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 discriminant -3
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.0553962164023\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.624095613.1

$q$-expansion

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
37.e even 6 1 inner
111.h 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.h.a 2
3.b odd 2 1 CM 111.1.h.a 2
4.b odd 2 1 1776.1.bq.a 2
5.b even 2 1 2775.1.w.a 2
5.c odd 4 2 2775.1.bb.a 4
9.c even 3 1 2997.1.o.a 2
9.c even 3 1 2997.1.v.a 2
9.d odd 6 1 2997.1.o.a 2
9.d odd 6 1 2997.1.v.a 2
12.b even 2 1 1776.1.bq.a 2
15.d odd 2 1 2775.1.w.a 2
15.e even 4 2 2775.1.bb.a 4
37.e even 6 1 inner 111.1.h.a 2
111.h odd 6 1 inner 111.1.h.a 2
148.j odd 6 1 1776.1.bq.a 2
185.l even 6 1 2775.1.w.a 2
185.r odd 12 2 2775.1.bb.a 4
333.k even 6 1 2997.1.o.a 2
333.o odd 6 1 2997.1.v.a 2
333.t even 6 1 2997.1.v.a 2
333.v odd 6 1 2997.1.o.a 2
444.p even 6 1 1776.1.bq.a 2
555.ba odd 6 1 2775.1.w.a 2
555.bh even 12 2 2775.1.bb.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.h.a 2 1.a even 1 1 trivial
111.1.h.a 2 3.b odd 2 1 CM
111.1.h.a 2 37.e even 6 1 inner
111.1.h.a 2 111.h odd 6 1 inner
1776.1.bq.a 2 4.b odd 2 1
1776.1.bq.a 2 12.b even 2 1
1776.1.bq.a 2 148.j odd 6 1
1776.1.bq.a 2 444.p even 6 1
2775.1.w.a 2 5.b even 2 1
2775.1.w.a 2 15.d odd 2 1
2775.1.w.a 2 185.l even 6 1
2775.1.w.a 2 555.ba odd 6 1
2775.1.bb.a 4 5.c odd 4 2
2775.1.bb.a 4 15.e even 4 2
2775.1.bb.a 4 185.r odd 12 2
2775.1.bb.a 4 555.bh even 12 2
2997.1.o.a 2 9.c even 3 1
2997.1.o.a 2 9.d odd 6 1
2997.1.o.a 2 333.k even 6 1
2997.1.o.a 2 333.v odd 6 1
2997.1.v.a 2 9.c even 3 1
2997.1.v.a 2 9.d odd 6 1
2997.1.v.a 2 333.o odd 6 1
2997.1.v.a 2 333.t even 6 1

Hecke kernels

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

Hecke characteristic polynomials

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