Properties

Label 111.1.i.a
Level 111
Weight 1
Character orbit 111.i
Analytic conductor 0.055
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM disc. -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 111 = 3 \cdot 37 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 111.i (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0553962164023\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.4107.1
Artin image size \(18\)
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.36963.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{7} + \zeta_{6}^{2} q^{9} + \zeta_{6}^{2} q^{12} + \zeta_{6} q^{13} + \zeta_{6}^{2} q^{16} -2 \zeta_{6} q^{19} -\zeta_{6}^{2} q^{21} + \zeta_{6}^{2} q^{25} + q^{27} -\zeta_{6}^{2} q^{28} - q^{31} + q^{36} + q^{37} -\zeta_{6}^{2} q^{39} - q^{43} + q^{48} -\zeta_{6}^{2} q^{52} + 2 \zeta_{6}^{2} q^{57} -2 \zeta_{6} q^{61} - q^{63} + q^{64} + \zeta_{6} q^{67} - q^{73} + q^{75} + 2 \zeta_{6}^{2} q^{76} + \zeta_{6} q^{79} -\zeta_{6} q^{81} - q^{84} + \zeta_{6}^{2} q^{91} + \zeta_{6} q^{93} - q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - q^{4} + q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} - q^{4} + q^{7} - q^{9} - q^{12} + q^{13} - q^{16} - 2q^{19} + q^{21} - q^{25} + 2q^{27} + q^{28} - 2q^{31} + 2q^{36} + 2q^{37} + q^{39} - 2q^{43} + 2q^{48} + q^{52} - 2q^{57} - 2q^{61} - 2q^{63} + 2q^{64} + q^{67} - 2q^{73} + 2q^{75} - 2q^{76} + q^{79} - q^{81} - 2q^{84} - q^{91} + q^{93} - 2q^{97} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).

\(n\) \(38\) \(76\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 0 0.500000 0.866025i 0 −0.500000 0.866025i 0
47.1 0 −0.500000 0.866025i −0.500000 0.866025i 0 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 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
37.c Even 1 yes
111.i Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(111, [\chi])\).