# Properties

 Label 3087.1.bj.a Level $3087$ Weight $1$ Character orbit 3087.bj Analytic conductor $1.541$ Analytic rank $0$ Dimension $12$ Projective image $D_{14}$ CM discriminant -3 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3087 = 3^{2} \cdot 7^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3087.bj (of order $$42$$, degree $$12$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.54061369400$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{21})$$ Defining polynomial: $$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$$ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 441) Projective image: $$D_{14}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{14} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{42}^{16} q^{4} +O(q^{10})$$ q - z^16 * q^4 $$q - \zeta_{42}^{16} q^{4} + (\zeta_{42}^{18} - \zeta_{42}^{12}) q^{13} - \zeta_{42}^{11} q^{16} + (\zeta_{42}^{20} - \zeta_{42}^{8}) q^{19} - \zeta_{42}^{13} q^{25} + ( - \zeta_{42}^{19} - \zeta_{42}^{16}) q^{31} + (\zeta_{42}^{17} - \zeta_{42}^{14}) q^{37} + ( - \zeta_{42}^{12} + \zeta_{42}^{3}) q^{43} + (\zeta_{42}^{13} - \zeta_{42}^{7}) q^{52} + (\zeta_{42}^{17} + \zeta_{42}^{14}) q^{61} - \zeta_{42}^{6} q^{64} + (\zeta_{42}^{10} + \zeta_{42}^{4}) q^{67} + (\zeta_{42}^{4} + \zeta_{42}) q^{73} + (\zeta_{42}^{15} - \zeta_{42}^{3}) q^{76} + ( - \zeta_{42}^{5} + \zeta_{42}^{2}) q^{79} + ( - \zeta_{42}^{15} - \zeta_{42}^{6}) q^{97} +O(q^{100})$$ q - z^16 * q^4 + (z^18 - z^12) * q^13 - z^11 * q^16 + (z^20 - z^8) * q^19 - z^13 * q^25 + (-z^19 - z^16) * q^31 + (z^17 - z^14) * q^37 + (-z^12 + z^3) * q^43 + (z^13 - z^7) * q^52 + (z^17 + z^14) * q^61 - z^6 * q^64 + (z^10 + z^4) * q^67 + (z^4 + z) * q^73 + (z^15 - z^3) * q^76 + (-z^5 + z^2) * q^79 + (-z^15 - z^6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - q^{4}+O(q^{10})$$ 12 * q - q^4 $$12 q - q^{4} + q^{16} + q^{25} + 5 q^{37} + 4 q^{43} - 7 q^{52} - 7 q^{61} + 2 q^{64} + 2 q^{67} + 2 q^{79}+O(q^{100})$$ 12 * q - q^4 + q^16 + q^25 + 5 * q^37 + 4 * q^43 - 7 * q^52 - 7 * q^61 + 2 * q^64 + 2 * q^67 + 2 * q^79

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3087\mathbb{Z}\right)^\times$$.

