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

Downloads

Learn more

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:

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

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