Properties

Label 936.1.bs.a
Level $936$
Weight $1$
Character orbit 936.bs
Analytic conductor $0.467$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -104
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 936.bs (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.467124851824\)
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: \( 1 \)
Twist minimal: yes
Projective image \(D_{9}\)
Projective field Galois closure of 9.1.62171080298496.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{18}^{3} q^{2} + \zeta_{18}^{8} q^{3} + \zeta_{18}^{6} q^{4} + ( -\zeta_{18}^{5} - \zeta_{18}^{7} ) q^{5} + \zeta_{18}^{2} q^{6} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{7} + q^{8} -\zeta_{18}^{7} q^{9} +O(q^{10})\) \( q -\zeta_{18}^{3} q^{2} + \zeta_{18}^{8} q^{3} + \zeta_{18}^{6} q^{4} + ( -\zeta_{18}^{5} - \zeta_{18}^{7} ) q^{5} + \zeta_{18}^{2} q^{6} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{7} + q^{8} -\zeta_{18}^{7} q^{9} + ( -\zeta_{18} + \zeta_{18}^{8} ) q^{10} -\zeta_{18}^{5} q^{12} + \zeta_{18}^{6} q^{13} + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{14} + ( \zeta_{18}^{4} + \zeta_{18}^{6} ) q^{15} -\zeta_{18}^{3} q^{16} + ( \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{17} -\zeta_{18} q^{18} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{20} + ( 1 + \zeta_{18}^{4} ) q^{21} + \zeta_{18}^{8} q^{24} + ( -\zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{25} + q^{26} + \zeta_{18}^{6} q^{27} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{28} + ( 1 - \zeta_{18}^{7} ) q^{30} -\zeta_{18}^{6} q^{31} + \zeta_{18}^{6} q^{32} + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{34} + ( -\zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{35} + \zeta_{18}^{4} q^{36} + ( -\zeta_{18} + \zeta_{18}^{8} ) q^{37} -\zeta_{18}^{5} q^{39} + ( -\zeta_{18}^{5} - \zeta_{18}^{7} ) q^{40} + ( -\zeta_{18}^{3} - \zeta_{18}^{7} ) q^{42} + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{43} + ( -\zeta_{18}^{3} - \zeta_{18}^{5} ) q^{45} + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{47} + \zeta_{18}^{2} q^{48} + ( -\zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{6} ) q^{49} + ( \zeta_{18}^{4} + \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{50} + ( -\zeta_{18}^{3} + \zeta_{18}^{4} ) q^{51} -\zeta_{18}^{3} q^{52} + q^{54} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{56} + ( -\zeta_{18} - \zeta_{18}^{3} ) q^{60} - q^{62} + ( -\zeta_{18}^{3} + \zeta_{18}^{8} ) q^{63} + q^{64} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{65} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{68} + ( 1 + \zeta_{18}^{2} + \zeta_{18}^{4} + \zeta_{18}^{6} ) q^{70} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{71} -\zeta_{18}^{7} q^{72} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{74} + ( 1 + \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{75} + \zeta_{18}^{8} q^{78} + ( -\zeta_{18} + \zeta_{18}^{8} ) q^{80} -\zeta_{18}^{5} q^{81} + ( -\zeta_{18} + \zeta_{18}^{6} ) q^{84} + ( 1 - \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{85} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{86} + ( \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{90} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{91} + \zeta_{18}^{5} q^{93} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{94} -\zeta_{18}^{5} q^{96} + ( 1 + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 3q^{2} - 3q^{4} + 6q^{8} + O(q^{10}) \) \( 6q - 3q^{2} - 3q^{4} + 6q^{8} - 3q^{13} - 3q^{15} - 3q^{16} + 6q^{21} - 3q^{25} + 6q^{26} - 3q^{27} + 6q^{30} + 3q^{31} - 3q^{32} - 6q^{35} - 3q^{42} - 3q^{45} - 3q^{49} - 3q^{50} - 3q^{51} - 3q^{52} + 6q^{54} - 3q^{60} - 6q^{62} - 3q^{63} + 6q^{64} + 3q^{70} + 6q^{75} - 3q^{84} + 3q^{85} - 3q^{90} + 6q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(-1\) \(\zeta_{18}^{6}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
