Properties

 Label 2151.1.f.a Level $2151$ Weight $1$ Character orbit 2151.f Analytic conductor $1.073$ Analytic rank $0$ Dimension $6$ Projective image $D_{9}$ CM discriminant -239 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2151 = 3^{2} \cdot 239$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2151.f (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$1.07348884217$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: yes Projective image $$D_{9}$$ Projective field Galois closure of 9.1.1733990286981681.1

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{2} -\zeta_{18}^{3} q^{3} + ( -\zeta_{18}^{3} - \zeta_{18}^{7} + \zeta_{18}^{8} ) q^{4} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{5} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{6} + ( \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{6} - \zeta_{18}^{7} ) q^{8} + \zeta_{18}^{6} q^{9} +O(q^{10})$$ $$q + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{2} -\zeta_{18}^{3} q^{3} + ( -\zeta_{18}^{3} - \zeta_{18}^{7} + \zeta_{18}^{8} ) q^{4} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{5} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{6} + ( \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{6} - \zeta_{18}^{7} ) q^{8} + \zeta_{18}^{6} q^{9} + ( -\zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{10} + ( -\zeta_{18}^{5} - \zeta_{18}^{7} ) q^{11} + ( -\zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{6} ) q^{12} + ( -\zeta_{18}^{5} - \zeta_{18}^{7} ) q^{15} + ( -\zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{5} + \zeta_{18}^{6} - \zeta_{18}^{7} ) q^{16} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{17} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{18} + ( 1 - \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - \zeta_{18}^{5} - \zeta_{18}^{7} ) q^{20} + ( 1 + \zeta_{18}^{2} + \zeta_{18}^{4} + \zeta_{18}^{6} ) q^{22} + ( 1 - \zeta_{18} - \zeta_{18}^{5} + \zeta_{18}^{6} ) q^{24} + ( \zeta_{18}^{4} + \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{25} + q^{27} + ( -\zeta_{18}^{5} - \zeta_{18}^{7} ) q^{29} + ( 1 + \zeta_{18}^{2} + \zeta_{18}^{4} + \zeta_{18}^{6} ) q^{30} + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{31} + ( 1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - \zeta_{18}^{5} + \zeta_{18}^{6} ) q^{32} + ( -\zeta_{18} + \zeta_{18}^{8} ) q^{33} + ( -\zeta_{18} + \zeta_{18}^{2} + 2 \zeta_{18}^{6} ) q^{34} + ( 1 + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{36} + ( 1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - \zeta_{18}^{5} + \zeta_{18}^{6} - \zeta_{18}^{7} + \zeta_{18}^{8} ) q^{40} + ( -\zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} + \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{44} + ( -\zeta_{18} + \zeta_{18}^{8} ) q^{45} + ( 1 - \zeta_{18} + \zeta_{18}^{4} - \zeta_{18}^{5} + \zeta_{18}^{8} ) q^{48} -\zeta_{18}^{3} q^{49} + ( -\zeta_{18} - 2 \zeta_{18}^{3} - \zeta_{18}^{5} - \zeta_{18}^{7} + \zeta_{18}^{8} ) q^{50} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{51} + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{54} + ( 2 + \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{55} + ( 1 + \zeta_{18}^{2} + \zeta_{18}^{4} + \zeta_{18}^{6} ) q^{58} + ( -\zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} + \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{60} + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{61} + ( \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{6} - \zeta_{18}^{7} ) q^{62} + ( 1 - \zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} + \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{64} + ( 1 - \zeta_{18}^{3} - \zeta_{18}^{5} - \zeta_{18}^{7} ) q^{66} + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{67} + ( 1 - 2 \zeta_{18} - 2 \zeta_{18}^{5} + \zeta_{18}^{6} ) q^{68} + 2 q^{71} + ( 1 - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{72} + ( 1 + \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{75} + ( 2 - \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} + \zeta_{18}^{6} - \zeta_{18}^{7} + \zeta_{18}^{8} ) q^{80} -\zeta_{18}^{3} q^{81} + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{83} + ( 1 + \zeta_{18}^{2} + \zeta_{18}^{4} + \zeta_{18}^{6} ) q^{85} + ( -\zeta_{18} + \zeta_{18}^{8} ) q^{87} + ( 1 - \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} - \zeta_{18}^{7} + \zeta_{18}^{8} ) q^{88} + ( 1 - \zeta_{18}^{3} - \zeta_{18}^{5} - \zeta_{18}^{7} ) q^{90} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{93} + ( 1 - \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} - \zeta_{18}^{7} + \zeta_{18}^{8} ) q^{96} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{98} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 3q^{3} - 3q^{4} - 6q^{8} - 3q^{9} + O(q^{10})$$ $$6q - 3q^{3} - 3q^{4} - 6q^{8} - 3q^{9} - 6q^{10} - 3q^{12} - 3q^{16} + 3q^{20} + 3q^{22} + 3q^{24} - 3q^{25} + 6q^{27} + 3q^{30} + 3q^{32} - 6q^{34} + 6q^{36} + 3q^{40} - 6q^{44} + 6q^{48} - 3q^{49} - 6q^{50} + 12q^{55} + 3q^{58} - 6q^{60} - 6q^{62} + 3q^{66} + 3q^{68} + 12q^{71} + 3q^{72} + 6q^{75} + 6q^{80} - 3q^{81} + 3q^{85} + 3q^{88} + 3q^{90} + 3q^{96} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times$$.

