# Properties

 Label 867.2.h.j Level $867$ Weight $2$ Character orbit 867.h Analytic conductor $6.923$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$867 = 3 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 867.h (of order $$8$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.92302985525$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{8})$$ Coefficient field: 16.0.1963501163244660295991296.1 Defining polynomial: $$x^{16} + 1889 x^{8} + 65536$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 51) Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} + \beta_{5} ) q^{2} + \beta_{7} q^{3} + ( 3 \beta_{8} + \beta_{9} ) q^{4} + ( -\beta_{14} - \beta_{15} ) q^{5} -\beta_{11} q^{6} + ( 5 \beta_{12} - \beta_{13} ) q^{8} + \beta_{12} q^{9} +O(q^{10})$$ $$q + ( \beta_{3} + \beta_{5} ) q^{2} + \beta_{7} q^{3} + ( 3 \beta_{8} + \beta_{9} ) q^{4} + ( -\beta_{14} - \beta_{15} ) q^{5} -\beta_{11} q^{6} + ( 5 \beta_{12} - \beta_{13} ) q^{8} + \beta_{12} q^{9} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{10} + ( -\beta_{10} + \beta_{11} ) q^{11} + ( -2 \beta_{14} - \beta_{15} ) q^{12} + ( -2 \beta_{8} + \beta_{9} ) q^{13} + ( -2 \beta_{3} - \beta_{5} ) q^{15} + 3 \beta_{4} q^{16} + \beta_{4} q^{18} + 3 \beta_{5} q^{19} + ( 4 \beta_{6} - 6 \beta_{7} ) q^{20} + 4 \beta_{14} q^{22} + ( 5 \beta_{10} - \beta_{11} ) q^{23} + ( \beta_{1} - 4 \beta_{2} ) q^{24} + ( -3 \beta_{12} + 3 \beta_{13} ) q^{25} + ( 2 \beta_{12} + 2 \beta_{13} ) q^{26} -\beta_{2} q^{27} + ( 2 \beta_{14} - 4 \beta_{15} ) q^{29} + ( -6 \beta_{8} - 2 \beta_{9} ) q^{30} + ( -2 \beta_{6} - 2 \beta_{7} ) q^{31} + ( -5 \beta_{3} - \beta_{5} ) q^{32} + ( -1 - \beta_{4} ) q^{33} + ( -3 \beta_{3} - \beta_{5} ) q^{36} -2 \beta_{6} q^{37} + 12 \beta_{8} q^{38} + ( 3 \beta_{14} - \beta_{15} ) q^{39} + ( 8 \beta_{10} + 6 \beta_{11} ) q^{40} + ( -\beta_{1} + \beta_{2} ) q^{41} -3 \beta_{13} q^{43} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{44} + ( \beta_{10} + \beta_{11} ) q^{45} + ( -4 \beta_{14} + 4 \beta_{15} ) q^{46} + ( 8 \beta_{8} + 2 \beta_{9} ) q^{47} + 3 \beta_{6} q^{48} -7 \beta_{3} q^{49} + ( 12 - 3 \beta_{4} ) q^{50} + 2 q^{52} + ( -6 \beta_{3} - 4 \beta_{5} ) q^{53} + \beta_{6} q^{54} + ( 4 \beta_{8} + \beta_{9} ) q^{55} + ( 3 \beta_{10} - 3 \beta_{11} ) q^{57} + ( 2 \beta_{1} - 16 \beta_{2} ) q^{58} + ( -4 \beta_{12} + 2 \beta_{13} ) q^{59} + ( -10 \beta_{12} + 4 \beta_{13} ) q^{60} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{61} -8 \beta_{10} q^{62} + ( -3 \beta_{8} + \beta_{9} ) q^{64} + ( -\beta_{6} - \beta_{7} ) q^{65} + 4 \beta_{3} q^{66} -4 q^{67} + ( 5 + \beta_{4} ) q^{69} + ( -4 \beta_{6} - 4 \beta_{7} ) q^{71} + ( -5 \beta_{8} - \beta_{9} ) q^{72} + ( 2 \beta_{14} + 4 \beta_{15} ) q^{73} + ( -8 \beta_{10} - 2 \beta_{11} ) q^{74} -3 \beta_{1} q^{75} + ( 12 \beta_{12} - 6 \beta_{13} ) q^{76} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{78} + ( 6 \beta_{10} - 6 \beta_{11} ) q^{79} + ( 12 \beta_{14} + 6 \beta_{15} ) q^{80} -\beta_{8} q^{81} + ( -2 \beta_{6} + 4 \beta_{7} ) q^{82} + ( -4 \beta_{3} + 2 \beta_{5} ) q^{83} -12 q^{86} + ( -2 \beta_{3} - 4 \beta_{5} ) q^{87} -4 \beta_{6} q^{88} + ( 2 \beta_{8} - 2 \beta_{9} ) q^{89} + ( 4 \beta_{14} + 2 \beta_{15} ) q^{90} + ( -2 \beta_{1} + 6 \beta_{2} ) q^{92} -2 \beta_{13} q^{93} + ( 16 \beta_{12} - 8 \beta_{13} ) q^{94} + ( 3 \beta_{1} - 9 \beta_{2} ) q^{95} + ( 4 \beta_{10} + \beta_{11} ) q^{96} + ( -8 \beta_{14} + 2 \beta_{15} ) q^{97} + ( -7 \beta_{8} - 7 \beta_{9} ) q^{98} + ( -\beta_{6} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 24q^{16} - 8q^{18} - 8q^{33} + 216q^{50} + 32q^{52} - 64q^{67} + 72q^{69} - 192q^{86} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 1889 x^{8} + 65536$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{9} + 1165 \nu$$$$)/1764$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{10} + 2929 \nu^{2}$$$$)/7056$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{8} + 724$$$$)/441$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{10} - 7589 \nu^{2}$$$$)/7056$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{11} + 2929 \nu^{3}$$$$)/7056$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{11} + 7589 \nu^{3}$$$$)/28224$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{12} + 2145 \nu^{4}$$$$)/12544$$ $$\beta_{9}$$ $$=$$ $$($$$$-29 \nu^{12} - 49661 \nu^{4}$$$$)/112896$$ $$\beta_{10}$$ $$=$$ $$($$$$-29 \nu^{13} - 49661 \nu^{5}$$$$)/451584$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{13} + 2145 \nu^{5}$$$$)/12544$$ $$\beta_{12}$$ $$=$$ $$($$$$65 \nu^{14} + 126881 \nu^{6}$$$$)/1806336$$ $$\beta_{13}$$ $$=$$ $$($$$$181 \nu^{14} + 325525 \nu^{6}$$$$)/1806336$$ $$\beta_{14}$$ $$=$$ $$($$$$-181 \nu^{15} - 325525 \nu^{7}$$$$)/7225344$$ $$\beta_{15}$$ $$=$$ $$($$$$65 \nu^{15} + 126881 \nu^{7}$$$$)/1806336$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 5 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{7} + 5 \beta_{6}$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{9} + 29 \beta_{8}$$ $$\nu^{5}$$ $$=$$ $$29 \beta_{11} + 36 \beta_{10}$$ $$\nu^{6}$$ $$=$$ $$-65 \beta_{13} + 181 \beta_{12}$$ $$\nu^{7}$$ $$=$$ $$181 \beta_{15} + 260 \beta_{14}$$ $$\nu^{8}$$ $$=$$ $$441 \beta_{4} - 724$$ $$\nu^{9}$$ $$=$$ $$1764 \beta_{2} - 1165 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-2929 \beta_{5} - 7589 \beta_{3}$$ $$\nu^{11}$$ $$=$$ $$11716 \beta_{7} - 7589 \beta_{6}$$ $$\nu^{12}$$ $$=$$ $$-19305 \beta_{9} - 49661 \beta_{8}$$ $$\nu^{13}$$ $$=$$ $$-49661 \beta_{11} - 77220 \beta_{10}$$ $$\nu^{14}$$ $$=$$ $$126881 \beta_{13} - 325525 \beta_{12}$$ $$\nu^{15}$$ $$=$$ $$-325525 \beta_{15} - 507524 \beta_{14}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/867\mathbb{Z}\right)^\times$$.