259.1
0.939693 0.342020i
−0.173648 + 0.984808i
−0.766044 0.642788i
0.939693 + 0.342020i
−0.173648 0.984808i
−0.766044 + 0.642788i
−0.500000 + 0.866025i −0.939693 0.342020i −0.500000 0.866025i 0.939693 + 1.62760i 0.766044 0.642788i −0.766044 + 1.32683i 1.00000 0.766044 + 0.642788i −1.87939
259.2 −0.500000 + 0.866025i 0.173648 + 0.984808i −0.500000 0.866025i −0.173648 0.300767i −0.939693 0.342020i 0.939693 1.62760i 1.00000 −0.939693 + 0.342020i 0.347296
259.3 −0.500000 + 0.866025i 0.766044 0.642788i −0.500000 0.866025i −0.766044 1.32683i 0.173648 + 0.984808i −0.173648 + 0.300767i 1.00000 0.173648 0.984808i 1.53209
571.1 −0.500000 0.866025i −0.939693 + 0.342020i −0.500000 + 0.866025i 0.939693 1.62760i 0.766044 + 0.642788i −0.766044 1.32683i 1.00000 0.766044 0.642788i −1.87939
571.2 −0.500000 0.866025i 0.173648 0.984808i −0.500000 + 0.866025i −0.173648 + 0.300767i −0.939693 + 0.342020i 0.939693 + 1.62760i 1.00000 −0.939693 0.342020i 0.347296
571.3 −0.500000 0.866025i 0.766044 + 0.642788i −0.500000 + 0.866025i −0.766044 + 1.32683i 0.173648 0.984808i −0.173648 0.300767i 1.00000 0.173648 + 0.984808i 1.53209
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 571.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
104.h odd 2 1 CM by \(\Q(\sqrt{-26}) \)
9.c even 3 1 inner
936.bs odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.1.bs.a 6
3.b odd 2 1 2808.1.bs.b 6
4.b odd 2 1 3744.1.ci.a 6
8.b even 2 1 3744.1.ci.b 6
8.d odd 2 1 936.1.bs.b yes 6
9.c even 3 1 inner 936.1.bs.a 6
9.d odd 6 1 2808.1.bs.b 6
13.b even 2 1 936.1.bs.b yes 6
24.f even 2 1 2808.1.bs.a 6
36.f odd 6 1 3744.1.ci.a 6
39.d odd 2 1 2808.1.bs.a 6
52.b odd 2 1 3744.1.ci.b 6
72.l even 6 1 2808.1.bs.a 6
72.n even 6 1 3744.1.ci.b 6
72.p odd 6 1 936.1.bs.b yes 6
104.e even 2 1 3744.1.ci.a 6
104.h odd 2 1 CM 936.1.bs.a 6
117.n odd 6 1 2808.1.bs.a 6
117.t even 6 1 936.1.bs.b yes 6
312.h even 2 1 2808.1.bs.b 6
468.bg odd 6 1 3744.1.ci.b 6
936.bs odd 6 1 inner 936.1.bs.a 6
936.bx even 6 1 3744.1.ci.a 6
936.cl even 6 1 2808.1.bs.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.1.bs.a 6 1.a even 1 1 trivial
936.1.bs.a 6 9.c even 3 1 inner
936.1.bs.a 6 104.h odd 2 1 CM
936.1.bs.a 6 936.bs odd 6 1 inner
936.1.bs.b yes 6 8.d odd 2 1
936.1.bs.b yes 6 13.b even 2 1
936.1.bs.b yes 6 72.p odd 6 1
936.1.bs.b yes 6 117.t even 6 1
2808.1.bs.a 6 24.f even 2 1
2808.1.bs.a 6 39.d odd 2 1
2808.1.bs.a 6 72.l even 6 1
2808.1.bs.a 6 117.n odd 6 1
2808.1.bs.b 6 3.b odd 2 1
2808.1.bs.b 6 9.d odd 6 1
2808.1.bs.b 6 312.h even 2 1
2808.1.bs.b 6 936.cl even 6 1
3744.1.ci.a 6 4.b odd 2 1
3744.1.ci.a 6 36.f odd 6 1
3744.1.ci.a 6 104.e even 2 1
3744.1.ci.a 6 936.bx even 6 1
3744.1.ci.b 6 8.b even 2 1
3744.1.ci.b 6 52.b odd 2 1
3744.1.ci.b 6 72.n even 6 1
3744.1.ci.b 6 468.bg odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 3 T_{5}^{4} + 2 T_{5}^{3} + 9 T_{5}^{2} + 3 T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(936, [\chi])\).

Hecke characteristic polynomials

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