# Properties

 Label 69.2.a.b Level $69$ Weight $2$ Character orbit 69.a Self dual yes Analytic conductor $0.551$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$69 = 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 69.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.550967773947$$ 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: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta q^{2} - q^{3} + 3 q^{4} + ( -1 + \beta ) q^{5} + \beta q^{6} + ( 1 + \beta ) q^{7} -\beta q^{8} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} - q^{3} + 3 q^{4} + ( -1 + \beta ) q^{5} + \beta q^{6} + ( 1 + \beta ) q^{7} -\beta q^{8} + q^{9} + ( -5 + \beta ) q^{10} + 4 q^{11} -3 q^{12} -2 \beta q^{13} + ( -5 - \beta ) q^{14} + ( 1 - \beta ) q^{15} - q^{16} + ( -5 + \beta ) q^{17} -\beta q^{18} + ( 5 + \beta ) q^{19} + ( -3 + 3 \beta ) q^{20} + ( -1 - \beta ) q^{21} -4 \beta q^{22} + q^{23} + \beta q^{24} + ( 1 - 2 \beta ) q^{25} + 10 q^{26} - q^{27} + ( 3 + 3 \beta ) q^{28} + 2 \beta q^{29} + ( 5 - \beta ) q^{30} + ( -2 - 2 \beta ) q^{31} + 3 \beta q^{32} -4 q^{33} + ( -5 + 5 \beta ) q^{34} + 4 q^{35} + 3 q^{36} + 2 \beta q^{37} + ( -5 - 5 \beta ) q^{38} + 2 \beta q^{39} + ( -5 + \beta ) q^{40} + ( -2 - 4 \beta ) q^{41} + ( 5 + \beta ) q^{42} + ( 1 - 3 \beta ) q^{43} + 12 q^{44} + ( -1 + \beta ) q^{45} -\beta q^{46} -4 q^{47} + q^{48} + ( -1 + 2 \beta ) q^{49} + ( 10 - \beta ) q^{50} + ( 5 - \beta ) q^{51} -6 \beta q^{52} + ( -3 - \beta ) q^{53} + \beta q^{54} + ( -4 + 4 \beta ) q^{55} + ( -5 - \beta ) q^{56} + ( -5 - \beta ) q^{57} -10 q^{58} + ( 4 - 4 \beta ) q^{59} + ( 3 - 3 \beta ) q^{60} + 2 \beta q^{61} + ( 10 + 2 \beta ) q^{62} + ( 1 + \beta ) q^{63} -13 q^{64} + ( -10 + 2 \beta ) q^{65} + 4 \beta q^{66} + ( 3 - \beta ) q^{67} + ( -15 + 3 \beta ) q^{68} - q^{69} -4 \beta q^{70} -8 q^{71} -\beta q^{72} + ( -2 + 4 \beta ) q^{73} -10 q^{74} + ( -1 + 2 \beta ) q^{75} + ( 15 + 3 \beta ) q^{76} + ( 4 + 4 \beta ) q^{77} -10 q^{78} + ( 3 + 3 \beta ) q^{79} + ( 1 - \beta ) q^{80} + q^{81} + ( 20 + 2 \beta ) q^{82} + 4 q^{83} + ( -3 - 3 \beta ) q^{84} + ( 10 - 6 \beta ) q^{85} + ( 15 - \beta ) q^{86} -2 \beta q^{87} -4 \beta q^{88} + ( 1 - \beta ) q^{89} + ( -5 + \beta ) q^{90} + ( -10 - 2 \beta ) q^{91} + 3 q^{92} + ( 2 + 2 \beta ) q^{93} + 4 \beta q^{94} + 4 \beta q^{95} -3 \beta q^{96} + ( 4 + 2 \beta ) q^{97} + ( -10 + \beta ) q^{98} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 6q^{4} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 6q^{4} - 2q^{5} + 2q^{7} + 2q^{9} - 10q^{10} + 8q^{11} - 