# Properties

 Label 855.2.be.a Level $855$ Weight $2$ Character orbit 855.be Analytic conductor $6.827$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$855 = 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 855.be (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.82720937282$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12} q^{2} + 2 \zeta_{12}^{2} q^{4} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{5} -4 \zeta_{12}^{3} q^{7} +O(q^{10})$$ $$q + 2 \zeta_{12} q^{2} + 2 \zeta_{12}^{2} q^{4} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{5} -4 \zeta_{12}^{3} q^{7} + ( -2 \zeta_{12} - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{10} + q^{11} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{13} + ( 8 - 8 \zeta_{12}^{2} ) q^{14} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} -2 \zeta_{12} q^{17} + ( 2 + 3 \zeta_{12}^{2} ) q^{19} + ( -2 - 4 \zeta_{12}^{3} ) q^{20} + 2 \zeta_{12} q^{22} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{23} + ( 4 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{25} + 4 q^{26} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{28} -9 \zeta_{12}^{2} q^{29} -7 q^{31} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{32} -4 \zeta_{12}^{2} q^{34} + ( -8 + 4 \zeta_{12} + 8 \zeta_{12}^{2} ) q^{35} + 2 \zeta_{12}^{3} q^{37} + ( 4 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{38} + ( 2 - 2 \zeta_{12}^{2} ) q^{41} + 2 \zeta_{12} q^{43} + 2 \zeta_{12}^{2} q^{44} + 12 q^{46} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{47} -9 q^{49} + ( 8 + 6 \zeta_{12}^{3} ) q^{50} + 4 \zeta_{12} q^{52} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{53} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{55} -18 \zeta_{12}^{3} q^{58} + ( -9 + 9 \zeta_{12}^{2} ) q^{59} + 7 \zeta_{12}^{2} q^{61} -14 \zeta_{12} q^{62} + 8 q^{64} + ( -4 + 2 \zeta_{12}^{3} ) q^{65} + ( -10 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{67} -4 \zeta_{12}^{3} q^{68} + ( -16 \zeta_{12} + 8 \zeta_{12}^{2} + 16 \zeta_{12}^{3} ) q^{70} + ( 1 - \zeta_{12}^{2} ) q^{71} + 10 \zeta_{12} q^{73} + ( -4 + 4 \zeta_{12}^{2} ) q^{74} + ( -6 + 10 \zeta_{12}^{2} ) q^{76} -4 \zeta_{12}^{3} q^{77} + ( 1 - \zeta_{12}^{2} ) q^{79} + ( -8 \zeta_{12} + 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{80} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{82} -6 \zeta_{12}^{3} q^{83} + ( 2 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{85} + 4 \zeta_{12}^{2} q^{86} + 11 \zeta_{12}^{2} q^{89} -8 \zeta_{12}^{2} q^{91} + 12 \zeta_{12} q^{92} -12 q^{94} + ( -5 - 4 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{95} + 6 \zeta_{12} q^{97} -18 \zeta_{12} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} - 2 q^{5} + O(q^{10})$$ $$4 q + 4 q^{4} - 2 q^{5} - 8 q^{10} + 4 q^{11} + 16 q^{14} + 8 q^{16} + 14 q^{19} - 8 q^{20} + 6 q^{25} + 16 q^{26} - 18 q^{29} - 28 q^{31} - 8 q^{34} - 16 q^{35} + 4 q^{41} + 4 q^{44} + 48 q^{46} - 36 q^{49} + 32 q^{50} - 2 q^{55} - 18 q^{59} + 14 q^{61} + 32 q^{64} - 16 q^{65} + 16 q^{70} + 2 q^{71} - 8 q^{74} - 4 q^{76} + 2 q^{79} + 8 q^{80} + 8 q^{85} + 8 q^{86} + 22 q^{89} - 16 q^{91} - 48 q^{94} - 16 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$$.

