Properties

Label 1620.2.i.d
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: no (minimal twist has level 540)
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} +O(q^{10})\) \( q -\zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} + ( -6 + 6 \zeta_{6} ) q^{11} + \zeta_{6} q^{13} - q^{19} -6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( -6 + 6 \zeta_{6} ) q^{29} -8 \zeta_{6} q^{31} - q^{35} -7 q^{37} + 6 \zeta_{6} q^{41} + ( 4 - 4 \zeta_{6} ) q^{43} + ( -12 + 12 \zeta_{6} ) q^{47} + 6 \zeta_{6} q^{49} -6 q^{53} + 6 q^{55} + ( -11 + 11 \zeta_{6} ) q^{61} + ( 1 - \zeta_{6} ) q^{65} + 7 \zeta_{6} q^{67} -6 q^{71} + 11 q^{73} + 6 \zeta_{6} q^{77} + ( 1 - \zeta_{6} ) q^{79} + ( -6 + 6 \zeta_{6} ) q^{83} -12 q^{89} + q^{91} + \zeta_{6} q^{95} + ( 13 - 13 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{5} + q^{7} + O(q^{10}) \) \( 2q - q^{5} + q^{7} - 6q^{11} + q^{13} - 2q^{19} - 6q^{23} - q^{25} - 6q^{29} - 8q^{31} - 2q^{35} - 14q^{37} + 6q^{41} + 4q^{43} - 12q^{47} + 6q^{49} - 12q^{53} + 12q^{55} - 11q^{61} + q^{65} + 7q^{67} - 12q^{71} + 22q^{73} + 6q^{77} + q^{79} - 6q^{83} - 24q^{89} + 2q^{91} + q^{95} + 13q^{97} + O(q^{100}) \)

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 0.500000 0.866025i 0 0 0
1081.1 0 0 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 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.d 2
3.b odd 2 1 1620.2.i.j 2
9.c even 3 1 540.2.a.e yes 1
9.c even 3 1 inner 1620.2.i.d 2
9.d odd 6 1 540.2.a.b 1
9.d odd 6 1 1620.2.i.j 2
36.f odd 6 1 2160.2.a.s 1
36.h even 6 1 2160.2.a.h 1
45.h odd 6 1 2700.2.a.k 1
45.j even 6 1 2700.2.a.m 1
45.k odd 12 2 2700.2.d.k 2
45.l even 12 2 2700.2.d.a 2
72.j odd 6 1 8640.2.a.br 1
72.l even 6 1 8640.2.a.bu 1
72.n even 6 1 8640.2.a.l 1
72.p odd 6 1 8640.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.a.b 1 9.d odd 6 1
540.2.a.e yes 1 9.c even 3 1
1620.2.i.d 2 1.a even 1 1 trivial
1620.2.i.d 2 9.c even 3 1 inner
1620.2.i.j 2 3.b odd 2 1
1620.2.i.j 2 9.d odd 6 1
2160.2.a.h 1 36.h even 6 1
2160.2.a.s 1 36.f odd 6 1
2700.2.a.k 1 45.h odd 6 1
2700.2.a.m 1 45.j even 6 1
2700.2.d.a 2 45.l even 12 2
2700.2.d.k 2 45.k odd 12 2
8640.2.a.l 1 72.n even 6 1
8640.2.a.s 1 72.p odd 6 1
8640.2.a.br 1 72.j odd 6 1
8640.2.a.bu 1 72.l even 6 1

Hecke kernels

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

\( T_{7}^{2} - T_{7} + 1 \)
\( T_{11}^{2} + 6 T_{11} + 36 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( 36 + 6 T + T^{2} \)
$13$ \( 1 - T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 36 + 6 T + T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( 64 + 8 T + T^{2} \)
$37$ \( ( 7 + T )^{2} \)
$41$ \( 36 - 6 T + T^{2} \)
$43$ \( 16 - 4 T + T^{2} \)
$47$ \( 144 + 12 T + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( 121 + 11 T + T^{2} \)
$67$ \( 49 - 7 T + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( ( -11 + T )^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( 36 + 6 T + T^{2} \)
$89$ \( ( 12 + T )^{2} \)
$97$ \( 169 - 13 T + T^{2} \)
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