Properties

Label 1620.2.i.e
Level $1620$
Weight $2$
Character orbit 1620.i
Analytic conductor $12.936$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -\zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} + ( -3 + 3 \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} + 5 q^{19} + 6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( 9 - 9 \zeta_{6} ) q^{29} -5 \zeta_{6} q^{31} -4 q^{35} + 2 q^{37} + 9 \zeta_{6} q^{41} + ( 10 - 10 \zeta_{6} ) q^{43} + ( 6 - 6 \zeta_{6} ) q^{47} -9 \zeta_{6} q^{49} -12 q^{53} + 3 q^{55} -9 \zeta_{6} q^{59} + ( 10 - 10 \zeta_{6} ) q^{61} + ( 4 - 4 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} + 3 q^{71} -4 q^{73} + 12 \zeta_{6} q^{77} + ( 4 - 4 \zeta_{6} ) q^{79} + ( -6 + 6 \zeta_{6} ) q^{83} -9 q^{89} + 16 q^{91} -5 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{5} + 4q^{7} + O(q^{10}) \) \( 2q - q^{5} + 4q^{7} - 3q^{11} + 4q^{13} + 10q^{19} + 6q^{23} - q^{25} + 9q^{29} - 5q^{31} - 8q^{35} + 4q^{37} + 9q^{41} + 10q^{43} + 6q^{47} - 9q^{49} - 24q^{53} + 6q^{55} - 9q^{59} + 10q^{61} + 4q^{65} - 2q^{67} + 6q^{71} - 8q^{73} + 12q^{77} + 4q^{79} - 6q^{83} - 18q^{89} + 32q^{91} - 5q^{95} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
541.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −0.500000 0.866025i 0 2.00000 3.46410i 0 0 0
1081.1 0 0 0 −0.500000 + 0.866025i 0 2.00000 + 3.46410i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.i.e 2
3.b odd 2 1 1620.2.i.l 2
9.c even 3 1 1620.2.a.d yes 1
9.c even 3 1 inner 1620.2.i.e 2
9.d odd 6 1 1620.2.a.a 1
9.d odd 6 1 1620.2.i.l 2
36.f odd 6 1 6480.2.a.y 1
36.h even 6 1 6480.2.a.m 1
45.h odd 6 1 8100.2.a.m 1
45.j even 6 1 8100.2.a.n 1
45.k odd 12 2 8100.2.d.g 2
45.l even 12 2 8100.2.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.a 1 9.d odd 6 1
1620.2.a.d yes 1 9.c even 3 1
1620.2.i.e 2 1.a even 1 1 trivial
1620.2.i.e 2 9.c even 3 1 inner
1620.2.i.l 2 3.b odd 2 1
1620.2.i.l 2 9.d odd 6 1
6480.2.a.m 1 36.h even 6 1
6480.2.a.y 1 36.f odd 6 1
8100.2.a.m 1 45.h odd 6 1
8100.2.a.n 1 45.j even 6 1
8100.2.d.b 2 45.l even 12 2
8100.2.d.g 2 45.k odd 12 2

Hecke kernels

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

\( T_{7}^{2} - 4 T_{7} + 16 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 16 - 4 T + T^{2} \)
$11$ \( 9 + 3 T + T^{2} \)
$13$ \( 16 - 4 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( -5 + T )^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 81 - 9 T + T^{2} \)
$31$ \( 25 + 5 T + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( 81 - 9 T + T^{2} \)
$43$ \( 100 - 10 T + T^{2} \)
$47$ \( 36 - 6 T + T^{2} \)
$53$ \( ( 12 + T )^{2} \)
$59$ \( 81 + 9 T + T^{2} \)
$61$ \( 100 - 10 T + T^{2} \)
$67$ \( 4 + 2 T + T^{2} \)
$71$ \( ( -3 + T )^{2} \)
$73$ \( ( 4 + T )^{2} \)
$79$ \( 16 - 4 T + T^{2} \)
$83$ \( 36 + 6 T + T^{2} \)
$89$ \( ( 9 + T )^{2} \)
$97$ \( 4 + 2 T + T^{2} \)
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