# Properties

 Label 143.1.d.a Level $143$ Weight $1$ Character orbit 143.d Self dual yes Analytic conductor $0.071$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -143 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$143 = 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 143.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.0713662968065$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.20449.1 Artin image: $D_5$ Artin field: Galois closure of 5.1.20449.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( -1 + \beta ) q^{3} + ( 1 - \beta ) q^{4} + ( 2 - \beta ) q^{6} -\beta q^{7} - q^{8} + ( 1 - \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( -1 + \beta ) q^{3} + ( 1 - \beta ) q^{4} + ( 2 - \beta ) q^{6} -\beta q^{7} - q^{8} + ( 1 - \beta ) q^{9} + q^{11} + ( -2 + \beta ) q^{12} + q^{13} - q^{14} + ( -2 + \beta ) q^{18} + ( -1 + \beta ) q^{19} - q^{21} + ( -1 + \beta ) q^{22} -\beta q^{23} + ( 1 - \beta ) q^{24} + q^{25} + ( -1 + \beta ) q^{26} - q^{27} + q^{28} + q^{32} + ( -1 + \beta ) q^{33} + ( 2 - \beta ) q^{36} + ( 2 - \beta ) q^{38} + ( -1 + \beta ) q^{39} -\beta q^{41} + ( 1 - \beta ) q^{42} + ( 1 - \beta ) q^{44} - q^{46} + \beta q^{49} + ( -1 + \beta ) q^{50} + ( 1 - \beta ) q^{52} + ( -1 + \beta ) q^{53} + ( 1 - \beta ) q^{54} + \beta q^{56} + ( 2 - \beta ) q^{57} + q^{63} + ( -1 + \beta ) q^{64} + ( 2 - \beta ) q^{66} - q^{69} + ( -1 + \beta ) q^{72} + ( -1 + \beta ) q^{73} + ( -1 + \beta ) q^{75} + ( -2 + \beta ) q^{76} -\beta q^{77} + ( 2 - \beta ) q^{78} - q^{82} -\beta q^{83} + ( -1 + \beta ) q^{84} - q^{88} -\beta q^{91} + q^{92} + ( -1 + \beta ) q^{96} + q^{98} + ( 1 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{3} + q^{4} + 3q^{6} - q^{7} - 2q^{8} + q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{3} + q^{4} + 3q^{6} - q^{7} - 2q^{8} + q^{9} + 2q^{11} - 3q^{12} + 2q^{13} - 2q^{14} - 3q^{18} - q^{19} - 2q^{21} - q^{22} - q^{23} + q^{24} + 2q^{25} - q^{26} - 2q^{27} + 2q^{28} + 2q^{32} - q^{33} + 3q^{36} + 3q^{38} - q^{39} - q^{41} + q^{42} + q^{44} - 2q^{46} + q^{49} - q^{50} + q^{52} - q^{53} + q^{54} + q^{56} + 3q^{57} + 2q^{63} - q^{64} + 3q^{66} - 2q^{69} - q^{72} - q^{73} - q^{75} - 3q^{76} - q^{77} + 3q^{78} - 2q^{82} - q^{83} - q^{84} - 2q^{88} - q^{91} + 2q^{92} - q^{96} + 2q^{98} + q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/143\mathbb{Z}\right)^\times$$.

 $$n$$ $$67$$ $$79$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
142.1
 −0.618034 1.61803
−1.61803 −1.61803 1.61803 0 2.61803 0.618034 −1.00000 1.61803 0
142.2 0.618034 0.618034 −0.618034 0 0.381966 −1.61803 −1.00000 −0.618034 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
143.d odd 2 1 CM by $$\Q(\sqrt{-143})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.1.d.a 2
3.b odd 2 1 1287.1.g.b 2
4.b odd 2 1 2288.1.m.b 2
5.b even 2 1 3575.1.h.f 2
5.c odd 4 2 3575.1.c.d 4
11.b odd 2 1 143.1.d.b yes 2
11.c even 5 2 1573.1.l.b 4
11.c even 5 2 1573.1.l.d 4
11.d odd 10 2 1573.1.l.a 4
11.d odd 10 2 1573.1.l.c 4
13.b even 2 1 143.1.d.b yes 2
13.c even 3 2 1859.1.i.b 4
13.d odd 4 2 1859.1.c.c 4
13.e even 6 2 1859.1.i.a 4
13.f odd 12 4 1859.1.k.c 8
33.d even 2 1 1287.1.g.a 2
39.d odd 2 1 1287.1.g.a 2
44.c even 2 1 2288.1.m.a 2
52.b odd 2 1 2288.1.m.a 2
55.d odd 2 1 3575.1.h.e 2
55.e even 4 2 3575.1.c.c 4
65.d even 2 1 3575.1.h.e 2
65.h odd 4 2 3575.1.c.c 4
143.d odd 2 1 CM 143.1.d.a 2
143.g even 4 2 1859.1.c.c 4
143.i odd 6 2 1859.1.i.b 4
143.k odd 6 2 1859.1.i.a 4
143.l odd 10 2 1573.1.l.b 4
143.l odd 10 2 1573.1.l.d 4
143.n even 10 2 1573.1.l.a 4
143.n even 10 2 1573.1.l.c 4
143.o even 12 4 1859.1.k.c 8
429.e even 2 1 1287.1.g.b 2
572.b even 2 1 2288.1.m.b 2
715.c odd 2 1 3575.1.h.f 2
715.q even 4 2 3575.1.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.1.d.a 2 1.a even 1 1 trivial
143.1.d.a 2 143.d odd 2 1 CM
143.1.d.b yes 2 11.b odd 2 1
143.1.d.b yes 2 13.b even 2 1
1287.1.g.a 2 33.d even 2 1
1287.1.g.a 2 39.d odd 2 1
1287.1.g.b 2 3.b odd 2 1
1287.1.g.b 2 429.e even 2 1
1573.1.l.a 4 11.d odd 10 2
1573.1.l.a 4 143.n even 10 2
1573.1.l.b 4 11.c even 5 2
1573.1.l.b 4 143.l odd 10 2
1573.1.l.c 4 11.d odd 10 2
1573.1.l.c 4 143.n even 10 2
1573.1.l.d 4 11.c even 5 2
1573.1.l.d 4 143.l odd 10 2
1859.1.c.c 4 13.d odd 4 2
1859.1.c.c 4 143.g even 4 2
1859.1.i.a 4 13.e even 6 2
1859.1.i.a 4 143.k odd 6 2
1859.1.i.b 4 13.c even 3 2
1859.1.i.b 4 143.i odd 6 2
1859.1.k.c 8 13.f odd 12 4
1859.1.k.c 8 143.o even 12 4
2288.1.m.a 2 44.c even 2 1
2288.1.m.a 2 52.b odd 2 1
2288.1.m.b 2 4.b odd 2 1
2288.1.m.b 2 572.b even 2 1
3575.1.c.c 4 55.e even 4 2
3575.1.c.c 4 65.h odd 4 2
3575.1.c.d 4 5.c odd 4 2
3575.1.c.d 4 715.q even 4 2
3575.1.h.e 2 55.d odd 2 1
3575.1.h.e 2 65.d even 2 1
3575.1.h.f 2 5.b even 2 1
3575.1.h.f 2 715.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} - 1$$ acting on $$S_{1}^{\mathrm{new}}(143, [\chi])$$.

## Hecke characteristic polynomials

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