Properties

Label 165.2.a.c
Level $165$
Weight $2$
Character orbit 165.a
Self dual yes
Analytic conductor $1.318$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + 2 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1} + 2\)

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
2.17009
−1.48119
0.311108
−2.70928 1.00000 5.34017 1.00000 −2.70928 1.07838 −9.04945 1.00000 −2.70928
1.2 −0.193937 1.00000 −1.96239 1.00000 −0.193937 3.35026 0.768452 1.00000 −0.193937
1.3 1.90321 1.00000 1.62222 1.00000 1.90321 −4.42864 −0.719004 1.00000 1.90321
\(n\): 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.c 3
3.b odd 2 1 495.2.a.e 3
4.b odd 2 1 2640.2.a.be 3
5.b even 2 1 825.2.a.k 3
5.c odd 4 2 825.2.c.g 6
7.b odd 2 1 8085.2.a.bk 3
11.b odd 2 1 1815.2.a.m 3
12.b even 2 1 7920.2.a.cj 3
15.d odd 2 1 2475.2.a.bb 3
15.e even 4 2 2475.2.c.r 6
33.d even 2 1 5445.2.a.z 3
55.d odd 2 1 9075.2.a.cf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.c 3 1.a even 1 1 trivial
495.2.a.e 3 3.b odd 2 1
825.2.a.k 3 5.b even 2 1
825.2.c.g 6 5.c odd 4 2
1815.2.a.m 3 11.b odd 2 1
2475.2.a.bb 3 15.d odd 2 1
2475.2.c.r 6 15.e even 4 2
2640.2.a.be 3 4.b odd 2 1
5445.2.a.z 3 33.d even 2 1
7920.2.a.cj 3 12.b even 2 1
8085.2.a.bk 3 7.b odd 2 1
9075.2.a.cf 3 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + T_{2}^{2} - 5 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(165))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 5 T + T^{2} + T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( 16 - 16 T + T^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( -8 - 12 T + 2 T^{2} + T^{3} \)
$17$ \( -184 - 52 T + 2 T^{2} + T^{3} \)
$19$ \( 160 - 16 T - 8 T^{2} + T^{3} \)
$23$ \( -128 - 64 T + T^{3} \)
$29$ \( -40 + 12 T + 10 T^{2} + T^{3} \)
$31$ \( 128 - 32 T - 8 T^{2} + T^{3} \)
$37$ \( ( 2 + T )^{3} \)
$41$ \( 8 + 44 T + 14 T^{2} + T^{3} \)
$43$ \( 400 - 80 T - 4 T^{2} + T^{3} \)
$47$ \( -128 - 32 T + 8 T^{2} + T^{3} \)
$53$ \( 8 - 52 T + 6 T^{2} + T^{3} \)
$59$ \( 320 - 16 T - 12 T^{2} + T^{3} \)
$61$ \( -248 - 52 T + 6 T^{2} + T^{3} \)
$67$ \( -64 - 48 T + 4 T^{2} + T^{3} \)
$71$ \( 128 - 32 T - 8 T^{2} + T^{3} \)
$73$ \( -344 + 4 T + 14 T^{2} + T^{3} \)
$79$ \( 800 - 64 T - 12 T^{2} + T^{3} \)
$83$ \( 16 - 120 T + T^{3} \)
$89$ \( -200 - 52 T + 10 T^{2} + T^{3} \)
$97$ \( -8 + 108 T - 22 T^{2} + T^{3} \)
show more
show less