM 891.2.g.a 4
33.d even 2 1 inner 891.2.g.a 4
36.f odd 6 1 528.2.b.a 2
36.h even 6 1 528.2.b.a 2
45.h odd 6 1 825.2.f.a 2
45.j even 6 1 825.2.f.a 2
45.k odd 12 2 825.2.d.a 4
45.l even 12 2 825.2.d.a 4
72.j odd 6 1 2112.2.b.e 2
72.l even 6 1 2112.2.b.f 2
72.n even 6 1 2112.2.b.e 2
72.p odd 6 1 2112.2.b.f 2
99.g even 6 1 33.2.d.a 2
99.g even 6 1 inner 891.2.g.a 4
99.h odd 6 1 33.2.d.a 2
99.h odd 6 1 inner 891.2.g.a 4
99.m even 15 4 363.2.f.c 8
99.n odd 30 4 363.2.f.c 8
99.o odd 30 4 363.2.f.c 8
99.p even 30 4 363.2.f.c 8
396.k even 6 1 528.2.b.a 2
396.o odd 6 1 528.2.b.a 2
495.o odd 6 1 825.2.f.a 2
495.r even 6 1 825.2.f.a 2
495.bd odd 12 2 825.2.d.a 4
495.bf even 12 2 825.2.d.a 4
792.s odd 6 1 2112.2.b.f 2
792.w even 6 1 2112.2.b.e 2
792.z even 6 1 2112.2.b.f 2
792.be odd 6 1 2112.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.d.a 2 9.c even 3 1
33.2.d.a 2 9.d odd 6 1
33.2.d.a 2 99.g even 6 1
33.2.d.a 2 99.h odd 6 1
363.2.f.c 8 99.m even 15 4
363.2.f.c 8 99.n odd 30 4
363.2.f.c 8 99.o odd 30 4
363.2.f.c 8 99.p even 30 4
528.2.b.a 2 36.f odd 6 1
528.2.b.a 2 36.h even 6 1
528.2.b.a 2 396.k even 6 1
528.2.b.a 2 396.o odd 6 1
825.2.d.a 4 45.k odd 12 2
825.2.d.a 4 45.l even 12 2
825.2.d.a 4 495.bd odd 12 2
825.2.d.a 4 495.bf even 12 2
825.2.f.a 2 45.h odd 6 1
825.2.f.a 2 45.j even 6 1
825.2.f.a 2 495.o odd 6 1
825.2.f.a 2 495.r even 6 1
891.2.g.a 4 1.a even 1 1 trivial
891.2.g.a 4 3.b odd 2 1 inner
891.2.g.a 4 9.c even 3 1 inner
891.2.g.a 4 9.d odd 6 1 inner
891.2.g.a 4 11.b odd 2 1 CM
891.2.g.a 4 33.d even 2 1 inner
891.2.g.a 4 99.g even 6 1 inner
891.2.g.a 4 99.h odd 6 1 inner
2112.2.b.e 2 72.j odd 6 1
2112.2.b.e 2 72.n even 6 1
2112.2.b.e 2 792.w even 6 1
2112.2.b.e 2 792.be odd 6 1
2112.2.b.f 2 72.l even 6 1
2112.2.b.f 2 72.p odd 6 1
2112.2.b.f 2 792.s odd 6 1
2112.2.b.f 2 792.z even 6 1

Hecke kernels

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

 $$T_{2}$$ $$T_{23}^{4} - 11 T_{23}^{2} + 121$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} + 4 T^{4} )^{2}$$
$3$ 
$5$ $$( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} )( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} )$$
$7$ $$( 1 + 7 T^{2} + 49 T^{4} )^{2}$$
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 + 13 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 17 T^{2} )^{4}$$
$19$ $$( 1 - 19 T^{2} )^{4}$$
$23$ $$( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} )( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} )$$
$29$ $$( 1 - 29 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 7 T + 37 T^{2} )^{4}$$
$41$ $$( 1 - 41 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 43 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} )( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} )$$
$53$ $$( 1 - 6 T + 53 T^{2} )^{2}( 1 + 6 T + 53 T^{2} )^{2}$$
$59$ $$( 1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4} )( 1 + 15 T + 166 T^{2} + 885 T^{3} + 3481 T^{4} )$$
$61$ $$( 1 + 61 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 3 T + 71 T^{2} )^{2}( 1 + 3 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 73 T^{2} )^{4}$$
$79$ $$( 1 + 79 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 83 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 9 T + 89 T^{2} )^{2}( 1 + 9 T + 89 T^{2} )^{2}$$
$97$ $$( 1 + 17 T + 192 T^{2} + 1649 T^{3} + 9409 T^{4} )^{2}$$