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 $4$

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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:

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

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