Properties

Label 116.1.d.b
Level 116
Weight 1
Character orbit 116.d
Self dual yes
Analytic conductor 0.058
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM discriminant -116
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 116 = 2^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 116.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0578915414654\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.116.1
Artin image $S_3$
Artin field Galois closure of 3.1.116.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} - q^{10} - q^{11} - q^{12} - q^{13} + q^{15} + q^{16} + 2q^{19} - q^{20} - q^{22} - q^{24} - q^{26} + q^{27} + q^{29} + q^{30} - q^{31} + q^{32} + q^{33} + 2q^{38} + q^{39} - q^{40} - q^{43} - q^{44} - q^{47} - q^{48} + q^{49} - q^{52} - q^{53} + q^{54} + q^{55} - 2q^{57} + q^{58} + q^{60} - q^{62} + q^{64} + q^{65} + q^{66} + 2q^{76} + q^{78} - q^{79} - q^{80} - q^{81} - q^{86} - q^{87} - q^{88} + q^{93} - q^{94} - 2q^{95} - q^{96} + q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\) \(59\) \(89\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1
0
1.00000 −1.00000 1.00000 −1.00000 −1.00000 0 1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
116.d odd 2 1 CM by \(\Q(\sqrt{-29}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 116.1.d.b yes 1
3.b odd 2 1 1044.1.g.a 1
4.b odd 2 1 116.1.d.a 1
5.b even 2 1 2900.1.g.a 1
5.c odd 4 2 2900.1.e.a 2
8.b even 2 1 1856.1.h.c 1
8.d odd 2 1 1856.1.h.a 1
12.b even 2 1 1044.1.g.b 1
20.d odd 2 1 2900.1.g.d 1
20.e even 4 2 2900.1.e.b 2
29.b even 2 1 116.1.d.a 1
29.c odd 4 2 3364.1.b.a 2
29.d even 7 6 3364.1.h.a 6
29.e even 14 6 3364.1.h.b 6
29.f odd 28 12 3364.1.j.f 12
87.d odd 2 1 1044.1.g.b 1
116.d odd 2 1 CM 116.1.d.b yes 1
116.e even 4 2 3364.1.b.a 2
116.h odd 14 6 3364.1.h.a 6
116.j odd 14 6 3364.1.h.b 6
116.l even 28 12 3364.1.j.f 12
145.d even 2 1 2900.1.g.d 1
145.h odd 4 2 2900.1.e.b 2
232.b odd 2 1 1856.1.h.c 1
232.g even 2 1 1856.1.h.a 1
348.b even 2 1 1044.1.g.a 1
580.e odd 2 1 2900.1.g.a 1
580.o even 4 2 2900.1.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.1.d.a 1 4.b odd 2 1
116.1.d.a 1 29.b even 2 1
116.1.d.b yes 1 1.a even 1 1 trivial
116.1.d.b yes 1 116.d odd 2 1 CM
1044.1.g.a 1 3.b odd 2 1
1044.1.g.a 1 348.b even 2 1
1044.1.g.b 1 12.b even 2 1
1044.1.g.b 1 87.d odd 2 1
1856.1.h.a 1 8.d odd 2 1
1856.1.h.a 1 232.g even 2 1
1856.1.h.c 1 8.b even 2 1
1856.1.h.c 1 232.b odd 2 1
2900.1.e.a 2 5.c odd 4 2
2900.1.e.a 2 580.o even 4 2
2900.1.e.b 2 20.e even 4 2
2900.1.e.b 2 145.h odd 4 2
2900.1.g.a 1 5.b even 2 1
2900.1.g.a 1 580.e odd 2 1
2900.1.g.d 1 20.d odd 2 1
2900.1.g.d 1 145.d even 2 1
3364.1.b.a 2 29.c odd 4 2
3364.1.b.a 2 116.e even 4 2
3364.1.h.a 6 29.d even 7 6
3364.1.h.a 6 116.h odd 14 6
3364.1.h.b 6 29.e even 14 6
3364.1.h.b 6 116.j odd 14 6
3364.1.j.f 12 29.f odd 28 12
3364.1.j.f 12 116.l even 28 12

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(116, [\chi])\).

Hecke characteristic polynomials

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