 $$n$$ $$479$$ $$1441$$ $$\chi(n)$$ $$-\zeta_{18}^{3}$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
238.1
 0.939693 − 0.342020i −0.766044 − 0.642788i −0.173648 + 0.984808i 0.939693 + 0.342020i −0.766044 + 0.642788i −0.173648 − 0.984808i
−0.766044 1.32683i −0.500000 + 0.866025i −0.673648 + 1.16679i 0.939693 1.62760i 1.53209 0 0.532089 −0.500000 0.866025i −2.87939
238.2 −0.173648 0.300767i −0.500000 + 0.866025i 0.439693 0.761570i −0.766044 + 1.32683i 0.347296 0 −0.652704 −0.500000 0.866025i 0.532089
238.3 0.939693 + 1.62760i −0.500000 + 0.866025i −1.26604 + 2.19285i −0.173648 + 0.300767i −1.87939 0 −2.87939 −0.500000 0.866025i −0.652704
1672.1 −0.766044 + 1.32683i −0.500000 0.866025i −0.673648 1.16679i 0.939693 + 1.62760i 1.53209 0 0.532089 −0.500000 + 0.866025i −2.87939
1672.2 −0.173648 + 0.300767i −0.500000 0.866025i 0.439693 + 0.761570i −0.766044 1.32683i 0.347296 0 −0.652704 −0.500000 + 0.866025i 0.532089
1672.3 0.939693 1.62760i −0.500000 0.866025i −1.26604 2.19285i −0.173648 0.300767i −1.87939 0 −2.87939 −0.500000 + 0.866025i −0.652704
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1672.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
239.b odd 2 1 CM by $$\Q(\sqrt{-239})$$
9.c even 3 1 inner
2151.f odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2151.1.f.a 6
9.c even 3 1 inner 2151.1.f.a 6
239.b odd 2 1 CM 2151.1.f.a 6
2151.f odd 6 1 inner 2151.1.f.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2151.1.f.a 6 1.a even 1 1 trivial
2151.1.f.a 6 9.c even 3 1 inner
2151.1.f.a 6 239.b odd 2 1 CM
2151.1.f.a 6 2151.f odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 3 T_{2}^{4} + 2 T_{2}^{3} + 9 T_{2}^{2} + 3 T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2151, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$3$ $$( 1 + T + T^{2} )^{3}$$
$5$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$( 1 - 3 T + T^{3} )^{2}$$
$19$ $$T^{6}$$
$23$ $$T^{6}$$
$29$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$31$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$37$ $$T^{6}$$
$41$ $$T^{6}$$
$43$ $$T^{6}$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$67$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$71$ $$( -2 + T )^{6}$$
$73$ $$T^{6}$$
$79$ $$T^{6}$$
$83$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$89$ $$T^{6}$$
$97$ $$T^{6}$$