Properties

Label 1620.2.a.j
Level $1620$
Weight $2$
Character orbit 1620.a
Self dual yes
Analytic conductor $12.936$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Defining polynomial: \(x^{3} - x^{2} - 5 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 180)
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.

\(f(q)\) \(=\) \( q + q^{5} + ( 1 - \beta_{1} ) q^{7} +O(q^{10})\) \( q + q^{5} + ( 1 - \beta_{1} ) q^{7} + \beta_{2} q^{11} + ( 2 + \beta_{2} ) q^{13} -\beta_{2} q^{17} + ( 2 - \beta_{2} ) q^{19} + ( 1 + \beta_{1} ) q^{23} + q^{25} + ( 1 - 2 \beta_{1} ) q^{29} + ( 2 - \beta_{2} ) q^{31} + ( 1 - \beta_{1} ) q^{35} + ( 4 + 2 \beta_{1} - \beta_{2} ) q^{37} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{41} + ( 2 + \beta_{2} ) q^{43} + ( -5 + \beta_{1} - \beta_{2} ) q^{47} + ( 6 + \beta_{2} ) q^{49} + ( 2 + 2 \beta_{1} ) q^{53} + \beta_{2} q^{55} + ( -2 - 2 \beta_{1} ) q^{59} + ( 7 + 2 \beta_{1} - \beta_{2} ) q^{61} + ( 2 + \beta_{2} ) q^{65} + ( 3 + \beta_{1} + \beta_{2} ) q^{67} + ( -8 - 2 \beta_{1} + \beta_{2} ) q^{71} + 8 q^{73} + ( -2 - 2 \beta_{1} + 3 \beta_{2} ) q^{77} + 2 q^{79} + ( 7 + \beta_{1} ) q^{83} -\beta_{2} q^{85} + 3 q^{89} + ( -4 \beta_{1} + 3 \beta_{2} ) q^{91} + ( 2 - \beta_{2} ) q^{95} + ( 6 - 2 \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{5} + 3q^{7} + O(q^{10}) \) \( 3q + 3q^{5} + 3q^{7} + 6q^{13} + 6q^{19} + 3q^{23} + 3q^{25} + 3q^{29} + 6q^{31} + 3q^{35} + 12q^{37} - 3q^{41} + 6q^{43} - 15q^{47} + 18q^{49} + 6q^{53} - 6q^{59} + 21q^{61} + 6q^{65} + 9q^{67} - 24q^{71} + 24q^{73} - 6q^{77} + 6q^{79} + 21q^{83} + 9q^{89} + 6q^{95} + 18q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 5 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 4 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + 2 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{2} + 2 \beta_{1} + 11\)\()/3\)

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
2.51414
−2.08613
0.571993
0 0 0 1.00000 0 −3.83502 0 0 0
1.2 0 0 0 1.00000 0 2.73419 0 0 0
1.3 0 0 0 1.00000 0 4.10083 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

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 1620.2.a.j 3
3.b odd 2 1 1620.2.a.i 3
4.b odd 2 1 6480.2.a.bw 3
5.b even 2 1 8100.2.a.u 3
5.c odd 4 2 8100.2.d.o 6
9.c even 3 2 540.2.i.b 6
9.d odd 6 2 180.2.i.b 6
12.b even 2 1 6480.2.a.bt 3
15.d odd 2 1 8100.2.a.v 3
15.e even 4 2 8100.2.d.p 6
36.f odd 6 2 2160.2.q.i 6
36.h even 6 2 720.2.q.k 6
45.h odd 6 2 900.2.i.c 6
45.j even 6 2 2700.2.i.c 6
45.k odd 12 4 2700.2.s.c 12
45.l even 12 4 900.2.s.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.i.b 6 9.d odd 6 2
540.2.i.b 6 9.c even 3 2
720.2.q.k 6 36.h even 6 2
900.2.i.c 6 45.h odd 6 2
900.2.s.c 12 45.l even 12 4
1620.2.a.i 3 3.b odd 2 1
1620.2.a.j 3 1.a even 1 1 trivial
2160.2.q.i 6 36.f odd 6 2
2700.2.i.c 6 45.j even 6 2
2700.2.s.c 12 45.k odd 12 4
6480.2.a.bt 3 12.b even 2 1
6480.2.a.bw 3 4.b odd 2 1
8100.2.a.u 3 5.b even 2 1
8100.2.a.v 3 15.d odd 2 1
8100.2.d.o 6 5.c odd 4 2
8100.2.d.p 6 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1620))\):

\( T_{7}^{3} - 3 T_{7}^{2} - 15 T_{7} + 43 \)
\( T_{11}^{3} - 24 T_{11} + 36 \)
\( T_{17}^{3} - 24 T_{17} - 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( 43 - 15 T - 3 T^{2} + T^{3} \)
$11$ \( 36 - 24 T + T^{3} \)
$13$ \( 76 - 12 T - 6 T^{2} + T^{3} \)
$17$ \( -36 - 24 T + T^{3} \)
$19$ \( 4 - 12 T - 6 T^{2} + T^{3} \)
$23$ \( -9 - 15 T - 3 T^{2} + T^{3} \)
$29$ \( 279 - 69 T - 3 T^{2} + T^{3} \)
$31$ \( 4 - 12 T - 6 T^{2} + T^{3} \)
$37$ \( 436 - 36 T - 12 T^{2} + T^{3} \)
$41$ \( 81 - 81 T + 3 T^{2} + T^{3} \)
$43$ \( 76 - 12 T - 6 T^{2} + T^{3} \)
$47$ \( 27 + 39 T + 15 T^{2} + T^{3} \)
$53$ \( -72 - 60 T - 6 T^{2} + T^{3} \)
$59$ \( 72 - 60 T + 6 T^{2} + T^{3} \)
$61$ \( 409 + 63 T - 21 T^{2} + T^{3} \)
$67$ \( 151 - 21 T - 9 T^{2} + T^{3} \)
$71$ \( -324 + 108 T + 24 T^{2} + T^{3} \)
$73$ \( ( -8 + T )^{3} \)
$79$ \( ( -2 + T )^{3} \)
$83$ \( -243 + 129 T - 21 T^{2} + T^{3} \)
$89$ \( ( -3 + T )^{3} \)
$97$ \( 424 + 36 T - 18 T^{2} + T^{3} \)
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