# Properties

 Label 855.2.c.d Level $855$ Weight $2$ Character orbit 855.c Analytic conductor $6.827$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$855 = 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 855.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.82720937282$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.16516096.1 Defining polynomial: $$x^{6} + 9 x^{4} + 13 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( -\beta_{3} + \beta_{5} ) q^{5} + ( \beta_{3} + \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{3} + \beta_{4} ) q^{8} +O(q^{10})$$ $$q -\beta_{5} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( -\beta_{3} + \beta_{5} ) q^{5} + ( \beta_{3} + \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{3} + \beta_{4} ) q^{8} + ( 2 - 2 \beta_{1} - \beta_{5} ) q^{10} + ( \beta_{3} - \beta_{4} ) q^{11} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{13} + ( 2 - 2 \beta_{2} ) q^{14} + ( -1 - \beta_{3} + \beta_{4} ) q^{16} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{17} + q^{19} + ( -1 + \beta_{2} - 2 \beta_{4} ) q^{20} + ( 4 \beta_{1} + 2 \beta_{5} ) q^{22} + 2 \beta_{1} q^{23} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{25} + ( -2 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{26} + ( -2 \beta_{1} - 4 \beta_{5} ) q^{28} + 6 q^{29} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{31} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{32} + ( -2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{34} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} ) q^{35} + ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{37} -\beta_{5} q^{38} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{40} + ( -2 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{41} + ( \beta_{3} + \beta_{4} ) q^{43} + ( 2 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{44} + ( -2 - 2 \beta_{3} + 2 \beta_{4} ) q^{46} + ( 4 \beta_{1} - \beta_{3} - \beta_{4} + 4 \beta_{5} ) q^{47} + ( 1 + \beta_{3} - \beta_{4} ) q^{49} + ( -4 \beta_{3} - \beta_{5} ) q^{50} + ( -3 \beta_{1} + \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{52} + ( -\beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{53} + ( -5 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{55} + ( -6 + 2 \beta_{3} - 2 \beta_{4} ) q^{56} -6 \beta_{5} q^{58} + ( 4 + 2 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{61} + ( -6 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{62} + ( 1 - 3 \beta_{2} ) q^{64} + ( 4 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{65} + ( 3 \beta_{1} - 5 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} ) q^{67} + ( -6 \beta_{1} - 6 \beta_{5} ) q^{68} + ( 2 - 2 \beta_{2} + 4 \beta_{4} ) q^{70} + ( -8 + 2 \beta_{2} ) q^{71} + ( -2 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{73} + ( -6 + 3 \beta_{2} - \beta_{3} + \beta_{4} ) q^{74} + ( -1 + \beta_{2} ) q^{76} + ( -4 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{77} + ( 4 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{79} + ( 5 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{80} + ( -6 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{82} + ( -2 \beta_{1} + 4 \beta_{5} ) q^{83} + ( 3 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{85} + ( 2 - 2 \beta_{2} ) q^{86} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{88} + ( 2 - 8 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{89} + ( -4 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{91} + ( -4 \beta_{1} - 2 \beta_{5} ) q^{92} + ( 6 - 2 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} ) q^{94} + ( -\beta_{3} + \beta_{5} ) q^{95} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{97} + ( 4 \beta_{1} + \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 8 q^{4} + q^{5} + O(q^{10})$$ $$6 q - 8 q^{4} + q^{5} + 12 q^{10} - 2 q^{11} + 16 q^{14} - 4 q^{16} + 6 q^{19} - 10 q^{20} + 3 q^{25} - 8 q^{26} + 36 q^{29} + 8 q^{34} + 3 q^{35} + 8 q^{40} - 12 q^{41} + 20 q^{44} - 8 q^{46} + 4 q^{49} + 4 q^{50} - 33 q^{55} - 40 q^{56} + 20 q^{59} - 14 q^{61} + 12 q^{64} + 20 q^{65} + 20 q^{70} - 52 q^{71} - 40 q^{74} - 8 q^{76} + 24 q^{79} + 32 q^{80} + 13 q^{85} + 16 q^{86} + 24 q^{89} - 24 q^{91} + 48 q^{94} + q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 9 x^{4} + 13 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} + 8 \nu^{2} + 5$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + \nu^{4} + 10 \nu^{3} + 6 \nu^{2} + 19 \nu - 1$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} + 10 \nu^{3} - 6 \nu^{2} + 19 \nu + 1$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} - 8 \nu^{3} - 7 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} + \beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} - 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-8 \beta_{4} + 8 \beta_{3} - 6 \beta_{2} + 19$$ $$\nu^{5}$$ $$=$$ $$-10 \beta_{5} - 8 \beta_{4} - 8 \beta_{3} + 41 \beta_{1}$$