 $$n$$ $$290$$ $$292$$ $$\chi(n)$$ $$1$$ $$\beta_{12}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
688.1
 −1.44269 + 0.597580i 1.44269 − 0.597580i 2.36657 − 0.980264i −2.36657 + 0.980264i −1.44269 − 0.597580i 1.44269 + 0.597580i 2.36657 + 0.980264i −2.36657 − 0.980264i −0.980264 + 2.36657i 0.980264 − 2.36657i 0.597580 − 1.44269i −0.597580 + 1.44269i −0.980264 − 2.36657i 0.980264 + 2.36657i 0.597580 + 1.44269i −0.597580 − 1.44269i
−1.10418 + 1.10418i −0.382683 + 0.923880i 0.438447i −0.518807 0.214897i −0.597580 1.44269i 0 −1.72424 1.72424i −0.707107 0.707107i 0.810145 0.335573i
688.2 −1.10418 + 1.10418i 0.382683 0.923880i 0.438447i 0.518807 + 0.214897i 0.597580 + 1.44269i 0 −1.72424 1.72424i −0.707107 0.707107i −0.810145 + 0.335573i
688.3 1.81129 1.81129i −0.382683 + 0.923880i 4.56155i 3.29045 + 1.36295i 0.980264 + 2.36657i 0 −4.63972 4.63972i −0.707107 0.707107i 8.42865 3.49126i
688.4 1.81129 1.81129i 0.382683 0.923880i 4.56155i −3.29045 1.36295i −0.980264 2.36657i 0 −4.63972 4.63972i −0.707107 0.707107i −8.42865 + 3.49126i
712.1 −1.10418 1.10418i −0.382683 0.923880i 0.438447i −0.518807 + 0.214897i −0.597580 + 1.44269i 0 −1.72424 + 1.72424i −0.707107 + 0.707107i 0.810145 + 0.335573i
712.2 −1.10418 1.10418i 0.382683 + 0.923880i 0.438447i 0.518807 0.214897i 0.597580 1.44269i 0 −1.72424 + 1.72424i −0.707107 + 0.707107i −0.810145 0.335573i
712.3 1.81129 + 1.81129i −0.382683 0.923880i 4.56155i 3.29045 1.36295i 0.980264 2.36657i 0 −4.63972 + 4.63972i −0.707107 + 0.707107i 8.42865 + 3.49126i
712.4 1.81129 + 1.81129i 0.382683 + 0.923880i 4.56155i −3.29045 + 1.36295i −0.980264 + 2.36657i 0 −4.63972 + 4.63972i −0.707107 + 0.707107i −8.42865 3.49126i
733.1 −1.81129 1.81129i −0.923880 + 0.382683i 4.56155i −1.36295 3.29045i 2.36657 + 0.980264i 0 4.63972 4.63972i 0.707107 0.707107i −3.49126 + 8.42865i
733.2 −1.81129 1.81129i 0.923880 0.382683i 4.56155i 1.36295 + 3.29045i −2.36657 0.980264i 0 4.63972 4.63972i 0.707107 0.707107i 3.49126 8.42865i
733.3 1.10418 + 1.10418i −0.923880 + 0.382683i 0.438447i 0.214897 + 0.518807i −1.44269 0.597580i 0 1.72424 1.72424i 0.707107 0.707107i −0.335573 + 0.810145i
733.4 1.10418 + 1.10418i 0.923880 0.382683i 0.438447i −0.214897 0.518807i 1.44269 + 0.597580i 0 1.72424 1.72424i 0.707107 0.707107i 0.335573 0.810145i
757.1 −1.81129 + 1.81129i −0.923880 0.382683i 4.56155i −1.36295 + 3.29045i 2.36657 0.980264i 0 4.63972 + 4.63972i 0.707107 + 0.707107i −3.49126 8.42865i
757.2 −1.81129 + 1.81129i 0.923880 + 0.382683i 4.56155i 1.36295 3.29045i −2.36657 + 0.980264i 0 4.63972 + 4.63972i 0.707107 + 0.707107i 3.49126 + 8.42865i
757.3 1.10418 1.10418i −0.923880 0.382683i 0.438447i 0.214897 0.518807i −1.44269 + 0.597580i 0 1.72424 + 1.72424i 0.707107 + 0.707107i −0.335573 0.810145i
