Properties

Label 912.1.cb.a
Level $912$
Weight $1$
Character orbit 912.cb
Analytic conductor $0.455$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 912.cb (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(6\)
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: no (minimal twist has level 228)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.88042790804544.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{18}^{8} q^{3} + ( -\zeta_{18}^{2} - \zeta_{18}^{4} ) q^{7} -\zeta_{18}^{7} q^{9} +O(q^{10})\) \( q -\zeta_{18}^{8} q^{3} + ( -\zeta_{18}^{2} - \zeta_{18}^{4} ) q^{7} -\zeta_{18}^{7} q^{9} + ( -\zeta_{18}^{5} + \zeta_{18}^{6} ) q^{13} + \zeta_{18}^{3} q^{19} + ( -\zeta_{18} - \zeta_{18}^{3} ) q^{21} -\zeta_{18} q^{25} -\zeta_{18}^{6} q^{27} + ( \zeta_{18} + \zeta_{18}^{5} ) q^{31} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{37} + ( -\zeta_{18}^{4} + \zeta_{18}^{5} ) q^{39} + ( \zeta_{18}^{3} + \zeta_{18}^{7} ) q^{43} + ( \zeta_{18}^{4} + \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{49} + \zeta_{18}^{2} q^{57} + ( 1 + \zeta_{18}^{8} ) q^{61} + ( -1 - \zeta_{18}^{2} ) q^{63} + ( -\zeta_{18}^{6} - \zeta_{18}^{8} ) q^{67} + ( -\zeta_{18}^{3} + \zeta_{18}^{4} ) q^{73} - q^{75} + ( -1 + \zeta_{18}^{7} ) q^{79} -\zeta_{18}^{5} q^{81} + ( -1 + \zeta_{18} + \zeta_{18}^{7} - \zeta_{18}^{8} ) q^{91} + ( 1 + \zeta_{18}^{4} ) q^{93} -\zeta_{18}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + O(q^{10}) \) \( 6q - 3q^{13} + 3q^{19} - 3q^{21} + 3q^{27} + 3q^{43} - 3q^{49} + 6q^{61} - 6q^{63} + 3q^{67} - 3q^{73} - 6q^{75} - 6q^{79} - 6q^{91} + 6q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(\zeta_{18}^{4}\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.766044 + 0.642788i
−0.766044 0.642788i
−0.173648 + 0.984808i
−0.173648 0.984808i
0.939693 0.342020i
0.939693 + 0.342020i
0 −0.766044 0.642788i 0 0 0 0.766044 + 1.32683i 0 0.173648 + 0.984808i 0
161.1 0 −0.766044 + 0.642788i 0 0 0 0.766044 1.32683i 0 0.173648 0.984808i 0
593.1 0 −0.173648 0.984808i 0 0 0 0.173648 0.300767i 0 −0.939693 + 0.342020i 0
689.1 0 −0.173648 + 0.984808i 0 0 0 0.173648 + 0.300767i 0 −0.939693 0.342020i 0
785.1 0 0.939693 + 0.342020i 0 0 0 −0.939693 + 1.62760i 0 0.766044 + 0.642788i 0
833.1 0 0.939693 0.342020i 0 0 0 −0.939693 1.62760i 0 0.766044 0.642788i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 833.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
19.e even 9 1 inner
57.l odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.1.cb.a 6
3.b odd 2 1 CM 912.1.cb.a 6
4.b odd 2 1 228.1.s.a 6
8.b even 2 1 3648.1.cr.a 6
8.d odd 2 1 3648.1.cr.b 6
12.b even 2 1 228.1.s.a 6
19.e even 9 1 inner 912.1.cb.a 6
24.f even 2 1 3648.1.cr.b 6
24.h odd 2 1 3648.1.cr.a 6
57.l odd 18 1 inner 912.1.cb.a 6
76.l odd 18 1 228.1.s.a 6
152.t even 18 1 3648.1.cr.a 6
152.u odd 18 1 3648.1.cr.b 6
228.v even 18 1 228.1.s.a 6
456.bh odd 18 1 3648.1.cr.a 6
456.bu even 18 1 3648.1.cr.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.1.s.a 6 4.b odd 2 1
228.1.s.a 6 12.b even 2 1
228.1.s.a 6 76.l odd 18 1
228.1.s.a 6 228.v even 18 1
912.1.cb.a 6 1.a even 1 1 trivial
912.1.cb.a 6 3.b odd 2 1 CM
912.1.cb.a 6 19.e even 9 1 inner
912.1.cb.a 6 57.l odd 18 1 inner
3648.1.cr.a 6 8.b even 2 1
3648.1.cr.a 6 24.h odd 2 1
3648.1.cr.a 6 152.t even 18 1
3648.1.cr.a 6 456.bh odd 18 1
3648.1.cr.b 6 8.d odd 2 1
3648.1.cr.b 6 24.f even 2 1
3648.1.cr.b 6 152.u odd 18 1
3648.1.cr.b 6 456.bu even 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(912, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 1 - T^{3} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 1 - 3 T + 9 T^{2} - 2 T^{3} + 3 T^{4} + T^{6} \)
$11$ \( T^{6} \)
$13$ \( 1 + 6 T + 12 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} \)
$17$ \( T^{6} \)
$19$ \( ( 1 - T + T^{2} )^{3} \)
$23$ \( T^{6} \)
$29$ \( T^{6} \)
$31$ \( 1 - 3 T + 9 T^{2} - 2 T^{3} + 3 T^{4} + T^{6} \)
$37$ \( ( 1 - 3 T + T^{3} )^{2} \)
$41$ \( T^{6} \)
$43$ \( 1 - 6 T + 12 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6} \)
$47$ \( T^{6} \)
$53$ \( T^{6} \)
$59$ \( T^{6} \)
$61$ \( 1 - 3 T + 12 T^{2} - 19 T^{3} + 15 T^{4} - 6 T^{5} + T^{6} \)
$67$ \( 1 - 6 T + 12 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6} \)
$71$ \( T^{6} \)
$73$ \( 1 + 6 T + 12 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} \)
$79$ \( 1 + 3 T + 12 T^{2} + 19 T^{3} + 15 T^{4} + 6 T^{5} + T^{6} \)
$83$ \( T^{6} \)
$89$ \( T^{6} \)
$97$ \( 1 - T^{3} + T^{6} \)
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