# Properties

 Label 867.2.e.d Level $867$ Weight $2$ Character orbit 867.e Analytic conductor $6.923$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 51) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{2} q^{2} -\zeta_{8}^{3} q^{3} + q^{4} -\zeta_{8} q^{6} -4 \zeta_{8} q^{7} -3 \zeta_{8}^{2} q^{8} -\zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{8}^{2} q^{2} -\zeta_{8}^{3} q^{3} + q^{4} -\zeta_{8} q^{6} -4 \zeta_{8} q^{7} -3 \zeta_{8}^{2} q^{8} -\zeta_{8}^{2} q^{9} -4 \zeta_{8} q^{11} -\zeta_{8}^{3} q^{12} -2 q^{13} + 4 \zeta_{8}^{3} q^{14} - q^{16} - q^{18} + 4 \zeta_{8}^{2} q^{19} -4 q^{21} + 4 \zeta_{8}^{3} q^{22} + 4 \zeta_{8} q^{23} -3 \zeta_{8} q^{24} + 5 \zeta_{8}^{2} q^{25} + 2 \zeta_{8}^{2} q^{26} -\zeta_{8} q^{27} -4 \zeta_{8} q^{28} + 4 \zeta_{8}^{3} q^{31} -5 \zeta_{8}^{2} q^{32} -4 q^{33} -\zeta_{8}^{2} q^{36} -8 \zeta_{8}^{3} q^{37} + 4 q^{38} + 2 \zeta_{8}^{3} q^{39} -8 \zeta_{8} q^{41} + 4 \zeta_{8}^{2} q^{42} -4 \zeta_{8}^{2} q^{43} -4 \zeta_{8} q^{44} -4 \zeta_{8}^{3} q^{46} + 8 q^{47} + \zeta_{8}^{3} q^{48} + 9 \zeta_{8}^{2} q^{49} + 5 q^{50} -2 q^{52} -6 \zeta_{8}^{2} q^{53} + \zeta_{8}^{3} q^{54} + 12 \zeta_{8}^{3} q^{56} + 4 \zeta_{8} q^{57} -12 \zeta_{8}^{2} q^{59} -8 \zeta_{8} q^{61} + 4 \zeta_{8} q^{62} + 4 \zeta_{8}^{3} q^{63} -7 q^{64} + 4 \zeta_{8}^{2} q^{66} + 12 q^{67} + 4 q^{69} -12 \zeta_{8}^{3} q^{71} -3 q^{72} -8 \zeta_{8} q^{74} + 5 \zeta_{8} q^{75} + 4 \zeta_{8}^{2} q^{76} + 16 \zeta_{8}^{2} q^{77} + 2 \zeta_{8} q^{78} + 4 \zeta_{8} q^{79} - q^{81} + 8 \zeta_{8}^{3} q^{82} -12 \zeta_{8}^{2} q^{83} -4 q^{84} -4 q^{86} + 12 \zeta_{8}^{3} q^{88} + 10 q^{89} + 8 \zeta_{8} q^{91} + 4 \zeta_{8} q^{92} + 4 \zeta_{8}^{2} q^{93} -8 \zeta_{8}^{2} q^{94} -5 \zeta_{8} q^{96} + 16 \zeta_{8}^{3} q^{97} + 9 q^{98} + 4 \zeta_{8}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} + O(q^{10})$$ $$4q + 4q^{4} - 8q^{13} - 4q^{16} - 4q^{18} - 16q^{21} - 16q^{33} + 16q^{38} + 32q^{47} + 20q^{50} - 8q^{52} - 28q^{64} + 48q^{67} + 16q^{69} - 12q^{72} - 4q^{81} - 16q^{84} - 16q^{86} + 40q^{89} + 36q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$290$$ $$292$$ $$\chi(n)$$ $$1$$ $$-\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
616.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
1.00000i −0.707107 + 0.707107i 1.00000 0 0.707107 + 0.707107i 2.82843 + 2.82843i 3.00000i 1.00000i 0
616.2 1.00000i 0.707107 0.707107i 1.00000 0 −0.707107 0.707107i −2.82843 2.82843i 3.00000i 1.00000i 0
829.1 1.00000i −0.707107 0.707107i 1.00000 0 0.707107 0.707107i 2.82843 2.82843i 3.00000i 1.00000i 0
829.2 1.00000i 0.707107 + 0.707107i 1.00000 0 −0.707107 + 0.707107i −2.82843 + 2.82843i 3.00000i 1.00000i 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
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.d 4
17.b even 2 1 inner 867.2.e.d 4
17.c even 4 2 inner 867.2.e.d 4
17.d even 8 2 51.2.d.b 2
17.d even 8 1 867.2.a.a 1
17.d even 8 1 867.2.a.b 1
17.e odd 16 8 867.2.h.d 8
51.g odd 8 2 153.2.d.a 2
51.g odd 8 1 2601.2.a.i 1
51.g odd 8 1 2601.2.a.j 1
68.g odd 8 2 816.2.c.c 2
85.k odd 8 1 1275.2.d.b 2
85.k odd 8 1 1275.2.d.d 2
85.m even 8 2 1275.2.g.a 2
85.n odd 8 1 1275.2.d.b 2
85.n odd 8 1 1275.2.d.d 2
136.o even 8 2 3264.2.c.e 2
136.p odd 8 2 3264.2.c.d 2
204.p even 8 2 2448.2.c.j 2

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$1 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$256 + T^{4}$$
$11$ $$256 + T^{4}$$
$13$ $$( 2 + T )^{4}$$
$17$ $$T^{4}$$
$19$ $$( 16 + T^{2} )^{2}$$
$23$ $$256 + T^{4}$$
$29$ $$T^{4}$$
$31$ $$256 + T^{4}$$
$37$ $$4096 + T^{4}$$
$41$ $$4096 + T^{4}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$( -8 + T )^{4}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( 144 + T^{2} )^{2}$$
$61$ $$4096 + T^{4}$$
$67$ $$( -12 + T )^{4}$$
$71$ $$20736 + T^{4}$$
$73$ $$T^{4}$$
$79$ $$256 + T^{4}$$
$83$ $$( 144 + T^{2} )^{2}$$
$89$ $$( -10 + T )^{4}$$
$97$ $$65536 + T^{4}$$