757.4 1.10418 1.10418i 0.923880 + 0.382683i 0.438447i −0.214897 + 0.518807i 1.44269 0.597580i 0 1.72424 + 1.72424i 0.707107 + 0.707107i 0.335573 + 0.810145i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 757.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.h.j 16
17.b even 2 1 inner 867.2.h.j 16
17.c even 4 2 inner 867.2.h.j 16
17.d even 8 4 inner 867.2.h.j 16
17.e odd 16 1 51.2.a.b 2
17.e odd 16 1 867.2.a.f 2
17.e odd 16 2 867.2.d.c 4
17.e odd 16 4 867.2.e.f 8
51.i even 16 1 153.2.a.e 2
51.i even 16 1 2601.2.a.t 2
68.i even 16 1 816.2.a.m 2
85.o even 16 1 1275.2.b.d 4
85.p odd 16 1 1275.2.a.n 2
85.r even 16 1 1275.2.b.d 4
119.p even 16 1 2499.2.a.o 2
136.q odd 16 1 3264.2.a.bl 2
136.s even 16 1 3264.2.a.bg 2
187.m even 16 1 6171.2.a.p 2
204.t odd 16 1 2448.2.a.v 2
221.y odd 16 1 8619.2.a.q 2
255.be even 16 1 3825.2.a.s 2
357.be odd 16 1 7497.2.a.v 2
408.bg odd 16 1 9792.2.a.cz 2
408.bm even 16 1 9792.2.a.cy 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.a.b 2 17.e odd 16 1
153.2.a.e 2 51.i even 16 1
816.2.a.m 2 68.i even 16 1
867.2.a.f 2 17.e odd 16 1
867.2.d.c 4 17.e odd 16 2
867.2.e.f 8 17.e odd 16 4
867.2.h.j 16 1.a even 1 1 trivial
867.2.h.j 16 17.b even 2 1 inner
867.2.h.j 16 17.c even 4 2 inner
867.2.h.j 16 17.d even 8 4 inner
1275.2.a.n 2 85.p odd 16 1
1275.2.b.d 4 85.o even 16 1
1275.2.b.d 4 85.r even 16 1
2448.2.a.v 2 204.t odd 16 1
2499.2.a.o 2 119.p even 16 1
2601.2.a.t 2 51.i even 16 1
3264.2.a.bg 2 136.s even 16 1
3264.2.a.bl 2 136.q odd 16 1
3825.2.a.s 2 255.be even 16 1
6171.2.a.p 2 187.m even 16 1
7497.2.a.v 2 357.be odd 16 1
8619.2.a.q 2 221.y odd 16 1
9792.2.a.cy 2 408.bm even 16 1
9792.2.a.cz 2 408.bg odd 16 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(867, [\chi])$$:

 $$T_{2}^{8} + 49 T_{2}^{4} + 256$$ $$T_{5}^{16} + 25889 T_{5}^{8} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 256 + 49 T^{4} + T^{8} )^{2}$$
$3$ $$( 1 + T^{8} )^{2}$$
$5$ $$256 + 25889 T^{8} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$65536 + 1889 T^{8} + T^{16}$$
$13$ $$( 4 + 21 T^{2} + T^{4} )^{4}$$
$17$ $$T^{16}$$
$19$ $$( 1679616 + 3969 T^{4} + T^{8} )^{2}$$
$23$ $$4294967296 + 3437249 T^{8} + T^{16}$$
$29$ $$( 21381376 + T^{8} )^{2}$$
$31$ $$4294967296 + 483584 T^{8} + T^{16}$$
$37$ $$4294967296 + 483584 T^{8} + T^{16}$$
$41$ $$256 + 25889 T^{8} + T^{16}$$
$43$ $$( 1679616 + 3969 T^{4} + T^{8} )^{2}$$
$47$ $$( 1024 + 132 T^{2} + T^{4} )^{4}$$
$53$ $$( 7311616 + 22816 T^{4} + T^{8} )^{2}$$
$59$ $$( 4096 + 2576 T^{4} + T^{8} )^{2}$$
$61$ $$16777216 + 47988992 T^{8} + T^{16}$$
$67$ $$( 4 + T )^{16}$$
$71$ $$281474976710656 + 123797504 T^{8} + T^{16}$$
$73$ $$53459728531456 + 505946624 T^{8} + T^{16}$$
$79$ $$184884258895036416 + 3172794624 T^{8} + T^{16}$$
$83$ $$( 4096 + 6928 T^{4} + T^{8} )^{2}$$
$89$ $$( 64 + 52 T^{2} + T^{4} )^{4}$$
$97$ $$1099511627776 + 234324224 T^{8} + T^{16}$$