Properties

Label 111.1.h.a
Level 111
Weight 1
Character orbit 111.h
Analytic conductor 0.055
Analytic rank 0
Dimension 2
Projective image \(D_{6}\)
CM disc. -3
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 111 = 3 \cdot 37 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 111.h (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 \(D_{6}\)
Projective field Galois closure of 6.2.624095613.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\zeta_{6} q^{3} \) \( + \zeta_{6} q^{4} \) \( -\zeta_{6} q^{7} \) \( + \zeta_{6}^{2} q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{6} q^{3} \) \( + \zeta_{6} q^{4} \) \( -\zeta_{6} q^{7} \) \( + \zeta_{6}^{2} q^{9} \) \( -\zeta_{6}^{2} q^{12} \) \( + ( -1 + \zeta_{6}^{2} ) q^{13} \) \( + \zeta_{6}^{2} q^{16} \) \( + \zeta_{6}^{2} q^{21} \) \( -\zeta_{6}^{2} q^{25} \) \(+ q^{27}\) \( -\zeta_{6}^{2} q^{28} \) \( + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{31} \) \(- q^{36}\) \(+ q^{37}\) \( + ( 1 + \zeta_{6} ) q^{39} \) \( + ( \zeta_{6} + \zeta_{6}^{2} ) q^{43} \) \(+ q^{48}\) \( + ( -1 - \zeta_{6} ) q^{52} \) \(+ q^{63}\) \(- q^{64}\) \( -\zeta_{6} q^{67} \) \(+ q^{73}\) \(- q^{75}\) \( + ( -1 + \zeta_{6}^{2} ) q^{79} \) \( -\zeta_{6} q^{81} \) \(- q^{84}\) \( + ( 1 + \zeta_{6} ) q^{91} \) \( + ( -1 + \zeta_{6}^{2} ) q^{93} \) \( + ( \zeta_{6} + \zeta_{6}^{2} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut q^{12} \) \(\mathstrut -\mathstrut 3q^{13} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut -\mathstrut q^{21} \) \(\mathstrut +\mathstrut q^{25} \) \(\mathstrut +\mathstrut 2q^{27} \) \(\mathstrut +\mathstrut q^{28} \) \(\mathstrut -\mathstrut 2q^{36} \) \(\mathstrut +\mathstrut 2q^{37} \) \(\mathstrut +\mathstrut 3q^{39} \) \(\mathstrut +\mathstrut 2q^{48} \) \(\mathstrut -\mathstrut 3q^{52} \) \(\mathstrut +\mathstrut 2q^{63} \) \(\mathstrut -\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut q^{67} \) \(\mathstrut +\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut 2q^{75} \) \(\mathstrut -\mathstrut 3q^{79} \) \(\mathstrut -\mathstrut q^{81} \) \(\mathstrut -\mathstrut 2q^{84} \) \(\mathstrut +\mathstrut 3q^{91} \) \(\mathstrut -\mathstrut 3q^{93} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
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
101.1 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.e Even 1 yes
111.h Odd 1 yes

Hecke kernels

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