Properties

Label 111.1.d.a
Level 111
Weight 1
Character orbit 111.d
Self dual yes
Analytic conductor 0.055
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -3, -111, 37
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.0553962164023\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-3}, \sqrt{37})\)
Artin image $D_4$
Artin field Galois closure of 4.0.333.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - q^{4} - 2q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} - q^{4} - 2q^{7} + q^{9} - q^{12} + q^{16} - 2q^{21} - q^{25} + q^{27} + 2q^{28} - q^{36} + q^{37} + q^{48} + 3q^{49} - 2q^{63} - q^{64} - 2q^{67} - 2q^{73} - q^{75} + q^{81} + 2q^{84} + O(q^{100}) \)

Character values

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

\(n\) \(38\) \(76\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
110.1
0
0 1.00000 −1.00000 0 0 −2.00000 0 1.00000 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.b even 2 1 RM by \(\Q(\sqrt{37}) \)
111.d odd 2 1 CM by \(\Q(\sqrt{-111}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 111.1.d.a 1
3.b odd 2 1 CM 111.1.d.a 1
4.b odd 2 1 1776.1.n.a 1
5.b even 2 1 2775.1.h.a 1
5.c odd 4 2 2775.1.b.a 2
9.c even 3 2 2997.1.n.b 2
9.d odd 6 2 2997.1.n.b 2
12.b even 2 1 1776.1.n.a 1
15.d odd 2 1 2775.1.h.a 1
15.e even 4 2 2775.1.b.a 2
37.b even 2 1 RM 111.1.d.a 1
111.d odd 2 1 CM 111.1.d.a 1
148.b odd 2 1 1776.1.n.a 1
185.d even 2 1 2775.1.h.a 1
185.h odd 4 2 2775.1.b.a 2
333.n odd 6 2 2997.1.n.b 2
333.q even 6 2 2997.1.n.b 2
444.g even 2 1 1776.1.n.a 1
555.b odd 2 1 2775.1.h.a 1
555.n even 4 2 2775.1.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.d.a 1 1.a even 1 1 trivial
111.1.d.a 1 3.b odd 2 1 CM
111.1.d.a 1 37.b even 2 1 RM
111.1.d.a 1 111.d odd 2 1 CM
1776.1.n.a 1 4.b odd 2 1
1776.1.n.a 1 12.b even 2 1
1776.1.n.a 1 148.b odd 2 1
1776.1.n.a 1 444.g even 2 1
2775.1.b.a 2 5.c odd 4 2
2775.1.b.a 2 15.e even 4 2
2775.1.b.a 2 185.h odd 4 2
2775.1.b.a 2 555.n even 4 2
2775.1.h.a 1 5.b even 2 1
2775.1.h.a 1 15.d odd 2 1
2775.1.h.a 1 185.d even 2 1
2775.1.h.a 1 555.b odd 2 1
2997.1.n.b 2 9.c even 3 2
2997.1.n.b 2 9.d odd 6 2
2997.1.n.b 2 333.n odd 6 2
2997.1.n.b 2 333.q even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{1}^{\mathrm{new}}(111, [\chi])\).

Hecke Characteristic Polynomials

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