# Properties

 Label 43.2.a.b Level $43$ Weight $2$ Character orbit 43.a Self dual yes Analytic conductor $0.343$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$43$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 43.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.343356728692$$ 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 Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -\beta q^{3} + ( 2 - \beta ) q^{5} -2 q^{6} + ( -2 + \beta ) q^{7} -2 \beta q^{8} - q^{9} +O(q^{10})$$ $$q + \beta q^{2} -\beta q^{3} + ( 2 - \beta ) q^{5} -2 q^{6} + ( -2 + \beta ) q^{7} -2 \beta q^{8} - q^{9} + ( -2 + 2 \beta ) q^{10} + ( -1 + 2 \beta ) q^{11} + ( 1 + 2 \beta ) q^{13} + ( 2 - 2 \beta ) q^{14} + ( 2 - 2 \beta ) q^{15} -4 q^{16} + ( 5 + 2 \beta ) q^{17} -\beta q^{18} + ( -2 - 2 \beta ) q^{19} + ( -2 + 2 \beta ) q^{21} + ( 4 - \beta ) q^{22} + ( 1 - 4 \beta ) q^{23} + 4 q^{24} + ( 1 - 4 \beta ) q^{25} + ( 4 + \beta ) q^{26} + 4 \beta q^{27} + 3 \beta q^{29} + ( -4 + 2 \beta ) q^{30} -3 q^{31} + ( -4 + \beta ) q^{33} + ( 4 + 5 \beta ) q^{34} + ( -6 + 4 \beta ) q^{35} -6 \beta q^{37} + ( -4 - 2 \beta ) q^{38} + ( -4 - \beta ) q^{39} + ( 4 - 4 \beta ) q^{40} + ( -1 - 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{42} + q^{43} + ( -2 + \beta ) q^{45} + ( -8 + \beta ) q^{46} + 6 q^{47} + 4 \beta q^{48} + ( -1 - 4 \beta ) q^{49} + ( -8 + \beta ) q^{50} + ( -4 - 5 \beta ) q^{51} + ( 11 - 2 \beta ) q^{53} + 8 q^{54} + ( -6 + 5 \beta ) q^{55} + ( -4 + 4 \beta ) q^{56} + ( 4 + 2 \beta ) q^{57} + 6 q^{58} + ( -2 + 2 \beta ) q^{59} + ( 4 + 3 \beta ) q^{61} -3 \beta q^{62} + ( 2 - \beta ) q^{63} + 8 q^{64} + ( -2 + 3 \beta ) q^{65} + ( 2 - 4 \beta ) q^{66} + ( 1 + 6 \beta ) q^{67} + ( 8 - \beta ) q^{69} + ( 8 - 6 \beta ) q^{70} + ( -6 - 2 \beta ) q^{71} + 2 \beta q^{72} + ( -12 + 3 \beta ) q^{73} -12 q^{74} + ( 8 - \beta ) q^{75} + ( 6 - 5 \beta ) q^{77} + ( -2 - 4 \beta ) q^{78} + ( 2 - 2 \beta ) q^{79} + ( -8 + 4 \beta ) q^{80} -5 q^{81} + ( -4 - \beta ) q^{82} + ( 9 + 4 \beta ) q^{83} + ( 6 - \beta ) q^{85} + \beta q^{86} -6 q^{87} + ( -8 + 2 \beta ) q^{88} + ( -6 - 3 \beta ) q^{89} + ( 2 - 2 \beta ) q^{90} + ( 2 - 3 \beta ) q^{91} + 3 \beta q^{93} + 6 \beta q^{94} -2 \beta q^{95} + ( -1 - 2 \beta ) q^{97} + ( -8 - \beta ) q^{98} + ( 1 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{5} - 4q^{6} - 4q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 