## Character values

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

 $$n$$ $$172$$ $$191$$ $$496$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
514.1
 1.30397i − 2.68667i − 0.285442i 0.285442i 2.68667i − 1.30397i
2.41987i 0 −3.85577 2.07772 + 0.826491i 0 3.18676i 4.49073i 0 2.00000 5.02781i
514.2 1.82254i 0 −1.32164 −1.94827 + 1.09737i 0 1.45033i 1.23634i 0 2.00000 + 3.55080i
514.3 0.906968i 0 1.17741 0.370556 + 2.20515i 0 2.59637i 2.88181i 0 2.00000 0.336083i
514.4 0.906968i 0 1.17741 0.370556 2.20515i 0 2.59637i 2.88181i 0 2.00000 + 0.336083i
514.5 1.82254i 0 −1.32164 −1.94827 1.09737i 0 1.45033i 1.23634i 0 2.00000 3.55080i
514.6 2.41987i 0 −3.85577 2.07772 0.826491i 0 3.18676i 4.49073i 0 2.00000 + 5.02781i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 514.6 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.c.d 6
3.b odd 2 1 95.2.b.b 6
5.b even 2 1 inner 855.2.c.d 6
5.c odd 4 2 4275.2.a.br 6
12.b even 2 1 1520.2.d.h 6
15.d odd 2 1 95.2.b.b 6
15.e even 4 2 475.2.a.j 6
57.d even 2 1 1805.2.b.e 6
60.h even 2 1 1520.2.d.h 6
60.l odd 4 2 7600.2.a.ck 6
285.b even 2 1 1805.2.b.e 6
285.j odd 4 2 9025.2.a.bx 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.b 6 3.b odd 2 1
95.2.b.b 6 15.d odd 2 1
475.2.a.j 6 15.e even 4 2
855.2.c.d 6 1.a even 1 1 trivial
855.2.c.d 6 5.b even 2 1 inner
1520.2.d.h 6 12.b even 2 1
1520.2.d.h 6 60.h even 2 1
1805.2.b.e 6 57.d even 2 1
1805.2.b.e 6 285.b even 2 1
4275.2.a.br 6 5.c odd 4 2
7600.2.a.ck 6 60.l odd 4 2
9025.2.a.bx 6 285.j odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(855, [\chi])$$:

 $$T_{2}^{6} + 10 T_{2}^{4} + 27 T_{2}^{2} + 16$$ $$T_{11}^{3} + T_{11}^{2} - 16 T_{11} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 27 T^{2} + 10 T^{4} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$125 - 25 T - 5 T^{2} + 2 T^{3} - T^{4} - T^{5} + T^{6}$$
$7$ $$144 + 104 T^{2} + 19 T^{4} + T^{6}$$
$11$ $$( -12 - 16 T + T^{2} + T^{3} )^{2}$$
$13$ $$576 + 236 T^{2} + 28 T^{4} + T^{6}$$
$17$ $$5184 + 1008 T^{2} + 59 T^{4} + T^{6}$$
$19$ $$( -1 + T )^{6}$$
$23$ $$64 + 208 T^{2} + 36 T^{4} + T^{6}$$
$29$ $$( -6 + T )^{6}$$
$31$ $$( 128 - 56 T + T^{3} )^{2}$$
$37$ $$1296 + 764 T^{2} + 56 T^{4} + T^{6}$$
$41$ $$( 24 - 44 T + 6 T^{2} + T^{3} )^{2}$$
$43$ $$144 + 104 T^{2} + 19 T^{4} + T^{6}$$
$47$ $$85264 + 7464 T^{2} + 187 T^{4} + T^{6}$$
$53$ $$64 + 2476 T^{2} + 156 T^{4} + T^{6}$$
$59$ $$( 48 + 8 T - 10 T^{2} + T^{3} )^{2}$$
$61$ $$( -776 - 104 T + 7 T^{2} + T^{3} )^{2}$$
$67$ $$484416 + 28556 T^{2} + 340 T^{4} + T^{6}$$
$71$ $$( 432 + 200 T + 26 T^{2} + T^{3} )^{2}$$
$73$ $$5184 + 1616 T^{2} + 131 T^{4} + T^{6}$$
$79$ $$( 32 - 8 T - 12 T^{2} + T^{3} )^{2}$$
$83$ $$141376 + 11728 T^{2} + 228 T^{4} + T^{6}$$
$89$ $$( 3456 - 284 T - 12 T^{2} + T^{3} )^{2}$$
$97$ $$576 + 236 T^{2} + 28 T^{4} + T^{6}$$