# Properties

 Label 116.1.d.a 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

# Related objects

## 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 $D_6$ Artin field Galois closure of 6.0.53824.1

## $q$-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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.a 1
3.b odd 2 1 1044.1.g.b 1
4.b odd 2 1 116.1.d.b yes 1
5.b even 2 1 2900.1.g.d 1
5.c odd 4 2 2900.1.e.b 2
8.b even 2 1 1856.1.h.a 1
8.d odd 2 1 1856.1.h.c 1
12.b even 2 1 1044.1.g.a 1
20.d odd 2 1 2900.1.g.a 1
20.e even 4 2 2900.1.e.a 2
29.b even 2 1 116.1.d.b yes 1
29.c odd 4 2 3364.1.b.a 2
29.d even 7 6 3364.1.h.b 6
29.e even 14 6 3364.1.h.a 6
29.f odd 28 12 3364.1.j.f 12
87.d odd 2 1 1044.1.g.a 1
116.d odd 2 1 CM 116.1.d.a 1
116.e even 4 2 3364.1.b.a 2
116.h odd 14 6 3364.1.h.b 6
116.j odd 14 6 3364.1.h.a 6
116.l even 28 12 3364.1.j.f 12
145.d even 2 1 2900.1.g.a 1
145.h odd 4 2 2900.1.e.a 2
232.b odd 2 1 1856.1.h.a 1
232.g even 2 1 1856.1.h.c 1
348.b even 2 1 1044.1.g.b 1
580.e odd 2 1 2900.1.g.d 1
580.o even 4 2 2900.1.e.b 2

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

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