# Properties

 Label 156.1.s.a Level $156$ Weight $1$ Character orbit 156.s Analytic conductor $0.078$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 156.s (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.0778541419707$$ 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: $$D_{6}$$ Projective field: Galois closure of 6.0.160398576.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{3} + ( -1 - \zeta_{6} ) q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{3} + ( -1 - \zeta_{6} ) q^{7} + \zeta_{6}^{2} q^{9} -\zeta_{6}^{2} q^{13} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{21} - q^{25} - q^{27} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{31} + q^{39} -\zeta_{6}^{2} q^{43} + ( 1 + \zeta_{6} + \zeta_{6}^{2} ) q^{49} + \zeta_{6}^{2} q^{61} + ( 1 - \zeta_{6}^{2} ) q^{63} + ( 1 - \zeta_{6}^{2} ) q^{67} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{73} -\zeta_{6} q^{75} + q^{79} -\zeta_{6} q^{81} + ( -1 + \zeta_{6}^{2} ) q^{91} + ( -1 + \zeta_{6}^{2} ) q^{93} + ( -1 - \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - 3q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} - 3q^{7} - q^{9} + q^{13} - 2q^{25} - 2q^{27} + 2q^{39} + q^{43} + 2q^{49} - q^{61} + 3q^{63} + 3q^{67} - q^{75} + 2q^{79} - q^{81} - 3q^{91} - 3q^{93} - 3q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/156\mathbb{Z}\right)^\times$$.

 $$n$$ $$53$$ $$79$$ $$145$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{6}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 0.866025i 0 0 0 −1.50000 + 0.866025i 0 −0.500000 0.866025i 0
101.1 0 0.500000 + 0.866025i 0 0 0 −1.50000 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 $$\Q(\sqrt{-3})$$
13.e even 6 1 inner
39.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.1.s.a 2
3.b odd 2 1 CM 156.1.s.a 2
4.b odd 2 1 624.1.cb.a 2
5.b even 2 1 3900.1.ca.b 2
5.c odd 4 2 3900.1.br.b 4
8.b even 2 1 2496.1.cb.a 2
8.d odd 2 1 2496.1.cb.b 2
12.b even 2 1 624.1.cb.a 2
13.b even 2 1 2028.1.s.a 2
13.c even 3 1 2028.1.g.a 2
13.c even 3 1 2028.1.s.a 2
13.d odd 4 2 2028.1.o.b 4
13.e even 6 1 inner 156.1.s.a 2
13.e even 6 1 2028.1.g.a 2
13.f odd 12 2 2028.1.d.c 2
13.f odd 12 2 2028.1.o.b 4
15.d odd 2 1 3900.1.ca.b 2
15.e even 4 2 3900.1.br.b 4
24.f even 2 1 2496.1.cb.b 2
24.h odd 2 1 2496.1.cb.a 2
39.d odd 2 1 2028.1.s.a 2
39.f even 4 2 2028.1.o.b 4
39.h odd 6 1 inner 156.1.s.a 2
39.h odd 6 1 2028.1.g.a 2
39.i odd 6 1 2028.1.g.a 2
39.i odd 6 1 2028.1.s.a 2
39.k even 12 2 2028.1.d.c 2
39.k even 12 2 2028.1.o.b 4
52.i odd 6 1 624.1.cb.a 2
65.l even 6 1 3900.1.ca.b 2
65.r odd 12 2 3900.1.br.b 4
104.p odd 6 1 2496.1.cb.b 2
104.s even 6 1 2496.1.cb.a 2
156.r even 6 1 624.1.cb.a 2
195.y odd 6 1 3900.1.ca.b 2
195.bf even 12 2 3900.1.br.b 4
312.ba even 6 1 2496.1.cb.b 2
312.bg odd 6 1 2496.1.cb.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.1.s.a 2 1.a even 1 1 trivial
156.1.s.a 2 3.b odd 2 1 CM
156.1.s.a 2 13.e even 6 1 inner
156.1.s.a 2 39.h odd 6 1 inner
624.1.cb.a 2 4.b odd 2 1
624.1.cb.a 2 12.b even 2 1
624.1.cb.a 2 52.i odd 6 1
624.1.cb.a 2 156.r even 6 1
2028.1.d.c 2 13.f odd 12 2
2028.1.d.c 2 39.k even 12 2
2028.1.g.a 2 13.c even 3 1
2028.1.g.a 2 13.e even 6 1
2028.1.g.a 2 39.h odd 6 1
2028.1.g.a 2 39.i odd 6 1
2028.1.o.b 4 13.d odd 4 2
2028.1.o.b 4 13.f odd 12 2
2028.1.o.b 4 39.f even 4 2
2028.1.o.b 4 39.k even 12 2
2028.1.s.a 2 13.b even 2 1
2028.1.s.a 2 13.c even 3 1
2028.1.s.a 2 39.d odd 2 1
2028.1.s.a 2 39.i odd 6 1
2496.1.cb.a 2 8.b even 2 1
2496.1.cb.a 2 24.h odd 2 1
2496.1.cb.a 2 104.s even 6 1
2496.1.cb.a 2 312.bg odd 6 1
2496.1.cb.b 2 8.d odd 2 1
2496.1.cb.b 2 24.f even 2 1
2496.1.cb.b 2 104.p odd 6 1
2496.1.cb.b 2 312.ba even 6 1
3900.1.br.b 4 5.c odd 4 2
3900.1.br.b 4 15.e even 4 2
3900.1.br.b 4 65.r odd 12 2
3900.1.br.b 4 195.bf even 12 2
3900.1.ca.b 2 5.b even 2 1
3900.1.ca.b 2 15.d odd 2 1
3900.1.ca.b 2 65.l even 6 1
3900.1.ca.b 2 195.y odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(156, [\chi])$$.

## Hecke characteristic polynomials

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