# Properties

 Label 165.2.a.b Level $165$ Weight $2$ Character orbit 165.a Self dual yes Analytic conductor $1.318$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 165.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.31753163335$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
−1.73205 1.00000 1.00000 −1.00000 −1.73205 2.00000 1.73205 1.00000 1.73205
1.2 1.73205 1.00000 1.00000 −1.00000 1.73205 2.00000 −1.73205 1.00000 −1.73205
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$1$$

## 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 165.2.a.b 2
3.b odd 2 1 495.2.a.c 2
4.b odd 2 1 2640.2.a.x 2
5.b even 2 1 825.2.a.e 2
5.c odd 4 2 825.2.c.c 4
7.b odd 2 1 8085.2.a.bd 2
11.b odd 2 1 1815.2.a.i 2
12.b even 2 1 7920.2.a.bz 2
15.d odd 2 1 2475.2.a.r 2
15.e even 4 2 2475.2.c.n 4
33.d even 2 1 5445.2.a.s 2
55.d odd 2 1 9075.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.b 2 1.a even 1 1 trivial
495.2.a.c 2 3.b odd 2 1
825.2.a.e 2 5.b even 2 1
825.2.c.c 4 5.c odd 4 2
1815.2.a.i 2 11.b odd 2 1
2475.2.a.r 2 15.d odd 2 1
2475.2.c.n 4 15.e even 4 2
2640.2.a.x 2 4.b odd 2 1
5445.2.a.s 2 33.d even 2 1
7920.2.a.bz 2 12.b even 2 1
8085.2.a.bd 2 7.b odd 2 1
9075.2.a.bh 2 55.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(165))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-8 - 4 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$-8 - 4 T + T^{2}$$
$23$ $$-48 + T^{2}$$
$29$ $$-12 + T^{2}$$
$31$ $$-32 + 8 T + T^{2}$$
$37$ $$-44 - 4 T + T^{2}$$
$41$ $$-12 + T^{2}$$
$43$ $$-44 - 4 T + T^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$-12 + 12 T + T^{2}$$
$59$ $$-48 + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$( -8 + T )^{2}$$
$71$ $$-192 + T^{2}$$
$73$ $$-104 - 4 T + T^{2}$$
$79$ $$88 + 20 T + T^{2}$$
$83$ $$132 - 24 T + T^{2}$$
$89$ $$-12 + 12 T + T^{2}$$
$97$ $$( 10 + T )^{2}$$