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 discriminant -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 \(6\), degree \(2\), minimal)

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: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.4107.1
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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
37.c even 3 1 inner
111.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 111.1.i.a 2
3.b odd 2 1 CM 111.1.i.a 2
4.b odd 2 1 1776.1.bw.a 2
5.b even 2 1 2775.1.z.a 2
5.c odd 4 2 2775.1.x.a 4
9.c even 3 1 2997.1.l.a 2
9.c even 3 1 2997.1.u.a 2
9.d odd 6 1 2997.1.l.a 2
9.d odd 6 1 2997.1.u.a 2
12.b even 2 1 1776.1.bw.a 2
15.d odd 2 1 2775.1.z.a 2
15.e even 4 2 2775.1.x.a 4
37.c even 3 1 inner 111.1.i.a 2
111.i odd 6 1 inner 111.1.i.a 2
148.i odd 6 1 1776.1.bw.a 2
185.n even 6 1 2775.1.z.a 2
185.s odd 12 2 2775.1.x.a 4
333.g even 3 1 2997.1.l.a 2
333.h even 3 1 2997.1.u.a 2
333.l odd 6 1 2997.1.u.a 2
333.u odd 6 1 2997.1.l.a 2
444.t even 6 1 1776.1.bw.a 2
555.w odd 6 1 2775.1.z.a 2
555.bi even 12 2 2775.1.x.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.i.a 2 1.a even 1 1 trivial
111.1.i.a 2 3.b odd 2 1 CM
111.1.i.a 2 37.c even 3 1 inner
111.1.i.a 2 111.i odd 6 1 inner
1776.1.bw.a 2 4.b odd 2 1
1776.1.bw.a 2 12.b even 2 1
1776.1.bw.a 2 148.i odd 6 1
1776.1.bw.a 2 444.t even 6 1
2775.1.x.a 4 5.c odd 4 2
2775.1.x.a 4 15.e even 4 2
2775.1.x.a 4 185.s odd 12 2
2775.1.x.a 4 555.bi even 12 2
2775.1.z.a 2 5.b even 2 1
2775.1.z.a 2 15.d odd 2 1
2775.1.z.a 2 185.n even 6 1
2775.1.z.a 2 555.w odd 6 1
2997.1.l.a 2 9.c even 3 1
2997.1.l.a 2 9.d odd 6 1
2997.1.l.a 2 333.g even 3 1
2997.1.l.a 2 333.u odd 6 1
2997.1.u.a 2 9.c even 3 1
2997.1.u.a 2 9.d odd 6 1
2997.1.u.a 2 333.h even 3 1
2997.1.u.a 2 333.l odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(111, [\chi])\).

Hecke characteristic polynomials

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