# Properties

 Label 111.1.d.b Level $111$ Weight $1$ Character orbit 111.d Self dual yes Analytic conductor $0.055$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -111 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.4107.1 Artin image: $D_8$ Artin field: Galois closure of 8.0.4102893.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} - q^{3} + q^{4} + \beta q^{5} + \beta q^{6} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} - q^{3} + q^{4} + \beta q^{5} + \beta q^{6} + q^{9} -2 q^{10} - q^{12} -\beta q^{15} - q^{16} -\beta q^{17} -\beta q^{18} + \beta q^{20} + \beta q^{23} + q^{25} - q^{27} -\beta q^{29} + 2 q^{30} + \beta q^{32} + 2 q^{34} + q^{36} - q^{37} + \beta q^{45} -2 q^{46} + q^{48} - q^{49} -\beta q^{50} + \beta q^{51} + \beta q^{54} + 2 q^{58} -\beta q^{59} -\beta q^{60} - q^{64} -\beta q^{68} -\beta q^{69} + \beta q^{74} - q^{75} -\beta q^{80} + q^{81} -2 q^{85} + \beta q^{87} + \beta q^{89} -2 q^{90} + \beta q^{92} -\beta q^{96} + \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{4} + 2q^{9} - 4q^{10} - 2q^{12} - 2q^{16} + 2q^{25} - 2q^{27} + 4q^{30} + 4q^{34} + 2q^{36} - 2q^{37} - 4q^{46} + 2q^{48} - 2q^{49} + 4q^{58} - 2q^{64} - 2q^{75} + 2q^{81} - 4q^{85} - 4q^{90} + O(q^{100})$$

## 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
 1.41421 −1.41421
−1.41421 −1.00000 1.00000 1.41421 1.41421 0 0 1.00000 −2.00000
110.2 1.41421 −1.00000 1.00000 −1.41421 −1.41421 0 0 1.00000 −2.00000
 $$n$$: 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
111.d odd 2 1 CM by $$\Q(\sqrt{-111})$$
3.b odd 2 1 inner
37.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

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