Properties

Label 912.1.ce.b
Level $912$
Weight $1$
Character orbit 912.ce
Analytic conductor $0.455$
Analytic rank $0$
Dimension $6$
Projective image $D_{18}$
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.ce (of order \(18\), degree \(6\), 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: yes
Projective image: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} + \cdots)\)

$q$-expansion

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
−0.766044 0.642788i
−0.766044 + 0.642788i
0.939693 + 0.342020i
0.939693 0.342020i
−0.173648 0.984808i
−0.173648 + 0.984808i
0 0.766044 0.642788i 0 0 0 −1.11334 0.642788i 0 0.173648 0.984808i 0
287.1 0 0.766044 + 0.642788i 0 0 0 −1.11334 + 0.642788i 0 0.173648 + 0.984808i 0
383.1 0 −0.939693 + 0.342020i 0 0 0 −0.592396 + 0.342020i 0 0.766044 0.642788i 0
431.1 0 −0.939693 0.342020i 0 0 0 −0.592396 0.342020i 0 0.766044 + 0.642788i 0
527.1 0 0.173648 0.984808i 0 0 0 1.70574 0.984808i 0 −0.939693 0.342020i 0
623.1 0 0.173648 + 0.984808i 0 0 0 1.70574 + 0.984808i 0 −0.939693 + 0.342020i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 623.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
76.k even 18 1 inner
228.u 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.ce.b yes 6
3.b odd 2 1 CM 912.1.ce.b yes 6
4.b odd 2 1 912.1.ce.a 6
8.b even 2 1 3648.1.cu.a 6
8.d odd 2 1 3648.1.cu.b 6
12.b even 2 1 912.1.ce.a 6
19.f odd 18 1 912.1.ce.a 6
24.f even 2 1 3648.1.cu.b 6
24.h odd 2 1 3648.1.cu.a 6
57.j even 18 1 912.1.ce.a 6
76.k even 18 1 inner 912.1.ce.b yes 6
152.s odd 18 1 3648.1.cu.b 6
152.v even 18 1 3648.1.cu.a 6
228.u odd 18 1 inner 912.1.ce.b yes 6
456.bj even 18 1 3648.1.cu.b 6
456.bt odd 18 1 3648.1.cu.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.1.ce.a 6 4.b odd 2 1
912.1.ce.a 6 12.b even 2 1
912.1.ce.a 6 19.f odd 18 1
912.1.ce.a 6 57.j even 18 1
912.1.ce.b yes 6 1.a even 1 1 trivial
912.1.ce.b yes 6 3.b odd 2 1 CM
912.1.ce.b yes 6 76.k even 18 1 inner
912.1.ce.b yes 6 228.u odd 18 1 inner
3648.1.cu.a 6 8.b even 2 1
3648.1.cu.a 6 24.h odd 2 1
3648.1.cu.a 6 152.v even 18 1
3648.1.cu.a 6 456.bt odd 18 1
3648.1.cu.b 6 8.d odd 2 1
3648.1.cu.b 6 24.f even 2 1
3648.1.cu.b 6 152.s odd 18 1
3648.1.cu.b 6 456.bj even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 3 T_{7}^{4} + 9 T_{7}^{2} + 9 T_{7} + 3 \) acting on \(S_{1}^{\mathrm{new}}(912, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 1 + T^{3} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 3 + 9 T + 9 T^{2} - 3 T^{4} + T^{6} \)
$11$ \( T^{6} \)
$13$ \( 3 + 6 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} \)
$17$ \( T^{6} \)
$19$ \( ( 1 - T + T^{2} )^{3} \)
$23$ \( T^{6} \)
$29$ \( T^{6} \)
$31$ \( 1 - 3 T + 9 T^{2} - 2 T^{3} + 3 T^{4} + T^{6} \)
$37$ \( 3 + 9 T^{2} + 6 T^{4} + T^{6} \)
$41$ \( T^{6} \)
$43$ \( 3 + 6 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} \)
$47$ \( T^{6} \)
$53$ \( T^{6} \)
$59$ \( T^{6} \)
$61$ \( 1 + 3 T + 12 T^{2} + 19 T^{3} + 15 T^{4} + 6 T^{5} + T^{6} \)
$67$ \( 1 - 6 T + 12 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6} \)
$71$ \( T^{6} \)
$73$ \( 1 - 6 T + 12 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6} \)
$79$ \( 1 - 3 T + 12 T^{2} - 19 T^{3} + 15 T^{4} - 6 T^{5} + T^{6} \)
$83$ \( T^{6} \)
$89$ \( T^{6} \)
$97$ \( 27 + 9 T^{3} + T^{6} \)
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