6q^{12} - 10q^{14} + 2q^{15} - 2q^{16} - 10q^{17} + 10q^{19} - 6q^{20} - 2q^{21} + 2q^{23} + 2q^{25} + 20q^{26} - 2q^{27} + 6q^{28} + 10q^{30} - 4q^{31} - 8q^{33} - 10q^{34} + 8q^{35} + 6q^{36} - 10q^{38} - 10q^{40} - 4q^{41} + 10q^{42} + 2q^{43} + 24q^{44} - 2q^{45} - 8q^{47} + 2q^{48} - 2q^{49} + 20q^{50} + 10q^{51} - 6q^{53} - 8q^{55} - 10q^{56} - 10q^{57} - 20q^{58} + 8q^{59} + 6q^{60} + 20q^{62} + 2q^{63} - 26q^{64} - 20q^{65} + 6q^{67} - 30q^{68} - 2q^{69} - 16q^{71} - 4q^{73} - 20q^{74} - 2q^{75} + 30q^{76} + 8q^{77} - 20q^{78} + 6q^{79} + 2q^{80} + 2q^{81} + 40q^{82} + 8q^{83} - 6q^{84} + 20q^{85} + 30q^{86} + 2q^{89} - 10q^{90} - 20q^{91} + 6q^{92} + 4q^{93} + 8q^{97} - 20q^{98} + 8q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
−2.23607 −1.00000 3.00000 1.23607 2.23607 3.23607 −2.23607 1.00000 −2.76393
1.2 2.23607 −1.00000 3.00000 −3.23607 −2.23607 −1.23607 2.23607 1.00000 −7.23607
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$23$$ $$-1$$

## 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 69.2.a.b 2
3.b odd 2 1 207.2.a.c 2
4.b odd 2 1 1104.2.a.m 2
5.b even 2 1 1725.2.a.ba 2
5.c odd 4 2 1725.2.b.o 4
7.b odd 2 1 3381.2.a.t 2
8.b even 2 1 4416.2.a.bm 2
8.d odd 2 1 4416.2.a.bg 2
11.b odd 2 1 8349.2.a.i 2
12.b even 2 1 3312.2.a.bb 2
15.d odd 2 1 5175.2.a.bk 2
23.b odd 2 1 1587.2.a.i 2
69.c even 2 1 4761.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.a.b 2 1.a even 1 1 trivial
207.2.a.c 2 3.b odd 2 1
1104.2.a.m 2 4.b odd 2 1
1587.2.a.i 2 23.b odd 2 1
1725.2.a.ba 2 5.b even 2 1
1725.2.b.o 4 5.c odd 4 2
3312.2.a.bb 2 12.b even 2 1
3381.2.a.t 2 7.b odd 2 1
4416.2.a.bg 2 8.d odd 2 1
4416.2.a.bm 2 8.b even 2 1
4761.2.a.v 2 69.c even 2 1
5175.2.a.bk 2 15.d odd 2 1
8349.2.a.i 2 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 5$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(69))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-5 + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-4 + 2 T + T^{2}$$
$7$ $$-4 - 2 T + T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$-20 + T^{2}$$
$17$ $$20 + 10 T + T^{2}$$
$19$ $$20 - 10 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$-20 + T^{2}$$
$31$ $$-16 + 4 T + T^{2}$$
$37$ $$-20 + T^{2}$$
$41$ $$-76 + 4 T + T^{2}$$
$43$ $$-44 - 2 T + T^{2}$$
$47$ $$( 4 + T )^{2}$$
$53$ $$4 + 6 T + T^{2}$$
$59$ $$-64 - 8 T + T^{2}$$
$61$ $$-20 + T^{2}$$
$67$ $$4 - 6 T + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$-76 + 4 T + T^{2}$$
$79$ $$-36 - 6 T + T^{2}$$
$83$ $$( -4 + T )^{2}$$
$89$ $$-4 - 2 T + T^{2}$$
$97$ $$-4 - 8 T + T^{2}$$