# Properties

 Label 867.2.h.d Level $867$ Weight $2$ Character orbit 867.h Analytic conductor $6.923$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ 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 primitive root of unity $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{16}^{2} q^{2} -\zeta_{16}^{3} q^{3} -\zeta_{16}^{4} q^{4} -\zeta_{16}^{5} q^{6} -4 \zeta_{16} q^{7} -3 \zeta_{16}^{6} q^{8} + \zeta_{16}^{6} q^{9} +O(q^{10})$$ $$q + \zeta_{16}^{2} q^{2} -\zeta_{16}^{3} q^{3} -\zeta_{16}^{4} q^{4} -\zeta_{16}^{5} q^{6} -4 \zeta_{16} q^{7} -3 \zeta_{16}^{6} q^{8} + \zeta_{16}^{6} q^{9} + 4 \zeta_{16}^{5} q^{11} + \zeta_{16}^{7} q^{12} -2 \zeta_{16}^{4} q^{13} -4 \zeta_{16}^{3} q^{14} + q^{16} - q^{18} -4 \zeta_{16}^{2} q^{19} + 4 \zeta_{16}^{4} q^{21} + 4 \zeta_{16}^{7} q^{22} + 4 \zeta_{16}^{5} q^{23} -3 \zeta_{16} q^{24} + 5 \zeta_{16}^{6} q^{25} -2 \zeta_{16}^{6} q^{26} + \zeta_{16} q^{27} + 4 \zeta_{16}^{5} q^{28} -4 \zeta_{16}^{3} q^{31} -5 \zeta_{16}^{2} q^{32} + 4 q^{33} + \zeta_{16}^{2} q^{36} -8 \zeta_{16}^{3} q^{37} -4 \zeta_{16}^{4} q^{38} + 2 \zeta_{16}^{7} q^{39} -8 \zeta_{16} q^{41} + 4 \zeta_{16}^{6} q^{42} + 4 \zeta_{16}^{6} q^{43} + 4 \zeta_{16} q^{44} + 4 \zeta_{16}^{7} q^{46} + 8 \zeta_{16}^{4} q^{47} -\zeta_{16}^{3} q^{48} + 9 \zeta_{16}^{2} q^{49} -5 q^{50} -2 q^{52} + 6 \zeta_{16}^{2} q^{53} + \zeta_{16}^{3} q^{54} + 12 \zeta_{16}^{7} q^{56} + 4 \zeta_{16}^{5} q^{57} -12 \zeta_{16}^{6} q^{59} + 8 \zeta_{16} q^{61} -4 \zeta_{16}^{5} q^{62} -4 \zeta_{16}^{7} q^{63} -7 \zeta_{16}^{4} q^{64} + 4 \zeta_{16}^{2} q^{66} -12 q^{67} + 4 q^{69} -12 \zeta_{16}^{3} q^{71} + 3 \zeta_{16}^{4} q^{72} -8 \zeta_{16}^{5} q^{74} + 5 \zeta_{16} q^{75} + 4 \zeta_{16}^{6} q^{76} -16 \zeta_{16}^{6} q^{77} -2 \zeta_{16} q^{78} -4 \zeta_{16}^{5} q^{79} -\zeta_{16}^{4} q^{81} -8 \zeta_{16}^{3} q^{82} -12 \zeta_{16}^{2} q^{83} + 4 q^{84} -4 q^{86} + 12 \zeta_{16}^{3} q^{88} -10 \zeta_{16}^{4} q^{89} + 8 \zeta_{16}^{5} q^{91} + 4 \zeta_{16} q^{92} + 4 \zeta_{16}^{6} q^{93} + 8 \zeta_{16}^{6} q^{94} + 5 \zeta_{16}^{5} q^{96} -16 \zeta_{16}^{7} q^{97} + 9 \zeta_{16}^{4} q^{98} -4 \zeta_{16}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{16} - 8q^{18} + 32q^{33} - 40q^{50} - 16q^{52} - 96q^{67} + 32q^{69} + 32q^{84} - 32q^{86} + O(q^{100})$$

