# Properties

 Label 876.1.br.a Level $876$ Weight $1$ Character orbit 876.br Analytic conductor $0.437$ Analytic rank $0$ Dimension $12$ Projective image $D_{36}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$876 = 2^{2} \cdot 3 \cdot 73$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 876.br (of order $$36$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.437180951067$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{36})$$ Defining polynomial: $$x^{12} - x^{6} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{36}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{36} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{36}^{15} q^{3} + ( \zeta_{36}^{10} + \zeta_{36}^{11} ) q^{7} -\zeta_{36}^{12} q^{9} +O(q^{10})$$ $$q -\zeta_{36}^{15} q^{3} + ( \zeta_{36}^{10} + \zeta_{36}^{11} ) q^{7} -\zeta_{36}^{12} q^{9} + ( \zeta_{36}^{2} - \zeta_{36}^{11} ) q^{13} + ( -\zeta_{36}^{13} + \zeta_{36}^{15} ) q^{19} + ( \zeta_{36}^{7} + \zeta_{36}^{8} ) q^{21} + \zeta_{36}^{17} q^{25} -\zeta_{36}^{9} q^{27} + ( \zeta_{36} + \zeta_{36}^{6} ) q^{31} + ( -\zeta_{36} - \zeta_{36}^{7} ) q^{37} + ( -\zeta_{36}^{8} - \zeta_{36}^{17} ) q^{39} + ( -\zeta_{36}^{5} + \zeta_{36}^{16} ) q^{43} + ( -\zeta_{36}^{2} - \zeta_{36}^{3} - \zeta_{36}^{4} ) q^{49} + ( -\zeta_{36}^{10} + \zeta_{36}^{12} ) q^{57} + ( -1 - \zeta_{36}^{14} ) q^{61} + ( \zeta_{36}^{4} + \zeta_{36}^{5} ) q^{63} + ( -\zeta_{36}^{6} - \zeta_{36}^{16} ) q^{67} + \zeta_{36}^{9} q^{73} + \zeta_{36}^{14} q^{75} + ( \zeta_{36}^{9} + \zeta_{36}^{13} ) q^{79} -\zeta_{36}^{6} q^{81} + ( \zeta_{36}^{3} + \zeta_{36}^{4} + \zeta_{36}^{12} + \zeta_{36}^{13} ) q^{91} + ( \zeta_{36}^{3} - \zeta_{36}^{16} ) q^{93} + ( -\zeta_{36}^{10} + \zeta_{36}^{14} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 6q^{9} + O(q^{10})$$ $$12q + 6q^{9} + 6q^{31} - 6q^{57} - 12q^{61} - 6q^{67} - 6q^{81} - 6q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/876\mathbb{Z}\right)^\times$$.

 $$n$$ $$293$$ $$439$$ $$589$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\zeta_{36}^{17}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 −0.984808 − 0.173648i −0.642788 + 0.766044i −0.984808 + 0.173648i −0.342020 − 0.939693i 0.342020 + 0.939693i 0.984808 − 0.173648i 0.642788 − 0.766044i 0.984808 + 0.173648i −0.642788 − 0.766044i 0.342020 − 0.939693i −0.342020 + 0.939693i 0.642788 + 0.766044i
0 −0.866025 + 0.500000i 0 0 0 0.168372 + 0.0451151i 0 0.500000 0.866025i 0
269.1 0 0.866025 0.500000i 0 0 0 0.218763 0.816436i 0 0.500000 0.866025i 0
317.1 0 −0.866025 0.500000i 0 0 0 0.168372 0.0451151i 0 0.500000 + 0.866025i 0
353.1 0 0.866025 0.500000i 0 0 0 0.296905 1.10806i 0 0.500000 0.866025i 0
377.1 0 −0.866025 + 0.500000i 0 0 0 1.58248 + 0.424024i 0 0.500000 0.866025i 0
413.1 0 0.866025 + 0.500000i 0 0 0 −0.515668 1.92450i 0 0.500000 + 0.866025i 0
461.1 0 −0.866025 + 0.500000i 0 0 0 −1.75085 0.469139i 0 0.500000 0.866025i 0
473.1 0 0.866025 0.500000i 0 0 0 −0.515668 + 1.92450i 0 0.500000 0.866025i 0
749.1 0 0.866025 + 0.500000i 0 0 0 0.218763 + 0.816436i 0 0.500000 + 0.866025i 0
797.1 0 −0.866025 0.500000i 0 0 0 1.58248 0.424024i 0 0.500000 + 0.866025i 0
809.1 0 0.866025 + 0.500000i 0 0 0 0.296905 + 1.10806i 0 0.500000 + 0.866025i 0
857.1 0 −0.866025 0.500000i 0 0 0 −1.75085 + 0.469139i 0 0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 857.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
73.k even 36 1 inner
219.v odd 36 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 876.1.br.a 12
3.b odd 2 1 CM 876.1.br.a 12
4.b odd 2 1 3504.1.fm.a 12
12.b even 2 1 3504.1.fm.a 12
73.k even 36 1 inner 876.1.br.a 12
219.v odd 36 1 inner 876.1.br.a 12
292.u odd 36 1 3504.1.fm.a 12
876.bq even 36 1 3504.1.fm.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
876.1.br.a 12 1.a even 1 1 trivial
876.1.br.a 12 3.b odd 2 1 CM
876.1.br.a 12 73.k even 36 1 inner
876.1.br.a 12 219.v odd 36 1 inner
3504.1.fm.a 12 4.b odd 2 1
3504.1.fm.a 12 12.b even 2 1
3504.1.fm.a 12 292.u odd 36 1
3504.1.fm.a 12 876.bq even 36 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(876, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$( 1 - T^{2} + T^{4} )^{3}$$
$5$ $$T^{12}$$
$7$ $$1 - 12 T + 45 T^{2} - 52 T^{3} + 69 T^{4} - 18 T^{5} + 2 T^{6} + 12 T^{7} - 9 T^{8} - 2 T^{9} + T^{12}$$
$11$ $$T^{12}$$
$13$ $$64 + 32 T^{3} + 8 T^{6} + 4 T^{9} + T^{12}$$
$17$ $$T^{12}$$
$19$ $$1 + 3 T^{2} + 69 T^{4} + 8 T^{6} - 6 T^{8} - 3 T^{10} + T^{12}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$1 - 12 T + 51 T^{2} - 70 T^{3} + 75 T^{4} - 114 T^{5} + 140 T^{6} - 126 T^{7} + 90 T^{8} - 50 T^{9} + 21 T^{10} - 6 T^{11} + T^{12}$$
$37$ $$729 + 27 T^{6} + T^{12}$$
$41$ $$T^{12}$$
$43$ $$1 + 12 T + 45 T^{2} + 52 T^{3} + 69 T^{4} + 18 T^{5} + 2 T^{6} - 12 T^{7} - 9 T^{8} + 2 T^{9} + T^{12}$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$( 3 + 9 T + 18 T^{2} + 21 T^{3} + 15 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$67$ $$( 3 + 9 T + 9 T^{2} + 6 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} )^{2}$$
$71$ $$T^{12}$$
$73$ $$( 1 + T^{2} )^{6}$$
$79$ $$1 - 15 T^{2} + 60 T^{4} + 35 T^{6} + 21 T^{8} + 6 T^{10} + T^{12}$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$( 3 + 9 T + 9 T^{2} - 3 T^{4} + T^{6} )^{2}$$