# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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