# Properties

 Label 43.2.c.a Level $43$ Weight $2$ Character orbit 43.c Analytic conductor $0.343$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$43$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 43.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.343356728692$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( -1 + \zeta_{6} ) q^{3} - q^{4} + ( 1 - \zeta_{6} ) q^{5} + ( -1 + \zeta_{6} ) q^{6} -3 \zeta_{6} q^{7} -3 q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + q^{2} + ( -1 + \zeta_{6} ) q^{3} - q^{4} + ( 1 - \zeta_{6} ) q^{5} + ( -1 + \zeta_{6} ) q^{6} -3 \zeta_{6} q^{7} -3 q^{8} + 2 \zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{10} + ( 1 - \zeta_{6} ) q^{12} + 5 \zeta_{6} q^{13} -3 \zeta_{6} q^{14} + \zeta_{6} q^{15} - q^{16} -3 \zeta_{6} q^{17} + 2 \zeta_{6} q^{18} + ( -1 + \zeta_{6} ) q^{19} + ( -1 + \zeta_{6} ) q^{20} + 3 q^{21} + ( 7 - 7 \zeta_{6} ) q^{23} + ( 3 - 3 \zeta_{6} ) q^{24} + 4 \zeta_{6} q^{25} + 5 \zeta_{6} q^{26} -5 q^{27} + 3 \zeta_{6} q^{28} -3 \zeta_{6} q^{29} + \zeta_{6} q^{30} + ( -5 + 5 \zeta_{6} ) q^{31} + 5 q^{32} -3 \zeta_{6} q^{34} -3 q^{35} -2 \zeta_{6} q^{36} + ( 9 - 9 \zeta_{6} ) q^{37} + ( -1 + \zeta_{6} ) q^{38} -5 q^{39} + ( -3 + 3 \zeta_{6} ) q^{40} -10 q^{41} + 3 q^{42} + ( -7 + 6 \zeta_{6} ) q^{43} + 2 q^{45} + ( 7 - 7 \zeta_{6} ) q^{46} -8 q^{47} + ( 1 - \zeta_{6} ) q^{48} + ( -2 + 2 \zeta_{6} ) q^{49} + 4 \zeta_{6} q^{50} + 3 q^{51} -5 \zeta_{6} q^{52} + ( 5 - 5 \zeta_{6} ) q^{53} -5 q^{54} + 9 \zeta_{6} q^{56} -\zeta_{6} q^{57} -3 \zeta_{6} q^{58} + 12 q^{59} -\zeta_{6} q^{60} + 13 \zeta_{6} q^{61} + ( -5 + 5 \zeta_{6} ) q^{62} + ( 6 - 6 \zeta_{6} ) q^{63} + 7 q^{64} + 5 q^{65} + ( 3 - 3 \zeta_{6} ) q^{67} + 3 \zeta_{6} q^{68} + 7 \zeta_{6} q^{69} -3 q^{70} + \zeta_{6} q^{71} -6 \zeta_{6} q^{72} -11 \zeta_{6} q^{73} + ( 9 - 9 \zeta_{6} ) q^{74} -4 q^{75} + ( 1 - \zeta_{6} ) q^{76} -5 q^{78} + 5 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -10 q^{82} + ( -9 + 9 \zeta_{6} ) q^{83} -3 q^{84} -3 q^{85} + ( -7 + 6 \zeta_{6} ) q^{86} + 3 q^{87} + ( 1 - \zeta_{6} ) q^{89} + 2 q^{90} + ( 15 - 15 \zeta_{6} ) q^{91} + ( -7 + 7 \zeta_{6} ) q^{92} -5 \zeta_{6} q^{93} -8 q^{94} + \zeta_{6} q^{95} + ( -5 + 5 \zeta_{6} ) q^{96} -2 q^{97} + ( -2 + 2 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - q^{3} - 2 q^{4} + q^{5} - q^{6} - 3 q^{7} - 6 q^{8} + 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{2} - q^{3} - 2 q^{4} + q^{5} - q^{6} - 3 q^{7} - 6 q^{8} + 2 q^{9} + q^{10} + q^{12} + 