Properties

Label 1620.2.a.g
Level $1620$
Weight $2$
Character orbit 1620.a
Self dual yes
Analytic conductor $12.936$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q - q^{5} + (\beta - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + (\beta - 1) q^{7} - \beta q^{11} + ( - 2 \beta + 2) q^{13} + (\beta - 3) q^{17} + (2 \beta - 1) q^{19} - 2 \beta q^{23} + q^{25} + (\beta - 6) q^{29} + ( - 4 \beta - 1) q^{31} + ( - \beta + 1) q^{35} + (3 \beta - 1) q^{37} + (3 \beta - 6) q^{41} + (\beta + 5) q^{43} + ( - \beta - 3) q^{47} + ( - 2 \beta - 3) q^{49} + ( - \beta - 9) q^{53} + \beta q^{55} + (\beta - 6) q^{59} - 4 q^{61} + (2 \beta - 2) q^{65} + ( - 6 \beta - 4) q^{67} + (3 \beta - 6) q^{71} + (3 \beta + 5) q^{73} + (\beta - 3) q^{77} + (6 \beta - 4) q^{79} + (7 \beta - 3) q^{83} + ( - \beta + 3) q^{85} - 3 \beta q^{89} + (4 \beta - 8) q^{91} + ( - 2 \beta + 1) q^{95} + ( - \beta - 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{7} + 4 q^{13} - 6 q^{17} - 2 q^{19} + 2 q^{25} - 12 q^{29} - 2 q^{31} + 2 q^{35} - 2 q^{37} - 12 q^{41} + 10 q^{43} - 6 q^{47} - 6 q^{49} - 18 q^{53} - 12 q^{59} - 8 q^{61} - 4 q^{65} - 8 q^{67} - 12 q^{71} + 10 q^{73} - 6 q^{77} - 8 q^{79} - 6 q^{83} + 6 q^{85} - 16 q^{91} + 2 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
0 0 0 −1.00000 0 −2.73205 0 0 0
1.2 0 0 0 −1.00000 0 0.732051 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.g 2
3.b odd 2 1 1620.2.a.h yes 2
4.b odd 2 1 6480.2.a.bh 2
5.b even 2 1 8100.2.a.t 2
5.c odd 4 2 8100.2.d.m 4
9.c even 3 2 1620.2.i.n 4
9.d odd 6 2 1620.2.i.m 4
12.b even 2 1 6480.2.a.bp 2
15.d odd 2 1 8100.2.a.s 2
15.e even 4 2 8100.2.d.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.g 2 1.a even 1 1 trivial
1620.2.a.h yes 2 3.b odd 2 1
1620.2.i.m 4 9.d odd 6 2
1620.2.i.n 4 9.c even 3 2
6480.2.a.bh 2 4.b odd 2 1
6480.2.a.bp 2 12.b even 2 1
8100.2.a.s 2 15.d odd 2 1
8100.2.a.t 2 5.b even 2 1
8100.2.d.l 4 15.e even 4 2
8100.2.d.m 4 5.c odd 4 2

Hecke kernels

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

\( T_{7}^{2} + 2T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 3 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 3 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 11 \) Copy content Toggle raw display
$23$ \( T^{2} - 12 \) Copy content Toggle raw display
$29$ \( T^{2} + 12T + 33 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 47 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 26 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 22 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$53$ \( T^{2} + 18T + 78 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 33 \) Copy content Toggle raw display
$61$ \( (T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 8T - 92 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T + 9 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T - 2 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T - 92 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T - 138 \) Copy content Toggle raw display
$89$ \( T^{2} - 27 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
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