Properties

Label 117.1.j.a
Level 117
Weight 1
Character orbit 117.j
Analytic conductor 0.058
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 117.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0583906064781\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.6591.1
Artin image $C_4\wr C_2$
Artin field Galois closure of 8.0.1601613.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + i q^{4} + ( -1 - i ) q^{7} +O(q^{10})\) \( q + i q^{4} + ( -1 - i ) q^{7} -i q^{13} - q^{16} + ( -1 + i ) q^{19} + i q^{25} + ( 1 - i ) q^{28} + ( 1 - i ) q^{31} + ( 1 + i ) q^{37} + i q^{49} + q^{52} -i q^{64} + ( -1 + i ) q^{67} + ( -1 - i ) q^{73} + ( -1 - i ) q^{76} + ( -1 + i ) q^{91} + ( 1 - i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} + O(q^{10}) \) \( 2q - 2q^{7} - 2q^{16} - 2q^{19} + 2q^{28} + 2q^{31} + 2q^{37} + 2q^{52} - 2q^{67} - 2q^{73} - 2q^{76} - 2q^{91} + 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-i\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
1.00000i
1.00000i
0 0 1.00000i 0 0 −1.00000 + 1.00000i 0 0 0
109.1 0 0 1.00000i 0 0 −1.00000 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.1.j.a 2
3.b odd 2 1 CM 117.1.j.a 2
4.b odd 2 1 1872.1.bd.a 2
5.b even 2 1 2925.1.s.a 2
5.c odd 4 1 2925.1.t.a 2
5.c odd 4 1 2925.1.t.b 2
9.c even 3 2 1053.1.bb.a 4
9.d odd 6 2 1053.1.bb.a 4
12.b even 2 1 1872.1.bd.a 2
13.b even 2 1 1521.1.j.b 2
13.c even 3 2 1521.1.bd.c 4
13.d odd 4 1 inner 117.1.j.a 2
13.d odd 4 1 1521.1.j.b 2
13.e even 6 2 1521.1.bd.b 4
13.f odd 12 2 1521.1.bd.b 4
13.f odd 12 2 1521.1.bd.c 4
15.d odd 2 1 2925.1.s.a 2
15.e even 4 1 2925.1.t.a 2
15.e even 4 1 2925.1.t.b 2
39.d odd 2 1 1521.1.j.b 2
39.f even 4 1 inner 117.1.j.a 2
39.f even 4 1 1521.1.j.b 2
39.h odd 6 2 1521.1.bd.b 4
39.i odd 6 2 1521.1.bd.c 4
39.k even 12 2 1521.1.bd.b 4
39.k even 12 2 1521.1.bd.c 4
52.f even 4 1 1872.1.bd.a 2
65.f even 4 1 2925.1.t.a 2
65.g odd 4 1 2925.1.s.a 2
65.k even 4 1 2925.1.t.b 2
117.y odd 12 2 1053.1.bb.a 4
117.z even 12 2 1053.1.bb.a 4
156.l odd 4 1 1872.1.bd.a 2
195.j odd 4 1 2925.1.t.b 2
195.n even 4 1 2925.1.s.a 2
195.u odd 4 1 2925.1.t.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.1.j.a 2 1.a even 1 1 trivial
117.1.j.a 2 3.b odd 2 1 CM
117.1.j.a 2 13.d odd 4 1 inner
117.1.j.a 2 39.f even 4 1 inner
1053.1.bb.a 4 9.c even 3 2
1053.1.bb.a 4 9.d odd 6 2
1053.1.bb.a 4 117.y odd 12 2
1053.1.bb.a 4 117.z even 12 2
1521.1.j.b 2 13.b even 2 1
1521.1.j.b 2 13.d odd 4 1
1521.1.j.b 2 39.d odd 2 1
1521.1.j.b 2 39.f even 4 1
1521.1.bd.b 4 13.e even 6 2
1521.1.bd.b 4 13.f odd 12 2
1521.1.bd.b 4 39.h odd 6 2
1521.1.bd.b 4 39.k even 12 2
1521.1.bd.c 4 13.c even 3 2
1521.1.bd.c 4 13.f odd 12 2
1521.1.bd.c 4 39.i odd 6 2
1521.1.bd.c 4 39.k even 12 2
1872.1.bd.a 2 4.b odd 2 1
1872.1.bd.a 2 12.b even 2 1
1872.1.bd.a 2 52.f even 4 1
1872.1.bd.a 2 156.l odd 4 1
2925.1.s.a 2 5.b even 2 1
2925.1.s.a 2 15.d odd 2 1
2925.1.s.a 2 65.g odd 4 1
2925.1.s.a 2 195.n even 4 1
2925.1.t.a 2 5.c odd 4 1
2925.1.t.a 2 15.e even 4 1
2925.1.t.a 2 65.f even 4 1
2925.1.t.a 2 195.u odd 4 1
2925.1.t.b 2 5.c odd 4 1
2925.1.t.b 2 15.e even 4 1
2925.1.t.b 2 65.k even 4 1
2925.1.t.b 2 195.j odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(117, [\chi])\).

Hecke characteristic polynomials

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