4q^{5} - 4q^{6} - 4q^{7} - 2q^{9} - 4q^{10} - 2q^{11} + 2q^{13} + 4q^{14} + 4q^{15} - 8q^{16} + 10q^{17} - 4q^{19} - 4q^{21} + 8q^{22} + 2q^{23} + 8q^{24} + 2q^{25} + 8q^{26} - 8q^{30} - 6q^{31} - 8q^{33} + 8q^{34} - 12q^{35} - 8q^{38} - 8q^{39} + 8q^{40} - 2q^{41} + 8q^{42} + 2q^{43} - 4q^{45} - 16q^{46} + 12q^{47} - 2q^{49} - 16q^{50} - 8q^{51} + 22q^{53} + 16q^{54} - 12q^{55} - 8q^{56} + 8q^{57} + 12q^{58} - 4q^{59} + 8q^{61} + 4q^{63} + 16q^{64} - 4q^{65} + 4q^{66} + 2q^{67} + 16q^{69} + 16q^{70} - 12q^{71} - 24q^{73} - 24q^{74} + 16q^{75} + 12q^{77} - 4q^{78} + 4q^{79} - 16q^{80} - 10q^{81} - 8q^{82} + 18q^{83} + 12q^{85} - 12q^{87} - 16q^{88} - 12q^{89} + 4q^{90} + 4q^{91} - 2q^{97} - 16q^{98} + 2q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
−1.41421 1.41421 0 3.41421 −2.00000 −3.41421 2.82843 −1.00000 −4.82843
1.2 1.41421 −1.41421 0 0.585786 −2.00000 −0.585786 −2.82843 −1.00000 0.828427
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$43$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.2.a.b 2
3.b odd 2 1 387.2.a.h 2
4.b odd 2 1 688.2.a.f 2
5.b even 2 1 1075.2.a.i 2
5.c odd 4 2 1075.2.b.f 4
7.b odd 2 1 2107.2.a.b 2
8.b even 2 1 2752.2.a.l 2
8.d odd 2 1 2752.2.a.m 2
11.b odd 2 1 5203.2.a.f 2
12.b even 2 1 6192.2.a.bd 2
13.b even 2 1 7267.2.a.b 2
15.d odd 2 1 9675.2.a.bf 2
43.b odd 2 1 1849.2.a.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.a.b 2 1.a even 1 1 trivial
387.2.a.h 2 3.b odd 2 1
688.2.a.f 2 4.b odd 2 1
1075.2.a.i 2 5.b even 2 1
1075.2.b.f 4 5.c odd 4 2
1849.2.a.f 2 43.b odd 2 1
2107.2.a.b 2 7.b odd 2 1
2752.2.a.l 2 8.b even 2 1
2752.2.a.m 2 8.d odd 2 1
5203.2.a.f 2 11.b odd 2 1
6192.2.a.bd 2 12.b even 2 1
7267.2.a.b 2 13.b even 2 1
9675.2.a.bf 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(43))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$2 - 4 T + T^{2}$$
$7$ $$2 + 4 T + T^{2}$$
$11$ $$-7 + 2 T + T^{2}$$
$13$ $$-7 - 2 T + T^{2}$$
$17$ $$17 - 10 T + T^{2}$$
$19$ $$-4 + 4 T + T^{2}$$
$23$ $$-31 - 2 T + T^{2}$$
$29$ $$-18 + T^{2}$$
$31$ $$( 3 + T )^{2}$$
$37$ $$-72 + T^{2}$$
$41$ $$-7 + 2 T + T^{2}$$
$43$ $$( -1 + T )^{2}$$
$47$ $$( -6 + T )^{2}$$
$53$ $$113 - 22 T + T^{2}$$
$59$ $$-4 + 4 T + T^{2}$$
$61$ $$-2 - 8 T + T^{2}$$
$67$ $$-71 - 2 T + T^{2}$$
$71$ $$28 + 12 T + T^{2}$$
$73$ $$126 + 24 T + T^{2}$$
$79$ $$-4 - 4 T + T^{2}$$
$83$ $$49 - 18 T + T^{2}$$
$89$ $$18 + 12 T + T^{2}$$
$97$ $$-7 + 2 T + T^{2}$$