## Character values

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

 $$n$$ $$290$$ $$292$$ $$\chi(n)$$ $$1$$ $$\zeta_{16}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
688.1
 0.923880 − 0.382683i −0.923880 + 0.382683i 0.923880 + 0.382683i −0.923880 − 0.382683i −0.382683 + 0.923880i 0.382683 − 0.923880i −0.382683 − 0.923880i 0.382683 + 0.923880i
0.707107 0.707107i −0.382683 + 0.923880i 1.00000i 0 0.382683 + 0.923880i −3.69552 + 1.53073i 2.12132 + 2.12132i −0.707107 0.707107i 0
688.2 0.707107 0.707107i 0.382683 0.923880i 1.00000i 0 −0.382683 0.923880i 3.69552 1.53073i 2.12132 + 2.12132i −0.707107 0.707107i 0
712.1 0.707107 + 0.707107i −0.382683 0.923880i 1.00000i 0 0.382683 0.923880i −3.69552 1.53073i 2.12132 2.12132i −0.707107 + 0.707107i 0
712.2 0.707107 + 0.707107i 0.382683 + 0.923880i 1.00000i 0 −0.382683 + 0.923880i 3.69552 + 1.53073i 2.12132 2.12132i −0.707107 + 0.707107i 0
733.1 −0.707107 0.707107i −0.923880 + 0.382683i 1.00000i 0 0.923880 + 0.382683i 1.53073 3.69552i −2.12132 + 2.12132i 0.707107 0.707107i 0
733.2 −0.707107 0.707107i 0.923880 0.382683i 1.00000i 0 −0.923880 0.382683i −1.53073 + 3.69552i −2.12132 + 2.12132i 0.707107 0.707107i 0
757.1 −0.707107 + 0.707107i −0.923880 0.382683i 1.00000i 0 0.923880 0.382683i 1.53073 + 3.69552i −2.12132 2.12132i 0.707107 + 0.707107i 0
757.2 −0.707107 + 0.707107i 0.923880 + 0.382683i 1.00000i 0 −0.923880 + 0.382683i −1.53073 3.69552i −2.12132 2.12132i 0.707107 + 0.707107i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 757.2 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.d 8
17.b even 2 1 inner 867.2.h.d 8
17.c even 4 2 inner 867.2.h.d 8
17.d even 8 4 inner 867.2.h.d 8
17.e odd 16 2 51.2.d.b 2
17.e odd 16 1 867.2.a.a 1
17.e odd 16 1 867.2.a.b 1
17.e odd 16 4 867.2.e.d 4
51.i even 16 2 153.2.d.a 2
51.i even 16 1 2601.2.a.i 1
51.i even 16 1 2601.2.a.j 1
68.i even 16 2 816.2.c.c 2
85.o even 16 1 1275.2.d.b 2
85.o even 16 1 1275.2.d.d 2
85.p odd 16 2 1275.2.g.a 2
85.r even 16 1 1275.2.d.b 2
85.r even 16 1 1275.2.d.d 2
136.q odd 16 2 3264.2.c.e 2
136.s even 16 2 3264.2.c.d 2
204.t odd 16 2 2448.2.c.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b 2 17.e odd 16 2
153.2.d.a 2 51.i even 16 2
816.2.c.c 2 68.i even 16 2
867.2.a.a 1 17.e odd 16 1
867.2.a.b 1 17.e odd 16 1
867.2.e.d 4 17.e odd 16 4
867.2.h.d 8 1.a even 1 1 trivial
867.2.h.d 8 17.b even 2 1 inner
867.2.h.d 8 17.c even 4 2 inner
867.2.h.d 8 17.d even 8 4 inner
1275.2.d.b 2 85.o even 16 1
1275.2.d.b 2 85.r even 16 1
1275.2.d.d 2 85.o even 16 1
1275.2.d.d 2 85.r even 16 1
1275.2.g.a 2 85.p odd 16 2
2448.2.c.j 2 204.t odd 16 2
2601.2.a.i 1 51.i even 16 1
2601.2.a.j 1 51.i even 16 1
3264.2.c.d 2 136.s even 16 2
3264.2.c.e 2 136.q odd 16 2

## Hecke kernels

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

 $$T_{2}^{4} + 1$$ $$T_{5}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$1 + T^{8}$$
$5$ $$T^{8}$$
$7$ $$65536 + T^{8}$$
$11$ $$65536 + T^{8}$$
$13$ $$( 4 + T^{2} )^{4}$$
$17$ $$T^{8}$$
$19$ $$( 256 + T^{4} )^{2}$$
$23$ $$65536 + T^{8}$$
$29$ $$T^{8}$$
$31$ $$65536 + T^{8}$$
$37$ $$16777216 + T^{8}$$
$41$ $$16777216 + T^{8}$$
$43$ $$( 256 + T^{4} )^{2}$$
$47$ $$( 64 + T^{2} )^{4}$$
$53$ $$( 1296 + T^{4} )^{2}$$
$59$ $$( 20736 + T^{4} )^{2}$$
$61$ $$16777216 + T^{8}$$
$67$ $$( 12 + T )^{8}$$
$71$ $$429981696 + T^{8}$$
$73$ $$T^{8}$$
$79$ $$65536 + T^{8}$$
$83$ $$( 20736 + T^{4} )^{2}$$
$89$ $$( 100 + T^{2} )^{4}$$
$97$ $$4294967296 + T^{8}$$