# Properties

 Label 867.2.e.f Level $867$ Weight $2$ Character orbit 867.e Analytic conductor $6.923$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.92302985525$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.5473632256.1 Defining polynomial: $$x^{8} + 49 x^{4} + 256$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 51) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + \beta_{7} q^{3} + ( -2 - \beta_{4} ) q^{4} + ( \beta_{6} + 2 \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{2} ) q^{6} + ( 4 \beta_{3} - \beta_{5} ) q^{8} -\beta_{3} q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{2} + \beta_{7} q^{3} + ( -2 - \beta_{4} ) q^{4} + ( \beta_{6} + 2 \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{2} ) q^{6} + ( 4 \beta_{3} - \beta_{5} ) q^{8} -\beta_{3} q^{9} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{10} + \beta_{1} q^{11} + ( -\beta_{6} - 3 \beta_{7} ) q^{12} + ( -3 + \beta_{4} ) q^{13} + ( -\beta_{3} + \beta_{5} ) q^{15} + 3 \beta_{4} q^{16} -\beta_{4} q^{18} + ( 3 \beta_{3} + 3 \beta_{5} ) q^{19} + ( -4 \beta_{6} - 10 \beta_{7} ) q^{20} -4 \beta_{7} q^{22} + ( \beta_{1} - 4 \beta_{2} ) q^{23} + ( -\beta_{1} - 5 \beta_{2} ) q^{24} + 3 \beta_{5} q^{25} + ( -4 \beta_{3} - 2 \beta_{5} ) q^{26} + \beta_{2} q^{27} + ( -4 \beta_{6} - 2 \beta_{7} ) q^{29} + ( -4 - 2 \beta_{4} ) q^{30} -2 \beta_{6} q^{31} + ( -4 \beta_{3} + \beta_{5} ) q^{32} + ( 1 - \beta_{4} ) q^{33} + ( 2 \beta_{3} - \beta_{5} ) q^{36} + ( 2 \beta_{6} + 2 \beta_{7} ) q^{37} -12 q^{38} + ( \beta_{6} - 2 \beta_{7} ) q^{39} + ( -6 \beta_{1} - 14 \beta_{2} ) q^{40} + ( \beta_{1} + 2 \beta_{2} ) q^{41} + ( 3 \beta_{3} + 3 \beta_{5} ) q^{43} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{44} + ( \beta_{1} + 2 \beta_{2} ) q^{45} + 4 \beta_{6} q^{46} + ( 6 + 2 \beta_{4} ) q^{47} + ( 3 \beta_{6} + 3 \beta_{7} ) q^{48} -7 \beta_{3} q^{49} + ( -12 - 3 \beta_{4} ) q^{50} + 2 q^{52} + ( 2 \beta_{3} - 4 \beta_{5} ) q^{53} + ( -\beta_{6} - \beta_{7} ) q^{54} + ( -3 - \beta_{4} ) q^{55} + 3 \beta_{1} q^{57} + ( -2 \beta_{1} - 18 \beta_{2} ) q^{58} + ( -2 \beta_{3} + 2 \beta_{5} ) q^{59} + ( 6 \beta_{3} - 4 \beta_{5} ) q^{60} + ( -2 \beta_{1} - 6 \beta_{2} ) q^{61} -8 \beta_{2} q^{62} + ( -4 + \beta_{4} ) q^{64} -\beta_{6} q^{65} + 4 \beta_{3} q^{66} + 4 q^{67} + ( 5 - \beta_{4} ) q^{69} + 4 \beta_{6} q^{71} + ( 4 + \beta_{4} ) q^{72} + ( -4 \beta_{6} - 6 \beta_{7} ) q^{73} + ( 2 \beta_{1} + 10 \beta_{2} ) q^{74} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{75} + ( 6 \beta_{3} - 6 \beta_{5} ) q^{76} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{78} -6 \beta_{1} q^{79} + ( 6 \beta_{6} + 18 \beta_{7} ) q^{80} - q^{81} + ( -2 \beta_{6} - 6 \beta_{7} ) q^{82} + ( -6 \beta_{3} - 2 \beta_{5} ) q^{83} -12 q^{86} + ( -2 \beta_{3} - 4 \beta_{5} ) q^{87} + ( 4 \beta_{6} + 4 \beta_{7} ) q^{88} + ( -4 + 2 \beta_{4} ) q^{89} + ( -2 \beta_{6} - 6 \beta_{7} ) q^{90} + ( 2 \beta_{1} + 8 \beta_{2} ) q^{92} + ( -2 \beta_{3} - 2 \beta_{5} ) q^{93} + ( -8 \beta_{3} + 8 \beta_{5} ) q^{94} + ( 3 \beta_{1} + 12 \beta_{2} ) q^{95} + ( \beta_{1} + 5 \beta_{2} ) q^{96} + ( 2 \beta_{6} - 6 \beta_{7} ) q^{97} -7 \beta_{4} q^{98} -\beta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 20q^{4} + O(q^{10})$$ $$8q - 20q^{4} - 20q^{13} + 12q^{16} - 4q^{18} - 40q^{30} + 4q^{33} - 96q^{38} + 56q^{47} - 108q^{50} + 16q^{52} - 28q^{55} - 28q^{64} + 32q^{67} + 36q^{69} + 36q^{72} - 8q^{81} - 96q^{86} - 24q^{89} - 28q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 49 x^{4} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 29 \nu$$$$)/36$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 65 \nu^{2}$$$$)/144$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + 29$$$$)/9$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{6} - 181 \nu^{2}$$$$)/144$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} + 65 \nu^{3}$$$$)/144$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} + 181 \nu^{3}$$$$)/576$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 5 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{7} + 5 \beta_{6}$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{4} - 29$$ $$\nu^{5}$$ $$=$$ $$36 \beta_{2} - 29 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-65 \beta_{5} - 181 \beta_{3}$$ $$\nu^{7}$$ $$=$$ $$260 \beta_{7} - 181 \beta_{6}$$

