Properties

 Label 111.1.i.a Level $111$ Weight $1$ Character orbit 111.i Analytic conductor $0.055$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -3 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$111 = 3 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 111.i (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.0553962164023$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.4107.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.36963.1

$q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{7} + \zeta_{6}^{2} q^{9} + \zeta_{6}^{2} q^{12} + \zeta_{6} q^{13} + \zeta_{6}^{2} q^{16} -2 \zeta_{6} q^{19} -\zeta_{6}^{2} q^{21} + \zeta_{6}^{2} q^{25} + q^{27} -\zeta_{6}^{2} q^{28} - q^{31} + q^{36} + q^{37} -\zeta_{6}^{2} q^{39} - q^{43} + q^{48} -\zeta_{6}^{2} q^{52} + 2 \zeta_{6}^{2} q^{57} -2 \zeta_{6} q^{61} - q^{63} + q^{64} + \zeta_{6} q^{67} - q^{73} + q^{75} + 2 \zeta_{6}^{2} q^{76} + \zeta_{6} q^{79} -\zeta_{6} q^{81} - q^{84} + \zeta_{6}^{2} q^{91} + \zeta_{6} q^{93} - q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} - q^{4} + q^{7} - q^{9} + O(q^{10})$$ $$2q - q^{3} - q^{4} + q^{7} - q^{9} - q^{12} + q^{13} - q^{16} - 2q^{19} + q^{21} - q^{25} + 2q^{27} + q^{28} - 2q^{31} + 2q^{36} + 2q^{37} + q^{39} - 2q^{43} + 2q^{48} + q^{52} - 2q^{57} - 2q^{61} - 2q^{63} + 2q^{64} + q^{67} - 2q^{73} + 2q^{75} - 2q^{76} + q^{79} - q^{81} - 2q^{84} - q^{91} + q^{93} - 2q^{97} + O(q^{100})$$

Character values

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

 $$n$$ $$38$$ $$76$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 0 0.500000 0.866025i 0 −0.500000 0.866025i 0
47.1 0 −0.500000 0.866025i −0.500000 0.866025i 0 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 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.c even 3 1 inner
111.i odd 6 1 inner

Twists

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

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

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(111, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 - T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$1 - T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$( 1 + T )^{2}$$
$37$ $$( -1 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$( 1 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$4 + 2 T + T^{2}$$
$67$ $$1 - T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 1 + T )^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( 1 + T )^{2}$$