# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$867 = 3 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 867.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.92302985525$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 51) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

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

## Character values

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

 $$n$$ $$290$$ $$292$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 − 1.56155i 1.56155i 2.56155i − 2.56155i
−1.56155 1.00000i 0.438447 0.561553i 1.56155i 0 2.43845 −1.00000 0.876894i
577.2 −1.56155 1.00000i 0.438447 0.561553i 1.56155i 0 2.43845 −1.00000 0.876894i
577.3 2.56155 1.00000i 4.56155 3.56155i 2.56155i 0 6.56155 −1.00000 9.12311i
577.4 2.56155 1.00000i 4.56155 3.56155i 2.56155i 0 6.56155 −1.00000 9.12311i
 $$n$$: 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
17.b even 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 4$$ acting on $$S_{2}^{\mathrm{new}}(867, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -4 - T + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$4 + 13 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$16 + 9 T^{2} + T^{4}$$
$13$ $$( 2 - 5 T + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( -36 + 3 T + T^{2} )^{2}$$
$23$ $$256 + 49 T^{2} + T^{4}$$
$29$ $$( 68 + T^{2} )^{2}$$
$31$ $$256 + 36 T^{2} + T^{4}$$
$37$ $$256 + 36 T^{2} + T^{4}$$
$41$ $$4 + 13 T^{2} + T^{4}$$
$43$ $$( -36 - 3 T + T^{2} )^{2}$$
$47$ $$( 32 + 14 T + T^{2} )^{2}$$
$53$ $$( -52 + 8 T + T^{2} )^{2}$$
$59$ $$( -8 + 6 T + T^{2} )^{2}$$
$61$ $$64 + 84 T^{2} + T^{4}$$
$67$ $$( -4 + T )^{4}$$
$71$ $$4096 + 144 T^{2} + T^{4}$$
$73$ $$2704 + 168 T^{2} + T^{4}$$
$79$ $$20736 + 324 T^{2} + T^{4}$$
$83$ $$( 8 - 10 T + T^{2} )^{2}$$
$89$ $$( -8 - 6 T + T^{2} )^{2}$$
$97$ $$1024 + 132 T^{2} + T^{4}$$