Properties

Label 1216.1.bi.a
Level $1216$
Weight $1$
Character orbit 1216.bi
Analytic conductor $0.607$
Analytic rank $0$
Dimension $12$
Projective image $D_{18}$
CM discriminant -8
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1216.bi (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.606863055362\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
Defining polynomial: \(x^{12} - x^{6} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + ( \zeta_{36}^{13} + \zeta_{36}^{15} ) q^{3} + ( -\zeta_{36}^{8} - \zeta_{36}^{10} - \zeta_{36}^{12} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{36}^{13} + \zeta_{36}^{15} ) q^{3} + ( -\zeta_{36}^{8} - \zeta_{36}^{10} - \zeta_{36}^{12} ) q^{9} + ( \zeta_{36} - \zeta_{36}^{11} ) q^{11} -\zeta_{36}^{8} q^{17} -\zeta_{36}^{7} q^{19} + \zeta_{36}^{4} q^{25} + ( \zeta_{36}^{3} + \zeta_{36}^{5} + \zeta_{36}^{7} + \zeta_{36}^{9} ) q^{27} + ( \zeta_{36}^{6} + \zeta_{36}^{8} + \zeta_{36}^{14} + \zeta_{36}^{16} ) q^{33} + ( \zeta_{36}^{12} - \zeta_{36}^{16} ) q^{41} -\zeta_{36}^{11} q^{43} + \zeta_{36}^{6} q^{49} + ( \zeta_{36}^{3} + \zeta_{36}^{5} ) q^{51} + ( \zeta_{36}^{2} + \zeta_{36}^{4} ) q^{57} + ( -\zeta_{36}^{3} - \zeta_{36}^{13} ) q^{59} + ( \zeta_{36}^{3} + \zeta_{36}^{17} ) q^{67} + ( -1 + \zeta_{36}^{10} ) q^{73} + ( -\zeta_{36} + \zeta_{36}^{17} ) q^{75} + ( -1 - \zeta_{36}^{2} - \zeta_{36}^{4} - \zeta_{36}^{6} + \zeta_{36}^{16} ) q^{81} + ( \zeta_{36} + \zeta_{36}^{5} ) q^{83} + ( -\zeta_{36}^{10} - \zeta_{36}^{16} ) q^{89} + ( -\zeta_{36}^{6} + \zeta_{36}^{10} ) q^{97} + ( -\zeta_{36} - \zeta_{36}^{3} - \zeta_{36}^{5} - \zeta_{36}^{9} - \zeta_{36}^{11} - \zeta_{36}^{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 6q^{9} + O(q^{10}) \) \( 12q + 6q^{9} + 6q^{33} - 6q^{41} + 6q^{49} - 12q^{73} - 18q^{81} - 6q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(-\zeta_{36}^{16}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
0.342020 0.939693i
−0.342020 + 0.939693i
0.984808 0.173648i
−0.984808 + 0.173648i
−0.642788 + 0.766044i
0.642788 0.766044i
0.342020 + 0.939693i
−0.342020 0.939693i
−0.642788 0.766044i
0.642788 + 0.766044i
0.984808 + 0.173648i
−0.984808 0.173648i
0 −0.118782 + 0.673648i 0 0 0 0 0 0.500000 + 0.181985i 0
33.2 0 0.118782 0.673648i 0 0 0 0 0 0.500000 + 0.181985i 0
97.1 0 −1.50881 1.26604i 0 0 0 0 0 0.500000 + 2.83564i 0
97.2 0 1.50881 + 1.26604i 0 0 0 0 0 0.500000 + 2.83564i 0
545.1 0 −1.20805 0.439693i 0 0 0 0 0 0.500000 + 0.419550i 0
545.2 0 1.20805 + 0.439693i 0 0 0 0 0 0.500000 + 0.419550i 0
737.1 0 −0.118782 0.673648i 0 0 0 0 0 0.500000 0.181985i 0
737.2 0 0.118782 + 0.673648i 0 0 0 0 0 0.500000 0.181985i 0
801.1 0 −1.20805 + 0.439693i 0 0 0 0 0 0.500000 0.419550i 0
801.2 0 1.20805 0.439693i 0 0 0 0 0 0.500000 0.419550i 0
865.1 0 −1.50881 + 1.26604i 0 0 0 0 0 0.500000 2.83564i 0
865.2 0 1.50881 1.26604i 0 0 0 0 0 0.500000 2.83564i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 865.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
19.f odd 18 1 inner
76.k even 18 1 inner
152.s odd 18 1 inner
152.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.1.bi.a 12
4.b odd 2 1 inner 1216.1.bi.a 12
8.b even 2 1 inner 1216.1.bi.a 12
8.d odd 2 1 CM 1216.1.bi.a 12
19.f odd 18 1 inner 1216.1.bi.a 12
76.k even 18 1 inner 1216.1.bi.a 12
152.s odd 18 1 inner 1216.1.bi.a 12
152.v even 18 1 inner 1216.1.bi.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.1.bi.a 12 1.a even 1 1 trivial
1216.1.bi.a 12 4.b odd 2 1 inner
1216.1.bi.a 12 8.b even 2 1 inner
1216.1.bi.a 12 8.d odd 2 1 CM
1216.1.bi.a 12 19.f odd 18 1 inner
1216.1.bi.a 12 76.k even 18 1 inner
1216.1.bi.a 12 152.s odd 18 1 inner
1216.1.bi.a 12 152.v even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 9 + 27 T^{2} + 9 T^{4} - 24 T^{6} + 18 T^{8} - 3 T^{10} + T^{12} \)
$5$ \( T^{12} \)
$7$ \( T^{12} \)
$11$ \( 1 - 9 T^{2} + 75 T^{4} - 52 T^{6} + 27 T^{8} - 6 T^{10} + T^{12} \)
$13$ \( T^{12} \)
$17$ \( ( 1 - T^{3} + T^{6} )^{2} \)
$19$ \( 1 - T^{6} + T^{12} \)
$23$ \( T^{12} \)
$29$ \( T^{12} \)
$31$ \( T^{12} \)
$37$ \( T^{12} \)
$41$ \( ( 3 + 6 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$43$ \( 1 - T^{6} + T^{12} \)
$47$ \( T^{12} \)
$53$ \( T^{12} \)
$59$ \( 9 + 36 T^{4} + 30 T^{6} - 3 T^{10} + T^{12} \)
$61$ \( T^{12} \)
$67$ \( 9 + 27 T^{2} + 9 T^{4} - 24 T^{6} + 18 T^{8} - 3 T^{10} + T^{12} \)
$71$ \( T^{12} \)
$73$ \( ( 1 + 3 T + 12 T^{2} + 19 T^{3} + 15 T^{4} + 6 T^{5} + T^{6} )^{2} \)
$79$ \( T^{12} \)
$83$ \( 1 - 9 T^{2} + 75 T^{4} - 52 T^{6} + 27 T^{8} - 6 T^{10} + T^{12} \)
$89$ \( ( 27 + 9 T^{3} + T^{6} )^{2} \)
$97$ \( ( 3 + 9 T + 9 T^{2} + 6 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} )^{2} \)
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