 $$n$$ $$172$$ $$191$$ $$496$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−1.73205 + 1.00000i 0 1.00000 1.73205i 1.23205 1.86603i 0 4.00000i 0 0 −0.267949 + 4.46410i
64.2 1.73205 1.00000i 0 1.00000 1.73205i −2.23205 + 0.133975i 0 4.00000i 0 0 −3.73205 + 2.46410i
334.1 −1.73205 1.00000i 0 1.00000 + 1.73205i 1.23205 + 1.86603i 0 4.00000i 0 0 −0.267949 4.46410i
334.2 1.73205 + 1.00000i 0 1.00000 + 1.73205i −2.23205 0.133975i 0 4.00000i 0 0 −3.73205 2.46410i
 $$n$$: 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
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.be.a 4
3.b odd 2 1 95.2.i.a 4
5.b even 2 1 inner 855.2.be.a 4
15.d odd 2 1 95.2.i.a 4
15.e even 4 1 475.2.e.a 2
15.e even 4 1 475.2.e.c 2
19.c even 3 1 inner 855.2.be.a 4
57.f even 6 1 1805.2.b.a 2
57.h odd 6 1 95.2.i.a 4
57.h odd 6 1 1805.2.b.b 2
95.i even 6 1 inner 855.2.be.a 4
285.n odd 6 1 95.2.i.a 4
285.n odd 6 1 1805.2.b.b 2
285.q even 6 1 1805.2.b.a 2
285.v even 12 1 475.2.e.a 2
285.v even 12 1 475.2.e.c 2
285.v even 12 1 9025.2.a.b 1
285.v even 12 1 9025.2.a.i 1
285.w odd 12 1 9025.2.a.a 1
285.w odd 12 1 9025.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.a 4 3.b odd 2 1
95.2.i.a 4 15.d odd 2 1
95.2.i.a 4 57.h odd 6 1
95.2.i.a 4 285.n odd 6 1
475.2.e.a 2 15.e even 4 1
475.2.e.a 2 285.v even 12 1
475.2.e.c 2 15.e even 4 1
475.2.e.c 2 285.v even 12 1
855.2.be.a 4 1.a even 1 1 trivial
855.2.be.a 4 5.b even 2 1 inner
855.2.be.a 4 19.c even 3 1 inner
855.2.be.a 4 95.i even 6 1 inner
1805.2.b.a 2 57.f even 6 1
1805.2.b.a 2 285.q even 6 1
1805.2.b.b 2 57.h odd 6 1
1805.2.b.b 2 285.n odd 6 1
9025.2.a.a 1 285.w odd 12 1
9025.2.a.b 1 285.v even 12 1
9025.2.a.i 1 285.v even 12 1
9025.2.a.j 1 285.w odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 4 T_{2}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(855, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 4 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 + 10 T - T^{2} + 2 T^{3} + T^{4}$$
$7$ $$( 16 + T^{2} )^{2}$$
$11$ $$( -1 + T )^{4}$$
$13$ $$16 - 4 T^{2} + T^{4}$$
$17$ $$16 - 4 T^{2} + T^{4}$$
$19$ $$( 19 - 7 T + T^{2} )^{2}$$
$23$ $$1296 - 36 T^{2} + T^{4}$$
$29$ $$( 81 + 9 T + T^{2} )^{2}$$
$31$ $$( 7 + T )^{4}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( 4 - 2 T + T^{2} )^{2}$$
$43$ $$16 - 4 T^{2} + T^{4}$$
$47$ $$1296 - 36 T^{2} + T^{4}$$
$53$ $$256 - 16 T^{2} + T^{4}$$
$59$ $$( 81 + 9 T + T^{2} )^{2}$$
$61$ $$( 49 - 7 T + T^{2} )^{2}$$
$67$ $$10000 - 100 T^{2} + T^{4}$$
$71$ $$( 1 - T + T^{2} )^{2}$$
$73$ $$10000 - 100 T^{2} + T^{4}$$
$79$ $$( 1 - T + T^{2} )^{2}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$( 121 - 11 T + T^{2} )^{2}$$
$97$ $$1296 - 36 T^{2} + T^{4}$$