# Properties

 Label 867.2.a.a Level $867$ Weight $2$ Character orbit 867.a Self dual yes Analytic conductor $6.923$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$867 = 3 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 867.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.92302985525$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 51) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 −1.00000 −1.00000 0 1.00000 −4.00000 3.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$17$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.a.a 1
3.b odd 2 1 2601.2.a.i 1
17.b even 2 1 867.2.a.b 1
17.c even 4 2 51.2.d.b 2
17.d even 8 4 867.2.e.d 4
17.e odd 16 8 867.2.h.d 8
51.c odd 2 1 2601.2.a.j 1
51.f odd 4 2 153.2.d.a 2
68.f odd 4 2 816.2.c.c 2
85.f odd 4 2 1275.2.d.d 2
85.i odd 4 2 1275.2.d.b 2
85.j even 4 2 1275.2.g.a 2
136.i even 4 2 3264.2.c.e 2
136.j odd 4 2 3264.2.c.d 2
204.l even 4 2 2448.2.c.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b 2 17.c even 4 2
153.2.d.a 2 51.f odd 4 2
816.2.c.c 2 68.f odd 4 2
867.2.a.a 1 1.a even 1 1 trivial
867.2.a.b 1 17.b even 2 1
867.2.e.d 4 17.d even 8 4
867.2.h.d 8 17.e odd 16 8
1275.2.d.b 2 85.i odd 4 2
1275.2.d.d 2 85.f odd 4 2
1275.2.g.a 2 85.j even 4 2
2448.2.c.j 2 204.l even 4 2
2601.2.a.i 1 3.b odd 2 1
2601.2.a.j 1 51.c odd 2 1
3264.2.c.d 2 136.j odd 4 2
3264.2.c.e 2 136.i even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(867))$$:

 $$T_{2} + 1$$ $$T_{5}$$ $$T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$4 + T$$
$11$ $$-4 + T$$
$13$ $$-2 + T$$
$17$ $$T$$
$19$ $$-4 + T$$
$23$ $$4 + T$$
$29$ $$T$$
$31$ $$-4 + T$$
$37$ $$8 + T$$
$41$ $$8 + T$$
$43$ $$-4 + T$$
$47$ $$8 + T$$
$53$ $$6 + T$$
$59$ $$-12 + T$$
$61$ $$8 + T$$
$67$ $$-12 + T$$
$71$ $$12 + T$$
$73$ $$T$$
$79$ $$4 + T$$
$83$ $$12 + T$$
$89$ $$10 + T$$
$97$ $$16 + T$$