 $$n$$ $$344$$ $$2404$$ $$\chi(n)$$ $$1$$ $$-\zeta_{42}^{20}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
460.1
 −0.988831 − 0.149042i 0.0747301 − 0.997204i −0.733052 + 0.680173i 0.365341 + 0.930874i 0.0747301 + 0.997204i −0.733052 − 0.680173i 0.826239 + 0.563320i 0.955573 + 0.294755i 0.955573 − 0.294755i −0.988831 + 0.149042i 0.365341 − 0.930874i 0.826239 − 0.563320i
0 0 0.733052 0.680173i 0 0 0 0 0 0
766.1 0 0 −0.365341 0.930874i 0 0 0 0 0 0
901.1 0 0 −0.826239 0.563320i 0 0 0 0 0 0
1207.1 0 0 −0.955573 0.294755i 0 0 0 0 0 0
1342.1 0 0 −0.365341 + 0.930874i 0 0 0 0 0 0
1648.1 0 0 −0.826239 + 0.563320i 0 0 0 0 0 0
1783.1 0 0 0.988831 + 0.149042i 0 0 0 0 0 0
2089.1 0 0 −0.0747301 + 0.997204i 0 0 0 0 0 0
2224.1 0 0 −0.0747301 0.997204i 0 0 0 0 0 0
2530.1 0 0 0.733052 + 0.680173i 0 0 0 0 0 0
2665.1 0 0 −0.955573 + 0.294755i 0 0 0 0 0 0
2971.1 0 0 0.988831 0.149042i 0 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2971.1 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 $$\Q(\sqrt{-3})$$
7.c even 3 1 inner
21.h odd 6 1 inner
49.f odd 14 1 inner
49.h odd 42 1 inner
147.k even 14 1 inner
147.o even 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3087.1.bj.a 12
3.b odd 2 1 CM 3087.1.bj.a 12
7.b odd 2 1 3087.1.bj.b 12
7.c even 3 1 3087.1.v.a 6
7.c even 3 1 inner 3087.1.bj.a 12
7.d odd 6 1 441.1.v.a 6
7.d odd 6 1 3087.1.bj.b 12
21.c even 2 1 3087.1.bj.b 12
21.g even 6 1 441.1.v.a 6
21.g even 6 1 3087.1.bj.b 12
21.h odd 6 1 3087.1.v.a 6
21.h odd 6 1 inner 3087.1.bj.a 12
49.e even 7 1 3087.1.bj.b 12
49.f odd 14 1 inner 3087.1.bj.a 12
49.g even 21 1 441.1.v.a 6
49.g even 21 1 3087.1.bj.b 12
49.h odd 42 1 3087.1.v.a 6
49.h odd 42 1 inner 3087.1.bj.a 12
63.i even 6 1 3969.1.bz.a 12
63.k odd 6 1 3969.1.bz.a 12
63.s even 6 1 3969.1.bz.a 12
63.t odd 6 1 3969.1.bz.a 12
147.k even 14 1 inner 3087.1.bj.a 12
147.l odd 14 1 3087.1.bj.b 12
147.n odd 42 1 441.1.v.a 6
147.n odd 42 1 3087.1.bj.b 12
147.o even 42 1 3087.1.v.a 6
147.o even 42 1 inner 3087.1.bj.a 12
441.y even 21 1 3969.1.bz.a 12
441.z even 21 1 3969.1.bz.a 12
441.bi odd 42 1 3969.1.bz.a 12
441.bm odd 42 1 3969.1.bz.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.1.v.a 6 7.d odd 6 1
441.1.v.a 6 21.g even 6 1
441.1.v.a 6 49.g even 21 1
441.1.v.a 6 147.n odd 42 1
3087.1.v.a 6 7.c even 3 1
3087.1.v.a 6 21.h odd 6 1
3087.1.v.a 6 49.h odd 42 1
3087.1.v.a 6 147.o even 42 1
3087.1.bj.a 12 1.a even 1 1 trivial
3087.1.bj.a 12 3.b odd 2 1 CM
3087.1.bj.a 12 7.c even 3 1 inner
3087.1.bj.a 12 21.h odd 6 1 inner
3087.1.bj.a 12 49.f odd 14 1 inner
3087.1.bj.a 12 49.h odd 42 1 inner
3087.1.bj.a 12 147.k even 14 1 inner
3087.1.bj.a 12 147.o even 42 1 inner
3087.1.bj.b 12 7.b odd 2 1
3087.1.bj.b 12 7.d odd 6 1
3087.1.bj.b 12 21.c even 2 1
3087.1.bj.b 12 21.g even 6 1
3087.1.bj.b 12 49.e even 7 1
3087.1.bj.b 12 49.g even 21 1
3087.1.bj.b 12 147.l odd 14 1
3087.1.bj.b 12 147.n odd 42 1
3969.1.bz.a 12 63.i even 6 1
3969.1.bz.a 12 63.k odd 6 1
3969.1.bz.a 12 63.s even 6 1
3969.1.bz.a 12 63.t odd 6 1
3969.1.bz.a 12 441.y even 21 1
3969.1.bz.a 12 441.z even 21 1
3969.1.bz.a 12 441.bi odd 42 1
3969.1.bz.a 12 441.bm odd 42 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{6} - 7T_{13}^{3} + 7T_{13} + 7$$ acting on $$S_{1}^{\mathrm{new}}(3087, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$T^{12}$$
$7$ $$T^{12}$$
$11$ $$T^{12}$$
$13$ $$(T^{6} - 7 T^{3} + 7 T + 7)^{2}$$
$17$ $$T^{12}$$
$19$ $$T^{12} - 7 T^{10} + 35 T^{8} - 84 T^{6} + \cdots + 49$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$T^{12} - 7 T^{10} + 35 T^{8} - 84 T^{6} + \cdots + 49$$
$37$ $$T^{12} - 5 T^{11} + 14 T^{10} - 29 T^{9} + \cdots + 1$$
$41$ $$T^{12}$$
$43$ $$(T^{6} - 2 T^{5} + 4 T^{4} - 8 T^{3} + 9 T^{2} + \cdots + 1)^{2}$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$T^{12} + 7 T^{11} + 28 T^{10} + 77 T^{9} + \cdots + 49$$
$67$ $$(T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1)^{2}$$
$71$ $$T^{12}$$
$73$ $$T^{12} - 7 T^{8} + 14 T^{7} + 14 T^{6} + \cdots + 49$$
$79$ $$(T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1)^{2}$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$(T^{6} + 7 T^{4} + 14 T^{2} + 7)^{2}$$