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 disc. -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 \(4\) and degree \(2\))

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: \( 1 \)
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.6591.1
Artin image size \(32\)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes
13.d Odd 1 yes
39.f Even 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(117, [\chi])\).