## Character values

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

 $$n$$ $$290$$ $$292$$ $$\chi(n)$$ $$1$$ $$-\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
616.1
 1.10418 + 1.10418i −1.10418 − 1.10418i −1.81129 − 1.81129i 1.81129 + 1.81129i −1.81129 + 1.81129i 1.81129 − 1.81129i 1.10418 − 1.10418i −1.10418 + 1.10418i
2.56155i −0.707107 + 0.707107i −4.56155 −2.51840 + 2.51840i 1.81129 + 1.81129i 0 6.56155i 1.00000i 6.45101 + 6.45101i
616.2 2.56155i 0.707107 0.707107i −4.56155 2.51840 2.51840i −1.81129 1.81129i 0 6.56155i 1.00000i −6.45101 6.45101i
616.3 1.56155i −0.707107 + 0.707107i −0.438447 0.397078 0.397078i −1.10418 1.10418i 0 2.43845i 1.00000i 0.620058 + 0.620058i
616.4 1.56155i 0.707107 0.707107i −0.438447 −0.397078 + 0.397078i 1.10418 + 1.10418i 0 2.43845i 1.00000i −0.620058 0.620058i
829.1 1.56155i −0.707107 0.707107i −0.438447 0.397078 + 0.397078i −1.10418 + 1.10418i 0 2.43845i 1.00000i 0.620058 0.620058i
829.2 1.56155i 0.707107 + 0.707107i −0.438447 −0.397078 0.397078i 1.10418 1.10418i 0 2.43845i 1.00000i −0.620058 + 0.620058i
829.3 2.56155i −0.707107 0.707107i −4.56155 −2.51840 2.51840i 1.81129 1.81129i 0 6.56155i 1.00000i 6.45101 6.45101i
829.4 2.56155i 0.707107 + 0.707107i −4.56155 2.51840 + 2.51840i −1.81129 + 1.81129i 0 6.56155i 1.00000i −6.45101 + 6.45101i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 829.4 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

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{4} + 9 T_{2}^{2} + 16$$ $$T_{5}^{8} + 161 T_{5}^{4} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 + 9 T^{2} + T^{4} )^{2}$$
$3$ $$( 1 + T^{4} )^{2}$$
$5$ $$16 + 161 T^{4} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$256 + 49 T^{4} + T^{8}$$
$13$ $$( 2 + 5 T + T^{2} )^{4}$$
$17$ $$T^{8}$$
$19$ $$( 1296 + 81 T^{2} + T^{4} )^{2}$$
$23$ $$65536 + 1889 T^{4} + T^{8}$$
$29$ $$( 4624 + T^{4} )^{2}$$
$31$ $$65536 + 784 T^{4} + T^{8}$$
$37$ $$65536 + 784 T^{4} + T^{8}$$
$41$ $$16 + 161 T^{4} + T^{8}$$
$43$ $$( 1296 + 81 T^{2} + T^{4} )^{2}$$
$47$ $$( 32 - 14 T + T^{2} )^{4}$$
$53$ $$( 2704 + 168 T^{2} + T^{4} )^{2}$$
$59$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$
$61$ $$4096 + 6928 T^{4} + T^{8}$$
$67$ $$( -4 + T )^{8}$$
$71$ $$16777216 + 12544 T^{4} + T^{8}$$
$73$ $$7311616 + 22816 T^{4} + T^{8}$$
$79$ $$429981696 + 63504 T^{4} + T^{8}$$
$83$ $$( 64 + 84 T^{2} + T^{4} )^{2}$$
$89$ $$( -8 + 6 T + T^{2} )^{4}$$
$97$ $$1048576 + 15376 T^{4} + T^{8}$$