5 q^{13} - 3 q^{14} + q^{15} - 2 q^{16} - 3 q^{17} + 2 q^{18} - q^{19} - q^{20} + 6 q^{21} + 7 q^{23} + 3 q^{24} + 4 q^{25} + 5 q^{26} - 10 q^{27} + 3 q^{28} - 3 q^{29} + q^{30} - 5 q^{31} + 10 q^{32} - 3 q^{34} - 6 q^{35} - 2 q^{36} + 9 q^{37} - q^{38} - 10 q^{39} - 3 q^{40} - 20 q^{41} + 6 q^{42} - 8 q^{43} + 4 q^{45} + 7 q^{46} - 16 q^{47} + q^{48} - 2 q^{49} + 4 q^{50} + 6 q^{51} - 5 q^{52} + 5 q^{53} - 10 q^{54} + 9 q^{56} - q^{57} - 3 q^{58} + 24 q^{59} - q^{60} + 13 q^{61} - 5 q^{62} + 6 q^{63} + 14 q^{64} + 10 q^{65} + 3 q^{67} + 3 q^{68} + 7 q^{69} - 6 q^{70} + q^{71} - 6 q^{72} - 11 q^{73} + 9 q^{74} - 8 q^{75} + q^{76} - 10 q^{78} + 5 q^{79} - q^{80} - q^{81} - 20 q^{82} - 9 q^{83} - 6 q^{84} - 6 q^{85} - 8 q^{86} + 6 q^{87} + q^{89} + 4 q^{90} + 15 q^{91} - 7 q^{92} - 5 q^{93} - 16 q^{94} + q^{95} - 5 q^{96} - 4 q^{97} - 2 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/43\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 −0.500000 + 0.866025i −1.00000 0.500000 0.866025i −0.500000 + 0.866025i −1.50000 2.59808i −3.00000 1.00000 + 1.73205i 0.500000 0.866025i
36.1 1.00000 −0.500000 0.866025i −1.00000 0.500000 + 0.866025i −0.500000 0.866025i −1.50000 + 2.59808i −3.00000 1.00000 1.73205i 0.500000 + 0.866025i
 $$n$$: 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
43.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.2.c.a 2
3.b odd 2 1 387.2.h.a 2
4.b odd 2 1 688.2.i.d 2
43.c even 3 1 inner 43.2.c.a 2
43.c even 3 1 1849.2.a.c 1
43.d odd 6 1 1849.2.a.a 1
129.f odd 6 1 387.2.h.a 2
172.g odd 6 1 688.2.i.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.c.a 2 1.a even 1 1 trivial
43.2.c.a 2 43.c even 3 1 inner
387.2.h.a 2 3.b odd 2 1
387.2.h.a 2 129.f odd 6 1
688.2.i.d 2 4.b odd 2 1
688.2.i.d 2 172.g odd 6 1
1849.2.a.a 1 43.d odd 6 1
1849.2.a.c 1 43.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(43, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$9 + 3 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$25 - 5 T + T^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$49 - 7 T + T^{2}$$
$29$ $$9 + 3 T + T^{2}$$
$31$ $$25 + 5 T + T^{2}$$
$37$ $$81 - 9 T + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$43 + 8 T + T^{2}$$
$47$ $$( 8 + T )^{2}$$
$53$ $$25 - 5 T + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$169 - 13 T + T^{2}$$
$67$ $$9 - 3 T + T^{2}$$
$71$ $$1 - T + T^{2}$$
$73$ $$121 + 11 T + T^{2}$$
$79$ $$25 - 5 T + T^{2}$$
$83$ $$81 + 9 T + T^{2}$$
$89$ $$1 - T + T^{2}$$
$97$ $$( 2 + T )^{2}$$